"Note that this doesn’t mean that Daniel will always lose the game but that he can lose by at most 1/12 the value of the game. If Nick doesn’t play optimally (N

"Note that this doesn’t mean that Daniel will always lose the game but that he can lose by at most 1/12 the value of the game. If Nick doesn’t play optimally (Nick doesn’t use his optimal mixed strategy), Daniel will most likely win!" I don't think this is correct. I think if Daniel plays his optimum strategy, Nick will get the same payoff no matter what he plays. I think this is a fairly general result, if one player is playing the optimal strategy, once the other player has eliminated options he should never play, it doesn't matter how his choices are distributed among the remaining options.