Non-Determinism and Nash Equilibria for Sequential Game over Partial Order

International audienceIn sequential games of traditional game theory, backward induction guarantees existence of Nash equilibrium by yielding a sub-game perfect equilibrium. But if payoffs range over a partially ordered set instead of the reals, then the backward induction predicate does no longer imply the Nash equilibrium predicate. Non-determinism is a solution: a suitable non-deterministic backward induction function returns a non-deterministic strategy profile which is a non-deterministic Nash equilibrium. The main notions and results in this article are constructive, conceptually simple and formalised in the proof assistant Coq

Acyclicity of Preferences, Nash Equilibria, and Subgame Perfect Equilibria: a Formal and Constructive Equivalence

In 1953, Kuhn showed that every sequential game has a Nash equilibrium by showing that a procedure, named ``backward induction'' in game theory, yields a Nash equilibrium. It actually yields Nash equilibria that define a proper subclass of Nash equilibria. In 1965, Selten named this proper subclass subgame perfect equilibria. In game theory, payoffs are rewards usually granted at the end of a game. Although traditional game theory mainly focuses on real-valued payoffs that are implicitly ordered by the usual total order over the reals, works of Simon or Blackwell already involved partially ordered payoffs. This paper generalises the notion of sequential game by replacing real-valued payoff functions with abstract atomic objects, called outcomes, and by replacing the usual total order over the reals with arbitrary binary relations over outcomes, called preferences. This introduces a general abstract formalism where Nash equilibrium, subgame perfect equilibrium, and ``backward induction'' can still be defined. This paper proves that the following three propositions are equivalent: 1) Preferences over the outcomes are acyclic. 2) Every sequential game has a Nash equilibrium. 3) Every sequential game has a subgame perfect equilibrium. The result is fully computer-certified using Coq. Beside the additional guarantee of correctness, the activity of formalisation using Coq also helps clearly identify the useful definitions and the main articulations of the proof

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

Connected Choice and the Brouwer Fixed Point Theorem

We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. Connected choice is the operation that finds a point in a non-empty connected closed set given by negative information. One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the same dimension. Another main result is that connected choice is complete for dimension greater than or equal to two in the sense that it is computably equivalent to Weak K\H{o}nig's Lemma. While we can present two independent proofs for dimension three and upwards that are either based on a simple geometric construction or a combinatorial argument, the proof for dimension two is based on a more involved inverse limit construction. The connected choice operation in dimension one is known to be equivalent to the Intermediate Value Theorem; we prove that this problem is not idempotent in contrast to the case of dimension two and upwards. We also prove that Lipschitz continuity with Lipschitz constants strictly larger than one does not simplify finding fixed points. Finally, we prove that finding a connectedness component of a closed subset of the Euclidean unit cube of any dimension greater or equal to one is equivalent to Weak K\H{o}nig's Lemma. In order to describe these results, we introduce a representation of closed subsets of the unit cube by trees of rational complexes.Comment: 36 page

An Ssreflect Tutorial

This document is a tutorial for ssreflect which is a proof language based on Coq. This tutorial is mostly dedicated to people who already know the basics of logic