# Working Papers

1. Index Theory of Strategic-Form Games with an Application to Extensive-Form Games (New version)

Whenever equivalent mixed strategies of a player are identified (topologically) in a normal-form game, the resulting space may not be a simplex anymore, but is a general polytope. We show that an index/degree theory of equilibria can be developed in full generality for games in which the strategy set of the players are general polytopes and their payoff functions are multiaffine. Index and degree theories work as a tool that helps identify equilibria that are robust to payoff perturbations of the game. Because the strategy set of each player is a result of the identification of equivalent mixed strategies, the resulting polytope is of lower dimension than the original mixed strategy simplices. This, together with an index theory, has algorithmic applications for checking for robustness of equilibria as well as finding equilibria in extensive-form games.

2. Information Spillover in a Bayesian Repeated Game (New version)

This paper considers an infinitely repeated three-player Bayesian game with lack of information on two sides, in which an informed player plays two zero-sum games simultaneously at each stage against two uninformed players. This is a generalization of Aumann, Maschler and Stearns (1995) two-player zero-sum one sided incomplete information model. Under a correlated prior, the informed player faces the problem of how to optimally disclose information among two uninformed players in order to maximize his long term average payoffs. The objective is to understand the adverse effects of "information spillover" from one game to the other in the equilibrium payoff set of the informed player. We provide conditions under which the informed player can fully overcome such adverse effects, and show that in some cases the adverse effects are unsurmountable and severe.

3. On Sustainable Equilibria (joint with S. Govindan and R. Laraki). (New version coming soon)

Proceedings 21st ACM Conference on Economics and Computation (EC’20).

Following the ideas laid out in Myerson (1996), Hofbauer (2000) defined an equilibrium of a game as sustainable if it can be made the unique equilibrium of a game obtained by deleting a subset of the strategies that are inferior replies to it, and then adding others. Hofbauer also formalized Myerson’s conjecture about the relationship between the sustainability of an equilibrium and its index: for generic games, an equilibrium is sustainable iff its index is +1. von Schemde and von Stengel (2008) proved this conjecture for bimatrix games. This paper shows that the conjecture is true for all finite games.

4. Finite Characterization of Perfect Equilibrium (with I. Callejas and S. Govindan) (New version - submitted)

Govindan and Klumpp (2002) provided a characterization of perfect equilibria using finite Lexicographic Probability Systems (LPSs). Their characterization was essentially finite in that they showed that there exists a finite bound on the number of levels in the LPS, but they did not compute it explicitly. In this note, we draw on two recent developments in Real Algebraic Geometry to compute this bound.

# Work in Progress

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5. Hyperstable Components are Essential

We show that in generic Extensive-Form games the hyperstable components of equilibria, as defined by Kohlberg and Mertens (1986), are the essential components of fixed-points of Nash-maps of the game. This results strengthens the result of Govindan and Wilson (2003) and characterizes hyperstable equilibria in terms of topological invariants of Nash-maps.

6. Lexicographic Probability Systems and Stable Equilibrium