**A** quick riddle from the Riddler

*In a two-person game, Abigail and Zian both choose between a and z. Abigail win one point with probability .9 if they choose (a,a) and with probability 1 if they choose (a,z), and two points with probability .4 if they choose (z,z) and with probability .6 if they choose (z,a). Find the optimal probabilities δ and ς of choosing a for both Abigail and Zian when δ is known to Zian.*

Since the average gain for Abigail is *δ*(1-.1*ς*)+2(1-*δ*)(.4+.2*ς*) the riddle sums up as solving the minmax problem

the solution in *ς* is either 0 or 1 depending on *δ* being smaller or larger than 12/22, which leads to this value as the expected gain. The saddlepoint is hardly visible in the above image. While *ς* is either 0 or 1 in the optimal setting, a constant choice of 1 or 0 would modify the optimal for *δ* except that Abigail must declare her value of *δ!*

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