We report on the formalization in Ssreflect/Coq of a number of concepts and results from algorithmic game theory, including potential games, smooth games, solution concepts such as Pure and Mixed Nash Equilibria, Coarse Correlated Equilibria, epsilonapproximate equilibria, and behavioral models of games such as bestresponse dynamics. We apply the formalization to prove Price of Stability bounds for, and convergence under bestresponse dynamics of, the Atomic Routing game, which has applications in computer networking. Our second application proves that Affine Congestion games are (5/3, 1/3)smooth, and therefore have Price of Anarchy 5/2. Our formalization is available online.
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From Nash Equilibria to Chain Recurrent Sets: An Algorithmic Solution Concept for Game Theory
In 1950, Nash proposed a natural equilibrium solution concept for games hence called Nash equilibrium, and proved that all finite games have at least one. The proof is through a simple yet ingenious application of Brouwer’s (or, in another version Kakutani’s) fixed point theorem, the most sophisticated result in his era’s topology—in fact, recent algorithmic work has established that Nash equilibria are computationally equivalent to fixed points. In this paper, we propose a new class of universal nonequilibrium solution concepts arising from an important theorem in the topology of dynamical systems that was unavailable to Nash. This approach starts with both a game and a learning dynamics, defined over mixed strategies. The Nash equilibria are fixpoints of the dynamics, but the system behavior is captured by an object far more general than the Nash equilibrium that is known in dynamical systems theory as chain recurrent set. Informally, once we focus on this solution concept—this notion of “the outcome of the game”—every game behaves like a potential game with the dynamics converging to these states. In other words, unlike Nash equilibria, this solution concept is algorithmic in the sense that it has a constructive proof of existence. We characterize this solution for simple benchmark games under replicator dynamics, arguably the best known evolutionary dynamics in game theory. For (weighted) potential games, the new concept coincides with the fixpoints/equilibria of the dynamics. However, in (variants of) zerosum games with fully mixed (i.e., interior) Nash equilibria, it covers the whole state space, as the dynamics satisfy specific information theoretic constants of motion. We discuss numerous novel computational, as well as structural, combinatorial questions raised by this chain recurrence conception of games.
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 Award ID(s):
 1763970
 NSFPAR ID:
 10094286
 Date Published:
 Journal Name:
 Entropy
 Volume:
 20
 Issue:
 10
 ISSN:
 10994300
 Page Range / eLocation ID:
 782
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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