Games between programs

One can set up a game between two programs generating single bits of output by for example taking the input at each step to be the concatenation of the historical sequences of outputs from the two programs. The pictures below show what happens if the programs operate by applying elementary cellular automaton rules t times to 2t+1 inputs. The plots on the left show cumulative scores in the Evens and Odds game; the array on the right indicates for each of the 256 possible rules the average number of wins it gets against each of the 256 rules. At some level considerable complexity is evident. But the rules that win most often typically seem to do so in rather simple ways.