We consider multi-player infinite duration games on graphs. A game is given by a
directed graph, with vertices partitioned in $k$ sets; Player $i$ controls
vertices in the ith set. Edges are coloured with colours in a set $C$, and each
player has an objective $W_i \subseteq C^\omega$. A strategy profile is an array
of $k$ strategies, $\bar{\sigma} = (\sigma_1,\dots, \sigma_k)$, one for each
player. It is a Nash equilibrium if no player has an incentive to change their
strategy, that is: If $\bar{\sigma}$ produces a losing outcome for Player $i$,
then Player $i$ cannot modify his strategy and obtain a winning outcome.
It is known that, under some mild hypothesis on the objectives $W_i$, every game
admits some Nash equilibrium. The proofs for this fact usually build a NE where
strategies use infinite memory, even for games with very simple objectives.
Question: Do all reachability/Büchi/parity games admit a Nash equilibrium in which all strategies are positional?