What happens to the Nash equilibria and minimax values of a strategic game, when we take its payoff table and modify all payoffs by raising them to the 3rd power?
My conjecture is that it depends:
- If, in the original game, all Nash equilibria and minimax points are attained using pure strategies alone - then a positive monotone transformation should have no effect on them. When everything is pure (deterministic), the utility function is ordinal and it is robust to monotone transformation.
- But if, in the original game, some Nash equilibria / minimax points are attained using mixed strategies - then a non-linear transformation might change them substantially.
For example, in the following zero-sum game:
4 0 0 4 3 3
The maxinim value is 3 and a maximin strategy for the row player is selecting the bottom row (the column player can guarantee at most 3 by mixing the two columns with equal probability).
But if we raise to the 3rd power:
64 00 00 64 27 27
Now the minimax value is 32, and a maximin strategy for the row player is mixing the two top rows.
So my question is: In what conditions does a positive monotone transformation on a game's payoff not change its maximin and Nash equilibrium strategies?