#no-regret

A player uses multiplicative weights, $x^{(t)}(a) \propto \exp\big(\eta \sum_{\tau < t} u^{(\tau)}(a)\big)$, starting from the uniform distribution. In every round, action $a^\ast$ earns at least $\varepsilon = 0.4$ more than action $a$, whatever the opponents do. With $\eta = 0.1$, what is the largest possible value of the ratio $x(a) / x(a^\ast)$ after 50 rounds?