Journal of Dynamics & Games
2015 , Volume 2 , Issue 2
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This paper studies learning in monotone Bayesian games with one-dimensional types and finitely many actions. Players switch between actions at a set of thresholds. A learning algorithm under which players adjust their strategies in the direction of better ones using payoffs received at similar signals to their current thresholds is examined. Convergence to equilibrium is shown in the case of supermodular games and potential games.
Smale's approach  to the classical two-players repeated Prisoner's Dilemma game is revisited here for $N$-players and Network games in the framework of Blackwell's approachability, stochastic approximations and differential inclusions.
In this paper we address a class of replicator dynamics, referred as polymatrix replicators, that contains well known classes of evolutionary game dynamics, such as the symmetric and asymmetric (or bimatrix) replicator equations, and some replicator equations for $n$-person games. Polymatrix replicators form a simple class of algebraic o.d.e.'s on prisms (products of simplexes), which describe the evolution of strategical behaviours within a population stratified in $p\geq 1$ social groups.
In the 80's Raymond Redheffer et al. developed a theory on the class of stably dissipative Lotka-Volterra systems. This theory is built around a reduction algorithm that ``infers'' the localization of the system' s attractor in some affine subspace. It was later proven that the dynamics on the attractor of such systems is always embeddable in a Hamiltonian Lotka-Volterra system.
In this paper we extend these results to polymatrix replicators.
We consider a chain of distributed systems governed by a degenerate parabolic equation, which satisfies a weak Hörmander type condition, with a control distributed over an open subdomain. In particular, we consider two objectives that we would like to accomplish. The first one being of a controllability type that consists of guaranteeing the terminal state to reach a target set starting from an initial condition; while the second one is keeping the state trajectory of the overall system close to a given reference trajectory over a finite time interval. We introduce the following framework. First, we partition the control subdomain into two disjoint open subdomains that are compatible with the strategy subspaces of the leader and that of the follower, respectively. Then, using the notion of Stackelberg's optimization (which is a hierarchical optimization framework), we provide a new result on the existence of optimal control strategies for such an optimization problem, where the follower (which corresponds to the second criterion) is required to respond optimally, in the sense of best-response correspondence to the strategy of the leader (which is associated to the controllability-type problem) so as to achieve the overall objectives. Finally, we remark on the implication of our result in assessing the influence of the target set on the strategy of the follower with respect to the direction of leader-follower (and vice-versa) information flow.
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