# Monotone transformation of a game

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?

• I don't quite understand your question (the last sentence in bold). Do you want to ask, "under what conditions does every positive monotone transformation not change..."? Or, are you looking for "circumstances" involving some relationship between the transformation and the original game... – usul Sep 8 '15 at 2:04
• Erel what is the source of your quote? – Giskard Sep 8 '15 at 7:38
• @denesp I didn't intend to quote anything - just to emphasize the question. – Erel Segal-Halevi Sep 8 '15 at 8:04