The Robot Javelin championships were ebroiled in scandal when it was discovered that Spears Robot was heading into the finals with a secret advantage. We were tasked with determining an updated strategy for Spears’ opponent, Java-lin, and ultimately Java-lin’s updated winning chances. How did we do it?

First we had to find the Nash equilibrium for the fair game, that is the game with no information leakage. Using symmetry and the indifference principle we can show that each robot throws again if their first throw is less than (√5 - 1)/2 = φ ~ 0.618034…, the golden ratio. Nice!

Next, we determine Spears Robot’s exploitation strategy using its informational advantage. It turns out the best thing for Spears to learn about its opponent’s first throw is whether it caused them to rethrow. (That is, whether their first throw is above or below φ.) When the throw is below, Spears knows it is up against a uniform distribution of the opponent’s second throw and so Spears should rethrow if its first throw is below 0.5. When the opponent’s first throw is above φ, another indifference calculation shows that Spears should re-throw on any score below (1 - φ/2) ~ 0.690983….

Finally, we have to turn the tables on Spears Robot and exploit its deviation from Nash! The threshold that Java-lin chooses to rethrow when below is 7/12 < φ. The exploitation happens in the interval [7/12, φ]. This is the interval where Java-lin would normally choose to rethrow, but because Spears Robot will assume this, Java-lin can keep its best throws in this interval and take advantage of Spears Robot keeping weak scores between 0.5 and φ.

Now that we know the counterstrategy we can compute its winning chances, which come to (229 - 60√5)/192 ~ 0.4939370904…. So Java-lin is almost able to even the chances to 50/50 with this counterstrategy!

Congrats to everyone who computed the winning probability!