Srihari Govindan

A global Newton method to compute Nash equilibria

Abstract A new algorithm is presented for computing Nash equilibria of finite games. Using Kohlberg and Mertens’ structure theorem we show that a homotopy method can be represented as a dynamical system and implemented by Smales global Newton method. The algorithm is outlined and computational experience is reported.

On Forward Induction

We examine Hillas and Kohlbergs conjecture that invariance to the addition of payoff-redundant strategies implies that a backward induction outcome survives deletion of strategies that are inferior replies to all equilibria with the same outcome. That is, invariance and backward induction imply forward induction. Although it suffices in simple games to interpret backward induction as a subgame-perfect or sequential equilibrium, to obtain general theorems we use a quasi-perfect equilibrium, viz. a sequential equilibrium in strategies that are admissible continuations from each information set. Using this version of backward induction, we prove the Hillas-Kohlberg conjecture for two-player extensive-form games with perfect recall. We also prove an analogous theorem for general games by interpreting backward induction as a proper equilibrium, since a proper equilibrium is equivalent to a quasi-perfect equilibrium of each extensive form with the same normal form, provided beliefs are justifed by perturbations invariant to inessential transformations of the extensive form. For a two-player game we prove that if a set of equilibria includes a proper equilibrium of every game with the same reduced normal form then it satisfies forward induction, i.e. it includes a proper equilibrium of the game after deleting strategies that are inferior replies to all equilibria in the set. We invoke slightly stronger versions of invariance and properness to handle nonlinearities in an N-player game.

Computing Nash equilibria by iterated polymatrix approximation

Abstract This article develops a new algorithm for computing Nash equilibria of N-player games. The algorithm approximates a game by a sequence of polymatrix games in which the players interact bilaterally. We provide sufficient conditions for local convergence to an equilibrium and report computational experience. The algorithm convergences globally and rapidly on test problems, although in theory it is not failsafe because it can stall on a set of codimension 1. But it can stall only at an approximate equilibrium with index +1, thus allowing a switch to the global Newton method, which is slower but can fail only on a set of codimension 2. Thus, the algorithm can be used to obtain a fast start for the more reliable global Newton method.

Direct Proofs of Generic Finiteness of Nash Equilibrium Outcomes

Using elementary techniques from semi-algebraic geometry, we give short proofs of two generic ̄niteness results for equilibria of ̄nite games. For each assignment of generic payo®s to a ̄xed normal form or to an extensive form with perfect recall, the Nash equilibria induce a ̄nite number of distributions over the possible outcomes.

Essential equilibria

The connected uniformly hyperstable sets of a finite game are shown to be precisely the essential components of Nash equilibria.

Structure theorems for game trees

Kohlberg and Mertens [Kohlberg, E. & Mertens, J. (1986) Econometrica 54, 1003–1039] proved that the graph of the Nash equilibrium correspondence is homeomorphic to its domain when the domain is the space of payoffs in normal-form games. A counterexample disproves the analog for the equilibrium outcome correspondence over the space of payoffs in extensive-form games, but we prove an analog when the space of behavior strategies is perturbed so that every path in the game tree has nonzero probability. Without such perturbations, the graph is the closure of the union of a finite collection of its subsets, each diffeomorphic to a corresponding path-connected open subset of the space of payoffs. As an application, we construct an algorithm for computing equilibria of an extensive-form game with a perturbed strategy space, and thus approximate equilibria of the unperturbed game.

A short proof of Harsanyi's purification theorem

Abstract A short proof of more general version of Harsanyis purification theorem is provided through an application of a powerful, yet intuitive, result from algebraic topology.

Robust nonexistence of equilibrium with incomplete markets

Abstract We construct a pure exchange economy with spot and real security markets for which there does not exist a competitive equilibrium. Moreover, we show that the problem of nonexistence is robust to small perturbations of the endowments of the consumers. The result is driven by a lack of strict convexity of preferences.

Maximal stable sets of two-player games

Abstract. If a connected component of perfect equilibria of a two-player game contains a stable set as defined by Mertens, then the component is itself stable. Thus the stable sets maximal under inclusion are connected components of perfect equilibria.

Symmetry and p-Stability

A symmetry of a game is a permutation of the player set and their strategy sets that leaves the payoff functions invariant. In this paper we introduce and discuss two relatively mild symmetry properties for set-valued solution concepts (that are equivalent when the solution concepts are single-valued) and show using examples that stable sets satisfy neither version. These examples also show that for every integer q, there exists a game with an equilibrium component of index q.