Evidential reasoning in the prisoner's dilemma

Classic game theory considers defection to be the rational choice in the Prisoner's Dilemma. Although defection maximizes a player's gain regardless of the opponent's choice, many players cooperate. We suggest that cooperation can be explained in part by expectations of reciprocal behavior, that is, by the belief that players facing the same situation probably will come to the same decision. In Experiment 1, expectations of reciprocity were experimentally manipulated. As predicted, cooperation increased monotonically with these expectations. In Experiment 2, experimentally manipulated expectations of opponent cooperation were not associated with higher rates of cooperation. These findings are interpreted in light of a general model of payoff maximization, and implications for other models (e.g., social value orientation) are discussed.     

Collective rationality in interactive decisions: evidence for team reasoning

Decision theory and game theory rest on a fundamental assumption that players seek to maximize their individual utilities, but in some interactive decisions it seems intuitively reasonable to aim to maximize the utility of the group of players as a whole. Such team reasoning requires collective preferences and a distinctive mode of reasoning from preferences to decisions. Findings from two experiments provide evidence for collective preferences and team reasoning. In lifelike vignettes (Experiment 1) and abstract games (Experiment 2) with certain structural properties, most players preferred team-reasoning strategies to strategies supporting unique Nash equilibria, although individually rational players should choose equilibrium strategies. These findings suggest that team reasoning predicts strategy choices more powerfully than orthodox game theory in some games.     

Inconsistencies in extensive games

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.     

What are impossible worlds?

In this paper, I argue for a particular conception of impossible worlds. Possible worlds, as traditionally understood, can be used in the analysis of propositions, the content of belief, the truth of counterfactuals, and so on. Yet possible worlds are not capable of differentiating propositions that are necessarily equivalent, making sense of the beliefs of agents who are not ideally rational, or giving truth values to counterfactuals with necessarily false antecedents. The addition of impossible worlds addresses these issues. The kinds of impossible worlds capable of performing this task are not mysterious sui generis entities, but sets of structured propositions that are themselves constructed out of possible worlds and relations. I also respond to a worry that these impossible worlds are unable to represent claims about the shape of modal space itself.     

Preface Writers are Consistent

The preface paradox does not show that it can be rational to have inconsistent beliefs, because preface writers do not have inconsistent beliefs. I argue, first, that a fully satisfactory solution to the preface paradox would have it that the preface writer's beliefs are consistent. The case here is on basic intuitive grounds, not the consequence of a theory of rationality or of belief. Second, I point out that there is an independently motivated theory of belief – sensitivism – which allows such a solution. I sketch a sensitivist account of the preface writer's (consistent) doxastic state.     

An Impossibility Theorem on Beliefs in Games

A paradox of self-reference in beliefs in games is identified, which yields a game-theoretic impossibility theorem akin to Russell’s Paradox. An informal version of the paradox is that the following configuration of beliefs is impossible:Ann believes that Bob assumes thatAnn believes that Bob’s assumption is wrongThis is formalized to show that any belief model of a certain kind must have a ‘hole.’ An interpretation of the result is that if the analyst’s tools are available to the players in a game, then there are statements that the players can think about but cannot assume. Connections are made to some questions in the foundations of game theory.Special Issue Ways of Worlds II. On Possible Worlds and Related Notions Edited by Vincent F. Hendricks and Stig Andur Pedersen     

Communist Conventions for Deductive Reasoning

In section 1, I develop epistemic communism, my view of the function of epistemically evaluative terms such as ‘rational’. The function is to support the coordination of our belief‐forming rules, which in turn supports the reliable acquisition of beliefs through testimony. This view is motivated by the existence of valid inferences that we hesitate to call rational. I defend the view against the worry that it fails to account for a function of evaluations within first‐personal deliberation. In the rest of the paper, I then argue, on the basis of epistemic communism, for a view about rationality itself. I set up the argument in section 2 by saying what a theory of rational deduction is supposed to do. I claim that such a theory would provide a necessary, sufficient, and explanatorily unifying condition for being a rational rule for inferring deductive consequences. I argue in section 3 that, given epistemic communism and the conventionality that it entails, there is no such theory. Nothing explains why certain rules for deductive reasoning are rational.     

WORLDS OR WORDS APART? WITTGENSTEIN ON UNDERSTANDING RELIGIOUS LANGUAGE

In this paper I develop an account of Wittgenstein's conception of what it is to understand religious language. I show that Wittgenstein's view undermines the idea that as regards religious faith only two options are possible – either adherence to a set of metaphysical beliefs (with certain ways of acting following from these beliefs) or passionate commitment to a ‘doctrineless’ form of life. I offer a defence of Wittgenstein's conception against Kai Nielsen's charges that Wittgenstein removes the ‘content’ from religious belief and renders the religious form of life ‘incommensurable’ with other domains of discourse, thus immunizing it against rational criticism.     

Belief in God: A game-theoretic paradox

Conclusions The Belief Game is a two-person, nonzero-sum game in which both players can do well [e.g., at (3, 4)] or badly [e.g., at (1,1)] simultaneously. The problem that occurs in the play of this game is that its rational outcome of (2, 3) is not only unappealing to both players, especially God, but also, paradoxically, there is an outcome, (3, 4), preferred by both players that is unattainable. Moreover, because God has a dominant strategy, His omniscience does not remedy the situation, though - less plausibly - if man possessed this quality, and God were aware of it, (3, 4) would be attainable.How reasonable is it to use the device of a simple game to argue that nonrevelation by God, and nonbelief by man, are rational strategies? Like any model of a complex reality, the Belief Game abstracts a great deal from the problem that confronts the thoughtful agnostic asking the most profound of existential questions. Yet, to the degree that belief in God is seen as a personal question, conceptualized in terms of a possible relationship one might have with one's Creator, it seems appropriate to try to model this relationship as a game. The most difficult question to answer, I suppose, is, if God exists, what are His preferences in such a game?I have argued that He would first like to be believed, but at the same time not reveal Himself. These goals, in my opinion, are consistent with the role He assumes in many biblical stories, although this is not to say that the Bible offers the final word on philosophical and theological matters in the modern world. Nevertheless, it seems to me to be a logical place from which to start, and the clues it offers on God's preferences seem not contradicted by contemporary events.Of course, the Belief Game supposes that God not only has preferences but makes choices as well. To many people today - myself included - these choices are not apparent. But if He does make them, and in particular chooses not to reveal Himself, I think the Belief Game helps us to understand why nonrevelation is rational. Furthermore, it gives us insight into why, given this choice by God, our own reasons for believing in Him - in a game-theoretic context - may be rendered tenuous.     

Manipulating Games by Sharing Information

We address the issue of manipulating games through communication. In the specific setting we consider (a variation of Boolean games), we assume there is some set of environment variables, the values of which are not directly accessible to players; the players have their own beliefs about these variables, and make decisions about what actions to perform based on these beliefs. The communication we consider takes the form of (truthful) announcements about the values of some environment variables; the effect of an announcement is the modification of the beliefs of the players who hear the announcement so that they accurately reflect the values of the announced variables. By choosing announcements appropriately, it is possible to perturb the game away from certain outcomes and towards others. We specifically focus on the issue of stabilisation: making announcements that transform a game from having no stable states to one that has stable configurations.     

Properties,laws, and worlds

Jonathan Schaffer argues against a necessary connection between properties and laws. He takes this to be a question of what possible worlds we ought to countenance in our best theories of modality, counterfactuals, etc. In doing so, he unfairly rigs the game in favor of contingentism. I argue that the necessitarian can resist Schaffer’s conclusion while accepting his key premise that our best theories of modality, counterfactuals, etc. require a very wide range of things called ‘possible worlds’. However, the necessitarian can and should insist that, in many cases, these worlds are not metaphysically possible. I will further argue that, having taken such a stance, the necessitarian has additional resources to respond to Schaffer’s other arguments against the view.     

Belief in free will as an adaptive,ungrounded belief

False beliefs and delusions are usually regarded negatively, especially in psychology and evolutionary biology. Recently, McKay and Dennett (2009b) have argued that there are ungrounded beliefs which confer benefits on individuals even if they are false. I propose to expand this class of beliefs to include the belief that one has free will, and I will defend the claim that this belief is advantageous, even if it is false. One derives one’s belief in control from one’s experience of control, which is generated by a set of cognitive systems termed “control systems.” While the control systems and the interpretive mechanism are useful in and of themselves, the belief in personal free will is adaptive because it directly leads to fitness-increasing behaviors. As such, we have good reason to regard the belief that one has free will as an adaptive, ungrounded belief. This paper will also suggest that further research on the possible distinction between belief in personal free will and belief in general free will may put us in a better position to understand recent, apparently contradictory data on individuals’ beliefs regarding free will and other related phenomena.     

Are creationists rational?

Creationism is usually regarded as an irrational set of beliefs. In this paper I propose that the best way to understand why individual learners settle on any mature set of beliefs is to see that as the developmental outcome of a series of “fast and frugal” boundedly rational inferences rather than as a rejection of reason. This applies to those whose views are opposed to science in general. A bounded rationality model of belief choices both serves to explain the fact that folk traditions tend to converge on “anti-modernity”, and to act as a default hypothesis, deviations from which we can use to identify other, arational, influences such as social psychological, economic and individual dispositions. I propose some educational and public policy strategies that might decrease the proportion of learners who find creationism and anti-science in general a rational choice.     

National security games

Issues that arise in using game theory to model national security problems are discussed, including positing nation-states as players, assuming that their decision makers act rationally and possess complete information, and modeling certain conflicts as two-person games. A generic two-person game called the Conflict Game, which captures strategic features of such variable-sum games as Chicken and Prisoners' Dilemma, is then analyzed. Unlike these classical games, however, the Conflict Game is a two-stage game in which each player can threaten to retaliate — and carry out this threat in the second stage — if its opponent chose noncooperation in the first stage.Conditions for the existence of different pure-strategy Nash equilibria, or stable outcomes, are found, and these results are extended to situations in which the players can select mixed strategies (i.e., make probabilistic threats or choices). Although the Conflict Game sheds light on the rational foundations underlying arms races, nuclear deterrence, and other strategic situations, more detailed assumptions are required to tie this generic game to specific conflicts.Steven J. Brams gratefully acknowledges the financial support of the National Science Foundation under Grant No. SES85-20154, the Sloan Foundation, and the Guggenheim Foundation.D. Marc Kilgour gratefully acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada under Grant No. A8974.     

Rationality and bounded information in repeated games, with application to the iterated Prisoner's Dilemma

Actions in a repeated game can in principle depend on all previous outcomes. Given this vast policy space, human players may often be forced to use heuristics that base actions on incomplete information, such as the outcomes of only the most recent trials. Here it is proven that such bounded rationality is often fully rational, in that the optimal policy based on some limited information about the game's history will be universally optimal (i.e., within the full policy space), provided that one's opponents are restricted to using this same information. It is then shown how this result allows explicit calculation of subgame-perfect equilibria (SPEs) for any repeated or stochastic game. The technique is applied to the iterated Prisoner's Dilemma for the case of 1-back memory. Two classes of SPEs are derived, which exhibit varying degrees of (individually rational) cooperation as a result of repeated interaction.     

A puzzle about epistemic akrasia

In this paper I will present a puzzle about epistemic akrasia, and I will use that puzzle to motivate accepting some non-standard views about the nature of epistemological judgment. The puzzle is that while it seems obvious that epistemic akrasia must be irrational, the claim that epistemic akrasia is always irrational amounts to the claim that a certain sort of justified false belief—a justified false belief about what one ought to believe—is impossible. But justified false beliefs seem to be possible in any domain, and it’s hard to see why beliefs about what one ought to believe should be an exception. I will argue that when we get clearer about what sort of psychological state epistemic akrasia is, we can resolve the puzzle in favor of the intuitive view that epistemic akrasia is always irrational.     

Game Theory and Rational Decision

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. This revised version was published online in July 2006 with corrections to the Cover Date.     

On fairness and learning agents in a bargaining model with uncertainty

The emergence of fair divisions in a repeated bargaining game is investigated in a computational model. Individuals of human societies often appeal to some norm of fairness in situations where an agreement over the division of a surplus is required. The employed framework consists of players alternating offers that describe possible ways to share a certain commodity. The players are allowed a limited number of offers to reach an agreement; if they fail to agree, the player who made the first offer, the lucky player, wins the whole lot at stake. Uncertainty is introduced in the process by randomly choosing the lucky player at the beginning of each iteration. In the experiments, the players acquired strategies by employing a variant of Q-learning, a reinforcement learning algorithm. Experiments were performed with different configurations of utility functions on the players’ preferences in taking actions in risky situations. Analysis of the results shows that the game theoretical model of a single shot of the bargaining game used in the experiments closely matches the outcomes obtained in the simulated framework, despite the differences in the quality of the players, who are assumed to be fully rational in the theoretical model. Learning agents that are timid toward risky situations manage to acquire strategies that lead to fair outcomes when playing against each other, but find themselves in a disadvantageous position when confronting bolder types.     

Rationality, Justification, and the Internalism/Externalism Debate

In this paper, I argue that what underlies internalism about justification is a rationalist conception of justification, not a deontological conception of justification, and I argue for the plausibility of this rationalist conception of justification. The rationalist conception of justification is the view that a justified belief is a belief that is held in a rational way; since we exercise our rationality through conscious deliberation, the rationalist conception holds that a belief is justified iff a relevant possible instance of conscious deliberation would endorse the belief. The importance of conscious deliberation stems from its role in guiding us in acquiring true beliefs: whereas the externalist holds that if we wish to acquire true beliefs, we have to begin by assuming that some of our usual methods of belief formation generally provide us with true beliefs, the internalist holds that if we form beliefs by conscious deliberation, we can be conscious of reasons for thinking that our beliefs are true. Conscious deliberation can make us conscious of reasons because it proceeds via rational intuitions. I argue that despite the fallibility of rational intuition, rational intuitions do enable us to become conscious of reasons for belief.

When psychology undermines beliefs

This paper attempts to specify the conditions under which a psychological explanation can undermine or debunk a set of beliefs. The focus will be on moral and religious beliefs, where a growing debate has emerged about the epistemic implications of cognitive science. Recent proposals by Joshua Greene and Paul Bloom will be taken as paradigmatic attempts to undermine beliefs with psychology. I will argue that a belief p may be undermined whenever: (i) p is evidentially based on an intuition which (ii) can be explained by a psychological mechanism that is (iii) unreliable for the task of believing p; and (iv) any other evidence for belief p is based on rationalization. I will also consider and defend two equally valid arguments for establishing unreliability: the redundancy argument and the argument from irrelevant factors. With this more specific understanding of debunking arguments, it is possible to develop new replies to some objections to psychological debunking arguments from both ethics and philosophy of religion.