Die Lehrveranstaltung behandelt die Grundlagen der nichtkooperativen Spieltheorie (interaktiven Entscheidungstheorie) und ihre Anwendungen auf. Die Spieltheorie ist eine mathematische Theorie, in der Entscheidungssituationen modelliert werden, in denen mehrere Beteiligte miteinander interagieren. Einführung in die. Spieltheorie von. Prof. Dr. Wolfgang Leininger und. PD Dr. Erwin Amann. Lehrstuhl Wirtschaftstheorie. Universität Dortmund. Postfach Experiments in Strategic Interactionpp. Applied mathematics Discrete mathematics Economics Games Decision theory Subdivisions of mathematics. Philosophically minded book of ra klassisch will want to pose a conceptual question right here: But you will casino lounge gaming club from earlier in this section that this is just what defines two moves as simultaneous. We can instead spieletheorie that teams fallenbrunnen casino often exogenously welded into being by complex interrelated psychological and institutional processes. 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 fifa 17 freundschaftsspiel geht nicht market competition with many players. The basic insight can 10bet login captured using a simple example. Gametree Odd Even 1. 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. In that round, it will be utility-maximizing for players to defect, since no punishment will be possible. Strategy and Market Structure: In fact, neither of us actually needs to be immoral to get this chain of mutual reasoning going; we need only think spieletheorie there is some possibility that the other might try tuchel spieler cheat on bargains.

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. 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.

This little thought experiment is plausible, but not ideal as an illustration because it is easily conflated with vague notions we might entertain about fusion of agency in the ideal of marriage—and it is important to distinguish the dynamics of preference conditionalization in teams of distinct agents from the simple collapse of individual agency.

So let us imagine a better example. Imagine a corporate Chairwoman consulting her risk-averse Board about whether they should pursue a dangerous hostile takeover bid.

Compare two possible procedures she might use: In both imagined processes there are, at the point of voting, sets of individual preferences to be aggregated by the vote.

But it is more likely that some preferences in the set generated by the second process were conditional on preferences of others.

A conditional preference as Stirling defines it is a preference that is influenced by information about the preferences of specified others.

This refers to the extent of controversy or discord to which a set of preferences, including a set of conditional preferences, would generate if equilibrium among them were implemented.

Members or leaders of teams do not always want to maximize concordance by engineering all internal games as Assurance or Hi-lo though they will always likely want to eliminate PDs.

For example, a manager might want to encourage a degree of competition among profit centers in a firm, while wanting the cost centers to identify completely with the team as a whole.

Stirling formally defines representation theorems for three kinds of ordered utility functions: These may be applied recursively, i. Stirling does not mention the work of Bacharach, so does not set his theory within the context of team reasoning or what we might reinterpret as team-centred choice.

We can then paraphrase his five constraints on aggregation as follows:. Influence may be set to zero, in which case the conditional preference ordering collapses to the categorical preference ordering to standard RPT.

A concordant ordering for a team must be determined by the social interactions of its sub-teams. This condition ensures that team preferences are not simply imposed on individual preferences.

Social influence relations are not reciprocal. This will likely look at first glance to be a strange restriction: But, as noted earlier, we need to keep conditional preference distinct from agent fusion, and this condition helps to do that.

More importantly, as a matter of mathematics it allows teams to be represented in directed graphs. The condition is not as restrictive, where modeling flexibility is concerned, as one might at first think, because it only bars us from representing an agent j influenced by another agent i from directly influencing i.

We are free to represent j as influencing k who in turn influences i. Concordant preference orderings are invariant under representational transformations that are equivalent with respect to information about conditional preferences.

If one sub-team prefers choice alternative A to B and all other sub-teams are indifferent between A and B, then the team does not prefer B to A.

Under these restrictions, Stirling proves an aggregation theorem which follows a general result for updating utility in light of new information that was developed by Abbas , Other Internet Resources.

Individual team members each calculate the team preference by aggregating conditional concordant preferences. Then the analyst applies marginalization.

This operation produces the non-conditional preferences of individual i ex post—that is, updated in light of her conditional concordant preferences and the information on which they are conditioned, namely, the conditional concordant preferences of the team.

Once all ex post preferences of agents have been calculated, the resulting games in which they are involved can be solved by standard analysis.

It provides a basis for formalization of team utility, which can be compared with any of the following: Situations of incomplete information can be solved using Byes-Nash or sequential equilibrium.

In some games, a player can improve her outcome by taking an action that makes it impossible for her to take what would be her best action in the corresponding simultaneous-move game.

Such actions are referred to as commitments , and they can serve as alternatives to external enforcement in games which would otherwise settle on Pareto-inefficient equilibria.

Consider the following hypothetical example which is not a PD. If we move simultaneously—you post a selling price and I independently give my agent an asking price—there will be no sale.

However, this move so far changes nothing. If you refuse to sell in the face of my threat, it is then not in my interest to carry it out, because in damaging you I also damage myself.

Since you know this you should ignore my threat. My threat is incredible , a case of cheap talk. However, I could make my threat credible by committing myself.

For example, I could sign a contract with some farmers promising to supply them with treated sewage fertilizer from my plant, but including an escape clause in the contract releasing me from my obligation only if I can double my lot size and so put it to some other use.

Now my threat is credible: Since you know this, you now have an incentive to sell me your land in order to escape its ruination. This sort of case exposes one of many fundamental differences between the logic of non-parametric and parametric maximization.

In parametric situations, an agent can never be made worse off by having more options. Even if a new option is worse than the options with which she began, she can just ignore it.

Another example will illustrate this, as well as the applicability of principles across game-types. Here we will build an imaginary situation that is not a PD—since only one player has an incentive to defect—but which is a social dilemma insofar as its NE in the absence of commitment is Pareto-inferior to an outcome that is achievable with a commitment device.

Suppose that two of us wish to poach a rare antelope from a national park in order to sell the trophy. One of us must flush the animal down towards the second person, who waits in a blind to shoot it and load it onto a truck.

You promise, of course, to share the proceeds with me. However, your promise is not credible. But now suppose I add the following opening move to the game.

Before our hunt, I rig out the truck with an alarm that can be turned off only by punching in a code. Only I know the code.

You, knowing this, now have an incentive to wait for me. What is crucial to notice here is that you prefer that I rig up the alarm, since this makes your promise to give me my share credible.

We may now combine our analysis of PDs and commitment devices in discussion of the application that first made game theory famous outside of the academic community.

The nuclear stand-off between the superpowers during the Cold War was exhaustively studied by the first generation of game theorists, many of whom worked for the US military.

See Poundstone for historical details. If one side launched a first strike, the other threatened to answer with a devastating counter-strike.

Game theorists objected that MAD was mad, because it set up a PD as a result of the fact that the reciprocal threats were incredible.

The reasoning behind this diagnosis went as follows. At that point, the American President finds his country already destroyed. Since the Russians can anticipate this, they should ignore the threat to retaliate and strike first.

Of course, the Americans are in an exactly symmetric position, so they too should strike first. What we should therefore expect, because it is the only NE of the game, is a race between the two powers to be the first to attack.

This game-theoretic analysis caused genuine consternation and fear on both sides during the Cold War, and is reputed to have produced some striking attempts at setting up strategic commitment devices.

They equipped a worldwide fleet of submarines with enough missiles to destroy the USSR. This made the reliability of their communications network less straightforward, and in so doing introduced an element of strategically relevant uncertainty.

The President probably could be less sure to be able to reach the submarines and cancel their orders to attack if any Soviet missile crossed the radar trigger line in Northern Canada.

Of course, the value of this in breaking symmetry depended on the Russians being aware of the potential problem. As a result, when an unequivocally mad American colonel launches missiles at Russia on his own accord, and the American President tries to convince his Soviet counterpart that the attack was unintended, the Russian Premier sheepishly tells him about the secret doomsday machine.

Now the two leaders can do nothing but watch in dismay as the world is blown up due to a game-theoretic mistake. The military game theorists were almost certainly mistaken to the extent that they modeled the Cold War as a one-shot PD in the first place.

Zero-sum games are a special case of constant-sum games, in which choices by players can neither increase nor decrease the available resources.

In zero-sum games the total benefit to all players in the game, for every combination of strategies, always adds to zero more informally, a player benefits only at the equal expense of others.

Other zero-sum games include matching pennies and most classical board games including Go and chess. Informally, in non-zero-sum games, a gain by one player does not necessarily correspond with a loss by another.

Constant-sum games correspond to activities like theft and gambling, but not to the fundamental economic situation in which there are potential gains from trade.

Sequential games or dynamic games are games where later players have some knowledge about earlier actions. This need not be perfect information about every action of earlier players; it might be very little knowledge.

The difference between simultaneous and sequential games is captured in the different representations discussed above.

Often, normal form is used to represent simultaneous games, while extensive form is used to represent sequential ones.

The transformation of extensive to normal form is one way, meaning that multiple extensive form games correspond to the same normal form.

Consequently, notions of equilibrium for simultaneous games are insufficient for reasoning about sequential games; see subgame perfection.

An important subset of sequential games consists of games of perfect information. A game is one of perfect information if all players know the moves previously made by all other players.

Most games studied in game theory are imperfect-information games. Many card games are games of imperfect information, such as poker and bridge.

Games of incomplete information can be reduced, however, to games of imperfect information by introducing " moves by nature ".

Games in which the difficulty of finding an optimal strategy stems from the multiplicity of possible moves are called combinatorial games.

Examples include chess and go. Games that involve imperfect information may also have a strong combinatorial character, for instance backgammon.

There is no unified theory addressing combinatorial elements in games. There are, however, mathematical tools that can solve particular problems and answer general questions.

Games of perfect information have been studied in combinatorial game theory , which has developed novel representations, e. These methods address games with higher combinatorial complexity than those usually considered in traditional or "economic" game theory.

A related field of study, drawing from computational complexity theory , is game complexity , which is concerned with estimating the computational difficulty of finding optimal strategies.

Research in artificial intelligence has addressed both perfect and imperfect information games that have very complex combinatorial structures like chess, go, or backgammon for which no provable optimal strategies have been found.

The practical solutions involve computational heuristics, like alpha-beta pruning or use of artificial neural networks trained by reinforcement learning , which make games more tractable in computing practice.

Games, as studied by economists and real-world game players, are generally finished in finitely many moves. Pure mathematicians are not so constrained, and set theorists in particular study games that last for infinitely many moves, with the winner or other payoff not known until after all those moves are completed.

The focus of attention is usually not so much on the best way to play such a game, but whether one player has a winning strategy. 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.

In the Downsian model, political candidates commit to ideologies on a one-dimensional policy space. Downs first shows how the political candidates will converge to the ideology preferred by the median voter if voters are fully informed, but then argues that voters choose to remain rationally ignorant which allows for candidate divergence.

Game Theory was applied in to the Cuban missile crisis during the presidency of John F. It has also been proposed that game theory explains the stability of any form of political government.

Taking the simplest case of a monarchy, for example, the king, being only one person, does not and cannot maintain his authority by personally exercising physical control over all or even any significant number of his subjects.

Sovereign control is instead explained by the recognition by each citizen that all other citizens expect each other to view the king or other established government as the person whose orders will be followed.

Coordinating communication among citizens to replace the sovereign is effectively barred, since conspiracy to replace the sovereign is generally punishable as a crime.

A game-theoretic explanation for democratic peace is that public and open debate in democracies sends clear and reliable information regarding their intentions to other states.

In contrast, it is difficult to know the intentions of nondemocratic leaders, what effect concessions will have, and if promises will be kept.

Thus there will be mistrust and unwillingness to make concessions if at least one of the parties in a dispute is a non-democracy. On the other hand, game theory predicts that two countries may still go to war even if their leaders are cognizant of the costs of fighting.

War may result from asymmetric information; two countries may have incentives to mis-represent the amount of military resources they have on hand, rendering them unable to settle disputes agreeably without resorting to fighting.

Moreover, war may arise because of commitment problems: Finally, war may result from issue indivisibilities. Wood thought this could be accomplished by making treaties with other nations to reduce greenhouse gas emissions.

Unlike those in economics, the payoffs for games in biology are often interpreted as corresponding to fitness.

In addition, the focus has been less on equilibria that correspond to a notion of rationality and more on ones that would be maintained by evolutionary forces.

Although its initial motivation did not involve any of the mental requirements of the Nash equilibrium , every ESS is a Nash equilibrium.

In biology, game theory has been used as a model to understand many different phenomena. It was first used to explain the evolution and stability of the approximate 1: Fisher suggested that the 1: Additionally, biologists have used evolutionary game theory and the ESS to explain the emergence of animal communication.

For example, the mobbing behavior of many species, in which a large number of prey animals attack a larger predator, seems to be an example of spontaneous emergent organization.

Biologists have used the game of chicken to analyze fighting behavior and territoriality.

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A hedonic game with 5 players that has empty core. Applications of Parity Games. Battle of the sexes - imperfect information.

Battle of the sexes - perfect information. BR fce pro 2h BoS. Chicken Two Pop Replicator Dynamics. Combined utility graph - product.

Complex systems organizational map. Cournot Theory of Wealth Cuban Missile Crisis Game Tree. Emanuel Uniform Price Auction.

EVIU time to drive to airport. Example Parity Game Solved. Exemplo do Paradoxo de Braess. Exhibit No 01 Alexander Rosenfeld.

Extensive form game 2. Extensive form game 3. Extensive form game 4. Extensive form of Stay Firm or Give In. Extensive-form tree with uncertainty.

Game of sprouts with n initial vertices, ending in minimum number of moves. Game Theory An extensive form game. Game theory author Mitsuo Suzuki.

Game Theory by Morton Davis - Flickr - brewbooks. Game theory schwimmbad kino. Myerson - Flickr - brewbooks. Game with no value.

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