Section: New Results
General Results in Game Theory
Our work on game theory is often motivated by applications to wireless networks but can often have a more general application.
In [38] , motivated by applications to multiantenna wireless networks, we propose a distributed and asynchronous algorithm for stochastic semidefinite programming This algorithm is a stochastic approximation of a continoustime matrix exponential scheme regularized by the addition of an entropylike term to the problem's objective function. We show that the resulting algorithm converges almost surely to an $\left(\u03f5\right)$approximation of the optimal solution requiring only an unbiased estimate of the gradient of the problem's stochastic objective.
As explained in the previous section, classical Nash equilibrium concepts become irrelevant in situations where the environment evolves over time. In [15] , we study one of the main concept of online learning and sequential decision problem known as regret minimization. Our objective is to provide a quick overview and a comprehensive introduction to online learning and game theory.
In practice, it is rarely reasonable to assume that players have access to the strategy of the others and implementing a best response can thus become cumbersome. Replicator dynamics is a fundamental approach in evolutionary game theory in which players adjust their strategies based on their actions’ cumulative payoffs over time – specifically, by playing mixed strategies that maximize their expected cumulative payoff.

In [19] , we investigate the impact of payoff shocks on the evolution of large populations of myopic players that employ simple strategy revision protocols such as the "imitation of success". In the noiseless case, this process is governed by the standard (deterministic) replicator dynamics; in the presence of noise however, the induced stochastic dynamics are different from previous versions of the stochastic replicator dynamics (such as the aggregateshocks model of Fudenberg and Harris, 1992). In this context, we show that strict equilibria are always stochastically asymptotically stable, irrespective of the magnitude of the shocks; on the other hand, in the highnoise regime, nonequilibrium states may also become stochastically asymptotically stable and dominated strategies may survive in perpetuity (they become extinct if the noise is low). Such behavior is eliminated if players are less myopic and revise their strategies based on their cumulative payoffs. In this case, we obtain a second order stochastic dynamical system whose attracting states coincide with the game's strict equilibria and where dominated strategies become extinct (a.s.), no matter the noise level.

In [13] , we study a new class of continuoustime learning dynamics consisting of a replicatorlike drift adjusted by a penalty term that renders the boundary of the game's strategy space repelling. These penaltyregulated dynamics are equivalent to players keeping an exponentially discounted aggregate of their ongoing payoffs and then using a smooth best response to pick an action based on these performance scores. Building on the duality with evolutionary game theory, we design a discretetime, payoffbased learning algorithm that converges to (arbitrarily precise) approximations of Nash equilibria in potential games. Moreover, the algorithm remains robust in the presence of stochastic perturbations and observation errors, and it does not require any synchronization between players, which is a very important property when applying such technique to traffic engineering.

In [18] , we investigate an other class of reinforcement learning dynamics in which the players strategy adjustment is regularized with a strongly convex penalty term. In contrast to the class of penalty functions used to define smooth best responses in models of stochastic fictitious play, the regularizers used in this paper need not be infinitely steep at the boundary of the simplex. Dropping this requirement gives rise to an important dichotomy between steep and nonsteep cases. In this general setting, our main results extend several properties of the replicator dynamics such as the elimination of dominated strategies, the asymptotic stability of strict Nash equilibria and the convergence of timeaveraged trajectories to interior Nash equilibria in zerosum games.

In [37] , we study a general class of gametheoretic learning dynamics in the presence of random payoff disturbances and observation noise, and we provide a unified framework that extends several rationality properties of the (stochastic) replicator dynamics and other game dynamics. In the unilateral case, we show that the stochastic dynamics under study lead to no regret, irrespective of the noise level. In the multiplayer case, we find that dominated strategies become extinct (a.s.) and strict Nash equilibria remain stochastically asymptotically stable – again, independently of the perturbations' magnitude. Finally, we establish an averaging principle for 2player games and we show that the empirical distribution of play converges to Nash equilibrium in zerosum games under any noise level.