arXiv subjects

Stéphane Le Roux

Publications and source records attributed to Stéphane Le Roux.

29 records · Page 2Linked to original sources

arXiv2016

A Semi-Potential for Finite and Infinite Sequential Games (Extended Abstract)

We consider a dynamical approach to sequential games. 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 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 sequential games 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 Delta^0_2 sets.

cs.GT↗

arXiv2016

Extending Finite Memory Determinacy to Multiplayer Games

We show that under some general conditions the finite memory determinacy of a class of two-player win/lose games played on finite graphs implies the existence of a Nash equilibrium built from finite memory strategies for the corresponding class of multi-player multi-outcome games. This generalizes a previous result by Brihaye, De Pril and Schewe. For most of our conditions we provide counterexamples showing that they cannot be dispensed with. Our proofs are generally constructive, that is, provide upper bounds for the memory required, as well as algorithms to compute the relevant winning strategies.

cs.GT↗

arXiv2016

Equilibria in multi-player multi-outcome infinite sequential games

We investigate the existence of certain types of equilibria (Nash, $\varepsilon$-Nash, subgame perfect, $\varepsilon$-subgame perfect, Pareto-optimal) in multi-player multi-outcome infinite sequential games. We use two fundamental approaches: one requires strong topological restrictions on the games, but produces very strong existence results. The other merely requires some very basic determinacy properties to still obtain some existence results. Both results are transfer results: starting with the existence of some equilibria for a small class of games, they allow us to conclude the existence of some type of equilibria for a larger class. To make the abstract results more concrete, we investigate as a special case infinite sequential games with real-valued payoff functions. Depending on the class of payoff functions (continuous, upper semi-continuous, Borel) and whether the game is zero-sum, we obtain various existence results for equilibria. Our results hold for games with two or up to countably many players.

cs.LO↗

arXiv2015

Finite choice, convex choice and finding roots

We investigate choice principles in the Weihrauch lattice for finite sets on the one hand, and convex sets on the other hand. Increasing cardinality and increasing dimension both correspond to increasing Weihrauch degrees. Moreover, we demonstrate that the dimension of convex sets can be characterized by the cardinality of finite sets encodable into them. Precisely, choice from an n+1 point set is reducible to choice from a convex set of dimension n, but not reducible to choice from a convex set of dimension n-1. Furthermore we consider searching for zeros of continuous functions. We provide an algorithm producing 3n real numbers containing all zeros of a continuous function with up to n local minima. This demonstrates that having finitely many zeros is a strictly weaker condition than having finitely many local extrema. We can prove 3n to be optimal.

math.LO↗

arXiv2014

Multi-scale metastable dynamics and the asymptotic stationary distribution of perturbed Markov chains

We consider a simple but important class of metastable discrete time Markov chains, which we call perturbed Markov chains. Basically, we assume that the transition matrices depend on a parameter $\varepsilon$, and converge as $\varepsilon$. We further assume that the chain is irreducible for $\varepsilon$ but may have several essential communicating classes when $\varepsilon$. This leads to metastable behavior, possibly on multiple time scales. For each of the relevant time scales, we derive two effective chains. The first one describes the (possibly irreversible) metastable dynamics, while the second one is reversible and describes metastable escape probabilities. Closed probabilistic expressions are given for the asymptotic transition probabilities of these chains, but we also show how to compute them in a fast and numerically stable way. As a consequence, we obtain efficient algorithms for computing the committor function and the limiting stationary distribution.

math.PR↗

arXiv2014

From winning strategy to Nash equilibrium

Game theory is usually considered applied mathematics, but a few game-theoretic results, such as Borel determinacy, were developed by mathematicians for mathematics in a broad sense. These results usually state determinacy, i.e. the existence of a winning strategy in games that involve two players and two outcomes saying who wins. In a multi-outcome setting, the notion of winning strategy is irrelevant yet usually replaced faithfully with the notion of (pure) Nash equilibrium. This article shows that every determinacy result over an arbitrary game structure, e.g. a tree, is transferable into existence of multi-outcome (pure) Nash equilibrium over the same game structure. The equilibrium-transfer theorem requires cardinal or order-theoretic conditions on the strategy sets and the preferences, respectively, whereas counter-examples show that every requirement is relevant, albeit possibly improvable. When the outcomes are finitely many, the proof provides an algorithm computing a Nash equilibrium without significant complexity loss compared to the two-outcome case. As examples of application, this article generalises Borel determinacy, positional determinacy of parity games, and finite-memory determinacy of Muller games.

math.LO↗

arXiv2008

Conversion/Preference Games

We introduce the concept of Conversion/Preference Games, or CP games for short. CP games generalize the standard notion of strategic games. First we exemplify the use of CP games. Second we formally introduce and define the CP-games formalism. Then we sketch two `real-life' applications, namely a connection between CP games and gene regulation networks, and the use of CP games to formalize implied information in Chinese Wall security. We end with a study of a particular fixed-point construction over CP games and of the resulting existence of equilibria in possibly infinite games.

cs.GT↗

arXiv2007

Discrete Nondeterminism and Nash Equilibria for Strategy-Based Games

Several notions of game enjoy a Nash-like notion of equilibrium without guarantee of existence. There are different ways of weakening a definition of Nash-like equilibrium in order to guarantee the existence of a weakened equilibrium. Nash's approach to the problem for strategic games is probabilistic, \textit{i.e.} continuous, and static. CP and BR approaches for CP and BR games are discrete and dynamic. This paper proposes an approach that lies between those two different approaches: a discrete and static approach. multi strategic games are introduced as a formalism that is able to express both sequential and simultaneous decision-making, which promises a good modelling power. multi strategic games are a generalisation of strategic games and sequential graph games that still enjoys a Cartesian product structure, \textit{i.e.} where agent actually choose their strategies. A pre-fixed point result allows guaranteeing existence of discrete and non deterministic equilibria. On the one hand, these equilibria can be computed with polynomial (low) complexity. On the other hand, they are effective in terms of recommendation, as shown by a numerical example.

cs.GT↗

arXiv2007

Graphs and Path Equilibria

The quest for optimal/stable paths in graphs has gained attention in a few practical or theoretical areas. To take part in this quest this chapter adopts an equilibrium-oriented approach that is abstract and general: it works with (quasi-arbitrary) arc-labelled digraphs, and it assumes very little about the structure of the sought paths and the definition of equilibrium, \textit{i.e.} optimality/stability. In this setting, this chapter presents a sufficient condition for equilibrium existence for every graph; it also presents a necessary condition for equilibrium existence for every graph. The necessary condition does not imply the sufficient condition a priori. However, the chapter pinpoints their logical difference and thus identifies what work remains to be done. Moreover, the necessary and the sufficient conditions coincide when the definition of optimality relates to a total order, which provides a full-equivalence property. These results are applied to network routing.

cs.GT↗

arXiv2007

Computable Closed Euclidean Subsets with and without Computable Points

The empty set of course contains no computable point. On the other hand, surprising results due to Zaslavskii, Tseitin, Kreisel, and Lacombe assert the existence of NON-empty co-r.e. closed sets devoid of computable points: sets which are `large' in the sense of positive Lebesgue measure. We observe that a certain size is in fact necessary: every non-empty co-r.e. closed real set without computable points has continuum cardinality. This leads us to investigate for various classes of computable real subsets whether they necessarily contain a (not necessarily effectively findable) computable point.

cs.LO↗

arXiv2007

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.

cs.DM↗