Delay Games with WMSO+U Winning Conditions
Delay games are two-player games of infinite duration in which one player may delay her moves to obtain a lookahead on her opponent’s moves. We consider delay games with winning conditions expressed in weak monadic second order logic with the unbounding quantifier, which is able to express (un)boundedness properties. We show that it is decidable whether the delaying player has a winning strategy using bounded lookahead and give a doubly-exponential upper bound on the necessary lookahead. In contrast, we show that bounded lookahead is not always sufficient: we present a game that can be won with unbounded lookahead, but not with bounded lookahead. Then, we consider such games with unbounded lookahead and show that the exact evolution of the lookahead is irrelevant: the winner is always the same, as long as the initial lookahead is large enough and the lookahead is unbounded.