Bargaining and Information Acquisition

THE research reported in this paper arose from a previous experimental study conducted by the authors.' That study involved bargains struck between two subjects who had opposing payoff functions and full information of one another's payoffs. By a flip of a coin one participant could choose a noncooperative outcome, unilaterally: the winner of the coin toss could simply choose an outcome which gave him $12 and left the other subject nothing, whether or not the loser agreed. However, if the two subjects cooperated, they could obtain from the experimenter $14, which could be split between the subjects in any mutually agreed-on manner. Cooperative game theory predicts that the subjects will cooperate and divide the rewards $13 to $1 (the Nash bargaining solution: an even division of the $2 gain from trade). Under no circumstances should the winner of the coin flip settle for less than $12, according to game

Two of the most important refinements of the Nash equilibrium concept for extensive form games with perfect recall are Selten's (1975) perfect equilibrium and Kreps and Wilson's (1982) more inclusive sequential equilibrium. These two equilibrium refinements are motivated in very different ways. Nonetheless, as Kreps and Wilson (1982, Section 7) point out, the two concepts lead to similar prescriptions for equilibrium play. For each particular game form, every perfect equilibrium is sequential. Moreover, for almost all assignments of payoffs to outcomes, almost all sequential equilibrium strategy profiles are perfect equilibrium profiles, and all sequential equilibrium outcomes are perfect equilibrium outcomes. We establish a stronger result: For almost all assignments of payoffs to outcomes, the sets of sequential and perfect equilibrium strategy profiles are identical. In other words, for almost all games each strategy profile which can be supported by beliefs satisfying the rationality requirement of sequential equilibrium can actually be supported by beliefs satisfying the stronger rationality requirement of perfect equilibrium. We obtain this result by exploiting the algebraic/geometric structure of these equilibrium correspondences, following from the fact that they are semi-algebraic sets; i.e., they are defined by finite systems of polynomial inequalities. That the perfect and sequential equilibrium correspondences have this semi-algebraic structure follows from a deep result from mathematical logic, the Tarski-Seidenberg Theorem; that this structure has important game-theoretic consequences follows from deep properties of semi-algebraic sets.

Refining Nash Equilibrium by Bayesian Iterative Conjectures Approach

In this paper, we examine relations among various refinements of a perfect equilibrium. First, we compare two refinements of a perfect equilibrium, a strictly perfect equilibrium (SPE) and a truly perfect equilibrium (TPE). We show that true perfectness implies strict perfectness and even strict properness. We also introduce the concept of a restrictive perfect equilibrium allowing only some or all sequences of totally mixed strategies satisfying a certain property. A proper equilibrium is an example of a restrictive perfect equilibrium. Then, true perfectness imposing the most severe robustness against perturbations implies any restrictive perfectness. This proves that every TPE is proper.

We conduct an experiment in which we auction the scarce rights to play the Proposer and Responder positions in subsequent ultimatum games. As a control treatment, we randomly allocate these rights and then charge exogenous participation fees according to the auction price sequences observed in the auction treatment. With endogenous selection into ultimatum games via auctions, we find that play converges to a session-specific Nash equilibrium and auction prices emerge which support this equilibrium by the principle of forward induction. With random assignment and exogenous participation fees, we find play also converges to a session-specific Nash equilibrium as predicted by the principle of loss avoidance. The Nash equilibrium observed within a session results in low ultimatum game offers, but the subgame perfect Nash equilibrium is never observed.

In this research we have developed experimental designs of the ultimatum game with supervised agents. This agents have unbiased and biased thinking depending on the case. We used Reinforcement Learning and Bucket Brigade to program the artficial agentes. We used simulations and behavior comparison to answer the following questions: Does artificial intelligence reach a perfect subgame equilibrium in the ultimatum game experiment? How would Artificial Intelligence behave in the Ultimatum Game experiment if biased thinking is included in it? This exploratory analysis showed one important result: artificial inteligence by itself doesn´t reach a perfect subgame equilibrium. Whereas, the experimental designs with biased thinking agents quickly converge to an equilibrium. Finally, we demonstrated that the agents with envy bias behaves the same as the ones with altruistic bias.

On the Behavior of Proposers in Ultimatum Games

In (Bonanno, 2013), a solution concept for extensive-form games, called perfect Bayesian equilibrium (PBE), was introduced and shown to be a strict refinement of subgame-perfect equilibrium; it was also shown that, in turn, sequential equilibrium (SE) is a strict refinement of PBE. In (Bonanno, 2016), the notion of PBE was used to provide a characterization of SE in terms of a strengthening of the two defining components of PBE (besides sequential rationality), namely AGM consistency and Bayes consistency. In this paper we explore the gap between PBE and SE by identifying solution concepts that lie strictly between PBE and SE; these solution concepts embody a notion of “conservative” belief revision. Furthermore, we provide a method for determining if a plausibility order on the set of histories is choice measurable, which is a necessary condition for a PBE to be a SE.

Bargaining under a deadline: evidence from the reverse ultimatum game

Refining Perfect Bayesian Equilibrium by Bayesian Iterative Conjectures Algorithm

Preface 1. Decision-Theoretic Foundations 1.1 Game Theory, Rationality, and Intelligence 1.2 Basic Concepts of Decision Theory 1.3 Axioms 1.4 The Expected-Utility Maximization Theorem 1.5 Equivalent Representations 1.6 Bayesian Conditional-Probability Systems 1.7 Limitations of the Bayesian Model 1.8 Domination 1.9 Proofs of the Domination Theorems Exercises 2. Basic Models 2.1 Games in Extensive Form 2.2 Strategic Form and the Normal Representation 2.3 Equivalence of Strategic-Form Games 2.4 Reduced Normal Representations 2.5 Elimination of Dominated Strategies 2.6 Multiagent Representations 2.7 Common Knowledge 2.8 Bayesian Games 2.9 Modeling Games with Incomplete Information Exercises 3. Equilibria of Strategic-Form Games 3.1 Domination and Ratonalizability 3.2 Nash Equilibrium 3.3 Computing Nash Equilibria 3.4 Significance of Nash Equilibria 3.5 The Focal-Point Effect 3.6 The Decision-Analytic Approach to Games 3.7 Evolution. Resistance. and Risk Dominance 3.8 Two-Person Zero-Sum Games 3.9 Bayesian Equilibria 3.10 Purification of Randomized Strategies in Equilibria 3.11 Auctions 3.12 Proof of Existence of Equilibrium 3.13 Infinite Strategy Sets Exercises 4. Sequential Equilibria of Extensive-Form Games 4.1 Mixed Strategies and Behavioral Strategies 4.2 Equilibria in Behavioral Strategies 4.3 Sequential Rationality at Information States with Positive Probability 4.4 Consistent Beliefs and Sequential Rationality at All Information States 4.5 Computing Sequential Equilibria 4.6 Subgame-Perfect Equilibria 4.7 Games with Perfect Information 4.8 Adding Chance Events with Small Probability 4.9 Forward Induction 4.10 Voting and Binary Agendas 4.11 Technical Proofs Exercises 5. Refinements of Equilibrium in Strategic Form 5.1 Introduction 5.2 Perfect Equilibria 5.3 Existence of Perfect and Sequential Equilibria 5.4 Proper Equilibria 5.5 Persistent Equilibria 5.6 Stable Sets 01 Equilibria 5.7 Generic Properties 5.8 Conclusions Exercises 6. Games with Communication 6.1 Contracts and Correlated Strategies 6.2 Correlated Equilibria 6.3 Bayesian Games with Communication 6.4 Bayesian Collective-Choice Problems and Bayesian Bargaining Problems 6.5 Trading Problems with Linear Utility 6.6 General Participation Constraints for Bayesian Games with Contracts 6.7 Sender-Receiver Games 6.8 Acceptable and Predominant Correlated Equilibria 6.9 Communication in Extensive-Form and Multistage Games Exercises Bibliographic Note 7. Repeated Games 7.1 The Repeated Prisoners Dilemma 7.2 A General Model of Repeated Garnet 7.3 Stationary Equilibria of Repeated Games with Complete State Information and Discounting 7.4 Repeated Games with Standard Information: Examples 7.5 General Feasibility Theorems for Standard Repeated Games 7.6 Finitely Repeated Games and the Role of Initial Doubt 7.7 Imperfect Observability of Moves 7.8 Repeated Wines in Large Decentralized Groups 7.9 Repeated Games with Incomplete Information 7.10 Continuous Time 7.11 Evolutionary Simulation of Repeated Games Exercises 8. Bargaining and Cooperation in Two-Person Games 8.1 Noncooperative Foundations of Cooperative Game Theory 8.2 Two-Person Bargaining Problems and the Nash Bargaining Solution 8.3 Interpersonal Comparisons of Weighted Utility 8.4 Transferable Utility 8.5 Rational Threats 8.6 Other Bargaining Solutions 8.7 An Alternating-Offer Bargaining Game 8.8 An Alternating-Offer Game with Incomplete Information 8.9 A Discrete Alternating-Offer Game 8.10 Renegotiation Exercises 9. Coalitions in Cooperative Games 9.1 Introduction to Coalitional Analysis 9.2 Characteristic Functions with Transferable Utility 9.3 The Core 9.4 The Shapkey Value 9.5 Values with Cooperation Structures 9.6 Other Solution Concepts 9.7 Colational Games with Nontransferable Utility 9.8 Cores without Transferable Utility 9.9 Values without Transferable Utility Exercises Bibliographic Note 10. Cooperation under Uncertainty 10.1 Introduction 10.2 Concepts of Efficiency 10.3 An Example 10.4 Ex Post Inefficiency and Subsequent Oilers 10.5 Computing Incentive-Efficient Mechanisms 10.6 Inscrutability and Durability 10.7 Mechanism Selection by an Informed Principal 10.8 Neutral Bargaining Solutions 10.9 Dynamic Matching Processes with Incomplete Information Exercises Bibliography Index

Evolution of fairness in the mixture of the Ultimatum Game and the Dictator Game

The dynamics of how fair offers come about in ultimatum game is studied via the method of agent-based modeling. Both fairness motive and adaptive learning are considered to be important in the fair behavior of human players in concerning literature. Here special attention is paid to situations where adaptive learning proposers encounter responders with either pure money concern or fairness motivation. The simulation results show that the interplay of adaptive learning participants yields a perfect sub-game equilibrium, but fair offers will be provided by proposers as long as a small proportion of responders play “tough” against unfair offer.

This chapter deals with games in extensive form. Here an explicit evolution of the interaction is given, describing precisely when each player plays, what actions are available and what information is available to each player when he makes a decision. We start with games with perfect information (such as chess) and prove Zermelo’s theorem for finite games. We then consider infinite games a la Gale–Stewart: we show that open games are determined and that under the axiom of choice, there exists an undetermined game. Next we introduce games with imperfect information and prove Kuhn’s theorem, which states that mixed and behavioral strategies are equivalent in games with perfect recall. We present the standard characterization of Nash equilibria in behavioral strategies and introduce the basic refinements of Nash equilibria in extensive-form games: subgame-perfection, Bayesian perfect and sequential equilibria, which impose rational behaviors not only on the equilibrium path but also off-path. We prove the existence of sequential equilibrium (Kreps and Wilson). For normal form games as in Chap. 4 we introduce the standard refinements of Nash equilibrium: perfect equilibrium (Selten) and proper equilibrium (Myerson). We prove that a proper equilibrium of a normal form game G induces a sequential equilibrium in every extensive-form game with perfect recall having G as normal form. Finally we discuss forward induction and stability (Kohlberg and Mertens).

The king of refinements of Nash equilibrium is trembling hand perfection. We show that it is NP-hard and Sqrt-Sum-hard to decide if a given pure strategy Nash equilibrium of a given three-player game in strategic form with integer payoffs is trembling hand perfect. Analogous results are shown for a number of other solution concepts, including proper equilibrium, (the strategy part of) sequential equilibrium, quasi-perfect equilibrium and CURB.The proofs all use a reduction from the problem of comparing the minmax value of a three-player game in strategic form to a given rational number. This problem was previously shown to be NP-hard by Borgs et al., while a Sqrt-Sum hardness result is given in this paper. The latter proof yields bounds on the algebraic degree of the minmax value of a three-player game that may be of independent interest.KeywordsNash EquilibriumPure StrategySolution ConceptBelief StructureSequential EquilibriumThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.