Yannick Viossat

Time Average Replicator and Best-Reply Dynamics

Using an explicit representation in terms of the logit map, we show, in a unilateral framework, that the time average of the replicator dynamics is a perturbed solution of the best-reply dynamics.

The replicator dynamics does not lead to correlated equilibria

It is shown that, under the replicator dynamics, all strategies played in correlated equilibrium may be eliminated, so that only strategies with zero marginal probability in all correlated equilibria survive. This occurs in particular in a family of 4×4 games built by adding a strategy to a Rock-Paper-Scissors game.

No-regret Dynamics and Fictitious Play

Potential based no-regret dynamics are shown to be related to fictitious play. Roughly, these are e-best reply dynamics where e is the maximal regret, which vanishes with time. This allows for alternative and sometimes much shorter proofs of known results on convergence of no-regret dynamics to the set of Nash equilibria.

Evolutionary beliefs and financial markets

Why do investors keep different opinions even though they learn from their own failures and successes? Why do investors keep different opinions even though they observe each other and learn from their relative failures and successes? We analyze beliefs dynamics when beliefs result from a very general learning process that favors beliefs leading to higher absolute or relative utility levels. We show that such a process converges to the Nash equilibrium in a game of strategic belief choices. The asymptotic beliefs are subjective and heterogeneous across the agents. Optimism (respectively overconfidence) as well as pessimism (respectively doubt) emerge from the learning process. Furthermore, we obtain a positive correlation between pessimism (respectively doubt) and risk tolerance. Under reasonable assumptions, beliefs exhibit a pessimistic bias and, as a consequence, the risk premium is higher than in a standard setting.

Evolutionary dynamics may eliminate all strategies used in correlated equilibrium

We show on a 4x4 example that many dynamics may eliminate all strategies used in correlated equilibria, and this for an open set of games. This holds for the best-response dynamics, the Brown-von Neumann-Nash dynamics and any monotonic or weakly sign-preserving dynamics satisfying some standard regularity conditions. For the replicator dynamics and the best-response dynamics, elimination of all strategies used in correlated equilibrium is shown to be robust to the addition of mixed strategies as new pure strategies.

Imitation dynamics with payoff shocks

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 aggregate-shocks model of Fudenberg and Harris in J Econ Theory 57(2):420–441, 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 high-noise regime, non-equilibrium 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 where non-equilibrium states are no longer attracting and dominated strategies become extinct (a.s.), no matter the noise level.

Evolutionary Strategic Beliefs and Financial Markets

We provide a discipline for belief formation through an evolutionary process which favors beliefs leading to higher utility levels at the Walrasian equilibrium. We show that such an evolutionary process converges to the Nash equilibrium in a game of strategic beliefs choices. The asymptotic beliefs are subjective and heterogeneous across the agents. Optimism (resp. overconfidence) as well as pessimism (resp. doubt) both emerge from the evolutionary process. Furthermore, we obtain a positive correlation between pessimism (resp. doubt) and risk-tolerance. We analyze the equilibrium characteristics. Under reasonable assumptions, the consensus belief is pessimistic and, as a consequence, the risk premium is higher than in a standard setting.