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Volume 1, 2014

Journal of Dynamics & Games

2014 , Volume 1 , Issue 3

Special issue in honor of the 60th birthday of Sylvain Sorin

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Preface: Special Issue in Honor of the 60th Birthday of Sylvain Sorin
Josef Hofbauer, Rida Laraki and Jérôme Renault
2014, 1(3): i-iv doi: 10.3934/jdg.2014.1.3i +[Abstract](513) +[PDF](220.5KB)
Abstract:
We have the immense pleasure of editing this special issue in honor of Sylvain Sorin. It follows the international conference ``GameS and StrategY in PariS,'' organized by the French school of mathematical game theory, that was also held in his honor. The conference took place at the Institut Henri Poincaré (IHP) in June 2012, included 21 plenary talks and attracted more than 150 participants.

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Asymptotic behavior of compositions of under-relaxed nonexpansive operators
Jean-Bernard Baillon, Patrick L. Combettes and Roberto Cominetti
2014, 1(3): 331-346 doi: 10.3934/jdg.2014.1.331 +[Abstract](420) +[PDF](330.0KB)
Abstract:
In general there exists no relationship between the fixed point sets of the composition and of the average of a family of nonexpansive operators in Hilbert spaces. In this paper, we establish an asymptotic principle connecting the cycles generated by under-relaxed compositions of nonexpansive operators to the fixed points of the average of these operators. In the special case when the operators are projectors onto closed convex sets, we prove a conjecture by De Pierro which has so far been established only for projections onto affine subspaces.
Policy improvement for perfect information additive reward and additive transition stochastic games with discounted and average payoffs
Matthew Bourque and T. E. S. Raghavan
2014, 1(3): 347-361 doi: 10.3934/jdg.2014.1.347 +[Abstract](549) +[PDF](401.6KB)
Abstract:
We give a policy improvement algorithm for additive reward, additive transition (ARAT) zero-sum two-player stochastic games for both discounted and average payoffs. The class of ARAT games includes perfect information games.
Pure and Random strategies in differential game with incomplete informations
Pierre Cardaliaguet, Chloé Jimenez and Marc Quincampoix
2014, 1(3): 363-375 doi: 10.3934/jdg.2014.1.363 +[Abstract](777) +[PDF](386.0KB)
Abstract:
We investigate a two players zero sum differential game with incomplete information on the initial state: The first player has a private information on the initial state while the second player knows only a probability distribution on the initial state. This could be view as a generalization to differential games of the famous Aumann-Maschler framework for repeated games. In an article of the first author, the existence of the value in random strategies was obtained for a finite number of initial conditions (the probability distribution is a finite combination of Dirac measures). The main novelty of the present work consists in : first extending the result on the existence of a value in random strategies for infinite number of initial conditions and second - and mainly - proving the existence of a value in pure strategies when the initial probability distribution is regular enough (without atoms).
Competing for customers in a social network
Pradeep Dubey, Rahul Garg and Bernard De Meyer
2014, 1(3): 377-409 doi: 10.3934/jdg.2014.1.377 +[Abstract](563) +[PDF](575.3KB)
Abstract:
Customers' proclivities to buy products often depend heavily on who else is buying the same product. This gives rise to non-cooperative games in which firms sell to customers located in a ``social network''. Nash Equilibrium (NE) in pure strategies exist in general. In the quasi-linear case, NE are unique. If there are no a priori biases between customers and firms, there is a cut-off level above which high cost firms are blockaded at an NE, while the rest compete uniformly throughout the network. Otherwise firms could end up as regional monopolies.
    The connectivity of a customer is related to the money firms spend on him. This becomes particularly transparent when externalities are dominant: NE can be characterized in terms of the invariant measures on the recurrent classes of the Markov chain underlying the social network.
    When cost functions of firms are convex, instead of just linear, NE need no longer be unique as we show via an example. But uniqueness is restored if there is enough competition between firms or if their valuations of clients are anonymous.
    Finally we develop a general model of nonlinear externalities and show that existence of NE remains intact.
Existence of the uniform value in zero-sum repeated games with a more informed controller
Fabien Gensbittel, Miquel Oliu-Barton and Xavier Venel
2014, 1(3): 411-445 doi: 10.3934/jdg.2014.1.411 +[Abstract](655) +[PDF](550.1KB)
Abstract:
We prove that in a two-player zero-sum repeated game where one of the players, say player $1$, is more informed than his opponent and controls the evolution of information on the state, the uniform value exists. This result extends previous results on Markov decision processes with partial observation (Rosenberg, Solan, Vieille [15]), and repeated games with an informed controller (Renault [14]). Our formal definition of a more informed player is more general than the inclusion of signals, allowing therefore for imperfect monitoring of actions. We construct an auxiliary stochastic game whose state space is the set of second order beliefs of player $2$ (beliefs about beliefs of player $1$ on the state variable of the original game) with perfect monitoring and we prove it has a value by using a result of Renault [14]. A key element in this work is to prove that player $1$ can use strategies of the auxiliary game in the original game in our general framework, from which we deduce that the value of the auxiliary game is also the value of our original game by using classical arguments.
A primal condition for approachability with partial monitoring
Shie Mannor, Vianney Perchet and Gilles Stoltz
2014, 1(3): 447-469 doi: 10.3934/jdg.2014.1.447 +[Abstract](354) +[PDF](744.0KB)
Abstract:
In approachability with full monitoring there are two types of conditions that are known to be equivalent for convex sets: a primal and a dual condition. The primal one is of the form: a set $\mathcal{C}$ is approachable if and only all containing half-spaces are approachable in the one-shot game. The dual condition is of the form: a convex set $\mathcal{C}$ is approachable if and only if it intersects all payoff sets of a certain form. We consider approachability in games with partial monitoring. In previous works [5,7] we provided a dual characterization of approachable convex sets and we also exhibited efficient strategies in the case where $\mathcal{C}$ is a polytope. In this paper we provide primal conditions on a convex set to be approachable with partial monitoring. They depend on a modified reward function and lead to approachability strategies based on modified payoff functions and that proceed by projections similarly to Blackwell's (1956) strategy. This is in contrast with previously studied strategies in this context that relied mostly on the signaling structure and aimed at estimating well the distributions of the signals received. Our results generalize classical results by Kohlberg [3] (see also [6]) and apply to games with arbitrary signaling structure as well as to arbitrary convex sets.
General limit value in dynamic programming
Jérôme Renault
2014, 1(3): 471-484 doi: 10.3934/jdg.2014.1.471 +[Abstract](672) +[PDF](385.9KB)
Abstract:
We consider a dynamic programming problem with arbitrary state space and bounded rewards. Is it possible to uniquely define a limit value for the problem, when the ``patience" of the decision-maker tends to infinity ? We consider, for each evaluation $\theta$ (a probability distribution over positive integers) the value function $v_{\theta}$ of the problem where the weight of any stage $t$ is given by $\theta_t$, and we investigate the uniform convergence of a sequence $(v_{\theta^k})_k$ when the ``impatience" of the evaluations vanishes, in the sense that $\sum_{t} | \theta^k_{t}-\theta^k_{t+1}| \rightarrow_{k \to \infty} 0.$ We prove that this uniform convergence happens if and only if the metric space $\{v_{\theta^k}, k\geq 1\}$ is totally bounded. Moreover there exists a particular function $v^*$, independent of the particular chosen sequence $({\theta^k})_k$, such that any limit point of such sequence of value functions is precisely $v^*$. The result applies in particular to discounted payoffs when the discount factor vanishes, as well as to average payoffs where the number of stages goes to infinity, and extends to models with stochastic transitions.
Local stability of strict equilibria under evolutionary game dynamics
William H. Sandholm
2014, 1(3): 485-495 doi: 10.3934/jdg.2014.1.485 +[Abstract](417) +[PDF](585.6KB)
Abstract:
We consider the stability of strict equilibrium under deterministic evolutionary game dynamics. We show that if the correlation between strategies' growth rates and payoffs is positive and bounded away from zero in a neighborhood of a strict equilibrium, then this equilibrium is locally stable.
A prequential test for exchangeable theories
Alvaro Sandroni and Eran Shmaya
2014, 1(3): 497-505 doi: 10.3934/jdg.2014.1.497 +[Abstract](468) +[PDF](311.5KB)
Abstract:
We construct a prequential test of probabilistic forecasts that does not reject correct forecasts when the data-generating processes is exchangeable and is not manipulable by a false forecaster.
Strong approachability
Barak Shani and Eilon Solan
2014, 1(3): 507-535 doi: 10.3934/jdg.2014.1.507 +[Abstract](394) +[PDF](9857.5KB)
Abstract:
We introduce the concept of strongly approachable sets in two-player repeated games with vector payoffs. A set in the payoff space is strongly approachable by a player if the player can guarantee that from a certain stage on the average payoff will be inside that set, regardless of the strategy that the other player implements. We provide sufficient conditions that ensure that a closed convex approachable set is also strongly approachable in the expected deterministic version of the game.
Game dynamics and Nash equilibria
Yannick Viossat
2014, 1(3): 537-553 doi: 10.3934/jdg.2014.1.537 +[Abstract](519) +[PDF](435.0KB)
Abstract:
There are games with a unique Nash equilibrium but such that, for almost all initial conditions, all strategies in the support of this equilibrium are eliminated by the replicator dynamics and the best-reply dynamics.

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