Equilibrium in a Halting Game
Even if you have a one-player game there is no computable equilibrium. Consider nature putting probability $1/2^i$ on program $i$. Any computable strategy will achieve some value strictly less than one or you could use it to solve the halting problem. But you can achieve any value less than one by the strategy that for some fixed sufficiently large $t$, simulates nature's program $j$ for $t$ steps and outputs the number of steps it takes program $j$ to halt if $j$ halts in $t$ steps, infinity otherwise.