This becomes problematic in interactive decisions, when individuals have only partial control over the outcomes, because expected utility maximization is undefined in the absence of assumptions about how the other participants will behave. Game theory therefore incorporates not only rationality but also common knowledge assumptions, enabling players to anticipate their co-players' strategies. Under these assumptions, disparate anomalies emerge. Instrumental rationality, conventionally interpreted, fails to explain intuitively obvious features of human interaction, yields predictions starkly at variance with experimental findings, and breaks down completely in certain cases.
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. 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 a 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. The paradox of backward induction is one of a family of paradoxes that arise if one builds possession and use of literally complete information into a concept of rationality.
Consider, by analogy, the stock market paradox that arises if we suppose that economically rational investment incorporates literally rational expectations: assume that no individual investor can beat the market in the long run because the market always knows everything the investor knows; then no one has incentive to gather knowledge about asset values; then no one will ever gather any such information and so from the assumption that the market knows everything it follows that the market cannot know anything!
As we will see in detail in various discussions below, most applications of game theory explicitly incorporate uncertainty and prospects for learning by players.
The extensive-form games with SPE that we looked at above are really conceptual tools to help us prepare concepts for application to situations where complete and perfect information is unusual. We cannot avoid the paradox if we think, as some philosophers and normative game theorists do, that one of the conceptual tools we want to use game theory to sharpen is a fully general idea of rationality itself.
But this is not a concern entertained by economists and other scientists who put game theory to use in empirical modeling. In real cases, 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 ; the same authors provide a non-technical overview of the issues in Fudenberg and Levine A first important distinction is between learning specific parameters between rounds of a repeated game see Section 4 with common players, and learning about general strategic expectations across different games.
The latter can include learning about players if the learner is updating expectations based on her models of types of players she recurrently encounters. A major difficulty for both players and modelers is that screening moves might be misinterpreted if players are also incentivized to make moves to signal information to one another see Section 4.
Finally, the discussion so far has assumed that all possible learning in a game is about the structure of the game itself.
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 is 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. The idea is sufficiently rich that its depths remain an open domain of investigation by game theorists. 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 the fugitive cares more 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. 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? 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. It is not. 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.
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. The Nobel laureate Thomas Schelling conjectured, and empirically demonstrated, that 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: those who slow down on yellows while others are rushing them will get rear-ended, while those who rush yellows in the other equilibrium will risk collision with those who jump off straightaway on greens.
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. Conditional game theory see Section 5 below provides promising resources for modeling cases such as this one, in which maintenance of coordination game equilibria likely must be supported by stable social norms, because players are anonymous and encounter regular opportunities to gain once-off advantages by defecting from supporting the prevailing equilibrium.
This work is currently ongoing. 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 international war on earth since , the Congo War of — Of course, it is not the case that most repeated games lead to disasters. In an influential monograph, Schelling uses game theory brilliantly to illuminate psychological features of human strategic interaction.
Binmore, K. Fun and games: A text on game theory. Lexington, MA: Heath. This is a basic text on mathematical game theory written by a leading game theorist. It presents mathematical aspects of the theory exceptionally clearly, and readers with a basic knowledge of school mathematics should be able to understand it.
Parts of it are far from elementary, making it interesting and informative even for readers with an intermediate-level understanding of game theory. Game theory: A very short introduction. Oxford: Oxford Univ. DOI: This very short introduction to the formal aspects of the theory outlines the basic ideas in an easily digestible form. Camerer, C. Behavioral game theory: Experiments in strategic interaction. Princeton, NJ: Princeton Univ. This is a magisterial review of almost the whole of behavioral game theory up to the early s.
This book covers many key topics in remarkable depth, and much of it is essentially psychological in flavor. Colman, A. Game theory and its applications in the social and biological sciences. London: Routledge. This monograph presents the basic ideas of game theory from a psychological perspective, reviews experimental evidence up to the mids, and discusses applications of game theory to voting, evolution of cooperation, and moral philosophy.
An appendix contains the most elementary available self-contained proof of the minimax theorem see Strategic Reasoning Before Game Theory. The first edition was published in Gibbons, R. A primer in game theory. A more orthodox and slightly simpler and shorter basic text on mathematical game theory than Binmore , widely prescribed in standard university courses and easily accessible to readers with a basic knowledge of school mathematics.
Luce, R.
0コメント