According to the standard objection to backward induction in games, its application depends on highly questionable assumptions about the players' expectations as regards future counterfactual game developments. It seems that, in order to make predictions needed for backward reasoning, the players must expect each player to act rationally at each node that in principle could be reached in the game, and also to expect that this confidence in the future rationality of the players would be kept by each player come (...) what may: even at the game-nodes that could only be reached by irrational play. Both expectations seem to be rather unreasonable: a player's initial disposition to rational behaviour may be weakened by a long stretch of irrational play on his part and, even more importantly, his initial confidence in the other players' future rationality may be undermined by an irrational play on their part. For different formulations of this objection see Binmore, Reny and, Bicchieri, Pettit and Sugden. and Aumann and.). (shrink)

The standard backward-induction reasoning in a game like the centipede assumes that the players maintain a common belief in rationality throughout the game. But that is a dubious assumption. Suppose the first player X didn't terminate the game in the first round; what would the second player Y think then? Since the backwards-induction argument says X should terminate the game, and it is supposed to be a sound argument, Y might be entitled to doubt X's rationality. Alternatively, Y might doubt (...) that X believes Y is rational, or that X believes Y believes X is rational, or Y might have some higher-order doubt. X’s deviant first move might cause a breakdown in common belief in rationality, therefore. Once that goes, the entire argument fails. The argument also assumes that the players act rationally at each stage of the game, even if this stage could not be reached by rational play. But it is also dubious to assume that past irrationality never exerts a corrupting influence on present play. However, the backwards-induction argument can be reconstructed for the centipede game on a more secure basis.1 It may be implausible to assume a common belief in rationality throughout the game, however the game might go, but the argument requires less than this. The standard idealisations in game theory certainly allow us to assume a common belief in rationality at the beginning of the game. They also allow us to assume this common belief persists so long as no one makes an irrational move. That is enough for the argument to go through. (shrink)

This commentary focuses on the parts of psychological game theory dealing with preference, as illustrated by team reasoning, and supports the conclusion that these theoretical notions do not contribute above and beyond existing theory in understanding social interaction. In particular, psychology and games are already bridged by a comprehensive, formal, and inherently psychological theory, interdependence theory (Kelley & Thibaut 1978; Kelley et al. 2003), which has been demonstrated to account for a wide variety of social interaction phenomena.

We present an axiomatic approach for a class of finite, extensive form games of perfect information that makes use of notions like “rationality at a node” and “knowledge at a node.” We distinguish between the game theorist's and the players' own “theory of the game.” The latter is a theory that is sufficient for each player to infer a certain sequence of moves, whereas the former is intended as a justification of such a sequence of moves. While in general the (...) game theorist's theory of the game is not and need not be axiomatized, the players' theory must be an axiomatic one, since we model players as analogous to automatic theorem provers that play the game by inferring (or computing) a sequence of moves. We provide the players with an axiomatic theory sufficient to infer a solution for the game (in our case, the backwards induction equilibrium), and prove its consistency. We then inquire what happens when the theory of the game is augmented with information that a move outside the inferred solution has occurred. We show that a theory that is sufficient for the players to infer a solution and still remains consistent in the face of deviations must be modular. By this we mean that players have distributed knowledge of it. Finally, we show that whenever the theory of the game is group-knowledge (or common knowledge) among the players (i.e., it is the same at each node), a deviation from the solution gives rise to inconsistencies and therefore forces a revision of the theory at later nodes. On the contrary, whenever a theory of the game is modular, a deviation from equilibrium play does not induce a revision of the theory. (shrink)

The Equal Weight View of disagreement says that if an agent sees that an epistemic peer disagrees with her about p, the agent should change her credence in p to half way between her initial credence, and the peer’s credence. But it is hard to believe the Equal Weight View for a surprising reason; not everyone believes it. And that means that if one did believe it, one would be required to lower one’s belief in it in light of this (...) peer disagreement. Brian Weatherson explores the options for how a proponent of the Equal Weight View might respond to this difficulty, and how this challenge fits into broader arguments against the Equal Weight View. (shrink)

The paper is essentially a short version Spohn "Strategic Rationality" which emphasizes in particular how the ideas developed there may be used to shed new light on the iterated prisoner's dilemma (and on iterated Newcomb's problem).

The paper attempts to rationalize cooperation in the one-shot prisoners' dilemma (PD). It starts by introducing (and preliminarily investigating) a new kind of equilibrium (differing from Aumann's correlated equilibria) according to which the players' actions may be correlated (sect. 2). In PD the Pareto-optimal among these equilibria is joint cooperation. Since these equilibria seem to contradict causal preconceptions, the paper continues with a standard analysis of the causal structure of decision situations (sect. 3). The analysis then raises to a reflexive (...) point of view according to which the agent integrates his own present and future decision situations into the causal picture of his situation (sect. 4). This reflexive structure is first applied to the toxin puzzle and then to Newcomb's problem, showing a way to rationalize drinking the toxin and taking only one box with-out assuming causal mystery (sect. 5). The latter result is finally extended to a rationalization of cooperation in PD (sect. 6). (shrink)

Rational choice theory enjoys unprecedented popularity and influence in the behavioral and social sciences, but it generates intractable problems when applied to socially interactive decisions. In individual decisions, instrumental rationality is defined in terms of expected utility maximization. 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. In particular, focal point selection in pure coordination games is inexplicable, though it is easily achieved in practice; the intuitively compelling payoff-dominance principle lacks rational justification; rationality in social dilemmas is self-defeating; a key solution concept for cooperative coalition games is frequently inapplicable; and rational choice in certain sequential games generates contradictions. In experiments, human players behave more cooperatively and receive higher payoffs than strict rationality would permit. Orthodox conceptions of rationality are evidently internally deficient and inadequate for explaining human interaction. Psychological game theory, based on nonstandard assumptions, is required to solve these problems, and some suggestions along these lines have already been put forward. Key Words: backward induction; Centipede game; common knowledge; cooperation; epistemic reasoning; game theory; payoff dominance; pure coordination game; rational choice theory; social dilemma. (shrink)

This is a comprehensive anthology of works concerning the nature of economics as a science, including classic texts and essays exploring specific branches and schools of economics. Apart from the classics, most of the selections in the third edition are new, as are the introduction and bibliography. No other anthology spans the whole field and offers a comprehensive introduction to questions about economic methodology.

There are two curious features about the backward induction argument (BIA) to the effect that repeated non-cooperation is the rational solution to the finite iterated prisoner’s dilemma (FIPD). First, however compelling the argument may seem, one remains hesitant either to recommend this solu- tion to players who are about to engage in cooperation or to explain cooperation as a deviation from rational play in real-life FIPD’s. Second, there seems to be a similarity between the BIA for the FIPD and the (...) surprise exam paradox (SEP) and one cannot help but wonder whether the former is indeed no more than an instance of the latter. I argue that there is an important difference between the BIA for the FIPD and the SEP, but that a comparison to the SEP can help us understand why the conclusion of the BIA for the FIPD strikes us as a counterintuitive solution to real-life FIPD’s. (shrink)

Many philosophers and game theorists have been struck by the thought that the backward induction argument (BIA) for the finite iterated pris- oner’s dilemma (FIPD) recommends a course of action which is grossly counter-intuitive and certainly contrary to the way in which people behave in real-life FIPD-situations (Luce and Raiffa 1957, Pettit and Sugden 1989, Bovens 1997).1 Yet the backwards induction argument puts itself forward as binding upon rational agents. What are we to conclude from this? Is it that people (...) in real-life FIPD-situations tend to act irrationally and that our own intuitions about what to do in such situations reveal us to be irra- tional? Alternatively, should we abandon game theory and decision theory as a guide to rationality? Or are there other ways in which the apparent disparity between the dictates of rationality and the reality of reasoning can be accommodated? (shrink)

I argue with my friends a lot. That is, I offer them reasons to believe all sorts of philosophical conclusions. Sadly, despite the quality of my arguments, and despite their apparent intelligence, they don’t always agree. They keep insisting on principles in the face of my wittier and wittier counterexamples, and they keep offering their own dull alleged counterexamples to my clever principles. What is a philosopher to do in these circumstances? (And I don’t mean get better friends.) One popular (...) answer these days is that I should, to some extent, defer to my friends. If I look at a batch of reasons and conclude p, and my equally talented friend reaches an incompatible conclusion q, I should revise my opinion so I’m now undecided between p and q. I should, in the preferred lingo, assign equal weight to my view as to theirs. This is despite the fact that I’ve looked at their reasons for concluding q and found them wanting. If I hadn’t, I would have already concluded q. The mere fact that a friend (from now on I’ll leave off the qualifier ‘equally talented and informed’, since all my friends satisfy that) reaches a contrary opinion should be reason to move me. Such a position is defended by Richard Feldman (2006a, 2006b), David Christensen (2007) and Adam Elga (forthcoming). This equal weight view, hereafter EW, is itself a philosophical position. And while some of my friends believe it, some of my friends do not. (Nor, I should add for your benefit, do I.) This raises an odd little dilemma. If EW is correct, then the fact that my friends disagree about it means that I shouldn’t be particularly confident that it is true, since EW says that I shouldn’t be too confident about any position on which my friends disagree. But, as I’ll argue below, to consistently implement EW, I have to be maximally confident that it is true. So to accept EW, I have to inconsistently both be very confident that it is true and not very confident that it is true. This seems like a problem, and a reason to not accept EW.. (shrink)

For contractarians, justice is the result of a rational bargain. The goal is to show that the rules of justice are consistent with rationality. The two most important bargaining theories of justice are David Gauthier’s and those that use the Nash’s bargaining solution. I argue that both of these approaches are fatally undermined by their reliance on a symmetry condition. Symmetry is a substantive constraint, not an implication of rationality. I argue that using symmetry to generate uniqueness undermines the goal (...) of bargaining theories of justice. (shrink)

For the past 20 years or so the literature on noncooperative games has been centered on the search for an equilibrium concept that expresses the notion of rational behavior in interactive situations. A basic tenet in this literature is that if a “rational solution” exists, it must be a Nash equilibrium. The consensus view, however, is that not all Nash equilibria can be accepted as rational solutions. Consider, for example, the game of Figure 1.

Backward induction was one of the earliest methods developed for solving finite sequential games with perfect information. It proved to be especially useful in the context of Tom Schelling’s ideas of credible versus incredible threats. BI can be also extended to solve complex games that include an infinite number of actions or an infinite number of periods. However, some more complex empirical or experimental predictions remain dramatically at odds with theoretical predictions obtained by BI. The primary example of such a (...) troublesome game is Centipede. The problems appear in other long games with sufficiently complex structure. BI also shares the problems of subgame perfect equilibrium and fails to eliminate certain unreasonable Nash equilibria. (shrink)

Deliberation about what to do in any context requires reasoning about what will or would happen in various alternative situations, including situations that the agent knows will never in fact be realized. In contexts that involve two or more agents who have to take account of each others' deliberation, the counterfactual reasoning may become quite complex. When I deliberate, I have to consider not only what the causal effects would be of alternative choices that I might make, but also what (...) other agents might believe about the potential effects of my choices, and how their alternative possible actions might affect my beliefs. Counterfactual possibilities are implicit in the models that game theorists and decision theorists have developed – in the alternative branches in the trees that model extensive form games and the different cells of the matrices of strategic form representations – but much of the reasoning about those possibilities remains in the informal commentary on and motivation for the models developed. Puzzlement is sometimes expressed by game theorists about the relevance of what happens in a game ‘off the equilibrium path’: of what would happen if what is both true and known by the players to be true were instead false. (shrink)

In its classical conception, game theory aspires to be a determinate decision theory for games, understood as elements of a structurally specified domain. Its aim is to determine for each game in the domain a complete solution to each player's decision problem, a solution valid for all real-world instantiations, regardless of context. "Permissiveness" would constrain the theory to designate as admissible for a player any conjecture consistent with the function's designation of admissible strategies for the other players. Given permissiveness and (...) other appropriate constraints, solution sets must contain only Nash equilibria and at least one pure-strategy equilibrium, and there is no solution to games in which no symmetry invariant set of pure-strategy equilibria forms a Cartesian product. These results imply that the classical program is unrealizable. Moreover, the program is implicitly committed to permissiveness, through its common-knowledge assumptions and its commitment to equilibrium. The resulting incoherence deeply undermines the classical conception in a way that consolidates a long series of contextualist criticisms. (shrink)

A cornerstone of game theory is backward induction, whereby players reason backward from the end of a game in extensive form to the beginning in order to determine what choices are rational at each stage of play. Truels, or three-person duels, are used to illustrate how the outcome can depend on (1) the evenness/oddness of the number of rounds (the parity problem) and (2) uncertainty about the endpoint of the game (the uncertainty problem). Since there is no known endpoint in (...) the latter case, an extension of the idea of backward induction is used to determine the possible outcomes. The parity problem highlights the lack of robustness of backward induction, but it poses no conflict between foundational principles. On the other hand, two conflicting views of the future underlie the uncertainty problem, depending on whether the number of rounds is bounded (the players invariably shoot from the start) or unbounded (they may all cooperate and never shoot, despite the fact that the truel will end with certainty and therefore be effectively bounded). Some real-life examples, in which destructive behavior sometimes occurred and sometimes did not, are used to illustrate these differences, and some ethical implications of the analysis are discussed. (shrink)

The backward induction controversy in game theory flared up and then practically ended within a decade – the 1990s. The protagonists, however, did not converge on an agreement about the source of the controversy. Why was this the case, if opposing sides had access to the same modelling techniques and empirical facts? In this paper I offer an explanation for this controversy and its unsettled end. The answer is not to be found in the modelling claims made by the opposing (...) protagonists, but in the tacit metaphors they operate under. Aristotle defined metaphor as giving a 'thing a name that belongs to something else' (Poetica, 1457b). The meaning of metaphors has not changed much since then – in contrast to models which are comparatively new, and still not well-understood, scientific tools. The controversy of backward induction in game theory provides a test bed for the explanatory power of metaphors. This paper frames the controversy in terms of metaphor choice to provide a common framework for the protagonists. This results in the identification of three different domains – mathematical logic, game theory and the world – each connected to the other via different metaphors. The controversy around backward induction is placed in, and tentatively explained by, this framework. (shrink)

Two philosophical questions arise about rationality in centipede games that are logically prior to attempts to apply the formal tools of game theory to this topic. First, given that the players have common knowledge of mutual rationality and common knowledge that they are each motivated solely to maximize their own profits, is there a backwards-induction argument that employs only familiar non-technical concepts about rationality, leads to the conclusion that the first player is rationally obligated to end the game at the (...) first step, is deductively valid, employs premises all of which are prima facie highly plausible, and is prima facie sound and )? Second, if there is such an argument, then is it actually sound, or is it instead defective somehow despite being prima facie sound? Addressing these two questions is our project. We present a backwards-induction argument that is prima facie sound; we argue that it is an instance of the notorious sorites paradox, and hence that the concepts of rational obligatoriness and rational permissibility are vague; and we briefly address the potential consequences of all this for the foundations of game theory and decision theory. (shrink)

In a conflict between two people, one person wants one thing and the other wants something else and they think they can't both have what they want. Suppose that what they want can only be the outcome of some joint action. Adam must do either y or z and Eve either y ' or z ' – here y -and- y ' would be one joint action, y -and- z ' would be another, and so on. Adam wants the outcome (...) of his doing z while Eve is doing y '. Eve wants the outcome of her doing z ' while Adam is doing y. Each thinks that these outcomes can't both be had. (shrink)

Equilibrium game theory borrows from neoclassical economics its rationality concept which it immediately puts to work in order to produce the basic results it needs for building an elaborate narrative of social interaction. This paper focuses on some recent objections to game theory's use of rationality assumptions in general, and of backward induction and subgame perfection in particular, and interprets them in the light of the postmodern critique of the grand meta-narratives which social theorists often rely on for social explanation. (...) The paper presents a defence of game theory which seeks to accommodate the postmodern critique. However, it goes on to show that such a defence is illegitimate and claims that the problem lies with the faulty conceptualisation of the main concept on which game theory rests: that of Reason. Having established the nature of the problem, it considers three alternative interpretations (Humean, postmodern and Hegelian) of why the problem resists logical solutions and of its significance for social theory. (shrink)

We represent the well-known surprise exam paradox in constructive and computable mathematics and offer solutions. One solution is based on Brouwer’s continuity principle in constructive mathematics, and the other involves type 2 Turing computability in classical mathematics. We also discuss the backward induction paradox for extensive form games in constructive logic.

According to a familiar argument, iterated prisoner's dilemmas of known finite lengths resolve for ideally rational and well-informed players: They would defect in the last round, anticipate this in the next to last round and so defect in it, and so on. But would they anticipate defections even if they had been cooperating? Not necessarily, say recent critics. These critics "lose" the backward-induction paradox by imposing indicative interpretations on rationality and information conditions. To regain it I propose subjunctive interpretations. To (...) solve it I stress that implications for ordinary imperfect players are limited. (shrink)

The backward induction argument purports to show that rational and suitably informed players will defect throughout a finite sequence of prisoner's dilemmas. It is supposed to be a useful argument for predicting how rational players will behave in a variety of interesting decision situations. Here, I lay out a set of assumptions defining a class of finite sequences of prisoner's dilemmas. Given these assumptions, I suggest how it might appear that backward induction succeeds and why it is actually fallacious. Then, (...) I go on to consider the consequences of adopting a stronger set of assumptions. Focusing my attention on stronger sets that, like the original, obey the informedness condition, I show that any supplementation of the original set that preserves informedness does so at the expense of forcing rational participants in prisoner's dilemma situations to have unexpected beliefs, ones that threaten the usefulness of backward induction. (shrink)

In the standard money pump, an agent with cyclical preferences can avoid exploitation if he shows foresight and solves his sequential decision problem using backward induction (BI). This way out is foreclosed in a modified money pump, which has been presented in Rabinowicz (2000). There, BI will lead the agent to behave in a self-defeating way. The present paper describes another sequential decision problem of this kind, the Centipede for an Intransitive Preferrer, which in some respects is even more striking (...) than the modified pump. In the new problem, the BI reasoning that implies self-defeating behavior does not rest on the controversial robustness assumption concerning beliefs in one's future rationality. This strengthens the claim that foresight cannot save the intransitive preferrer from a self-defeating course of action. (shrink)

In order to vindicate rational-choice theory as a mode of explaining social patterns in general - social patterns beyond the narrow range of economic behaviour - we have to recognize the legitimacy of explaining the resilience of certain patterns of behaviour: that is, explaining, not necessarily why they emerged or have been sustained, but why they are robust and reliable. And once we allow the legitimacy of explaining resilience, then we can see how functionalist theory may also serve us well (...) in social science; we lose the basis - the empty black box argumenton which the rational-choice critique of the theory has mostly been grounded. (shrink)

This article argues that a particular notion of rationality, more exactly a specific notion of legitimate inference, is presupposed by much work in the social sciences to their detriment. The author describes the notion of rationality he has in mind, explains why it is misguided, identifies where and how it affects social research, and illustrates why that research is weaker as a result. The notion of legitimate inference the author has in mind is one that believes inferences are guided by (...) principles that are formal, universal and a priori—that is, that make no substantive, domain-specific empirical assumptions about the world. The author briefly provides a variety of reasons to be skeptical about such principles. Those reasons extend from broad philosophical considerations to quasi-empirical evidence. The author then argues that this notion of inference is involved with both the practice and the content of the social sciences in various ways. In terms of practice, this notion of inference lies behind the way statistical testing is generally done in the social sciences. In terms of content, the author argues that this notion of rationality is invoked, for example, in game theory and in rational expectations macroeconomics. In all cases, use of the formal rationality notion leads social scientists to put more faith in their results than is warranted and thus is an obstacle to doing better social science. (shrink)

In certain finite extensive games with perfect information, Cristina Bicchieri (1989) derives a logical contradiction from the assumptions that players are rational and that they have common knowledge of the theory of the game. She argues that this may account for play outside the Nash equilibrium. She also claims that no inconsistency arises if the players have the minimal beliefs necessary to perform backward induction. We here show that another contradiction can be derived even with minimal beliefs, so there is (...) no paradox of common knowledge specifically. These inconsistencies do not make play outside Nash equilibrium plausible, but rather indicate that the epistemic specification must incorporate a system for belief revision. Whether rationality is common knowledge is not the issue. (shrink)

I discuss how simulating another agent can be useful in some game-theoretical situations, particularly iterated games such as the centipede game.

The paper argues that the standard decision theoretic account of strategies and their rationality or optimality is much too narrow, that strategies should rather condition future action to future decision situations (a point of view already developed in my Grundlagen der Entscheidungstheorie, sect. 4.4), that practical deliberation must therefore essentially rely on a relation of superiority and inferiority between possible future decision situations, that all this allows to substantially broaden the theory of practical rationality, that a long list of points (...) attended to in the literature can be subsumed under the broadened perspective (including a novel view on the iterated prisoner's dilemma and on iterated Newcomb's problem, which, however, is revised in Spohn (2003) "Dependency Equilibria and the Causal Structure of Decision and Game Situation"), and that the task to complete and systematize this list indeed forms a fruitful research programme. (shrink)