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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.

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    $\begingroup$ 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. $\endgroup$ – guest Nov 15 '18 at 14:42
  • $\begingroup$ @guest - I have a specific objective in mind with the question. A more readable proof of that particular theorem will meet the objective. $\endgroup$ – benblumsmith Nov 16 '18 at 3:21
  • $\begingroup$ 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. $\endgroup$ – jfkoehler Nov 19 '18 at 3:33
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In Saul Stahl's A Gentle Introduction to Game Theory there is a simple proof of the following statement on page 30:

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.

I would think that perhaps you could find a way to generalize these to the case you need for your student to the level of rigor you desire? Good luck.

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  • $\begingroup$ 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. $\endgroup$ – benblumsmith Nov 17 '18 at 2:17
  • $\begingroup$ Got it - well, this book is a bit more elementary perhaps than you want but should give you a flavor of such proofs. I'm sure there are many more other texts that do as well, it's just one I've taught from so I knew it had something relevant. $\endgroup$ – kcrisman Nov 19 '18 at 12:48
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    $\begingroup$ It's perfect. I can ask my students to try to extend the argument to more than 2 players. $\endgroup$ – benblumsmith Nov 20 '18 at 4:12

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