Naive fictitious play in an evolutionary environment

A fictitious play algorithm with a unit memory length within an evolutionary environment is considered. The aggregate average behavior model is proposed and analyzed. The existence, uniqueness and global asymptotic stability of equilibrium is proved for games with a cycling property. Also, a noisy version of the algorithm is considered, which gives rise to a system with a unique, globally asymptotically stable equilibrium for any game

Uniqueness Conditions for Point-Rationalizable

The unique point-rationalizable solution of a game is the unique Nash equilibrium. However, this solution has the additional advantage that it can be justified by the epistemic assumption that it is Common Knowledge of the players that only best responses are chosen. Thus, games with a unique point-rationalizable solution allow for a plausible explanation of equilibrium play in one-shot strategic situations, and it is therefore desireable to identify such games. In order to derive sufficient and necessary conditions for unique point-rationalizable solutions this paper adopts and generalizes the contraction-property approach of Moulin (1984) and of Bernheim (1984). Uniqueness results obtained in this paper are derived under fairly general assumptions such as games with arbitrary metrizable strategy sets and are especially useful for complete and bounded, for compact, as well as for finite strategy sets. As a mathematical side result existence of a unique fixed point is proved under conditions that generalize a fixed point theorem due to Edelstein (1962).

Uniqueness conditions for point-rationalizable solutions of games with metrizable strategy sets

The unique point-rationalizable solution of a game is the unique Nash equilibrium. However, this solution has the additional advantage that it can be justified by the epistemic assumption that it is Common Knowledge of the players that only best responses are chosen. Thus, games with a unique point-rationalizable solution allow for a plausible explanation of equilibrium play in one-shot strategic situations, and it is therefore desireable to identify such games. In order to derive sufficient and necessary conditions for unique point-rationalizable solutions this paper adopts and generalizes the contraction-property approach of Moulin (1984) and of Bernheim (1984). Uniqueness results obtained in this paper are derived under fairly general assumptions such as games with arbitrary metrizable strategy sets and are especially useful for complete and bounded, for compact, as well as for finite strategy sets. As a mathematical side result existence of a unique fixed point is proved under conditions that generalize a fixed point theorem due to Edelstein (1962)

On the convergence problem in Mean Field Games: a two state model without uniqueness

We consider N-player and mean field games in continuous time over a finite horizon, where the position of each agent belongs to {-1,1}. If there is uniqueness of mean field game solutions, e.g. under monotonicity assumptions, then the master equation possesses a smooth solution which can be used to prove convergence of the value functions and of the feedback Nash equilibria of the N-player game, as well as a propagation of chaos property for the associated optimal trajectories. We study here an example with anti-monotonous costs, and show that the mean field game has exactly three solutions. We prove that the value functions converge to the entropy solution of the master equation, which in this case can be written as a scalar conservation law in one space dimension, and that the optimal trajectories admit a limit: they select one mean field game soution, so there is propagation of chaos. Moreover, viewing the mean field game system as the necessary conditions for optimality of a deterministic control problem, we show that the N-player game selects the optimizer of this problem

On the Single-Valuedness of the Pre-Kernel

Based on results given in the recent book by Meinhardt (2013), which presents a dual characterization of the pre-kernel by a finite union of solution sets of a family of quadratic and convex objective functions, we could derive some results related to the uniqueness of the pre-kernel. Rather than extending the knowledge of game classes for which the pre-kernel consists of a single point, we apply a different approach. We select a game from an arbitrary game class with a single pre-kernel element satisfying the non-empty interior condition of a payoff equivalence class, and then establish that the set of related and linear independent games which are derived from this pre-kernel point of the default game replicates this point also as its sole pre-kernel element. In the proof we apply results and techniques employed in the above work. Namely, we prove in a first step that the linear mapping of a pre-kernel element into a specific vector subspace of balanced excesses is a singleton. Secondly, that there cannot exist a different and non-transversal vector subspace of balanced excesses in which a linear transformation of a pre-kernel element can be mapped. Furthermore, we establish that on the restricted subset on the game space that is constituted by the convex hull of the default and the set of related games, the pre-kernel correspondence is single-valued, and therefore continuous. Finally, we provide sufficient conditions that preserve the pre-nucleolus property for related games even when the default game has not a single pre-kernel point.Comment: 24 pages, 2 table