4 Presentations

• 
  08:30 AM - 08:55 AM

  Consistency of decompositions for equivalent games

  • Nikolaos Pnevmatikos, presenter, University Paris 2, Panthéon-Assas
  • Joseph Abdou, University Paris 1, Panthéon-Sorbonne
  • Marco Scarsini, LUISS University
  • Xavier Venel, University Paris 1, Panthéon-Sorbonne

  Potential games are an interesting class of games that admit pure Nash equilibria and behave well with respect to the most common learning procedures. Some games, although they are not potential games, are close—in a sense to be made precise—to some potential game. It is therefore interesting to examine whether their equilibria are close to the equilibria of the potential game. With this in mind in their seminal paper Candogan, Menache, Ozdaglar and Parrilo (2011) were able to show that the class of strategic-form games having a fixed set of players and fixed sets of actions for each player is a linear space that can be decomposed into the orthogonal sum of three components, called the potential, harmonic and non-strategic component. Games in the harmonic component have a completely mixed equilibrium where all players mix uniformly over their actions; games in the non-strategic component are such that the payoff of each player is not affected by her own action, but only by other players’ actions. To achieve this decomposition Candogan et al. (2011) associate to each game a graph where vertices are action profiles and edges connect profiles that differ only for the action of one player. The analysis is then carried out by studying flows on graphs and using the Helmholtz decomposition theorem.
  The decomposition of Candogan et al. (2011) refers to games having all the same set of players and the same set of actions for each player. In their construction nothing connects the decomposition of a specific game G with the decomposition of another game G′ that is obtained from G adding an action to the set of feasible actions for player i. One may argue that this is reasonable, since the two games live in linear spaces of different dimension and the new game with an extra action may have equilibria that are very different from the ones in the original game, so, in general, the two games may have very little in common. In some situations, though, the two games are indeed strongly related. For instance, consider the case where the payoffs corresponding to the new action are just a replica of the payoffs of another action. In this case, from a strategic viewpoint, the two games G and G′ are actually the same game and every equilibrium in G′ can be mapped to an equilibrium in G. It would be reasonable to expect that the decomposition of G and G′ be strongly related. Unfortunately this is not the case. The question that we want to address in this paper is the following: is it possible to conceive a procedure that produces decompositions of G and G′ that are coherent? More generally, we will be looking at decomposition procedures that respect the game-theoretic structure of the problem. The fact that the components in the decomposition of Candogan et al. (2011) are orthogonal is a consequence of a suitable choice of inner product on the space of games with fixed players and action sets. This inner product is the same for every dimension and is defined using a uniform distribution. Here we will consider a class of distributions that will induce a class of inner products, that will in turn produce a class of decompositions. We will show that, given a decomposition of the game G based on the measure μ, we can find a new measure μ′ such that the μ′-decomposition of G′ is coherent with the μ-decomposition of G.

• 
  08:55 AM - 09:20 AM

  Expectational Stability in Aggregative Games

  • Luciana Fiorini, presenter, University of Western Australia
  • Wilfredo Maldonado, Catholic University of Brasilia.
  • Richard Cornes, deceased

  Using the replacement function associated with aggregative games, we analyze the expectational dynamics of the aggregate strategy of the game. We can interpret the Nash equilibrium of the game as the rational expectations equilibrium (REE) of the system, and we examine the expectational stability of the REE. We characterize local stability in terms of fundamentals and the REE itself. We illustrate the results through well-known aggregative games (Cournot games, Bertrand competition with differentiated goods, rent seeking games, and the public goods provision game) and analyze their global expectational dynamics.

• 
  09:20 AM - 09:45 AM

  Game Theory with Information: Witsenhausen Intrinsic Model

  • Daniel Kadnikov, presenter, ENPC, CERMICS
  • Michel De Lara, ENPC, CERMICS
  • Jean-Philippe Chancelier, ENPC, CERMICS

  In a context of competition, information -- who knows what and before whom -- plays a crucial role. Here, we concentrate on three models where the concept of information is present: Kuhn's extensive tree model (K-model), Alos-Ferrer, Ritzberger infinite tree model (AFR-model) and Witsenhausen model (W-model).

  The model proposed by Witsenhausen has the following main ingredients: a set of agents taking their decisions from a decision space, Nature taking decisions in a sample space, configuration space which is the product of the decision space by sample space equipped with a measure, information fields that are sigma-fields on the configuration space and, as usual, strategies that are mappings from configurations to actions which are measurable w.r.t. information fields. W-model deals with information in stochastic control problems, in all generality. For example, it allows to look at a problem without a priori knowing the order in which decisions were made by agents. In the subclass made of causal systems, there is at least one ordering in which agents take their decisions consistently with the given information sigma-fields.

  As pointed out by Witsenhausen, the difficulties in specifying the information structure of a game were faced and overcome in the early days of Game Theory through the introduction of extensive form games. The extensive form is the most richly structured way to describe game situations. The definition of the extensive form, that is now standard in most of the literature on Game Theory, is due to Kuhn, who modified the earlier definition used by von Neumann and Morgenstern. In his model Kuhn uses the language of graph theory to define four main ingredients of the game: players, game tree, information sets and strategies. The infinite tree AFR-model, as implied by its name, generalizes K-model with its finite tree to any possible tree one can imagine: infinite (repeated games), transfinite (long cheap talk) and even continuous (stochastic and differential games). To tame the zoo of infinite trees, authors use the language of set theory constructing not trees, but posets, thus elaborating the most general framework to describe tree structures in games existing up to now.

  We study whether AFR and W-models have the same potential to model games. First, we embed the subclass of causal W-models into the AFR-formalism by building an AFR-tree and translating definitions of information and strategies from W-formalism to AFR-formalism. Second, we will move in the opposite direction.

• 
  09:45 AM - 10:10 AM

  Equilibrium refinement in signaling games as truth conditions of counterfactuals

  • Christina Pawlowitsch, presenter, Paris 2, Panthéon-Assas

  In a game in extensive form - a game given by a tree - what can be an equilibrium typically
  depends on what players would do at a point in the game "off the equilibrium path," that is, a
  point in the game that can in principle be reached but that is not reached in the equilibrium under
  study. Equilibrium renement based on restrictions on beliefs "off the equilibrium path" can be
  related to Lewis's (1973) account of counterfactuals. In signaling games with two states of
  the world, two signals, and two actions in response to signals, "forward induction" (Govindan
  and Wilson 2009), which for this class of games coincides with "divinity" (Banks and Sobel
  1987), is equivalent to Lewis's accessibility condition relying on the similarity between the
  actual world and other possible worlds. The results are illustrated in a game-theoretic
  model of communicative implicatures driven by politeness.