Greedy Galois Games

We show that two duelers with similar, lousy shooting skills (a.k.a. Galois duelers) will choose to take turns firing in accordance with the famous Thue-Morse sequence if they greedily demand their chances to fire as soon as the other’s a priori probability of winning exceeds their own. This contrasts with a result from the approximation theory of complex functions that says what more patient duelers would do, if they really cared about being as fair as possible. We note a consequent interpretation of the ThueMorse sequence in terms of certain expansions in fractional bases close to, but greater than, 1. Two players, Alice and Bob, are in a duel. They take turns firing at each other. However, both are Galois duelers, i.e., terrible shots, and equally so. On the other hand, they are deeply committed to fairness, and therefore they make the following deal. Before a single firearm is discharged, they draw up a firing sequence, i.e., the sequence of turns they will take, according to the following “greedy” rules. Alice shoots first. Bob then shoots as many times as he needs to obtain a probability of winning that exceeds the probability that Alice has won so far. Then Alice shoots again, until her a priori probability of having won exceeds Bob’s. Bob shoots next following the same rule, and so on until someone finally shuffles off his/her mortal coil. To illustrate, suppose the duelers’ hitting probability is 1/3. Alice shoots first, so her probability of winning by the end of round 0 is 1/3. Bob’s probability of winning so far is zero, so he shoots next. For Bob to win in round 1, Alice has to have missed in round 0, and Bob has to hit. Therefore, Bob’s probability of having won by the end of round 1 is (2/3)(1/3) = 2/9. This is still less than 1/3, so Bob shoots again in round 2. For Bob to win in round 2, he must survive Alice’s initial shot, miss in round 1, and hit in round 2. Hence his probability of ∗This work was funded in part by NSF grant DMS-1001370. 1Famously, the prodigal algebraist and Republican Radical Évariste Galois lost a duel over a lover on May 30, 1832, dying the next day.