# Accessible written proof of the Nash Indifference Theorem (game theory)?

In game theory, the Nash Indifference Theorem states that if a mixed strategy $$A$$ is a best response to a mixed strategy $$B$$, then every pure strategy in the support of $$A$$ is also a best response to $$B$$ (with the same expected payoff).

I am looking for a written proof of this theorem that could be read by an undergraduate with a high school but not a college level math background. Are you aware of such a proof?

This document contains a proof (section 3, on pp. 18-19) but the sigma notation, double subscripts, and terse proof style will all be very hard for the students I have in mind.

I am not familiar with many standard texts in Game Theory; I'm hoping you may be aware of some that prove this theorem in a wordier and less notationally intense way.

• Why does it have to be that exact theorem and why proof (as opposed to explanations, application)? Maybe just having him read a popular book or article on overall topic of game theory would be better near term objective. – guest Nov 15 '18 at 14:42
• @guest - I have a specific objective in mind with the question. A more readable proof of that particular theorem will meet the objective. – benblumsmith Nov 16 '18 at 3:21
• Hi. I have done thing in calc when discussing basic optimization problems here. As long as a student has a basic understanding of where a quadratics maximum occurs, and you explain some basics of utility, this is an easy way in. Not sure if this helps. – jfkoehler Nov 19 '18 at 3:33

Theorem [3.]5. In any $$2\times 2$$ zero-sum game, if one players employs a fixed [possibly mixed] strategy, then the opponent has an optimal counterstrategy that is pure.
The extension to the non zero-sum case is an exercise later in the book. Naturally, this is weaker than what you are looking for, but maybe this will be enough. I don't really see a way to rigorously prove such statements without sigma notation in the $$n\times m$$ case, though probably there are nice graphical intuitions. Stahl proves the $$2\times 2$$ case of the existence of a (mixed) Nash equilibrium at the end of the book in that way.
• Thank you, this is perfect. Incidentally, I think the students in question would be able to parse something like $p_1a_1 + p_2a_2 + \dots + p_na_n$. It's just $\Sigma$ and double subscripts and reading proofs in general would be a lot all at once. – benblumsmith Nov 17 '18 at 2:17