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1.5.4 About the Solution Concepts > 1.5.4.12 Comments on the Concept of Maxmin ... - Pg. 37

1.5 More about the Scope of Game Theory 37 1.5.4.12 Comments on the Concept of Maxmin Strategy Profiles By definition, a Nash equilibrium is a profile of strategies which is stable to sin- gle deviations. In this sense, the concept of Nash equilibria possesses some stability properties. As mentioned in Sec. 1.5.1, players do not always (or may not necessarily want to) maximize their utility. Many people who arrive much earlier than their train or bus leaves could maximize their expected utility by decreasing the margin they take to catch their train or bus with high probability. Nash equilibria do not capture the notion of risk. By contrast, the maxmin solution is a very simple concept, which employs the latter notion (more advanced concepts and different approaches of risk aversion exist, e.g., see Driesen (2010)). Let us illustrate this concept with a simple example (Laraki and Zamir, 2003). Example 41 Consider the game described by Table 1.6: player 1 chooses the row and player 2 chooses the column. The unique pure Nash equilibrium of this matrix game is the action profile (r 3 , c 2 ). However, if for some reason, player 2 chooses column c 1 , player 1 gets a very bad utility (namely -100). As a consequence, playing at the Nash equilibrium is very dangerous from the standpoint of player 1. Such a player may want to play the action r 1 which provides him with a utility equal to 2 whatever the other player does.