Payoffs for Mixed Strategies

https://www2.bc.edu/~sonmezt/E308SL7.pdf

Further, one could apply the ideas presented in the above pdf, to a one-shot simultaneous game. (i.e. a simultaneous game played only once)
Assume a 2-person game where the goal of the game is to win.

1. Lets assume a one-shot game. In this situation, what if the Payoff Matrix has no obvious Nash Equilibria i.e. at least 1 player finds that its best to change the decision. What is the best strategy in this situation. One way to think is as follows
1. Assume that the game is a repeated game. Now if one does not want to be exploited, the strategy is to mix up one's decisions. How should this mixing be done ?  In order to minimize the payoffs of the other person, it is best to seek the minimum of the maximum payoffs the other player -- why so ? (have a look at the article), and in doing so one finds the mixed strategy.
2. Once one knows the mixed strategy, choose the decision with the highest yeild in the long-run as the decision for the one-shot game. 

Thus, one can think through the decisions for games that have no Nash Equilibria by thinking through a repeated-game and then using that as a decision in a one-shot game. Of course, if the game does have Nash Equilibria then those are the rational choices.

Also, of course that real life interactions are very rarely one-shot i.e. most are repeat-games, and that is why many other factors start coming into the picture. (including ethics and what not). As an example have a look at this which suggests how and why social-diversity may in fact lead to cooperation and altruism.