Normative and Descriptive Game Theory

The paper is an addition to the intensionalist approach to decision theory, with emphasis on game theoretic modelling. Extensionality in games is an a priori requirement that players exhibit the same behavior in all algebraically equivalent games on pain of irrationality. Intensionalism denies that it is always irrational to play differently in differently represented but algebraically equivalent versions of a game. I offer a framework to integrate game non-extensionality with the more familiar idea of linguistic non-extensionality from philosophy of language, (...) followed by applications of it based on toy examples of well-known game models. I argue that the notion of what I call “Intensional Nash Equilibrium” is, in effect, very useful in understanding human decision-making. (shrink)

History based models suggest a process-based approach to epistemic and temporal reasoning. In this work, we introduce preferences to history based models. Motivated by game theoretical observations, we discuss how preferences can dynamically be updated in history based models. Following, we consider arrow update logic and event calculus, and give history based models for these logics. This allows us to relate dynamic logics of history based models to a broader framework.

The paper has a twofold aim. On the one hand, it provides what appears to be the first game-theoretic modeling of Napoleon’s last campaign, which ended dramatically on 18 June 1815 at Waterloo. It is specifically concerned with the decision Napoleon made on 17 June 1815 to detach part of his army against the Prussians he had defeated, though not destroyed, on 16 June at Ligny. Military historians agree that this decision was crucial but disagree about whether it was rational. (...) Hypothesizing a zero-sum game between Napoleon and Blücher, and computing its solution, we show that it could have been a cautious strategy on the former's part to divide his army, a conclusion which runs counter to the charges of misjudgement commonly heard since Clausewitz. On the other hand, the paper addresses methodological issues. We defend its case study against the objections of irrelevance that have been raised elsewhere against “analytic narratives”, and conclude that military campaigns provide an opportunity for successful application of the formal theories of rational choice. Generalizing the argument, we finally investigate the conflict between narrative accounts – the historians' standard mode of expression – and mathematical modeling. (shrink)

Games have a complex, and seemingly paradoxical structure: they are both competitive and cooperative, and the competitive element is required for the cooperative element to work out. They are mechanisms for transforming competition into cooperation. Several contemporary philosophers of sport have located the primary mechanism of conversion in the mental attitudes of the players. I argue that these views cannot capture the phenomenological complexity of game-play, nor the difficulty and moral complexity of achieving cooperation through game-play. In this paper, I (...) present a different account of the relationship between competition and cooperation. My view is a distributed view of the conversion: success depends on a large number of features. First, the players must achieve the right motivational state: playing for the sake of the struggle, rather than to win. Second, successful transformation depends on a large number of extra-mental features, including good game design, and social and institutional features. (shrink)

When an agent’s motivation is sensitive to how his supervisor thinks about the agent’s competence, the supervisor has to take into account both informational and expressive contents of her message to the agent. This paper shows that the supervisor can credibly express her trust in the agent’s ability only by being un- clear about what to do. Suggesting what to do, i.e., “directives,” could reveal the supervisor’s “distrust” and reduce the agent’s equilibrium effort level even though it provides useful information (...) about the decision environment. There is also an equilibrium in which directives are neutral in expressive content. However, it is shown that neologism proofness favors equilibria in which directives are double- edged swords. (shrink)

The chapter reviews recent theoretical and empirical developments concerning the theory of team reasoning in game theoretic interactions.

Universal social institutions, such as marriage, commons management and property, have emerged independently in radically different cultures. This requires explanation. As Boyer and Petersen point out ‘in a purely localist framework would have to constitute massively improbable coincidences’ . According to Boyer and Petersen, those institutions emerged naturally out of genetically wired behavioural dispositions, such as marriage out of mating strategies and borders out of territorial behaviour. While I agree with Boyer and Petersen that ‘unnatural’ institutions cannot thrive, this one-sided (...) explanation of universal social institutions in terms of genetic human nature is unsatisfactory. Drawing on the literature on multi-level selection and gene-culture coevolution, I argue that universal social institutions are first and foremost the products of cultural selection. They occupy fitness peaks in the landscape of cultural possibilities, much in the same way that biological adaptations occupy fitness peaks in the landscape of biological possibilities. To show this, I use game-theory. By modelling the domains of social interaction in which marriage, commons management, and property emerged as Prisoner’s dilemma situations, it becomes clear how an institutional framework allows the group to move to an interactive equilibrium with a larger payoff. Institutions do so by incentivising all parties to adopt a cooperative strategy. They are culturally selected ways of optimising genetically constrained domains of human social interaction. (shrink)

Standard models in epistemic game theory make strong assumptions about agents’ knowledge of their own beliefs. Agents are typically assumed to be introspectively omniscient: if an agent believes an event with probability p, she is certain that she believes it with probability p. This paper investigates the extent to which this assumption can be relaxed while preserving some standard epistemic results. Geanakoplos (1989) claims to provide an Agreement Theorem using the “truth” axiom, together with the property of balancedness, a significant (...) relaxation of introspective omniscience. I provide an example which shows that Geanakoplos’s statement is incorrect. I then introduce a new property, local balancedness, which allows us both to correct Geanakoplos’s result, and to extend it to cases where the truth axiom may fail. I exploit this general Agreement Theorem to provide novel epistemic conditions for correlated and Nash equilibrium, both of which relax the assumption of introspective omniscience. In all three cases, the results are also extended to infinite state spaces. (This is Chapter 5 of my 2014 Oxford DPhil (PhD) thesis.). (shrink)

Although trust and reciprocity are ubiquitous in social exchange, their neurobiological substrate remains largely unknown. Here, we investigated the effect of damage to the ventromedial prefrontal cortex (vmPFC)—a brain region critical for valuing social information—on individuals’ decisions in a trust game and in a risk game. In the trust game, one player, the investor, is endowed with a sum of money, which she can keep or invest. The amount she decides to invest is tripled and sent to the other player, (...) the trustee, who then decides what fraction to return to the investor. In separate runs, ten patients with focal bilateral damage to the vmPFC and control participants made decision while playing in the role of either investor or trustee with different anonymous counterparts in each run. A risk game was also included in which the investor faced exactly the same decisions as in the trust game, but a random device (i.e., a computer), not another player, determined the final payoffs. Results showed that vmPFC patients’ investments were not modulated by the type of opponent player (e.g., human vs. computer) present in the environment. Thus, vmPFC patients showed comparable risk-taking preferences both in social (trust game) and nonsocial (risk game) contexts. In stark contrast, control participants were less willing to take risk and invest when they believed that they were interacting with people than a computer. Furthermore, when acted as trustee, vmPFC patients made lower back transfers toward investors, thereby showing less reciprocity behavior. Taken together, these results indicate that social valuation and emotion subserved by vmPFC have a critical role in trusting and reciprocity decisions. The present findings support the hypothesis that vmPFC damage may impair affective systems specifically designed for mediating social transaction with other individuals. (shrink)

Most definitions of cooperation provide sufficient but not necessary conditions. This paper describes a form of minimal cooperation, corresponding to mass actions implying many agents, such as demonstrations. It characterizes its intentional, epistemic, strategic, and teleological aspects, mostly obtained from weakening classical concepts. The rationality of minimal cooperation turns out to be part of its definition, whereas it is usually considered as an optional though desirable feature. Game-theoretic concepts thus play an important role in its definition. The paper concludes by (...) answering concrete questions about what should and should not be called cooperation. (shrink)

We conceive of a player in dynamic games as a set of agents, which are assigned the distinct tasks of reasoning and node-specific choices. The notion of agent connectedness measuring the sequential stability of a player over time is then modeled in an extended type-based epistemic framework. Moreover, we provide an epistemic foundation for backward induction in terms of agent connectedness. Besides, it is argued that the epistemic independence assumption underlying backward induction is stronger than usually presumed.

In recent years, the human ability to reasoning about mental states of others in order to explain and predict their behavior has come to be a highly active area of research. Researchers from a wide range of fields { from biology and psychology through linguistics to game theory and logic{ contribute new ideas and results. This interdisciplinary workshop, collocated with the Thirteenth International Conference on Theoretical Aspects of Rationality and Knowledge (TARK XIII), aims to shed light on models of social (...) reasoning that take into account realistic resource bounds. People reason about other people's mental states in order to understand and predict the others' behavior. This capability to reason about others' knowledge, beliefs and intentions is often referred to as theory of mind. Idealized rational agents are capable of recursion in their social reasoning, and can reason about phenomena like common knowledge. Such idealized social reasoning has been modeled by modal logics such as epistemic logic and BDI (belief, desire, intention) logics and by epistemic game theory. However, in real-world situations, many people seem to lose track of such recursive social reasoning after only a few levels. The workshop provides a forum for researchers that attempt to analyze, understand and model how resource-bounded agents reason about other minds. (shrink)

The mathematical tools of game theory are frequently used in the social sciences and economic consultancy. But how do they explain social phenomena and support prescriptive judgments? And is the use of game theory really necessary? I analyze the logical form of explanatory and prescriptive game theoretical statements, and argue for two claims: (1) explanatory game theory can and should be reduced to rational choice theory in all cases; and (2) prescriptive game theory gives bad advice in some cases, is (...) reducible to rational choice theory in other cases, while it makes no sense in yet other cases. (shrink)

The ultimatum game is a simple bargaining situation where the behavior of people frequently contradicts the optimal strategy according to classical game theory. Thus, according to many scholars, the commonly observed behavior should be considered irrational. We argue that this putative irrationality stems from a wrong conception of metanormativity (the study of norms about the establishment of norms). After discussing different metanormative conceptions, we defend a Quinean, naturalistic approach to the evaluation of norms. After reviewing empirical literature on the ultimatum (...) game, we argue that the common behavior in the ultimatum game is rational and justified. We therefore suggest that the norms of economic rationality should be amended. (shrink)

This article explores some issues having to do with the use of experimental results from one‐shot games to reach conclusions about the existence of social preferences that are taken to figure in the explanation of cooperation in repeated interactions in real life. †To contact the author, please write to: Division of Humanities and Social Sciences, California Institute of Technology, Pasadena, CA 91125; e‐mail: [email protected].

Gintis assumes the behavioral (=social) sciences are in disarray, and so proposes a theory for their unification. Examination of the unity of the physical sciences reveals he misunderstands the unity of science in general, and so fails to see that the social sciences are already unified with the physical sciences. Another explanation of the differences between them is outlined. (Published Online April 27 2007).

The various behavioral disciplines model human behavior in distinct and incompatible ways. Yet, recent theoretical and empirical developments have created the conditions for rendering coherent the areas of overlap of the various behavioral disciplines. The analytical tools deployed in this task incorporate core principles from several behavioral disciplines. The proposed framework recognizes evolutionary theory, covering both genetic and cultural evolution, as the integrating principle of behavioral science. Moreover, if decision theory and game theory are broadened to encompass other-regarding preferences, they (...) become capable of modeling all aspects of decision making, including those normally considered “psychological,” “sociological,” or “anthropological.” The mind as a decision-making organ then becomes the organizing principle of psychology. (Published Online April 27 2007) Key Words: behavioral game theory; behavioral science; evolutionary theory; experimental psychology; gene-culture coevolution; rational actor model; socialization. (shrink)

Can social phenomena be understood by analyzing their parts? Contemporary economic theory often assumes that they can. The methodology of constructing models which trace the behavior of perfectly rational agents in idealized environments rests on the premise that such models, while restricted, help us isolate tendencies, that is, the stable separate effects of economic causes that can be used to explain and predict economic phenomena. In this paper, I question both the claim that models in economics supply claims about tendencies (...) and also the view that economics, when successful, necessarily follows this method. When economics licenses successful policy interventions, as it did in the case of the Federal Communications Commission spectrum auctions, its method is not to study tendencies but rather to study the phenomenon as a whole. Key Words: economic models • tendencies • economic experiments • policy making • John Stuart Mill. (shrink)

Game theory is a branch of economics that uses powerful mathematical models to predict what agents ought to do when interacting with other agents strategically. Bounded rationality is a sub-field of game theory that sets out to explain why, in some interesting cases, people don't act according their utility maximizing strategies, as described by game theory. Interactive Epistemology is formal tool used by Game Theorists and computer scientists to model interactive cases of knowledge. This interesting and useful tool has been (...) previously ignored by philosophers. I'd like to introduce philosophers to interactive epistemology. After doing so, I'll go on to describe the way I've used this powerful formal tool in my own research, by giving some arguments about Bounded Epistemology, which is an analogue of Bounded Rationality, and, if I'm right, is explainable according to many, but not all, of the same models. Doing so, however, requires first setting out and explaining many of these concepts more fully. (shrink)

Teaching economics has been shown to encourage students to defect in a prisoner's dilemma game. However, can ethics training reverse that effect and promote cooperation? We conducted an experiment to answer this question. We found that students who had the ethics module had higher rates of cooperation than students without the ethics module, even after controlling for communication and other factors expected to affect cooperation. We conclude that the teaching of ethics can mitigate the possible adverse incentives of the prisoner's (...) dilemma, and, by implication, the adverse effects of economics and business training. (shrink)

One reason why humans don't behave according to standard game theoretical rationality is because it's not realistic to assume that everyone else is behaving rationally. An individual is expected to have psychological mechanisms that function to maximize his/her long-term payoffs in a world of potentially “irrational” individuals. Psychological decision theory has to be individualistic because individuals make decisions, not groups.

Game-theoretic explanations of behavior need supplementation to be descriptive; behavior has multiple causes, only some governed by traditional rationality. An evolutionarily informed theory of action countenances overlapping causal domains: neurobiological, psychological, and rational. Colman's discussion is insufficient because he neither evaluates learning models nor qualifies under what conditions his propositions hold. Still, inability to incorporate emotions in axiomatic models highlights the need for a comprehensive theory of functional rationality.

The primrose path and prisoner's dilemma paradigms may require cognitive (executive) control: The active maintenance of context representations in lateral prefrontal cortex to provide top-down support for specific behaviors in the face of short delays or stronger response tendencies. This perspective suggests further tests of whether altruism is a type of self-control, including brain imaging, induced affect, and dual-task studies.

Features of experimental design impose auxiliary hypotheses on experimenters. Hertwig & Ortmann rightly argue that the ways some variables are implemented in psychology cloud results, whereas the different implementations in economics provide for more robust results. However, not all design variables support this general conclusion. The repetition of trials may confuse results depending on what theory is being tested. We explore this in the case of simple bargaining games.

Hertwig and Ortmann imply that failure of many behavioral psychologists to observe several central methodological principles of experimental economics derives mainly from differences in disciplinary culture. I suggest that there are deeper philosophical causes, based (ironically) on a legacy of methodological individualism in psychology from which economists have substantially cured themselves through use of game theory. Psychologists often misidentify their objects of study by trying to wrench subjects out of their normal behavioral contexts in games.

While fairness is often mentioned as a determinant of ultimatum bargaining behavior, few data sets are available that can test theories that incorporate fairness considerations. This paper tests the reciprocal kindness theory in Rabin (1993 Incorporating fairness into game theory and economics, The American Economic Review 83: 1281-1302) as an application to the one-period ultimatum bargaining game. We report on data from 100 ultimatum games that vary the financial stakes of the game from 1 to 15. Responder behavior is strongly (...) in support of the kindness theory and proposer behavior weakly in support of it. Offer percentages and past offers influence behavior the most, whereas the size of the pie has a marginally significant effect on offer percentages. The data is more in support of reciprocal kindness than alternative theories of equal-split or learning behavior, although the data also weakly support a minimum percentage threshold hypothesis. As a whole, our results together with existing studies suggest that, for smaller stakes games, fairness considerations dominate monetary considerations. This has implications for more complicated naturally occurring bargaining environments in which the financial stakes can vary widely. (shrink)

Within the rational choice approach trust is usually analysed as a problem of choice under uncertainty. In a standard social situation in which trust plays a role a trustor A has to choose between trusting or mistrusting a trustee B. If he chooses to trust, B can either honour the trust given which will be of some advantage to both or exploit it no matter what the consequences for A are. A can in general protect himself against being exploited by (...) choosing not to trust. But then he forgoes the possible advantages of trustful co-operation. A´s problem typically is that he is not sufficiently informed about B´s incentives or motivation. He may, for instance, not know whether B possesses some favourable character traits or not. The paper uses the well known game of trust with imperfect information to illustrate the core of the problem. From a rational choice point of view the possibility of a trustful decision is determined by the beliefs of A concerning B´s preferences. Thus the crucial parameter in the model is a probability measure p that measures A´s subjective expectations that a rational B will co-operate. According to the standard rational choice account trust is either a consequence or simply the expression of the expectations of a trustee represented by p. I refer to this account of trust as the theory of trust as rational expectancy. I shall criticise this solely cognitive view of trust for neglecting some central aspects of trusting relationships. My argument draws on the distinction between trust and reliance. It will be based on some simple observations, in particular: 1. The perception of the choice problem by the trustor as well as his perception of the trustee and the trustee´s characteristics may depend on wether or not the trustor is in fact trusting or mistrusting. 2. The trustor´s evaluation of the possible outcomes may depend on wether or not the trustor is in fact trusting or mistrusting. 3. The motivation of the trustee will often depend on whether he is trusted or not. If my argument is sound, then - to some considerable extend - it is impossible to analyse trust problems solely in terms of options, preferences and believes. There simply are no preferences and believes of the interacting individuals which are prior to and independent of trust. Trust then must be understood as a cause or basis of a trustor´s expectations, and not as their consequence. I shall finish my considerations by pointing to some general implications for an account of trust as social capital. (shrink)

Human beings are peculiar. In laboratory experiments, they often cooperate in one-shot prisoners’ dilemmas, they frequently offer 1/2 and reject low offers in the ultimatum game, and they often bid 1/2 in the game of divide-the-cake All these behaviors are puzzling from the point of view of game theory. The first two are irrational, if utility is measured in a certain way.1 The last isn’t positively irrational, but it is no more rational than other possible actions, since there are infinitely (...) many other Nash equilibria besides the one in which both players bid 1/2. At the same time, these behaviors seem to indicate that people are sometimes inclined to be cooperative, fair, and just. In his stimulating new book, Brian Skyrms sets himself the task of showing why these inclinations evolved, or how they might have evolved, under the pressure of natural selection. The goal is not to justify our ethical intuitions, but to explain why we have them.2.. (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)

This paper reports an experiment which investigates a possible cognitive antecedent of event-splitting effects (ESEs) experimentally observed by Starmer and Sugden (1993) and Humphrey (1995) – the learning of absolute frequency of event category impacting on the learning of probability of event category – and reveals some evidence that it is responsible for observed ESEs. It is also suggested and empirically substantiated that stripped-down prospect theory will accurately predict ESEs in some decision making tasks, but will not perform well in (...) others. This contention, it is argued, is indicative of fundamental descriptive shortcomings in the economic conception of choice under uncertainty and may entail implications beyond the direct concerns of this paper. (shrink)

We generalize the concept of Nash equilibrium in mixed strategies for strategic form games to allow for ambiguity in the players' expectations. In contrast to other contributions, we model ambiguity by means of so-called lower probability measures or belief functions, which makes it possible to distinguish between a player's assessment of ambiguity and his attitude towards ambiguity. We also generalize the concept of trembling hand perfect equilibrium. Finally, we demonstrate that for certain attitudes towards ambiguity it is possible to explain (...) cooperation in the one-shot Prisoner's Dilemma in a way that is in accordance with some recent experimental findings. (shrink)

Donald Green and Ian Shapiro argue that rational choice scholarship in political science is excessively theory?driven: too few of its theoretical insights have been subjected to serious empirical scrutiny and survived. But rational choice theorizing has the potential to identify and correct logical inconsistencies and slippages. It is thus valuable even if the resulting theories are not tested empirically. When Green and Shapiro's argument concerning collective dilemmas and free riding is formalized, it turns out to be deeply flawed and in (...) many respects outright false. Their mistake is common enough: they misclassify a variety of collective dilemmas as prisoner's dilemmas. Because they misunderstand the theory of rational choice, Green and Shapiro allege that it is refuted by empirical findings that, in fact, support it. (shrink)

Many different arguments against the possibility of perfect rationality have appeared in the literature, and these target several different conceptions of perfect rationality. It is not clear how these different conceptions of perfect rationality are related, nor is it clear how the arguments showing their impossibility are related, and it is especially unclear what the impossibility results show when taken together. This paper gives an exposition of the different conceptions of perfect rationality, an the various sorts of argument against them; (...) clarifies which conceptions of perfect rationality are targeted by which arguments; and finally attempts to systematize the results available to date. (shrink)

This paper is part of a larger project defending of the foundations of microeconomics against recent criticisms by philosophers. Here, we undermine one source of these criticisms, arising from philosophers' disappointment with the performance of microeconomic tools, in particular game theory, when these are applied to normative decision theory. Hollis and Sugden have recently articulated such disappointment in a sophisticated way, and have argued on the basis of it that the economic conception of rationality is inadequate. We argue, however, that (...) their claim rests upon a misunderstanding of the concept of a game as it is used in microeconomics. (shrink)

David Gauthier claims that it can be rational to co-operate in a prisoner's dilemma if one has adopted a disposition constraining one's self from maximizing one's individual expected utility, i.e., a constrained maximizer disposition. But I claim cooperation cannot be both voluntary and constrained. In resolving this tension I ask what constrained maximizer dispositions might be. One possibility is that they are rationally acquired, irrevocable psychological mechanisms which determine but do not rationalize cooperation. Another possibility is that they are rationally (...) acquired preference-functions rationalizing cooperation as maximizing. I argue that if they are the first thing, then their adoption fails to make co-operation rational even if, as Gauthier also claims, actions are rational if they express rational dispositions. I then suggest that taking constrained maximizer dispositions to be things of the second sort would result in them being able to make co-operation rational, and that so-taking them therefore serves the bulk and spirit of Gauthier's larger claims, which I reconstruct accordingly. (shrink)

The game theoretic notion of best-response reasoning is sometimes criticized when its application produces multiple solutions of games, some of which seem less compelling than others. The recent development of the theory of team reasoning addresses this by suggesting that interacting players in games may sometimes reason as members of a team – a group of individuals who act together in the attainment of some common goal. A number of properties have been suggested for team-reasoning decision-makers’ goals to satisfy, but (...) a few formal representations have been discussed. In this paper we suggest a possible representation of these goals based on the notion of mutual advantage. We propose a method for measuring extents of individual and mutual advantage to the interacting decision-makers, and define team interests as the attainment of outcomes associated with maximum mutual advantage in the games they play. (shrink)

Ever since game theory has become a dominant mode of investigation in economics, critics have pointed out that it is a formally strong but empirically weak, if not empty, practice.1 We argue against the empirical irrelevance of game theory by investigating the architecture of game theoretic explanations more closely. In particular, we study the role of game models, and find that they assume the role of mediators as autonomous relaters of theory and phenomena. We further argue that stories play an (...) essential part in the mediating function of models: stories that recount the phenomena in such a way that it becomes compatible with the theoretical framework. Contrary to some claims, these stories are subject to evaluation by various criteria, some of which are criteria derived from intuitive judgment. These criteria provide the means to judge whether a model represents a phenomenon or not. Through good stories, phenomena can therefore be related to highly abstract models. Applied game theory, we conclude, has therefore empirical content by representing phenomena in an abstract fashion. (JEL B4,C7. Keywords: Economic methodology, Game theory.). (shrink)