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!**