Perhaps it would be fruitful to consider the two participants, who I’ll call Alpha and Omega, as finite computer programs, for which Omega has access to Alpha’s source code. Maybe Alpha also has access to Omega’s source code. Each of them chooses a number (from the natural numbers, the reals, [0,1], {0,1}, or whatever). It is common knowledge between them that Omega’s goal is to choose differently from Alpha, and Alpha’s goal is to choose the same as Omega.
Given various constraints on the computational or proof-theoretic capabilities of Alpha and Omega, under what circumstances does either player have a winning strategy?
If they each have access to a source of randomness, the game could be generalised to Omega trying to maximise the probability that they differ, and Alpha’s being to minimise that probability.
Perhaps it would be fruitful to consider the two participants, who I’ll call Alpha and Omega, as finite computer programs, for which Omega has access to Alpha’s source code. Maybe Alpha also has access to Omega’s source code. Each of them chooses a number (from the natural numbers, the reals, [0,1], {0,1}, or whatever). It is common knowledge between them that Omega’s goal is to choose differently from Alpha, and Alpha’s goal is to choose the same as Omega.
Given various constraints on the computational or proof-theoretic capabilities of Alpha and Omega, under what circumstances does either player have a winning strategy?
If they each have access to a source of randomness, the game could be generalised to Omega trying to maximise the probability that they differ, and Alpha’s being to minimise that probability.