A zero-sum game for two players allows a player to take no action during a turn. Can I reach optimal play (where both players always choose the best possible action in each turn) by the following scheme?
- Randomly choose move orders $A_{1}$: $O_{A1}, O_{A2}, ...$ for player A over turns $1,2,...$
- Find player B's optimal orders $B_{1}$: $O_{B1}, O_{B2}, ...$ for countering $A_{1}$ (with an integer linear programming solver). If $B_{1}$ makes some orders $O_{Ai}$ in $A_{1}$ invalid, pretend that player A takes no action during turn $i$. For example, capturing a piece can make $O_{A3}$ an invalid order. In this case, pretend $O_{A3}$ does not exist.
- Same as step 2 except we find player A's optimal orders $A_{2}$ that counters $B_{1}$.
- Repeat steps 2 and 3 until the orders of both sides don't change.
Optimal play, which is a Nash equilibrium, exists. The scheme looks like coordinated descent except for the bit that ignores invalid moves.
I ask this question because I am trying to use an integer linear programming solver instead of implementing minimax for finding optimal play.
