# Spieletheorie

10.02.2018

Einführung in die. Spieltheorie von. Prof. Dr. Wolfgang Leininger und. PD Dr. Erwin Amann. Lehrstuhl Wirtschaftstheorie. Universität Dortmund. Postfach Die Lehrveranstaltung behandelt die Grundlagen der nichtkooperativen Spieltheorie (interaktiven Entscheidungstheorie) und ihre Anwendungen auf. Die Spieltheorie ist ein Teilgebiet der Mathematik, das sich mit der Ein Spiel im Sinne der Spieltheorie ist eine Entscheidungssituation mit mehreren. Danach hat sich die Spieltheorie erst allmählich und seit überaus stürmisch als die beherrschende Methodik in den - traditionell normativ ausgerichteten - Wirtschaftswissenschaften sowie mehr und mehr auch in den sozialwissenschaftlichen Nachbardisziplinen durchgesetzt. Er ist überzeugter Europäer, ein echtes Zoon Politikon und interessiert sich für fast alles ausser Sport und selbst das manchmal. Was heisst das konkret? In einer Tabelle dargestellt sieht das so aus: Ein Spiel, das nach einmaliger Durchführung nicht wiederholt wird, wird als sogenanntes One-Shot-Game bezeichnet. Dieses Vorgehen kann nicht nur für "reine" Spiele, sondern auch für das Verhalten von Gruppen in Wirtschaft und Gesellschaft genutzt werden. Andere Begründungen für das Gleichgewicht basieren auf einleuchtenden Axiomen z. Dazu verfügt die Agentennormalform generell über so viele Spieler bzw. Man kann daher Gleichgewichte - und nur diese! Das Problem der spieltheoretischen Lösungksonzepte ist häufig nicht die Existenz von Gleichgewichten, sondern eine Vielfalt. Im Unterschied zur klassischen Entscheidungstheorie modelliert diese Theorie also Situationen, in denen der Erfolg des Einzelnen nicht nur vom eigenen Handeln, sondern auch von dem anderer abhängt interdependente Entscheidungssituation. Lineal, ein nicht programmierbarer Taschenrechner. Simon und Daniel Kahneman den Nobelpreis. Inferiore Strategien sind fragwürdig, weil es andere Strategien gibt, die sich auf vielfältige Verhaltensweisen s -i als beste Antwort erweisen. Sie sollten daher als Lösungsstrategien ausscheiden und - ähnlich wie dominierte Strategien - wiederholt eliminiert werden.### Spieletheorie Video

Game Theory & Philosophy: Why You Will Watch This - MONSTER BOXDie Spieltheorie modelliert die verschiedensten Situationen als ein Spiel. Im Spiel Gefangenendilemma sind die Spieler die beiden Gefangenen und ihre Aktionsmengen sind aussagen und schweigen.

In der Informatik versucht man, mit Hilfe von Suchstrategien und Heuristiken allgemein: Man spricht in diesem Zusammenhang vom first movers advantage bzw.

Unterschieden werden hierbei drei Begriffe: Darum wird in spieltheoretischen Modellen meist nicht von perfekter Information ausgegangen.

Spiele werden meist entweder in strategischer Normal- Form oder in extensiver Form beschrieben. Weiterhin ist noch die Agentennormalform zu nennen.

Gerecht wird diese Darstellungsform am ehesten solchen Spielen, bei denen alle Spieler ihre Strategien zeitgleich und ohne Kenntnis der Wahl der anderen Spieler festlegen.

Zur Veranschaulichung verwendet man meist eine Bimatrixform. Wer oder was ist eigentlich ein Spieler in einer gegebenen Situation? Die Agentennormalform beantwortet diese Frage so: Wichtige sind das Minimax-Gleichgewicht , das wiederholte Streichen dominierter Strategien sowie Teilspielperfektheit und in der kooperativen Spieltheorie der Core, der Nucleolus , die Verhandlungsmenge und die Imputationsmenge.

Damit ist eine reine Strategie der Spezialfall einer gemischten Strategie, in der immer dann, wenn die Aktionsmenge eines Spielers nichtleer ist, die gesamte Wahrscheinlichkeitsmasse auf eine einzige Aktion der Aktionsmenge gelegt wird.

Thus experimental subjects tend to cooperate at first in these circumstances, but learn after some number of rounds to defect.

The experimenter cannot infer that she has successfully induced a one-shot PD with her experimental setup until she sees this behavior stabilize.

If players of games realize that other players may need to learn game structures and equilibria from experience, this gives them reason to take account of what happens off the equilibrium paths of extensive-form games.

Of course, if a player fears that other players have not learned equilibrium, this may well remove her incentive to play an equilibrium strategy herself.

This raises a set of deep problems about social learning Fudenberg and Levine The crucial answer in the case of applications of game theory to interactions among people is that young people are socialized by growing up in networks of institutions , including cultural norms.

Most complex games that people play are already in progress among people who were socialized before them—that is, have learned game structures and equilibria Ross a.

Novices must then only copy those whose play appears to be expected and understood by others. Institutions and norms are rich with reminders, including homilies and easily remembered rules of thumb, to help people remember what they are doing Clark As noted in Section 2.

Given the complexity of many of the situations that social scientists study, we should not be surprised that mis-specification of models happens frequently.

Applied game theorists must do lots of learning, just like their subjects. Thus the paradox of backward induction is only apparent.

Unless players have experienced play at equilibrium with one another in the past, even if they are all economically rational and all believe this about one another, we should predict that they will attach some positive probability to the conjecture that understanding of game structures among some players is imperfect.

This then explains why people, even if they are economically rational agents, may often, or even usually, play as if they believe in trembling hands.

Learning of equilibria may take various forms for different agents and for games of differing levels of complexity and risk. Incorporating it into game-theoretic models of interactions thus introduces an extensive new set of technicalities.

For the most fully developed general theory, the reader is referred to Fudenberg and Levine It was said above that people might usually play as if they believe in trembling hands.

They must make and test conjectures about this from their social contexts. Sometimes, contexts are fixed by institutional rules.

In other markets, she might know she is expect to haggle, and know the rules for that too. Given the unresolved complex relationship between learning theory and game theory, the reasoning above might seem to imply that game theory can never be applied to situations involving human players that are novel for them.

Fortunately, however, we face no such impasse. In a pair of influential papers in the mid-to-late s, McKelvey and Palfrey , developed the solution concept of quantal response equilibrium QRE.

QRE is not a refinement of NE, in the sense of being a philosophically motivated effort to strengthen NE by reference to normative standards of rationality.

It is, rather, a method for calculating the equilibrium properties of choices made by players whose conjectures about possible errors in the choices of other players are uncertain.

QRE is thus standard equipment in the toolkit of experimental economists who seek to estimate the distribution of utility functions in populations of real people placed in situations modeled as games.

QRE would not have been practically serviceable in this way before the development of econometrics packages such as Stata TM allowed computation of QRE given adequately powerful observation records from interestingly complex games.

QRE is rarely utilized by behavioral economists, and is almost never used by psychologists, in analyzing laboratory data. But NE, though it is a minimalist solution concept in one sense because it abstracts away from much informational structure, is simultaneously a demanding empirical expectation if it imposed categorically that is, if players are expected to play as if they are all certain that all others are playing NE strategies.

Predicting play consistent with QRE is consistent with—indeed, is motivated by—the view that NE captures the core general concept of a strategic equilibrium.

NE defines a logical principle that is well adapted for disciplining thought and for conceiving new strategies for generic modeling of new classes of social phenomena.

For purposes of estimating real empirical data one needs to be able to define equilibrium statistically. QRE represents one way of doing this, consistently with the logic of NE.

We will see later that there is an alternative interpretation of mixing, not involving randomization at a particular information set; but we will start here from the coin-flipping interpretation and then build on it in Section 3.

Our river-crossing game from Section 1 exemplifies this. Symmetry of logical reasoning power on the part of the two players ensures that the fugitive can surprise the pursuer only if it is possible for him to surprise himself.

Suppose that we ignore rocks and cobras for a moment, and imagine that the bridges are equally safe.

He must then pre-commit himself to using whichever bridge is selected by this randomizing device. This fixes the odds of his survival regardless of what the pursuer does; but since the pursuer has no reason to prefer any available pure or mixed strategy, and since in any case we are presuming her epistemic situation to be symmetrical to that of the fugitive, we may suppose that she will roll a three-sided die of her own.

Note that if one player is randomizing then the other does equally well on any mix of probabilities over bridges, so there are infinitely many combinations of best replies.

However, each player should worry that anything other than a random strategy might be coordinated with some factor the other player can detect and exploit.

Since any non-random strategy is exploitable by another non-random strategy, in a zero-sum game such as our example, only the vector of randomized strategies is a NE.

Now let us re-introduce the parametric factors, that is, the falling rocks at bridge 2 and the cobras at bridge 3.

Suppose that Player 1, the fugitive, cares only about living or dying preferring life to death while the pursuer simply wishes to be able to report that the fugitive is dead, preferring this to having to report that he got away.

In other words, neither player cares about how the fugitive lives or dies. Suppose also for now that neither player gets any utility or disutility from taking more or less risk.

In this case, the fugitive simply takes his original randomizing formula and weights it according to the different levels of parametric danger at the three bridges.

She will be using her NE strategy when she chooses the mix of probabilities over the three bridges that makes the fugitive indifferent among his possible pure strategies.

The bridge with rocks is 1. Therefore, he will be indifferent between the two when the pursuer is 1. The cobra bridge is 1. Then the pursuer minimizes the net survival rate across any pair of bridges by adjusting the probabilities p1 and p2 that she will wait at them so that.

Now let f1, f2, f3 represent the probabilities with which the fugitive chooses each respective bridge. Then the fugitive finds his NE strategy by solving.

These two sets of NE probabilities tell each player how to weight his or her die before throwing it. Note the—perhaps surprising—result that the fugitive, though by hypothesis he gets no enjoyment from gambling, uses riskier bridges with higher probability.

We were able to solve this game straightforwardly because we set the utility functions in such a way as to make it zero-sum , or strictly competitive.

That is, every gain in expected utility by one player represents a precisely symmetrical loss by the other. However, this condition may often not hold.

Suppose now that the utility functions are more complicated. The pursuer most prefers an outcome in which she shoots the fugitive and so claims credit for his apprehension to one in which he dies of rockfall or snakebite; and she prefers this second outcome to his escape.

The fugitive prefers a quick death by gunshot to the pain of being crushed or the terror of an encounter with a cobra. Most of all, of course, he prefers to escape.

Suppose, plausibly, that fugitive cares much strongly about surviving than he does about getting killed one way rather than another. This is because utility does not denote a hidden psychological variable such as pleasure.

As we discussed in Section 2. How, then, can we model games in which cardinal information is relevant? Here, we will provide a brief outline of their ingenious technique for building cardinal utility functions out of ordinal ones.

It is emphasized that what follows is merely an outline , so as to make cardinal utility non-mysterious to you as a student who is interested in knowing about the philosophical foundations of game theory, and about the range of problems to which it can be applied.

Providing a manual you could follow in building your own cardinal utility functions would require many pages. Such manuals are available in many textbooks.

Suppose that we now assign the following ordinal utility function to the river-crossing fugitive:. We are supposing that his preference for escape over any form of death is stronger than his preferences between causes of death.

This should be reflected in his choice behaviour in the following way. In a situation such as the river-crossing game, he should be willing to run greater risks to increase the relative probability of escape over shooting than he is to increase the relative probability of shooting over snakebite.

Suppose we asked the fugitive to pick, from the available set of outcomes, a best one and a worst one. Now imagine expanding the set of possible prizes so that it includes prizes that the agent values as intermediate between W and L.

We find, for a set of outcomes containing such prizes, a lottery over them such that our agent is indifferent between that lottery and a lottery including only W and L.

In our example, this is a lottery that includes being shot and being crushed by rocks. Call this lottery T. What exactly have we done here? Furthermore, two agents in one game, or one agent under different sorts of circumstances, may display varying attitudes to risk.

Perhaps in the river-crossing game the pursuer, whose life is not at stake, will enjoy gambling with her glory while our fugitive is cautious.

Both agents, after all, can find their NE strategies if they can estimate the probabilities each will assign to the actions of the other. We can now fill in the rest of the matrix for the bridge-crossing game that we started to draw in Section 2.

If both players are risk-neutral and their revealed preferences respect ROCL, then we have enough information to be able to assign expected utilities, expressed by multiplying the original payoffs by the relevant probabilities, as outcomes in the matrix.

Suppose that the hunter waits at the cobra bridge with probability x and at the rocky bridge with probability y. Then, continuing to assign the fugitive a payoff of 0 if he dies and 1 if he escapes, and the hunter the reverse payoffs, our complete matrix is as follows:.

We can now read the following facts about the game directly from the matrix. No pair of pure strategies is a pair of best replies to the other.

But in real interactive choice situations, agents must often rely on their subjective estimations or perceptions of probabilities.

In one of the greatest contributions to twentieth-century behavioral and social science, Savage showed how to incorporate subjective probabilities, and their relationships to preferences over risk, within the framework of von Neumann-Morgenstern expected utility theory.

Then, just over a decade later, Harsanyi showed how to solve games involving maximizers of Savage expected utility. This is often taken to have marked the true maturity of game theory as a tool for application to behavioral and social science, and was recognized as such when Harsanyi joined Nash and Selten as a recipient of the first Nobel prize awarded to game theorists in As we observed in considering the need for people playing games to learn trembling hand equilibria and QRE, when we model the strategic interactions of people we must allow for the fact that people are typically uncertain about their models of one another.

This uncertainty is reflected in their choices of strategies. This game has four NE: Consider the fourth of these NE. The structure of the game incentivizes efforts by Player I to supply Player III with information that would open up her closed information set.

Player III should believe this information because the structure of the game shows that Player I has incentive to communicate it truthfully.

Theorists who think of game theory as part of a normative theory of general rationality, for example most philosophers, and refinement program enthusiasts among economists, have pursued a strategy that would identify this solution on general principles.

The relevant beliefs here are not merely strategic, as before, since they are not just about what players will do given a set of payoffs and game structures, but about what understanding of conditional probability they should expect other players to operate with.

What beliefs about conditional probability is it reasonable for players to expect from each other? A SE has two parts: Consider again the NE R, r 2 , r 3.

Suppose that Player III assigns pr 1 to her belief that if she gets a move she is at node The use of the consistency requirement in this example is somewhat trivial, so consider now a second case also taken from Kreps , p.

The idea of SE is hopefully now clear. We can apply it to the river-crossing game in a way that avoids the necessity for the pursuer to flip any coins of we modify the game a bit.

This requirement is captured by supposing that all strategy profiles be strictly mixed , that is, that every action at every information set be taken with positive probability.

You will see that this is just equivalent to supposing that all hands sometimes tremble, or alternatively that no expectations are quite certain.

A SE is said to be trembling-hand perfect if all strategies played at equilibrium are best replies to strategies that are strictly mixed.

You should also not be surprised to be told that no weakly dominated strategy can be trembling-hand perfect, since the possibility of trembling hands gives players the most persuasive reason for avoiding such strategies.

How can the non-psychological game theorist understand the concept of an NE that is an equilibrium in both actions and beliefs?

Multiple kinds of informational channels typically link different agents with the incentive structures in their environments. Some agents may actually compute equilibria, with more or less error.

Others may settle within error ranges that stochastically drift around equilibrium values through more or less myopic conditioned learning. Still others may select response patterns by copying the behavior of other agents, or by following rules of thumb that are embedded in cultural and institutional structures and represent historical collective learning.

Note that the issue here is specific to game theory, rather than merely being a reiteration of a more general point, which would apply to any behavioral science, that people behave noisily from the perspective of ideal theory.

In a given game, whether it would be rational for even a trained, self-aware, computationally well resourced agent to play NE would depend on the frequency with which he or she expected others to do likewise.

If she expects some other players to stray from NE play, this may give her a reason to stray herself. Instead of predicting that human players will reveal strict NE strategies, the experienced experimenter or modeler anticipates that there will be a relationship between their play and the expected costs of departures from NE.

Consequently, maximum likelihood estimation of observed actions typically identifies a QRE as providing a better fit than any NE.

Rather, she conjectures that they are agents, that is, that there is a systematic relationship between changes in statistical patterns in their behavior and some risk-weighted cardinal rankings of possible goal-states.

If the agents are people or institutionally structured groups of people that monitor one another and are incentivized to attempt to act collectively, these conjectures will often be regarded as reasonable by critics, or even as pragmatically beyond question, even if always defeasible given the non-zero possibility of bizarre unknown circumstances of the kind philosophers sometimes consider e.

The analyst might assume that all of the agents respond to incentive changes in accordance with Savage expected-utility theory, particularly if the agents are firms that have learned response contingencies under normatively demanding conditions of market competition with many players.

All this is to say that use of game theory does not force a scientist to empirically apply a model that is likely to be too precise and narrow in its specifications to plausibly fit the messy complexities of real strategic interaction.

A good applied game theorist should also be a well-schooled econometrician. However, games are often played with future games in mind, and this can significantly alter their outcomes and equilibrium strategies.

Our topic in this section is repeated games , that is, games in which sets of players expect to face each other in similar situations on multiple occasions.

This may no longer hold, however, if the players expect to meet each other again in future PDs. Imagine that four firms, all making widgets, agree to maintain high prices by jointly restricting supply.

That is, they form a cartel. This will only work if each firm maintains its agreed production quota. Typically, each firm can maximize its profit by departing from its quota while the others observe theirs, since it then sells more units at the higher market price brought about by the almost-intact cartel.

In the one-shot case, all firms would share this incentive to defect and the cartel would immediately collapse. However, the firms expect to face each other in competition for a long period.

In this case, each firm knows that if it breaks the cartel agreement, the others can punish it by underpricing it for a period long enough to more than eliminate its short-term gain.

Of course, the punishing firms will take short-term losses too during their period of underpricing. But these losses may be worth taking if they serve to reestablish the cartel and bring about maximum long-term prices.

One simple, and famous but not , contrary to widespread myth, necessarily optimal strategy for preserving cooperation in repeated PDs is called tit-for-tat.

This strategy tells each player to behave as follows:. A group of players all playing tit-for-tat will never see any defections.

Since, in a population where others play tit-for-tat, tit-for-tat is the rational response for each player, everyone playing tit-for-tat is a NE.

You may frequently hear people who know a little but not enough game theory talk as if this is the end of the story. There are two complications. First, the players must be uncertain as to when their interaction ends.

Suppose the players know when the last round comes. In that round, it will be utility-maximizing for players to defect, since no punishment will be possible.

Now consider the second-last round. In this round, players also face no punishment for defection, since they expect to defect in the last round anyway.

So they defect in the second-last round. But this means they face no threat of punishment in the third-last round, and defect there too.

We can simply iterate this backwards through the game tree until we reach the first round. Since cooperation is not a NE strategy in that round, tit-for-tat is no longer a NE strategy in the repeated game, and we get the same outcome—mutual defection—as in the one-shot PD.

Therefore, cooperation is only possible in repeated PDs where the expected number of repetitions is indeterminate. Of course, this does apply to many real-life games.

Note that in this context any amount of uncertainty in expectations, or possibility of trembling hands, will be conducive to cooperation, at least for awhile.

When people in experiments play repeated PDs with known end-points, they indeed tend to cooperate for awhile, but learn to defect earlier as they gain experience.

Now we introduce a second complication. Consider our case of the widget cartel. Suppose the players observe a fall in the market price of widgets.

Perhaps this is because a cartel member cheated. Or perhaps it has resulted from an exogenous drop in demand. If tit-for-tat players mistake the second case for the first, they will defect, thereby setting off a chain-reaction of mutual defections from which they can never recover, since every player will reply to the first encountered defection with defection, thereby begetting further defections, and so on.

If players know that such miscommunication is possible, they have incentive to resort to more sophisticated strategies.

In particular, they may be prepared to sometimes risk following defections with cooperation in order to test their inferences.

However, if they are too forgiving, then other players can exploit them through additional defections. In general, sophisticated strategies have a problem.

Because they are more difficult for other players to infer, their use increases the probability of miscommunication.

But miscommunication is what causes repeated-game cooperative equilibria to unravel in the first place. The complexities surrounding information signaling, screening and inference in repeated PDs help to intuitively explain the folk theorem , so called because no one is sure who first recognized it, that in repeated PDs, for any strategy S there exists a possible distribution of strategies among other players such that the vector of S and these other strategies is a NE.

Thus there is nothing special, after all, about tit-for-tat. Real, complex, social and political dramas are seldom straightforward instantiations of simple games such as PDs.

Hardin offers an analysis of two tragically real political cases, the Yugoslavian civil war of —95, and the Rwandan genocide, as PDs that were nested inside coordination games.

A coordination game occurs whenever the utility of two or more players is maximized by their doing the same thing as one another, and where such correspondence is more important to them than whatever it is, in particular, that they both do.

A standard example arises with rules of the road: In these circumstances, any strategy that is a best reply to any vector of mixed strategies available in NE is said to be rationalizable.

That is, a player can find a set of systems of beliefs for the other players such that any history of the game along an equilibrium path is consistent with that set of systems.

Pure coordination games are characterized by non-unique vectors of rationalizable strategies. In such situations, players may try to predict equilibria by searching for focal points , that is, features of some strategies that they believe will be salient to other players, and that they believe other players will believe to be salient to them.

Coordination was, indeed, the first topic of game-theoretic application that came to the widespread attention of philosophers.

In , the philosopher David Lewis published Convention , in which the conceptual framework of game-theory was applied to one of the fundamental issues of twentieth-century epistemology, the nature and extent of conventions governing semantics and their relationship to the justification of propositional beliefs.

The basic insight can be captured using a simple example. This insight, of course, well preceded Lewis; but what he recognized is that this situation has the logical form of a coordination game.

Thus, while particular conventions may be arbitrary, the interactive structures that stabilize and maintain them are not. Furthermore, the equilibria involved in coordinating on noun meanings appear to have an arbitrary element only because we cannot Pareto-rank them; but Millikan shows implicitly that in this respect they are atypical of linguistic coordinations.

In a city, drivers must coordinate on one of two NE with respect to their behaviour at traffic lights.

Either all must follow the strategy of rushing to try to race through lights that turn yellow or amber and pausing before proceeding when red lights shift to green, or all must follow the strategy of slowing down on yellows and jumping immediately off on shifts to green.

Both patterns are NE, in that once a community has coordinated on one of them then no individual has an incentive to deviate: However, the two equilibria are not Pareto-indifferent, since the second NE allows more cars to turn left on each cycle in a left-hand-drive jurisdiction, and right on each cycle in a right-hand jurisdiction, which reduces the main cause of bottlenecks in urban road networks and allows all drivers to expect greater efficiency in getting about.

Unfortunately, for reasons about which we can only speculate pending further empirical work and analysis, far more cities are locked onto the Pareto-inferior NE than on the Pareto-superior one.

While various arrangements might be NE in the social game of science, as followers of Thomas Kuhn like to remind us, it is highly improbable that all of these lie on a single Pareto-indifference curve.

These themes, strongly represented in contemporary epistemology, philosophy of science and philosophy of language, are all at least implicit applications of game theory.

The reader can find a broad sample of applications, and references to the large literature, in Nozick Most of the social and political coordination games played by people also have this feature.

Unfortunately for us all, inefficiency traps represented by Pareto-inferior NE are extremely common in them. And sometimes dynamics of this kind give rise to the most terrible of all recurrent human collective behaviors.

That is, in neither situation, on either side, did most people begin by preferring the destruction of the other to mutual cooperation.

However, the deadly logic of coordination, deliberately abetted by self-serving politicians, dynamically created PDs. Some individual Serbs Hutus were encouraged to perceive their individual interests as best served through identification with Serbian Hutu group-interests.

That is, they found that some of their circumstances, such as those involving competition for jobs, had the form of coordination games.

They thus acted so as to create situations in which this was true for other Serbs Hutus as well. Eventually, once enough Serbs Hutus identified self-interest with group-interest, the identification became almost universally correct , because 1 the most important goal for each Serb Hutu was to do roughly what every other Serb Hutu would, and 2 the most distinctively Serbian thing to do, the doing of which signalled coordination, was to exclude Croats Tutsi.

That is, strategies involving such exclusionary behavior were selected as a result of having efficient focal points. But the outcome is ghastly: Serbs and Croats Hutus and Tutsis seem progressively more threatening to each other as they rally together for self-defense, until both see it as imperative to preempt their rivals and strike before being struck.

If Hardin is right—and the point here is not to claim that he is , but rather to point out the worldly importance of determining which games agents are in fact playing—then the mere presence of an external enforcer NATO?

The Rwandan genocide likewise ended with a military solution, in this case a Tutsi victory. But this became the seed for the most deadly war on earth since , the Congo War of — Of course, it is not the case that most repeated games lead to disasters.

The biological basis of friendship in people and other animals is partly a function of the logic of repeated games. The importance of payoffs achievable through cooperation in future games leads those who expect to interact in them to be less selfish than temptation would otherwise encourage in present games.

The fact that such equilibria become more stable through learning gives friends the logical character of built-up investments, which most people take great pleasure in sentimentalizing.

Furthermore, cultivating shared interests and sentiments provides networks of focal points around which coordination can be increasingly facilitated.

More directly, her claim was that conventions are not merely the products of decisions of many individual people, as might be suggested by a theorist who modeled a convention as an equilibrium of an n-person game in which each player was a single person.

Similar concerns about allegedly individualistic foundations of game theory have been echoed by another philosopher, Martin Hollis and economists Robert Sugden , , and Michael Bacharach The explanation seems to require appeal to very strong forms of both descriptive and normative individualism.

As Binmore forcefully argued, and as most commentators seem subsequently to have acknowledged, this line of criticism confused game theory as mathematics with questions about which game theoretic models are most typically applicable to situations in which people find themselves.

At 3, players would be indifferent between cooperating and defecting. Then we get the following transformation of the game:. Thus if the players find this equilibrium, we should not say that they have played non-NE strategies in a PD.

Rather, we should say that the PD was the wrong model of their situation. But, Bacharach, Sugden and Gold argue, human game players will often or usually avoid framing situations in such a way that a one-shot PD is the right model of their circumstances.

Note that the welfare of the team might make a difference to cardinal payoffs without making enough of a difference to trump the lure of unilateral defection.

Suppose it bumped them up to 2. This point is important, since in experiments in which subjects play sequences of one-shot PDs not repeated PDs, since opponents in the experiments change from round to round , majorities of subjects begin by cooperating but learn to defect as the experiments progress.

The team reasoners then re-frame the situation to defend themselves. Individualistic reasoners and team reasoners are not claimed to be different types of people.

People, Bacharach maintains, flip back and forth between individualistic agency and participation in team agency. If they do come to such recognition, perhaps by finding a focal point, then the Pure Coordination game is transformed into the following game known as Hi-Lo:.

Crucially, here the transformation requires more than mere team reasoning. The players also need focal points to know which of the two Pure Coordination equilibria offers the less risky prospect for social stabilization Binmore In fact, Bacharach and his executors are interested in the relationship between Pure Coordination games and Hi-Lo games for a special reason.

At this point Bacharach and his friends adopt the philosophical reasoning of the refinement program. Therefore, they conclude, axioms for team reasoning should be built into refined foundations of game theory.

The non-psychological game theorist can propose a subtle shift of emphasis: To this extent their agency is partly or wholly—and perhaps stochastically—identified with these groups, and this will need to be reflected when we model their agency using utility functions.

Then we could better describe the theory we want as a theory of team-centred choice rather than as a theory of team reasoning. Note that this philosophical interpretation is consistent with the idea that some of our evidence, perhaps even our best evidence, for the existence of team-centred choice is psychological.

It is also consistent with the suggestion that the processes that flip people between individualized and team-centred agency are partly latent.

The point is simply that we need not follow Bacharach in thinking of game theory as a model of reasoning or rationality in order to be persuaded that he has identified a gap we would like to have formal resources to fill.

Members of such teams are under considerable social pressure to choose actions that maximize prospects for victory over actions that augment their personal statistics.

The problem with these examples is that they embed difficult identification problems with respect to the estimation of utility functions; a narrowly self-interested player who wants to be popular with fans might behave identically to a team-centred player.

Soldiers in battle conditions provide more persuasive examples. Though trying to convince soldiers to sacrifice their lives in the interests of their countries is often ineffective, most soldiers can be induced to take extraordinary risks in defense of their buddies, or when enemies directly menace their home towns and families.

It is easy to think of other kinds of teams with which most people plausibly identify some or most of the time: Strongly individualistic social theory tries to construct such teams as equilibria in games amongst individual people, but no assumption built into game theory or, for that matter, mainstream economic theory forces this perspective.

We can instead suppose that teams are often exogenously welded into being by complex interrelated psychological and institutional processes. This invites the game theorist to conceive of a mathematical mission that consists not in modeling team reasoning, but rather in modeling choice that is conditional on the existence of team dynamics.

The intuitive target Stirling has in mind is that of processes by which people derive their actual preferences partly on the basis of the comparative consequences for group welfare of different possible profiles of preferences that members could severally hypothetically reveal.

Let us develop the intuitive idea of preference conditionalization in more detail. People may often—perhaps typically—defer full resolution of their preferences until they get more information about the preferences of others who are their current or potential team-mates.

Stirling himself provides a simple arguably too simple example from Keeney and Raiffa , in which a farmer forms a clear preference among different climate conditions for a land purchase only after, and partly in light of, learning the preferences of his wife.

The existence of such strategies, for cleverly designed games, has important consequences in descriptive set theory. Much of game theory is concerned with finite, discrete games, that have a finite number of players, moves, events, outcomes, etc.

Many concepts can be extended, however. Continuous games allow players to choose a strategy from a continuous strategy set. The problem of finding an optimal strategy in a differential game is closely related to the optimal control theory.

In particular, there are two types of strategies: A particular case of differential games are the games with a random time horizon. Therefore, the players maximize the mathematical expectation of the cost function.

It was shown that the modified optimization problem can be reformulated as a discounted differential game over an infinite time interval.

Evolutionary game theory studies players who adjust their strategies over time according to rules that are not necessarily rational or farsighted.

Such rules may feature imitation, optimization or survival of the fittest. In the social sciences, such models typically represent strategic adjustment by players who play a game many times within their lifetime and, consciously or unconsciously, occasionally adjust their strategies.

Individual decision problems with stochastic outcomes are sometimes considered "one-player games". These situations are not considered game theoretical by some authors.

Although these fields may have different motivators, the mathematics involved are substantially the same, e. Stochastic outcomes can also be modeled in terms of game theory by adding a randomly acting player who makes "chance moves" " moves by nature ".

For some problems, different approaches to modeling stochastic outcomes may lead to different solutions. For example, the difference in approach between MDPs and the minimax solution is that the latter considers the worst-case over a set of adversarial moves, rather than reasoning in expectation about these moves given a fixed probability distribution.

The minimax approach may be advantageous where stochastic models of uncertainty are not available, but may also be overestimating extremely unlikely but costly events, dramatically swaying the strategy in such scenarios if it is assumed that an adversary can force such an event to happen.

General models that include all elements of stochastic outcomes, adversaries, and partial or noisy observability of moves by other players have also been studied.

The " gold standard " is considered to be partially observable stochastic game POSG , but few realistic problems are computationally feasible in POSG representation.

These are games the play of which is the development of the rules for another game, the target or subject game. Metagames seek to maximize the utility value of the rule set developed.

The theory of metagames is related to mechanism design theory. The term metagame analysis is also used to refer to a practical approach developed by Nigel Howard.

Subsequent developments have led to the formulation of confrontation analysis. These are games prevailing over all forms of society.

Pooling games are repeated plays with changing payoff table in general over an experienced path and their equilibrium strategies usually take a form of evolutionary social convention and economic convention.

Pooling game theory emerges to formally recognize the interaction between optimal choice in one play and the emergence of forthcoming payoff table update path, identify the invariance existence and robustness, and predict variance over time.

The theory is based upon topological transformation classification of payoff table update over time to predict variance and invariance, and is also within the jurisdiction of the computational law of reachable optimality for ordered system.

Mean field game theory is the study of strategic decision making in very large populations of small interacting agents.

This class of problems was considered in the economics literature by Boyan Jovanovic and Robert W. Rosenthal , in the engineering literature by Peter E.

The games studied in game theory are well-defined mathematical objects. To be fully defined, a game must specify the following elements: These equilibrium strategies determine an equilibrium to the game—a stable state in which either one outcome occurs or a set of outcomes occur with known probability.

Most cooperative games are presented in the characteristic function form, while the extensive and the normal forms are used to define noncooperative games.

The extensive form can be used to formalize games with a time sequencing of moves. Games here are played on trees as pictured here.

Here each vertex or node represents a point of choice for a player. The player is specified by a number listed by the vertex.

The lines out of the vertex represent a possible action for that player. The payoffs are specified at the bottom of the tree.

The extensive form can be viewed as a multi-player generalization of a decision tree. It involves working backward up the game tree to determine what a rational player would do at the last vertex of the tree, what the player with the previous move would do given that the player with the last move is rational, and so on until the first vertex of the tree is reached.

The game pictured consists of two players. The way this particular game is structured i. Suppose that Player 1 chooses U and then Player 2 chooses A: Player 1 then gets a payoff of "eight" which in real-world terms can be interpreted in many ways, the simplest of which is in terms of money but could mean things such as eight days of vacation or eight countries conquered or even eight more opportunities to play the same game against other players and Player 2 gets a payoff of "two".

The extensive form can also capture simultaneous-move games and games with imperfect information. To represent it, either a dotted line connects different vertices to represent them as being part of the same information set i.

See example in the imperfect information section. The normal or strategic form game is usually represented by a matrix which shows the players, strategies, and payoffs see the example to the right.

More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions. In the accompanying example there are two players; one chooses the row and the other chooses the column.

Each player has two strategies, which are specified by the number of rows and the number of columns. The payoffs are provided in the interior.

The first number is the payoff received by the row player Player 1 in our example ; the second is the payoff for the column player Player 2 in our example.

Suppose that Player 1 plays Up and that Player 2 plays Left. Then Player 1 gets a payoff of 4, and Player 2 gets 3. When a game is presented in normal form, it is presumed that each player acts simultaneously or, at least, without knowing the actions of the other.

If players have some information about the choices of other players, the game is usually presented in extensive form. Every extensive-form game has an equivalent normal-form game, however the transformation to normal form may result in an exponential blowup in the size of the representation, making it computationally impractical.

In games that possess removable utility, separate rewards are not given; rather, the characteristic function decides the payoff of each unity.

The balanced payoff of C is a basic function. Although there are differing examples that help determine coalitional amounts from normal games, not all appear that in their function form can be derived from such.

Formally, a characteristic function is seen as: N,v , where N represents the group of people and v: Such characteristic functions have expanded to describe games where there is no removable utility.

As a method of applied mathematics , game theory has been used to study a wide variety of human and animal behaviors.

It was initially developed in economics to understand a large collection of economic behaviors, including behaviors of firms, markets, and consumers.

The first use of game-theoretic analysis was by Antoine Augustin Cournot in with his solution of the Cournot duopoly. The use of game theory in the social sciences has expanded, and game theory has been applied to political, sociological, and psychological behaviors as well.

This work predates the name "game theory", but it shares many important features with this field. The developments in economics were later applied to biology largely by John Maynard Smith in his book Evolution and the Theory of Games.

In addition to being used to describe, predict, and explain behavior, game theory has also been used to develop theories of ethical or normative behavior and to prescribe such behavior.

Game-theoretic arguments of this type can be found as far back as Plato. The primary use of game theory is to describe and model how human populations behave.

This particular view of game theory has been criticized. It is argued that the assumptions made by game theorists are often violated when applied to real-world situations.

Game theorists usually assume players act rationally, but in practice, human behavior often deviates from this model. Game theorists respond by comparing their assumptions to those used in physics.

Thus while their assumptions do not always hold, they can treat game theory as a reasonable scientific ideal akin to the models used by physicists.

There is an ongoing debate regarding the importance of these experiments and whether the analysis of the experiments fully captures all aspects of the relevant situation.

Price , have turned to evolutionary game theory in order to resolve these issues. These models presume either no rationality or bounded rationality on the part of players.

Despite the name, evolutionary game theory does not necessarily presume natural selection in the biological sense. Evolutionary game theory includes both biological as well as cultural evolution and also models of individual learning for example, fictitious play dynamics.

Some scholars, like Leonard Savage , [ citation needed ] see game theory not as a predictive tool for the behavior of human beings, but as a suggestion for how people ought to behave.

This normative use of game theory has also come under criticism. Game theory is a major method used in mathematical economics and business for modeling competing behaviors of interacting agents.

This research usually focuses on particular sets of strategies known as "solution concepts" or "equilibria".

A common assumption is that players act rationally. In non-cooperative games, the most famous of these is the Nash equilibrium. A set of strategies is a Nash equilibrium if each represents a best response to the other strategies.

If all the players are playing the strategies in a Nash equilibrium, they have no unilateral incentive to deviate, since their strategy is the best they can do given what others are doing.

The payoffs of the game are generally taken to represent the utility of individual players. A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of a particular economic situation.

One or more solution concepts are chosen, and the author demonstrates which strategy sets in the presented game are equilibria of the appropriate type.

Naturally one might wonder to what use this information should be put. Economists and business professors suggest two primary uses noted above: The application of game theory to political science is focused in the overlapping areas of fair division , political economy , public choice , war bargaining , positive political theory , and social choice theory.

In each of these areas, researchers have developed game-theoretic models in which the players are often voters, states, special interest groups, and politicians.

Early examples of game theory applied to political science are provided by Anthony Downs. In his book An Economic Theory of Democracy , [53] he applies the Hotelling firm location model to the political process.

Toward a History of Game Theory. We might summarize the intuition behind all this as follows: This situation has dramatically changed, in ways we will examine as we go along, over the past six decades, as the framework has been deepened and generalized. The reader should note that these two uses of one lotto bay within the same discipline are technically unconnected. The existence of such strategies, for cleverly designed games, has important consequences in casino-kornwestheim set theory. What*spieletheorie*happened here intuitively is that Player I realizes that if he plays C refuse to confess chomutov eishockey node 1, then Player II will be able to maximize her utility by suckering him and playing D. We will present some examples of this interactive path selection, and detailed techniques for reasoning through these examples, after we have described a situation we can use a tree to model. Concordant preference orderings are invariant

**gog spiele weitergeben**representational transformations that are montagsspiel live with respect to information about conditional preferences. In the s this motivation of economists fit comfortably with the dominance of behaviourism and radical empiricism in myp2p.eu live and in the philosophy of science respectively. Here, no strategy strictly dominates another. Wichtige sind das Minimax-Gleichgewichtdas wiederholte Streichen dominierter Strategien sowie Teilspielperfektheit und in der kooperativen Spieltheorie der Starburst free spins, der Nucleolusdie Verhandlungsmenge und die Imputationsmenge.

## Spieletheorie - remarkable, very

Ein Gleichgewicht ist damit ein Strategien- Vektor. Broder cheap roger vivier shoes bei Kriegsspiele maschinen blech aus zweiter hand bei Warum man mit der US Flagge nicht das Bett beziehen darf gebrauchte verformungsmaschinen bei Mardi Gras gebrauchte schleif bei Spieltheorie einfach erklärt I: Die Spieltheorie erlaubt es, soziale Konfliktsituationen, die strategische Spiele genannt werden, facettenreich abzubilden und mathematisch streng zu lösen. Eine kleine Ankündigung zu einer grossen Frage: Oft kann man die stabilen Ergebnisse durch statische Stabilitätskonzepte charakterisieren. Unten finden sich einige Bemerkungen zu den Unterschieden. Einen solchen bietet ein Spiel. Man spricht in diesem Zusammenhang vom first movers advantage bzw. Veranstaltungszeiten und -ort Diese Veranstaltung wird nicht in jedem Semester angeboten. Die veränderte unberechenbare Streikstrategie war erfolgreich. Perfektes Erinnerungsvermögen ist das Wissen jedes Spielers über sämtliche Informationen, die ihm bereits in der Vergangenheit zugänglich waren. In der Informatik versucht man, mit Hilfe von Suchstrategien und Heuristiken allgemein: Sie bedient mannigfaltige Anwendungsfelder. Bei reinen Glücksspielen ist die Entscheidung des Spielers über die Teilnahme sowie über die Höhe des Einsatzes bereits die wichtigste. Heute box24 casino mobile es eine Fülle an Einführungen zur Spieltheorie, die kaum noch zu überblicken ist. Halten beide dicht, kann nur ein Indizienprozess geführt werden und sie werden mit je 2 Jahren davonkommen. Unstimmigkeiten über die Spielregeln, etwa, ob bei Mensch ärgere Dich le casino 770 est il fiable die Pflicht besteht, einen lavadome casino Kegel zu schlagen, wenn dies im betreffenden Zug möglich ist, oder ob bei Mau Mau eine gezogene Karte sofort gelegt*palina diamond city*darf, wenn sie passt, werden in der Regel als ernsthafte Störung betrachtet, wenn sie nicht vor dem Spiel comdirect top preis etf wurden. Excellent Post aber aber ich frage mich, wollen wissen, wenn Sie einen litte mehr zu diesem Thema zu schreiben Thema? Sie versucht dabei unter

**110 428,**das rationale Entscheidungsverhalten in sozialen Konfliktsituationen davon abzuleiten. Jeder Zug im Verlauf eines Spiels verlangt nach einem Spieler im Sinne eines unabhängigen Entscheiders, da die lokale Interessenlage einer Person oder Institution von Informationsbezirk zu Informationsbezirk divergieren kann. If tit-for-tat players mistake the second case for the first, they will defect, thereby setting off a chain-reaction of mutual defections from which they can never recover, since every player will reply to online casino no sticky bonus first encountered defection with defection, thereby begetting further defections, and

**110 428**on. Notice that Hobbes has not argued that tyranny is a desirable

*palina diamond city*in itself. They thus acted so as to create situations in which this was true for other Serbs Hutus as well. To this job we therefore now turn. Each terminal node corresponds to an outcome. People, Bacharach maintains, flip back and forth elitepartner premium testen individualistic agency and participation in team agency. Long before game theory had come along to show analysts how to think about this spinters of problem systematically, basketball em live im internet had occurred to some actual military leaders and influenced their strategies. Now imagine expanding the set of possible prizes easy withdrawal online casino that it lottoland.com seriös prizes that the agent values as intermediate between W and L. As a result, when set into what is intended to be a one-shot PD in the experimental laboratory, people tend to initially play as if the game were a single round of a repeated PD. It was said above that people might usually play as if they believe in trembling hands.

### spieletheorie - opinion you

Wenn man von der arbeitssparenden Form eines Spiels ausgeht, so muss eine entsprechende Strategie eines Spielers für jeden Informationsbezirk ein Zug ausgewählt werden. Man kann leicht zeigen, dass jedes Spiel, dessen Aktionsmengen endlich sind, ein Nash-Gleichgewicht in gemischten Strategien haben muss. Die Normalform beschränkt sich im Wesentlichen auf die A-priori- Strategiemengen der einzelnen Spieler und eine Auszahlungsfunktion als Funktion der gewählten Strategiekombinationen. Im Unterschied zur klassischen Entscheidungstheorie modelliert diese Theorie also Situationen, in denen der Erfolg des Einzelnen nicht nur vom eigenen Handeln, sondern auch von dem anderer abhängt interdependente Entscheidungssituation. Falls unterschiedliche Ergebnisse erzielt werden, gewinnt der zweite Spieler einen Euro. Durch die Nutzung dieser Website erklären Sie sich mit den Nutzungsbedingungen und der Datenschutzrichtlinie einverstanden. Eigentlicher Fokus spieltheoretischer Untersuchungen ist nicht die Verhaltensoptimierung in Gesellschaftsspielen. In der Spieltheorie gibt es dominierende und inferiore Strategien mit verschiedenen Strategievektoren. Standard ist das Spiel mit vollständiger Information sowie perfektem Erinnerungsvermögen.Ein Spiel im Sinne der Spieltheorie ist eine Entscheidungssituation mit mehreren Beteiligten, die sich mit ihren Entscheidungen gegenseitig beeinflussen.

Der Begriff Spieltheorie engl. Generell wird die nichtkooperative von der kooperativen Spieltheorie so unterschieden: Kooperative Spieltheorie ist als axiomatische Theorie von Koalitionsfunktionen charakteristischen Funktionen aufzufassen und ist auszahlungsorientiert.

Nichtkooperative Spieltheorie ist dagegen aktions- bzw. Aumann und John Forbes Nash Jr. Historischer Ausgangspunkt der Spieltheorie ist die Analyse des Homo oeconomicus , insbesondere durch Bernoulli , Bertrand , Cournot , Edgeworth , von Zeuthen und von Stackelberg.

Dieses Buch gilt auch heute noch als wegweisender Meilenstein. Simon und Daniel Kahneman den Nobelpreis. Maskin und Roger B.

Die Spieltheorie modelliert die verschiedensten Situationen als ein Spiel. Im Spiel Gefangenendilemma sind die Spieler die beiden Gefangenen und ihre Aktionsmengen sind aussagen und schweigen.

In der Informatik versucht man, mit Hilfe von Suchstrategien und Heuristiken allgemein: Man spricht in diesem Zusammenhang vom first movers advantage bzw.

Unterschieden werden hierbei drei Begriffe: Darum wird in spieltheoretischen Modellen meist nicht von perfekter Information ausgegangen.

Spiele werden meist entweder in strategischer Normal- Form oder in extensiver Form beschrieben. Weiterhin ist noch die Agentennormalform zu nennen.

Gerecht wird diese Darstellungsform am ehesten solchen Spielen, bei denen alle Spieler ihre Strategien zeitgleich und ohne Kenntnis der Wahl der anderen Spieler festlegen.

Zur Veranschaulichung verwendet man meist eine Bimatrixform. Wer oder was ist eigentlich ein Spieler in einer gegebenen Situation? The distinction between acting parametrically on a passive world and acting non-parametrically on a world that tries to act in anticipation of these actions is fundamental.

The values of all of these variables are independent of your plans and intentions, since the rock has no interests of its own and takes no actions to attempt to assist or thwart you.

Furthermore, his probable responses should be expected to visit costs upon you, which you would be wise to consider.

Finally, the relative probabilities of his responses will depend on his expectations about your probable responses to his responses.

The logical issues associated with the second sort of situation kicking the person as opposed to the rock are typically much more complicated, as a simple hypothetical example will illustrate.

Suppose first that you wish to cross a river that is spanned by three bridges. Assume that swimming, wading or boating across are impossible.

The first bridge is known to be safe and free of obstacles; if you try to cross there, you will succeed. The second bridge lies beneath a cliff from which large rocks sometimes fall.

The third is inhabited by deadly cobras. Now suppose you wish to rank-order the three bridges with respect to their preferability as crossing-points.

The first bridge is obviously best, since it is safest. To rank-order the other two bridges, you require information about their relative levels of danger.

Your reasoning here is strictly parametric because neither the rocks nor the cobras are trying to influence your actions, by, for example, concealing their typical patterns of behaviour because they know you are studying them.

It is obvious what you should do here: Now let us complicate the situation a bit. Your decision-making situation here is slightly more complicated, but it is still strictly parametric.

However, this is all you must decide, and your probability of a successful crossing is entirely up to you; the environment is not interested in your plans.

However, if we now complicate the situation by adding a non-parametric element, it becomes more challenging. Suppose that you are a fugitive of some sort, and waiting on the other side of the river with a gun is your pursuer.

She will catch and shoot you, let us suppose, only if she waits at the bridge you try to cross; otherwise, you will escape. As you reason through your choice of bridge, it occurs to you that she is over there trying to anticipate your reasoning.

It will seem that, surely, choosing the safe bridge straight away would be a mistake, since that is just where she will expect you, and your chances of death rise to certainty.

So perhaps you should risk the rocks, since these odds are much better. But wait … if you can reach this conclusion, your pursuer, who is just as rational and well-informed as you are, can anticipate that you will reach it, and will be waiting for you if you evade the rocks.

So perhaps you must take your chances with the cobras; that is what she must least expect. But, then, no … if she expects that you will expect that she will least expect this, then she will most expect it.

This dilemma, you realize with dread, is general: You appear to be trapped in indecision. All that might console you a bit here is that, on the other side of the river, your pursuer is trapped in exactly the same quandary, unable to decide which bridge to wait at because as soon as she imagines committing to one, she will notice that if she can find a best reason to pick a bridge, you can anticipate that same reason and then avoid her.

We know from experience that, in situations such as this, people do not usually stand and dither in circles forever.

However, until the s neither philosophers nor economists knew how to find it mathematically. As a result, economists were forced to treat non-parametric influences as if they were complications on parametric ones.

This is likely to strike the reader as odd, since, as our example of the bridge-crossing problem was meant to show, non-parametric features are often fundamental features of decision-making problems.

Classical economists, such as Adam Smith and David Ricardo, were mainly interested in the question of how agents in very large markets—whole nations—could interact so as to bring about maximum monetary wealth for themselves.

Economists always recognized that this set of assumptions is purely an idealization for purposes of analysis, not a possible state of affairs anyone could try or should want to try to attain.

But until the mathematics of game theory matured near the end of the s, economists had to hope that the more closely a market approximates perfect competition, the more efficient it will be.

No such hope, however, can be mathematically or logically justified in general; indeed, as a strict generalization the assumption was shown to be false as far back as the s.

This article is not about the foundations of economics, but it is important for understanding the origins and scope of game theory to know that perfectly competitive markets have built into them a feature that renders them susceptible to parametric analysis.

Because agents face no entry costs to markets, they will open shop in any given market until competition drives all profits to zero.

This implies that if production costs are fixed and demand is exogenous, then agents have no options about how much to produce if they are trying to maximize the differences between their costs and their revenues.

These production levels can be determined separately for each agent, so none need pay attention to what the others are doing; each agent treats her counterparts as passive features of the environment.

The other kind of situation to which classical economic analysis can be applied without recourse to game theory is that of a monopoly facing many customers.

However, both perfect and monopolistic competition are very special and unusual market arrangements. Prior to the advent of game theory, therefore, economists were severely limited in the class of circumstances to which they could neatly apply their models.

Philosophers share with economists a professional interest in the conditions and techniques for the maximization of human welfare.

In addition, philosophers have a special concern with the logical justification of actions, and often actions must be justified by reference to their expected outcomes.

One tradition in philosophy, utilitarianism, is based on the idea that all justifiable actions must be justified in this way. Without game theory, both of these problems resist analysis wherever non-parametric aspects are relevant.

In doing this, we will need to introduce, define and illustrate the basic elements and techniques of game theory.

To this job we therefore now turn. An economic agent is, by definition, an entity with preferences. Game theorists, like economists and philosophers studying rational decision-making, describe these by means of an abstract concept called utility.

This refers to some ranking, on some specified scale, of the subjective welfare or change in subjective welfare that an agent derives from an object or an event.

For example, we might evaluate the relative welfare of countries which we might model as agents for some purposes by reference to their per capita incomes, and we might evaluate the relative welfare of an animal, in the context of predicting and explaining its behavioral dispositions, by reference to its expected evolutionary fitness.

In the case of people, it is most typical in economics and applications of game theory to evaluate their relative welfare by reference to their own implicit or explicit judgments of it.

This is why we referred above to subjective welfare. Consider a person who adores the taste of pickles but dislikes onions.

She might be said to associate higher utility with states of the world in which, all else being equal, she consumes more pickles and fewer onions than with states in which she consumes more onions and fewer pickles.

However, economists in the early 20th century recognized increasingly clearly that their main interest was in the market property of decreasing marginal demand, regardless of whether that was produced by satiated individual consumers or by some other factors.

In the s this motivation of economists fit comfortably with the dominance of behaviourism and radical empiricism in psychology and in the philosophy of science respectively.

If this looks circular to you, it should: Like other tautologies occurring in the foundations of scientific theories, this interlocking recursive system of definitions is useful not in itself, but because it helps to fix our contexts of inquiry.

When such theorists say that agents act so as to maximize their utility, they want this to be part of the definition of what it is to be an agent, not an empirical claim about possible inner states and motivations.

Economists and others who interpret game theory in terms of RPT should not think of game theory as in any way an empirical account of the motivations of some flesh-and-blood actors such as actual people.

Rather, they should regard game theory as part of the body of mathematics that is used to model those entities which might or might not literally exist who consistently select elements from mutually exclusive action sets, resulting in patterns of choices, which, allowing for some stochasticity and noise, can be statistically modeled as maximization of utility functions.

On this interpretation, game theory could not be refuted by any empirical observations, since it is not an empirical theory in the first place.

Of course, observation and experience could lead someone favoring this interpretation to conclude that game theory is of little help in describing actual human behavior.

Some other theorists understand the point of game theory differently. They view game theory as providing an explanatory account of strategic reasoning.

These two very general ways of thinking about the possible uses of game theory are compatible with the tautological interpretation of utility maximization.

The philosophical difference is not idle from the perspective of the working game theorist, however. As we will see in a later section, those who hope to use game theory to explain strategic reasoning , as opposed to merely strategic behavior , face some special philosophical and practical problems.

Since game theory is a technology for formal modeling, we must have a device for thinking of utility maximization in mathematical terms.

Such a device is called a utility function. We will introduce the general idea of a utility function through the special case of an ordinal utility function.

Later, we will encounter utility functions that incorporate more information. Suppose that agent x prefers bundle a to bundle b and bundle b to bundle c.

We then map these onto a list of numbers, where the function maps the highest-ranked bundle onto the largest number in the list, the second-highest-ranked bundle onto the next-largest number in the list, and so on, thus:.

The only property mapped by this function is order. The magnitudes of the numbers are irrelevant; that is, it must not be inferred that x gets 3 times as much utility from bundle a as she gets from bundle c.

Thus we could represent exactly the same utility function as that above by. The numbers featuring in an ordinal utility function are thus not measuring any quantity of anything.

For the moment, however, we will need only ordinal functions. All situations in which at least one agent can only act to maximize his utility through anticipating either consciously, or just implicitly in his behavior the responses to his actions by one or more other agents is called a game.

Agents involved in games are referred to as players. If all agents have optimal actions regardless of what the others do, as in purely parametric situations or conditions of monopoly or perfect competition see Section 1 above we can model this without appeal to game theory; otherwise, we need it.

In literature critical of economics in general, or of the importation of game theory into humanistic disciplines, this kind of rhetoric has increasingly become a magnet for attack.

The reader should note that these two uses of one word within the same discipline are technically unconnected. Furthermore, original RPT has been specified over the years by several different sets of axioms for different modeling purposes.

Once we decide to treat rationality as a technical concept, each time we adjust the axioms we effectively modify the concept. Consequently, in any discussion involving economists and philosophers together, we can find ourselves in a situation where everyone uses the same word to refer to something different.

For readers new to economics, game theory, decision theory and the philosophy of action, this situation naturally presents a challenge.

We might summarize the intuition behind all this as follows: Economic rationality might in some cases be satisfied by internal computations performed by an agent, and she might or might not be aware of computing or having computed its conditions and implications.

In other cases, economic rationality might simply be embodied in behavioral dispositions built by natural, cultural or market selection. Each player in a game faces a choice among two or more possible strategies.

The significance of the italicized phrase here will become clear when we take up some sample games below. A crucial aspect of the specification of a game involves the information that players have when they choose strategies.

A board-game of sequential moves in which both players watch all the action and know the rules in common , such as chess, is an instance of such a game.

By contrast, the example of the bridge-crossing game from Section 1 above illustrates a game of imperfect information , since the fugitive must choose a bridge to cross without knowing the bridge at which the pursuer has chosen to wait, and the pursuer similarly makes her decision in ignorance of the choices of her quarry.

The difference between games of perfect and of imperfect information is related to though certainly not identical with! Let us begin by distinguishing between sequential-move and simultaneous-move games in terms of information.

It is natural, as a first approximation, to think of sequential-move games as being ones in which players choose their strategies one after the other, and of simultaneous-move games as ones in which players choose their strategies at the same time.

For example, if two competing businesses are both planning marketing campaigns, one might commit to its strategy months before the other does; but if neither knows what the other has committed to or will commit to when they make their decisions, this is a simultaneous-move game.

Chess, by contrast, is normally played as a sequential-move game: Chess can be turned into a simultaneous-move game if the players each call moves on a common board while isolated from one another; but this is a very different game from conventional chess.

It was said above that the distinction between sequential-move and simultaneous-move games is not identical to the distinction between perfect-information and imperfect-information games.

Explaining why this is so is a good way of establishing full understanding of both sets of concepts. As simultaneous-move games were characterized in the previous paragraph, it must be true that all simultaneous-move games are games of imperfect information.

However, some games may contain mixes of sequential and simultaneous moves. For example, two firms might commit to their marketing strategies independently and in secrecy from one another, but thereafter engage in pricing competition in full view of one another.

If the optimal marketing strategies were partially or wholly dependent on what was expected to happen in the subsequent pricing game, then the two stages would need to be analyzed as a single game, in which a stage of sequential play followed a stage of simultaneous play.

Whole games that involve mixed stages of this sort are games of imperfect information, however temporally staged they might be. Games of perfect information as the name implies denote cases where no moves are simultaneous and where no player ever forgets what has gone before.

As previously noted, games of perfect information are the logically simplest sorts of games. This is so because in such games as long as the games are finite, that is, terminate after a known number of actions players and analysts can use a straightforward procedure for predicting outcomes.

A player in such a game chooses her first action by considering each series of responses and counter-responses that will result from each action open to her.

She then asks herself which of the available final outcomes brings her the highest utility, and chooses the action that starts the chain leading to this outcome.

This process is called backward induction because the reasoning works backwards from eventual outcomes to present choice problems.

There will be much more to be said about backward induction and its properties in a later section when we come to discuss equilibrium and equilibrium selection.

For now, it has been described just so we can use it to introduce one of the two types of mathematical objects used to represent games: A game tree is an example of what mathematicians call a directed graph.

That is, it is a set of connected nodes in which the overall graph has a direction. We can draw trees from the top of the page to the bottom, or from left to right.

In the first case, nodes at the top of the page are interpreted as coming earlier in the sequence of actions.

In the case of a tree drawn from left to right, leftward nodes are prior in the sequence to rightward ones.

An unlabelled tree has a structure of the following sort:. The point of representing games using trees can best be grasped by visualizing the use of them in supporting backward-induction reasoning.

Just imagine the player or analyst beginning at the end of the tree, where outcomes are displayed, and then working backwards from these, looking for sets of strategies that describe paths leading to them.

We will present some examples of this interactive path selection, and detailed techniques for reasoning through these examples, after we have described a situation we can use a tree to model.

Trees are used to represent sequential games, because they show the order in which actions are taken by the players.

However, games are sometimes represented on matrices rather than trees. This is the second type of mathematical object used to represent games.

For example, it makes sense to display the river-crossing game from Section 1 on a matrix, since in that game both the fugitive and the hunter have just one move each, and each chooses their move in ignorance of what the other has decided to do.

Here, then, is part of the matrix:. Thus, for example, the upper left-hand corner above shows that when the fugitive crosses at the safe bridge and the hunter is waiting there, the fugitive gets a payoff of 0 and the hunter gets a payoff of 1.

Whenever the hunter waits at the bridge chosen by the fugitive, the fugitive is shot. These outcomes all deliver the payoff vector 0, 1.

You can find them descending diagonally across the matrix above from the upper left-hand corner. Whenever the fugitive chooses the safe bridge but the hunter waits at another, the fugitive gets safely across, yielding the payoff vector 1, 0.

These two outcomes are shown in the second two cells of the top row. All of the other cells are marked, for now , with question marks. The problem here is that if the fugitive crosses at either the rocky bridge or the cobra bridge, he introduces parametric factors into the game.

In these cases, he takes on some risk of getting killed, and so producing the payoff vector 0, 1 , that is independent of anything the hunter does.

In general, a strategic-form game could represent any one of several extensive-form games, so a strategic-form game is best thought of as being a set of extensive-form games.

Where order of play is relevant, the extensive form must be specified or your conclusions will be unreliable. The distinctions described above are difficult to fully grasp if all one has to go on are abstract descriptions.

Suppose that the police have arrested two people whom they know have committed an armed robbery together.

Unfortunately, they lack enough admissible evidence to get a jury to convict. They do , however, have enough evidence to send each prisoner away for two years for theft of the getaway car.

The chief inspector now makes the following offer to each prisoner: We can represent the problem faced by both of them on a single matrix that captures the way in which their separate choices interact; this is the strategic form of their game:.

Each cell of the matrix gives the payoffs to both players for each combination of actions. So, if both players confess then they each get a payoff of 2 5 years in prison each.

This appears in the upper-left cell. If neither of them confess, they each get a payoff of 3 2 years in prison each. This appears as the lower-right cell.

This appears in the upper-right cell. The reverse situation, in which Player II confesses and Player I refuses, appears in the lower-left cell.

Each player evaluates his or her two possible actions here by comparing their personal payoffs in each column, since this shows you which of their actions is preferable, just to themselves, for each possible action by their partner.

Player II, meanwhile, evaluates her actions by comparing her payoffs down each row, and she comes to exactly the same conclusion that Player I does.

Wherever one action for a player is superior to her other actions for each possible action by the opponent, we say that the first action strictly dominates the second one.

In the PD, then, confessing strictly dominates refusing for both players. Both players know this about each other, thus entirely eliminating any temptation to depart from the strictly dominated path.

Thus both players will confess, and both will go to prison for 5 years. The players, and analysts, can predict this outcome using a mechanical procedure, known as iterated elimination of strictly dominated strategies.

Player 1 can see by examining the matrix that his payoffs in each cell of the top row are higher than his payoffs in each corresponding cell of the bottom row.

Therefore, it can never be utility-maximizing for him to play his bottom-row strategy, viz. Now it is obvious that Player II will not refuse to confess, since her payoff from confessing in the two cells that remain is higher than her payoff from refusing.

So, once again, we can delete the one-cell column on the right from the game. We now have only one cell remaining, that corresponding to the outcome brought about by mutual confession.

Since the reasoning that led us to delete all other possible outcomes depended at each step only on the premise that both players are economically rational — that is, will choose strategies that lead to higher payoffs over strategies that lead to lower ones—there are strong grounds for viewing joint confession as the solution to the game, the outcome on which its play must converge to the extent that economic rationality correctly models the behavior of the players.

Had we begun by deleting the right-hand column and then deleted the bottom row, we would have arrived at the same solution. One of these respects is that all its rows and columns are either strictly dominated or strictly dominant.

In any strategic-form game where this is true, iterated elimination of strictly dominated strategies is guaranteed to yield a unique solution.

Later, however, we will see that for many games this condition does not apply, and then our analytic task is less straightforward. The reader will probably have noticed something disturbing about the outcome of the PD.

This is the most important fact about the PD, and its significance for game theory is quite general. For now, however, let us stay with our use of this particular game to illustrate the difference between strategic and extensive forms.

The reasoning behind this idea seems obvious: In fact, however, this intuition is misleading and its conclusion is false.

If Player I is convinced that his partner will stick to the bargain then he can seize the opportunity to go scot-free by confessing. Of course, he realizes that the same temptation will occur to Player II; but in that case he again wants to make sure he confesses, as this is his only means of avoiding his worst outcome.

But now suppose that the prisoners do not move simultaneously. This is the sort of situation that people who think non-communication important must have in mind.

Now Player II will be able to see that Player I has remained steadfast when it comes to her choice, and she need not be concerned about being suckered.

This gives us our opportunity to introduce game-trees and the method of analysis appropriate to them. First, however, here are definitions of some concepts that will be helpful in analyzing game-trees:.

Each terminal node corresponds to an outcome. These quick definitions may not mean very much to you until you follow them being put to use in our analyses of trees below.

It will probably be best if you scroll back and forth between them and the examples as we work through them. Player I is to commit to refusal first, after which Player II will reciprocate when the police ask for her choice.

Each node is numbered 1, 2, 3, … , from top to bottom, for ease of reference in discussion. Here, then, is the tree:.

Look first at each of the terminal nodes those along the bottom. These represent possible outcomes. Each of the structures descending from the nodes 1, 2 and 3 respectively is a subgame.

If the subgame descending from node 3 is played, then Player II will face a choice between a payoff of 4 and a payoff of 3. Consult the second number, representing her payoff, in each set at a terminal node descending from node 3.

II earns her higher payoff by playing D. We may therefore replace the entire subgame with an assignment of the payoff 0,4 directly to node 3, since this is the outcome that will be realized if the game reaches that node.

Now consider the subgame descending from node 2. Here, II faces a choice between a payoff of 2 and one of 0. She obtains her higher payoff, 2, by playing D.

We may therefore assign the payoff 2,2 directly to node 2. Now we move to the subgame descending from node 1.

This subgame is, of course, identical to the whole game; all games are subgames of themselves. Player I now faces a choice between outcomes 2,2 and 0,4.

Consulting the first numbers in each of these sets, he sees that he gets his higher payoff—2—by playing D. D is, of course, the option of confessing.

So Player I confesses, and then Player II also confesses, yielding the same outcome as in the strategic-form representation. What has happened here intuitively is that Player I realizes that if he plays C refuse to confess at node 1, then Player II will be able to maximize her utility by suckering him and playing D.

On the tree, this happens at node 3. This leaves Player I with a payoff of 0 ten years in prison , which he can avoid only by playing D to begin with.

He therefore defects from the agreement. This will often not be true of other games, however. As noted earlier in this section, sometimes we must represent simultaneous moves within games that are otherwise sequential.

We represent such games using the device of information sets. Consider the following tree:. The oval drawn around nodes b and c indicates that they lie within a common information set.

This means that at these nodes players cannot infer back up the path from whence they came; Player II does not know, in choosing her strategy, whether she is at b or c.

But you will recall from earlier in this section that this is just what defines two moves as simultaneous. We can thus see that the method of representing games as trees is entirely general.

If no node after the initial node is alone in an information set on its tree, so that the game has only one subgame itself , then the whole game is one of simultaneous play.

If at least one node shares its information set with another, while others are alone, the game involves both simultaneous and sequential play, and so is still a game of imperfect information.

Only if all information sets are inhabited by just one node do we have a game of perfect information. Following the general practice in economics, game theorists refer to the solutions of games as equilibria.

Philosophically minded readers will want to pose a conceptual question right here: Note that, in both physical and economic systems, endogenously stable states might never be directly observed because the systems in question are never isolated from exogenous influences that move and destabilize them.

In both classical mechanics and in economics, equilibrium concepts are tools for analysis , not predictions of what we expect to observe.

As we will see in later sections, it is possible to maintain this understanding of equilibria in the case of game theory. However, as we noted in Section 2.

For them, a solution to a game must be an outcome that a rational agent would predict using the mechanisms of rational computation alone. The interest of philosophers in game theory is more often motivated by this ambition than is that of the economist or other scientist.

A set of strategies is a NE just in case no player could improve her payoff, given the strategies of all other players in the game, by changing her strategy.

Notice how closely this idea is related to the idea of strict dominance: Now, almost all theorists agree that avoidance of strictly dominated strategies is a minimum requirement of economic rationality.

A player who knowingly chooses a strictly dominated strategy directly violates clause iii of the definition of economic agency as given in Section 2.

This implies that if a game has an outcome that is a unique NE, as in the case of joint confession in the PD, that must be its unique solution. We can specify one class of games in which NE is always not only necessary but sufficient as a solution concept.

These are finite perfect-information games that are also zero-sum. A zero-sum game in the case of a game involving just two players is one in which one player can only be made better off by making the other player worse off.

Tic-tac-toe is a simple example of such a game: We can put this another way: In tic-tac-toe, this is a draw. However, most games do not have this property.

For one thing, it is highly unlikely that theorists have yet discovered all of the possible problems. However, we can try to generalize the issues a bit.

First, there is the problem that in most non-zero-sum games, there is more than one NE, but not all NE look equally plausible as the solutions upon which strategically alert players would hit.

Consider the strategic-form game below taken from Kreps , p. This game has two NE: Note that no rows or columns are strictly dominated here.

But if Player I is playing s1 then Player II can do no better than t1, and vice-versa; and similarly for the s2-t2 pair.

If NE is our only solution concept, then we shall be forced to say that either of these outcomes is equally persuasive as a solution.

Note that this is not like the situation in the PD, where the socially superior situation is unachievable because it is not a NE. In the case of the game above, both players have every reason to try to converge on the NE in which they are better off.

Consider another example from Kreps , p. Here, no strategy strictly dominates another. So should not the players and the analyst delete the weakly dominated row s2?

When they do so, column t1 is then strictly dominated, and the NE s1-t2 is selected as the unique solution. However, as Kreps goes on to show using this example, the idea that weakly dominated strategies should be deleted just like strict ones has odd consequences.

Suppose we change the payoffs of the game just a bit, as follows:. Note that this game, again, does not replicate the logic of the PD.

There, it makes sense to eliminate the most attractive outcome, joint refusal to confess, because both players have incentives to unilaterally deviate from it, so it is not an NE.

This is not true of s2-t1 in the present game. If the possibility of departures from reliable economic rationality is taken seriously, then we have an argument for eliminating weakly dominated strategies: Player I thereby insures herself against her worst outcome, s2-t2.

Of course, she pays a cost for this insurance, reducing her expected payoff from 10 to 5. On the other hand, we might imagine that the players could communicate before playing the game and agree to play correlated strategies so as to coordinate on s2-t1, thereby removing some, most or all of the uncertainty that encourages elimination of the weakly dominated row s1, and eliminating s1-t2 as a viable solution instead!

Any proposed principle for solving games that may have the effect of eliminating one or more NE from consideration as solutions is referred to as a refinement of NE.

In the case just discussed, elimination of weakly dominated strategies is one possible refinement, since it refines away the NE s2-t1, and correlation is another, since it refines away the other NE, s1-t2, instead.

So which refinement is more appropriate as a solution concept? In principle, there seems to be no limit on the number of refinements that could be considered, since there may also be no limits on the set of philosophical intuitions about what principles a rational agent might or might not see fit to follow or to fear or hope that other players are following.

We now digress briefly to make a point about terminology. This reflected the fact the revealed preference approaches equate choices with economically consistent actions, rather than intending to refer to mental constructs.

However, this usage is likely to cause confusion due to the recent rise of behavioral game theory Camerer Applications also typically incorporate special assumptions about utility functions, also derived from experiments.

For example, players may be taken to be willing to make trade-offs between the magnitudes of their own payoffs and inequalities in the distribution of payoffs among the players.

We will turn to some discussion of behavioral game theory in Section 8. For the moment, note that this use of game theory crucially rests on assumptions about psychological representations of value thought to be common among people.

We mean by this the kind of game theory used by most economists who are not behavioral economists. They treat game theory as the abstract mathematics of strategic interaction, rather than as an attempt to directly characterize special psychological dispositions that might be typical in humans.

Non-psychological game theorists tend to take a dim view of much of the refinement program. This is for the obvious reason that it relies on intuitions about inferences that people should find sensible.

Like most scientists, non-psychological game theorists are suspicious of the force and basis of philosophical assumptions as guides to empirical and mathematical modeling.

Behavioral game theory, by contrast, can be understood as a refinement of game theory, though not necessarily of its solution concepts, in a different sense.

It motivates this restriction by reference to inferences, along with preferences, that people do find natural , regardless of whether these seem rational , which they frequently do not.

Non-psychological and behavioral game theory have in common that neither is intended to be normative—though both are often used to try to describe norms that prevail in groups of players, as well to explain why norms might persist in groups of players even when they appear to be less than fully rational to philosophical intuitions.

Let us therefore group non-psychological and behavioral game theorists together, just for purposes of contrast with normative game theorists, as descriptive game theorists.

Descriptive game theorists are often inclined to doubt that the goal of seeking a general theory of rationality makes sense as a project.

Institutions and evolutionary processes build many environments, and what counts as rational procedure in one environment may not be favoured in another.

On the other hand, an entity that does not at least stochastically i. To such entities game theory has no application in the first place.

This does not imply that non-psychological game theorists abjure all principled ways of restricting sets of NE to subsets based on their relative probabilities of arising.

In particular, non-psychological game theorists tend to be sympathetic to approaches that shift emphasis from rationality onto considerations of the informational dynamics of games.

We should perhaps not be surprised that NE analysis alone often fails to tell us much of applied, empirical interest about strategic-form games e.

Equilibrium selection issues are often more fruitfully addressed in the context of extensive-form games. In order to deepen our understanding of extensive-form games, we need an example with more interesting structure than the PD offers.

This game is not intended to fit any preconceived situation; it is simply a mathematical object in search of an application. If you are confused by this, remember that a strategy must tell a player what to do at every information set where that player has an action.

Since each player chooses between two actions at each of two information sets here, each player has four strategies in total.

The first letter in each strategy designation tells each player what to do if he or she reaches their first information set, the second what to do if their second information set is reached.

This is a bit puzzling, since if Player I reaches her second information set 7 in the extensive-form game, she would hardly wish to play L there; she earns a higher payoff by playing R at node 7.

In analyzing extensive-form games, however, we should care what happens off the path of play, because consideration of this is crucial to what happens on the path.

We are throwing away information relevant to game solutions if we ignore off-path outcomes, as mere NE analysis does.

Notice that this reason for doubting that NE is a wholly satisfactory equilibrium concept in itself has nothing to do with intuitions about rationality, as in the case of the refinement concepts discussed in Section 2.

Begin, again, with the last subgame, that descending from node 7. At node 5 II chooses R. Note that, as in the PD, an outcome appears at a terminal node— 4, 5 from node 7—that is Pareto superior to the NE.

Again, however, the dynamics of the game prevent it from being reached. It gives an outcome that yields a NE not just in the whole game but in every subgame as well.

This is a persuasive solution concept because, again unlike the refinements of Section 2. It does, however, assume that players not only know everything strategically relevant to their situation but also use all of that information.

But, as noted earlier, it is best to be careful not to confuse the general normative idea of rationality with computational power and the possession of budgets, in time and energy, to make the most of it.

An agent playing a subgame perfect strategy simply chooses, at every node she reaches, the path that brings her the highest payoff in the subgame emanating from that node.

A main value of analyzing extensive-form games for SPE is that this can help us to locate structural barriers to social optimization.

If our players wish to bring about the more socially efficient outcome 4,5 here, they must do so by redesigning their institutions so as to change the structure of the game.

The enterprise of changing institutional and informational structures so as to make efficient outcomes more likely in the games that agents that is, people, corporations, governments, etc.

The main techniques are reviewed in Hurwicz and Reiter , the first author of which was awarded the Nobel Prize for his pioneering work in the area.

Many readers, but especially philosophers, might wonder why, in the case of the example taken up in the previous section, mechanism design should be necessary unless players are morbidly selfish sociopaths.

This theme is explored with great liveliness and polemical force in Binmore , We have seen that in the unique NE of the PD, both players get less utility than they could have through mutual cooperation.

This may strike you, even if you are not a Kantian as it has struck many commentators as perverse. Surely, you may think, it simply results from a combination of selfishness and paranoia on the part of the players.

To begin with they have no regard for the social good, and then they shoot themselves in the feet by being too untrustworthy to respect agreements.

This way of thinking is very common in popular discussions, and badly mixed up. To dispel its influence, let us first introduce some terminology for talking about outcomes.

Welfare economists typically measure social good in terms of Pareto efficiency. Now, the outcome 3,3 that represents mutual cooperation in our model of the PD is clearly Pareto superior over mutual defection; at 3,3 both players are better off than at 2,2.

So it is true that PDs lead to inefficient outcomes. This was true of our example in Section 2. However, inefficiency should not be associated with immorality.

A utility function for a player is supposed to represent everything that player cares about , which may be anything at all. As we have described the situation of our prisoners they do indeed care only about their own relative prison sentences, but there is nothing essential in this.

What makes a game an instance of the PD is strictly and only its payoff structure. Thus we could have two Mother Theresa types here, both of whom care little for themselves and wish only to feed starving children.

But suppose the original Mother Theresa wishes to feed the children of Calcutta while Mother Juanita wishes to feed the children of Bogota.

Our saints are in a PD here, though hardly selfish or unconcerned with the social good. In that case, this must be reflected in their utility functions, and hence in their payoffs.

But all this shows is that not every possible situation is a PD; it does not show that selfishness is among the assumptions of game theory. Agents who wish to avoid inefficient outcomes are best advised to prevent certain games from arising; the defender of the possibility of Kantian rationality is really proposing that they try to dig themselves out of such games by turning themselves into different kinds of agents.

In general, then, a game is partly defined by the payoffs assigned to the players. In any application, such assignments should be based on sound empirical evidence.

Our last point above opens the way to a philosophical puzzle, one of several that still preoccupy those concerned with the logical foundations of game theory.

It can be raised with respect to any number of examples, but we will borrow an elegant one from C. Consider the following game:.

The NE outcome here is at the single leftmost node descending from node 8. To see this, backward induct again. At node 10, I would play L for a payoff of 3, giving II a payoff of 1.

II can do better than this by playing L at node 9, giving I a payoff of 0. I can do better than this by playing L at node 8; so that is what I does, and the game terminates without II getting to move.

A puzzle is then raised by Bicchieri along with other authors, including Binmore and Pettit and Sugden by way of the following reasoning.

But now we have the following paradox: Both players use backward induction to solve the game; backward induction requires that Player I know that Player II knows that Player I is economically rational; but Player II can solve the game only by using a backward induction argument that takes as a premise the failure of Player I to behave in accordance with economic rationality.

This is the paradox of backward induction. That is, a player might intend to take an action but then slip up in the execution and send the game down some other path instead.

In our example, Player II could reason about what to do at node 9 conditional on the assumption that Player I chose L at node 8 but then slipped.

Gintis points out that the apparent paradox does not arise merely from our supposing that both players are economically rational. It rests crucially on the additional premise that each player must know, and reasons on the basis of knowing, that the other player is economically rational.

A player has reason to consider out-of-equilibrium possibilities if she either believes that her opponent is economically rational but his hand may tremble or she attaches some nonzero probability to the possibility that he is not economically rational or she attaches some doubt to her conjecture about his utility function.

We will return to this issue in Section 7 below. The paradox of backward induction, like the puzzles raised by equilibrium refinement, is mainly a problem for those who view game theory as contributing to a normative theory of rationality specifically, as contributing to that larger theory the theory of strategic rationality.

This involves appeal to the empirical fact that actual agents, including people, must learn the equilibrium strategies of games they play, at least whenever the games are at all complicated.

What it means to say that people must learn equilibrium strategies is that we must be a bit more sophisticated than was indicated earlier in constructing utility functions from behavior in application of Revealed Preference Theory.

Instead of constructing utility functions on the basis of single episodes, we must do so on the basis of observed runs of behavior once it has stabilized , signifying maturity of learning for the subjects in question and the game in question.

As a result, when set into what is intended to be a one-shot PD in the experimental laboratory, people tend to initially play as if the game were a single round of a repeated PD.

The repeated PD has many Nash equilibria that involve cooperation rather than defection. Thus experimental subjects tend to cooperate at first in these circumstances, but learn after some number of rounds to defect.

The experimenter cannot infer that she has successfully induced a one-shot PD with her experimental setup until she sees this behavior stabilize.

If players of games realize that other players may need to learn game structures and equilibria from experience, this gives them reason to take account of what happens off the equilibrium paths of extensive-form games.

Of course, if a player fears that other players have not learned equilibrium, this may well remove her incentive to play an equilibrium strategy herself.

This raises a set of deep problems about social learning Fudenberg and Levine The crucial answer in the case of applications of game theory to interactions among people is that young people are socialized by growing up in networks of institutions , including cultural norms.

Most complex games that people play are already in progress among people who were socialized before them—that is, have learned game structures and equilibria Ross a.

Novices must then only copy those whose play appears to be expected and understood by others. Institutions and norms are rich with reminders, including homilies and easily remembered rules of thumb, to help people remember what they are doing Clark As noted in Section 2.

Given the complexity of many of the situations that social scientists study, we should not be surprised that mis-specification of models happens frequently.

Applied game theorists must do lots of learning, just like their subjects. Thus the paradox of backward induction is only apparent.

Unless players have experienced play at equilibrium with one another in the past, even if they are all economically rational and all believe this about one another, we should predict that they will attach some positive probability to the conjecture that understanding of game structures among some players is imperfect.

This then explains why people, even if they are economically rational agents, may often, or even usually, play as if they believe in trembling hands.

Learning of equilibria may take various forms for different agents and for games of differing levels of complexity and risk.

## 2 comments on “Spieletheorie”

## Natilar

10.02.2018 at 08:10Wacker, welche ausgezeichnete Antwort.