Exponentially fast convergence to (strict) equilibrium via hedging
This provides a solution for applications like data networks where rapid convergence is critical, though it is incremental as it builds on existing exponential weights algorithms.
The paper tackles the problem of fast convergence in N-person games with pure Nash equilibria, showing that the hedge variant of exponential weights achieves exponentially fast convergence to strict equilibrium with perfect information, and maintains local convergence with high probability under uncertainty.
Motivated by applications to data networks where fast convergence is essential, we analyze the problem of learning in generic N-person games that admit a Nash equilibrium in pure strategies. Specifically, we consider a scenario where players interact repeatedly and try to learn from past experience by small adjustments based on local - and possibly imperfect - payoff information. For concreteness, we focus on the so-called "hedge" variant of the exponential weights algorithm where players select an action with probability proportional to the exponential of the action's cumulative payoff over time. When players have perfect information on their mixed payoffs, the algorithm converges locally to a strict equilibrium and the rate of convergence is exponentially fast - of the order of $\mathcal{O}(\exp(-a\sum_{j=1}^{t}γ_{j}))$ where $a>0$ is a constant and $γ_{j}$ is the algorithm's step-size. In the presence of uncertainty, convergence requires a more conservative step-size policy, but with high probability, the algorithm remains locally convergent and achieves an exponential convergence rate.