Game Theory

In this paper we analyse in the framework of constructive mathematics (BISH) the validity of Farkas' lemma and related propositions, namely the Fredholm alternative for solvability of systems of linear equations, optimality criteria in linear programming, Stiemke's lemma and the Superhedging Duality from mathematical finance, and von Neumann's minimax theorem with application to constructive game theory.

We present a semantic game for Gödel logic and its extensions, where the players’ interaction stepwise reduces arbitrary claims about the relative order of truth degrees of complex formulas to atomic ones. The paper builds on a previously developed game for Gödel logic with projection operator in Fermüller et al., Information processing and management of uncertainty in knowledge-based systems, Springer, Cham, 2020, pp. 257–270). This game is extended to cover Gödel logic with involutive negations and constants, and then lifted to (...) a provability game using the concept of disjunctive strategies. Winning strategies in the provability game, with and without constants and involutive negations, turn out to correspond to analytic proofs in a version of \ :1941–1958, 2010) and in a sequent-of-relations calculus, Automated reasoning with analytic tableaux and related methods, Springer, Berlin, 1999, pp. 36–51) respectively. (shrink)

Methodological individualists often claim that any social phenomenon can ultimately be explained in terms of the actions and interactions of individuals. Any Nagelian version of methodological individualism requires that there be bridge laws that translate social statements into individualistic ones. We show that Nagelian individualism can be put to logical scrutiny by making the relevant social and individualistic languages fully explicit and mathematically precise. In particular, we prove that the social statement that a group of agents performs a deontically admissible (...) group action cannot be expressed in a well-established deontic logic of agency that models every combination of actions, omissions, abilities, and obligations of finitely many individual agents. (shrink)

The aim of this paper is to introduce the notion of a game for intuitionistic first-order Kripke models. We also establish links between notions presented here and the notions of logical equivalence and bounded bisimulation for intuitionistic first-order Kripke models, and the Ehrenfeucht–Fraïssé game for classical first-order structures.

Heidegger’s Being and Time - Irfan Ajvazi.

In a recent article, Ethan Bradley and Mark Navin (2021) argue that vaccine refusal is not akin to free riding. Here, I defend one connection between vaccine refusal and free riding and suggest that, when viewed in conjunction with their other arguments, this might constitute a reason to mandate Covid-19 vaccination.

This paper examines two strands of literature regarding economic models of cooperation. First, payoff transformation theories assume that people may not be exclusively motivated by self-interest, but also care about equality and fairness. Second, team reasoning theorists assume that people might reason from the perspective of the team, rather than an individualistic perspective. Can these two theories be unified? In contrast to the consensus among team reasoning theorists, I argue that team reasoning can be viewed as a particular type of (...) payoff transformation. However, I also demonstrate that many payoff transformations yield actions that team reasoning rules out. (shrink)

We characterize the determinacy of $F_\sigma $ games of length $\omega ^2$ in terms of determinacy assertions for short games. Specifically, we show that $F_\sigma $ games of length $\omega ^2$ are determined if, and only if, there is a transitive model of ${\mathsf {KP}}+{\mathsf {AD}}$ containing $\mathbb {R}$ and reflecting $\Pi _1$ facts about the next admissible set.As a consequence, one obtains that, over the base theory ${\mathsf {KP}} + {\mathsf {DC}} + ``\mathbb {R}$ exists,” determinacy for $F_\sigma $ (...) games of length $\omega ^2$ is stronger than ${\mathsf {AD}}$, but weaker than ${\mathsf {AD}} + \Sigma _1$ -separation. (shrink)

I envision new directions in the methodology of experimental games in the field of developmental, environmental and resource economics. Although there have been extensive discussions on experimenta...

Salience reasoning, many have argued, can help solve coordination problems, but only if such reasoning is supplemented by higher-order predictions, e.g. beliefs about what others believe yet others will choose. In this paper, I will argue that this line of reasoning is self-undermining. Higher-order behavioral predictions defeat salience-based behavioral predictions. To anchor my argument in the philosophical literature, I will develop it in response and opposition to the popular Lewisian model of salience reasoning in coordination games. This model imports the (...) problematic higher-order beliefs by way of a ‘symmetric reasoning’ constraint. In the second part of this paper, I will argue that a player may employ salience reasoning only if she suspends judgment about what others believe yet others will do. (shrink)

This book discusses the main shortcomings of the classical solution concept from noncooperative game theory and provides a comprehensive study of the more refined concepts that have been introduced to overcome these drawbacks. The plausibility of the assumptions underlying each such concept are discussed, desirable properties as well as deficiencies are illustrated, characterizations are derived and the relationships between the various concepts are studied. The first six chapters provide an informal discussion with many examples as well as a comprehensive overview (...) for normal form games. The remaining chapters are devoted to specific applications, illustrating the strength of the various concepts. (shrink)

Professor Zagare provides methods for analysing the structure of the game; considers zero and nonzero-sum games and the fundamental 'minimax theorem'; and investigates games with more than two players, including the possibility of coalitions between players.

Why do we use epistemic modals like 'might'? According to Factualism, the function of 'might' is to exchange information about state-of-affairs in the modal universe. As an alternative to Factualism, this paper offers a game-theoretic rationale for epistemic possibility operators in a Bayesian setting. The background picture is one whereby communication facilitates coordination, but coordination could fail if there's too much uncertainty, since the players' ability to share a belief is undermined. However, 'might' and related expressions can be used to (...) reveal one's uncertainty, and exploit this to coordinate despite the lack of a common epistemic ground. The final result is a way to articulate a non-Factualist view of epistemic possibility modals that builds on their standard semantics. (shrink)

The Nash counterfactual considers the question: what would happen were I to change my behaviour assuming no one else does. By contrast, the Kantian counterfactual considers the question: what would happen were everyone to deviate from some behaviour. We present a model that endogenizes the decision to engage in this type of Kantian reasoning. Autonomous agents using this moral framework receive psychic payoffs equivalent to the cooperate-cooperate payoff in Prisoner’s Dilemma regardless of the other player’s action. Moreover, if both interacting (...) agents play Prisoner’s Dilemma using this moral framework, their material outcomes are a Pareto improvement over the Nash equilibrium. (shrink)

Philosophical accounts of collective intentionality typically rely on members to form a personal intention of sorts, viewed as a mental state. This tendency is opposed by recent economic literature on team-directed reasoning, which focuses on the reasoning process leading up to the formation of the members’ intentions. Our formal analysis bridges these paradigms and criticizes the team- directed reasoning account on two counts: first, team-directed reasoning is supposed to transcend traditional game and decision theory by adopting a certain collectivistic reasoning (...) method. However, we show that team-directed reasoning yields the same action recommendations as a certain I-mode we-intention type. Accordingly, an important part of we-mode reasoning can be reduced to I-mode reasoning with certain preferences. Second, contrary to the claims of team-directed-reasoning theorists, we refute that team-directed reasoning surpasses pro-group intentions in selecting cooperatively rational solutions. That is, in some scenarios team-directed reasoning fails to guarantee successful cooperation whereas pro-group intentions succeed in doing so. We therefore propose to revise team-directed reasoning and introduce a third we-intention type, called participatory intentions. We prove that participatory intentions guarantee that a best group action is performed whenever either team-directed reasoning or pro-group intentions do. (shrink)

Cooperation has been analyzed primarily in the context of theories of collective intentionality. These discussions have primarily focused on interactions between pairs or small groups of agents who know one another personally. Cooperative game theory has also been used to argue for a form of cooperation in large unorganized groups. Here I consider a form of minimal cooperation that can arise among members of potentially large organized groups (e.g., corporate teams, committees, governmental bodies). I argue that members of organized groups (...) can be minimally cooperative in virtue of playing roles in an organizational structure and having a common goal. The minimal form of cooperation I argue for is not grounded in collective intentions involving symmetric mental states, special collective intentional modes, or joint commitments. More generally, I show how considering minimal cooperation in the context of organized groups provides an opportunity to reevaluate the extent to which the social world and social phenomena depend on internalist mental factors (e.g., intentions, beliefs) and externalist non-mental factors (e.g., documents, laws, job descriptions). The view of minimal cooperation among members of organized groups I offer provides support for an externalist rather than internalist theory of at least one social phenomenon. (shrink)

Games are meant to be fun, yet economists have successfully developed games that are less fun and less understood by participants especially in developing countries. This paper surveys failures in...

Deception has long been an important topic in philosophy. However, the traditional analysis of the concept, which requires that a deceiver intentionally cause her victim to have a false belief, rules out the possibility of much deception in the animal kingdom. Cognitively unsophisticated species, such as fireflies and butterflies, have simply evolved to mislead potential predators and/or prey. To capture such cases of “functional deception,” several researchers Machiavellian intelligence II, Cambridge University Press, Cambridge, pp 112–143, 1997; Searcy and Nowicki, The (...) evolution of animal communication, Princeton University Press, Princeton, 2005; Skyrms, Signals, Oxford University Press, Oxford 2010) have endorsed the broader view that deception only requires that a deceiver benefit from sending a misleading signal. Moreover, in order to facilitate game-theoretic study of deception in the context of Lewisian sender-receiver games, Brian Skyrms has proposed an influential formal analysis of this view. Such formal analyses have the potential to enhance our philosophical understanding of deception in humans as well as animals. However, as we argue in this paper, Skyrms’s analysis, as well as two recently proposed alternative analyses, are seriously flawed and can lead us to draw unwarranted conclusions about deception. (shrink)

In this paper, I investigate the betting behavior of a decision-maker who can influence the likelihood of the events upon which she is betting. In decision theory, this is best known as a situation of moral hazard. Focusing on a particularly simple case, I sketch the first systematic analysis of moral hazard in the canonical Savage framework. From the results of this analysis, I draw two philosophical conclusions. First, from an observational and a descriptive point of view, there need to (...) be no incompatibility between moral hazard and the Savage framework. This qualifies the incompatibility view, that is ubiquitous in decision theory. Second, in general, moral hazard is not sufficient to overcome the challenges posed by state-dependent utility to the behavioral identification of beliefs. This qualifies the sufficiency view, that is influential in decision theory. These two philosophical conclusions are the main contributions of my paper. (shrink)

In the dynamic choice literature, temptations are usually understood as temporary shifts in an agent’s preferences. What has been puzzling about these cases is that, on the one hand, an agent seems to do better by her own lights if she does not give into the temptation, and does so without engaging in costly commitment strategies. This seems to indicate that it is instrumentally irrational for her to give into temptation. On the other hand, resisting temptation also requires her to (...) act contrary to the preferences she has at the time of temptation. But that seems to be instrumentally irrational as well. I here consider the two most prominent types of argument why resisting temptation could nevertheless be instrumentally rational, namely two-tier and intra-personal cooperation arguments. I establish that the arguments either fail or are redundant. In particular, the arguments fail under the pervasive assumption in both decision theory and the wider literature on practical rationality that the agent’s preferences over the objects of choice are themselves the standard of instrumental rationality. And they either still fail or they become redundant when we give up that assumption. (shrink)

A complete probability guide of Hold'em Poker, this guide covers all possible gaming situations. The author focuses on the practical side of the presentation and use of the probabilities involved in Hold'em, while taking into account the subjective side of the probability-based criteria of each player's strategy.

This book presents not only the mathematical concept of probability, but also its philosophical aspects, the relativity of probability and its applications and even the psychology of probability. All explanations are made in a comprehensible manner and are supported with suggestive examples from nature and daily life, and even with challenging math paradoxes.

Over the past two decades, gamblers have begun taking mathematics into account more seriously than ever before. While probability theory is the only rigorous theory modeling the uncertainty, even though in idealized conditions, numerical probabilities are viewed not only as mere mathematical information, but also as a decision-making criterion, especially in gambling. This book presents the mathematics underlying the major games of chance and provides a precise account of the odds associated with all gaming events. It begins by explaining in (...) simple terms the meaning of the concept of probability for the layman and goes on to become an enlightening journey through the mathematics of chance, randomness and risk. It then continues with the basics of discrete probability, combinatorics and counting arguments for those interested in the supporting mathematics. These mathematic sections may be skipped by readers who do not have a minimal background in mathematics; these readers can skip directly to the Guide to Numerical Results to pick the odds and recommendations they need for the desired gaming situation. Doing so is possible due to the organization of that chapter, in which the results are listed at the end of each section, mostly in the form of tables. The chapter titled The Mathematics of Games of Chance presents these games not only as a good application field for probability theory, but also in terms of human actions where probability-based strategies can be tried to achieve favorable results. Through suggestive examples, the reader can see what are the experiments, events and probability fields in games of chance and how probability calculus works there. The main portion of this work is a collection of probability results for each type of game. Each game s section is packed with formulas and tables. Each section also contains a description of the game, a classification of the gaming events and the applicable probability calculations. The primary goal of this work is to allow the reader to quickly find the odds for a specific gaming situation, in order to improve his or her betting/gaming decisions. Every type of gaming event is tabulated in a logical, consistent and comprehensive manner. The complete methodology and complete or partial calculations are shown to teach players how to calculate probability for any situation, for every stage of the game for any game. Here, readers can find the real odds, returned by precise mathematical formulas and not by partial simulations that most software uses. Collections of odds are presented, as well as strategic recommendations based on those odds, where necessary, for each type of gaming situation. The book contains much new and original material that has not been published previously and provides great coverage of probabilities for the following games of chance: Dice, Slots, Roulette, Baccarat, Blackjack, Texas Hold em Poker, Lottery and Sport Bets. Most of games of chance are predisposed to probability-based decisions. This is why the approach is not an exclusively statistical one, but analytical: every gaming event is taken as an individual applied probability problem to solve. A special chapter defines the probability-based strategy and mathematically shows why such strategy is theoretically optimal.". (shrink)

Continuing his series of books on the mathematics of gambling, the author shows how a simple-rule game such as roulette is suited to a complex mathematical model whose applications generate improved betting systems that take into account a player's personal playing criteria. The book is both practical and theoretical, but is mainly devoted to the application of theory. About two-thirds of the content is lists of categories and sub-categories of improved betting systems, along with all the parameters that might stand (...) as the main objective criteria in a personal strategy - odds, profits and losses. The work contains new and original material not published before. The mathematical chapter describes complex bets, the profit function, the equivalence between bets and all their properties. All theoretical results are accompanied by suggestive concrete examples and can be followed by anyone with a minimal mathematical background because they involve only basic algebraic skills and set theory basics. The reader may also choose to skip the math and go directly to the sections containing applications, where he or she can pick desired numerical results from tables. The book offers no new so-called winning strategies, although it discusses them from a mathematical point of view. It does, however, offer improved betting systems and helps to organize a player's choices in roulette betting, according to mathematical facts and personal strategies. It is a must-have roulette handbook to be studied before placing your bets on the turn of either a European or American roulette wheel. (shrink)

This work is a complete mathematical guide to lottery games, covering all of the problems related to probability, combinatorics, and all parameters describing the lottery matrices, as well as the various playing systems. The mathematics sections describe the mathematical model of the lottery, which is in fact the essence of the lotto game. The applications of this model provide players with all the mathematical data regarding the parameters attached to the gaming events and personal playing systems. By applying these data, (...) one can find all the winning probabilities for the play with one line, and how these probabilities change with playing the various types of systems containing several lines, depending on their structure. Also, each playing system has a formula attached that provides the number of possible multiple prizes in various circumstances. Other mathematical parameters of the playing systems and the correlations between them are also presented. The generality of the mathematical model and of the obtained formulas allows their application for any existent lottery and any playing system. Each formula is followed by numerical results covering the most frequent lottery matrices worldwide and by multiple examples predominantly belonging to the 6/49 lottery. The listing of the numerical results in dozens of well-organized tables, along with instructions and examples of using them, makes possible the direct usage of this guide by players without a mathematical background. The author also discusses from a mathematical point of view the strategies of choosing involved in the lotto game. The book does not offer so-called winning strategies, but helps players to better organize their own playing systems and to confront their own convictions with the incontestable reality offered by the direct applications of the mathematical model of the lotto game. As a must-have handbook for any lottery player, this book offers essential information about the game itself and can provide the basis for gaming decisions of any kind. (shrink)

The expression "analytic narratives" is used to refer to a range of quite recent studies that lie on the boundaries between history, political science, and economics. These studies purport to explain specific historical events by combining the usual narrative approach of historians with the analytic tools that economists and political scientists draw from formal rational choice theories. Game theory, especially of the extensive form version, is currently prominent among these tools, but there is nothing inevitable about such a technical choice. (...) The chapter explains what analytic narratives are by reviewing the studies of the major book Analytic Narratives (1998), which are concerned with the workings of political institutions broadly speaking, as well as several cases drawn from military and security studies, which form an independent source of the analytic narratives literature. At the same time as it gradually develops a definition of analytic narratives, the chapter investigates how they fulfil one of their main purposes, which is to provide explanations of a better standing than those of traditional history. An important principle that will emerge in the course of the discussion is that narration is called upon not only to provide facts and problems, but also to contribute to the explanation itself. The chapter distinguishes between several expository schemes of analytic narratives according to the way they implement this principle. From all the arguments developed here, it seems clear that the current applications of analytic narratives do not exhaust their potential, and in particular that they deserve the attention of economic historians, if only because they are concerned with microeconomic interactions that are not currently their focus of attention. (shrink)

We confront the philosophical literature on fair division problems with axiomatic and game-theoretic work in economics. Firstly, we show that the proportionality method advocated in Curtis is not implied by a general principle of fairness, and that the proportional rule cannot be explicated axiomatically from that very principle. Secondly, we suggest that Broome’s notion of claims is too restrictive and that game-theoretic approaches can rectify this shortcoming. More generally, we argue that axiomatic and game-theoretic work in economics is an indispensable (...) ingredient of any theorizing about fair division problems and allocative justice. (shrink)

The bi-polar confrontation between the Soviet Union and the USA involved many leading game theorists from both sides of the Iron Curtain: Oskar Morgenstern, John von Neumann, Michael Intriligator, John Nash, Thomas Schelling and Steven Brams from the United States and Nikolay Vorob’ev, Leon A. Petrosyan, Elena B. Yanovskaya and Olga N. Bondareva from the Soviet Union. The formalization of game theory took place prior to the Cold War but the geopolitical confrontation hastened and shaped its evolution. In our article (...) we outline four similarities and differences between Western GT and Soviet GT: 1) the Iron Curtain resulted in a lagged evolution of GT in the Soviet Union; 2) Soviet GT focused more on operations research and issues of centralized planning; 3) the contemporary Western view on Soviet GT was biased and Soviet contributions, including works on dynamic stability, non-emptiness of the core and many refinements, suggest that Soviet GT was able to catch up to the Western level relatively fast; 4) international conferences, including Vilnius, 1971, fostered interaction between Soviet game theorists and their Western colleagues. In general, we consider the Cold War to be a positive environment for GT in the West and in the Soviet Union. (shrink)

The most important analytical tool in non-cooperative game theory is the concept of a Nash equilibrium, which is a collection of possibly mixed strategies, one for each player, with the property that each player's strategy is a best reply to the strategies of the other players. If we do not go into normative game theory, which concerns itself with the recommendation of strategies, and focus instead entirely on the positive theory of prediction, two alternative interpretations of the Nash equilibrium concept (...) are predominantly available.In the more traditional one, a Nash equilibrium is a prediction of actual play. A game may not have a Nash equilibrium in pure strategies, and a mixed strategy equilibrium may be difficult to incorporate into this interpretation if it involves the idea of actual randomization over equally good pure strategies. In another interpretation originating from Harsanyi, see also Rubinstein, and Aumann and Brandenburger, a Nash equilibrium is a ‘consistent’ collection of probabilistic expectations, conjectures, on the players. It is consistent in the sense that for each player each pure strategy, which has positive probability according to the conjecture about that player, is indeed a best reply to the conjectures about others. (shrink)

The pioneering contributions of Bewley, Gilboa and Schmeidler highlighted important weaknesses in the foundations of economics and game theory. The Bayesian methodology on which these fields are based does not answer such basic questions as what makes beliefs reasonable, or how agents should form beliefs and expectations. Providing the initial impetus for debating these issues is a contribution that will have the lasting value it deserves.

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)

Upon learning that John C. Harsanyi was awarded the Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel, in 1994, for his pioneering work in game theory, few economists probably questioned the appropriateness of that choice. The Budapest-born social scientist had already been recognized as a first-rank contributor to non-cooperative game theory for some time. However, as many readers of this journal will be aware, Harsanyi first contributed to welfare economics, not game theory. More importantly, he was (...) philosophically minded and accordingly has been “acknowledged as the most influential philosopher in economics”. 1 This is of some significance since, before Harsanyi became acquainted with economics around 1950, his main interest was philosophy and, to a lesser extent, sociology and psychology. Rather than an economist with philosophical leanings, Harsanyi was actually a philosopher turned economist. (shrink)

To a practitioner in the social sciences, game theory primarily helps to identify situations in which interdependent decisions are somehow problematic; solutions often require venturing into the social sciences. Game theory is usually about anticipating each other's choices; it can also cope with influencing other's choices. To a social scientist the great contribution of game theory is probably the payoff matrix, an accounting device comparable to the equals sign in algebra.