Infinite subgame perfect equilibrium in the Hausdorff difference hierarchy

Subgame perfect equilibria are specific Nash equilibria in perfect information games in extensive form. They are important because they relate to the rationality of the players. They always exist in infinite games with continuous real-valued payoffs, but may fail to exist even in simple games with slightly discontinuous payoffs. This article considers only games whose outcome functions are measurable in the Hausdorff difference hierarchy of the open sets (\textit{i.e.} $\Delta^0_2$ when in the Baire space), and it characterizes the families of linear preferences such that every game using these preferences has a subgame perfect equilibrium: the preferences without infinite ascending chains (of course), and such that for all players $a$ and $b$ and outcomes $x,y,z$ we have $\neg(z <_a y <_a x \,\wedge\, x <_b z <_b y)$. Moreover at each node of the game, the equilibrium constructed for the proof is Pareto-optimal among all the outcomes occurring in the subgame. Additional results for non-linear preferences are presented.Comment: The alternative definition of the difference hierarchy has changed slightl

Dynamics and Coalitions in Sequential Games

We consider N-player non-zero sum games played on finite trees (i.e., sequential games), in which the players have the right to repeatedly update their respective strategies (for instance, to improve the outcome wrt to the current strategy profile). This generates a dynamics in the game which may eventually stabilise to a Nash Equilibrium (as with Kukushkin's lazy improvement), and we argue that it is interesting to study the conditions that guarantee such a dynamics to terminate. We build on the works of Le Roux and Pauly who have studied extensively one such dynamics, namely the Lazy Improvement Dynamics. We extend these works by first defining a turn-based dynamics, proving that it terminates on subgame perfect equilibria, and showing that several variants do not terminate. Second, we define a variant of Kukushkin's lazy improvement where the players may now form coalitions to change strategies. We show how properties of the players' preferences on the outcomes affect the termination of this dynamics, and we thereby characterise classes of games where it always terminates (in particular two-player games).Comment: In Proceedings GandALF 2017, arXiv:1709.0176

Evolving Computability

We consider the degrees of non-computability (Weihrauch degrees) of finding winning strategies (or more generally, Nash equilibria) in infinite sequential games with certain winning sets (or more generally, outcome sets). In particular, we show that as the complexity of the winning sets increases in the difference hierarchy, the complexity of constructing winning strategies increases in the effective Borel hierarchy.Comment: An extended abstract of this work has appeared in the Proceedings of CiE 201

A Semi-Potential for Finite and Infinite Games in Extensive Form

We consider a dynamical approach to game in extensive forms. By restricting the convertibility relation over strategy profiles, we obtain a semi-potential (in the sense of Kukushkin), and we show that in finite games the corresponding restriction of better-response dynamics will converge to a Nash equilibrium in quadratic (finite) time. Convergence happens on a per-player basis, and even in the presence of players with cyclic preferences, the players with acyclic preferences will stabilize. Thus, we obtain a candidate notion for rationality in the presence of irrational agents. Moreover, the restriction of convertibility can be justified by a conservative updating of beliefs about the other players strategies.For infinite games in extensive form we can retain convergence to a Nash equilibrium (in some sense), if the preferences are given by continuous payoff functions; or obtain a transfinite convergence if the outcome sets of the game are Δ02-sets

Foundations of Software Science and Computation Structures

We study multi-player turn-based games played on (potentially infinite) directed graphs. An outcome is assigned to every play of the game. Each player has a preference relation on the set of outcomes which allows him to compare plays. We focus on the recently introduced notion of weak subgame perfect equilibrium (weak SPE). This is a variant of the classical notion of SPE, where players who deviate can only use strategies deviating from their initial strategy in a finite number of histories. Having an SPE in a game implies having a weak SPE but the contrary is generally false. We propose general conditions on the structure of the game graph and on the preference relations of the players that guarantee the existence of a weak SPE, that additionally is finite-memory. From this general result, we derive two large classes of games for which there always exists a weak SPE: (i) the games with a finite-range outcome function, and (ii) the games with a finite underlying graph and a prefix-independent outcome function. For the second class, we identify conditions on the preference relations that guarantee memoryless strategies for the weak SPE.Comment: 28 page

Computable Analysis and Game Theory: From Foundations to Applications

This body of research showcases several facets of the intersection between computer science and game theory. On the foundational side, we explore the obstructions to the computability of Nash equilibria in the setting of computable analysis. In particular, we study the Weihrauch degree of the problem of finding a Nash equilibrium for a multiplayer game in normal form. We conclude that the Weihrauch degree Nash for multiplayer games lies between AoUC∗[0,1] and AoUC⋄[0,1] (Theorem 5.3). As a slight detour, we also explore the demarcation between computable and non-computable computational problems pertaining to the verification of machine learning. We demonstrate that many verification questions are computable without the need to specify a machine learning framework (Section 7.2). As well as looking into the theory of learners, robustness and sparisty of training data. On the application side, we study the use of Hypergames in Cybersecurity. We look into cybersecurity AND/OR attack graphs and how we could turn them into a hypergame (8.1). Hyper Nash equilibria is not an ideal solution for these games, however, we propose a regret-minimisation based solution concept. In Section 8.2, we survey the area of Hypergames and their connection to cybersecurity, showing that even if there is a small overlap, the reach is limited. We suggest new research directions such as adaptive games, generalisation and transferability (Section 8.3)

Concurrent Games and Semi-Random Determinacy

Consider concurrent, infinite duration, two-player win/lose games played on graphs. If the winning condition satisfies some simple requirement, the existence of Player 1 winning (finite-memory) strategies is equivalent to the existence of winning (finite-memory) strategies in finitely many derived one-player games. Several classical winning conditions satisfy this simple requirement. Under an additional requirement on the winning condition, the non-existence of Player 1 winning strategies from all vertices is equivalent to the existence of Player 2 stochastic strategies almost-sure winning from all vertices. Only few classical winning conditions satisfy this additional requirement, but a fairness variant of omega-regular languages does

Essays on information and incentives

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Economics, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (p. 163-168).This thesis studies problems of belief and information formation of agents, and its effect on incentive provision in problems of experimental and mechanism design. Chapter 1 is based on joint work with Arun Chandrasekhar and Horacio Larreguy. In this chapter we present the results of an experiment we conducted in rural Karnataka, India, to get evidence on how agents learn from each other's actions in the context of a social network. Theory has mostly focused on two leading models of social learning on networks: Bayesian updating and local averaging (DeGroot rules of thumb) which can yield greatly divergent behavior; individuals employing local averaging rules of thumb often double-count information and, in our context, may not exhibit convergent behavior in the long run. We study experiments in which seven individuals are placed into a network, each with full knowledge of its structure. The participants attempt to learn the underlying (binary) state of the world. Individuals receive independent, identically distributed signals about the state in the first period only; thereafter, individuals make guesses about the underlying state of the world and these guesses are transmitted to their neighbors at the beginning of the following round. We consider various environments including incomplete information Bayesian models and provide evidence that individuals are best described by DeGroot models wherein they either take simple majority of opinions in their neighborhood Chapter 2 is based on joint work with Arun Chandrasekhar, and studies how researchers should design payment schemes when making experiments on repeated games, such as the game studied in Chapter 1. It is common for researchers studying repeated and dynamic games in a lab experiment to pay participants for all rounds or a randomly chosen round. We argue that these payment schemes typically implement different set of subgame perfect equilibria (SPE) outcomes than the target game. Specifically, paying a participant for a randomly chosen round (or for all rounds with even small amounts of curvature) makes the game such that early rounds matter more to the agent, by lowering discounted future payments. In addition, we characterize the mechanics of the problems induced by these payment methods. We are able to measure the extent and shape of the distortions. We also establish that a simple payment scheme, paying participants for the last (randomly occurring) round, implements the game. The result holds for any dynamic game with time separable utility and discounting. A partial converse holds: any payment scheme implementing the SPE should generically be history and time independent and only depend on the contemporaneous decision. Chapter 3 studies a different but related problem, in which agents now have imperfect information not about some state of nature, but rather about the behavior of other players, and how this affects policy making when the planner does not know what agents expects her to do. Specifically, I study the problem of a government with low credibility, who decides to make a reform to remove ex-post time inconsistent incentives due to lack of commitment. The government has to take a policy action, but has the ability to commit to limiting its discretionary power. If the public believed the reform solved this time inconsistency problem, the policy maker could achieve complete discretion. However, if the public does not believe the reform to be successful some discretion must be sacrificed in order to induce public trust. With repeated interactions, the policy maker can build reputation about her reformed incentives. However, equilibrium reputation dynamics are extremely sensitive to assumptions about the publics beliefs, particularly after unexpected events. To overcome this limitation, I study the optimal robust policy that implements public trust for all beliefs that are consistent with common knowledge of rationality. I focus on robustness to all extensive-form rationalizable beliefs and provide a characterization. I show that the robust policy exhibits both partial and permanent reputation building along its path, as well as endogenous transitory reputation losses. In addition, I demonstrate that almost surely the policy maker eventually convinces the public she does not face a time consistency problem and she is able to do this with an exponential arrival rate. This implies that as we consider more patient policy makers, the payoff of robust policies converge to the complete information benchmark. I finally explore how further restrictions on beliefs alter optimal policy and accelerate reputation building.by Juan P. Xandri Antuña.Ph.D