  This paper studies the last-iterate convergence properties of the exponential
weights algorithm with constant learning rates. We consider a repeated
interaction in discrete time, where each player uses an exponential weights
algorithm characterized by an initial mixed action and a fixed learning rate,
so that the mixed action profile $p^t$ played at stage $t$ follows an
homogeneous Markov chain. At first, we show that whenever a strict Nash
equilibrium exists, the probability to play a strict Nash equilibrium at the
next stage converges almost surely to 0 or 1. Secondly, we show that the limit
of $p^t$, whenever it exists, belongs to the set of ``Nash Equilibria with
Equalizing Payoffs''. Thirdly, we show that in strong coordination games, where
the payoff of a player is positive on the diagonal and 0 elsewhere, $p^t$
converges almost surely to one of the strict Nash equilibria. We conclude with
open questions.
