May’s challenge asked for the probability that a robot should adopt the discrete strategy in the Nash equilibrium solution of the newly structured Robot Updated Swimming Trials (or RUST for short). This equilibrium probability p has the property that if all opponents are choosing the discrete strategy with this probability, then the remaining competitor is indifferent to choose the discrete strategy or not. Interestingly, in order to solve for p we do not need to compute what the alternate continuous strategy is, we must only assume it would put nonzero weight on every race with probability 1. We leave the argument for this assumption to the reader.

By symmetry, the remaining competitor’s probability of winning must be exactly 1/3 whether they choose discrete or continuous. So we just have to compute the probability of winning with discrete or continuous with all other competitors using probability p, set it equal to 1/3, and then backsolve for p. In the image above, we show the derivation of how to solve for p by computing the win probability when using the continuous strategy (a similar computation could be done using the discrete strategy, but care is required to account for ALL the possible ways to win!).

The final result shows that the probability of choosing the discrete strategy in the Nash equilibrium is very high, p ~ 0.999560…

Congrats to this month’s solvers that were able to compute this tricky probability!