  We show that solving delay games with winning conditions given by
deterministic and nondeterministic weak Muller automata is 2EXPTIME-complete
respectively 3EXPTIME-complete. Furthermore, doubly and triply exponential
lookahead is necessary and sufficient to win such games. These results are the
first that show that the succinctness of the automata types used to specify the
winning conditions has an influence on the complexity of these problems.
