Abstract
In this article, I briefly describe why Game Theory is so important in order to have a better understanding of the social sciences. After summarising the main assumptions that are the building blocks of the theory, I review some aspects of non-cooperative theory, with an obvious focus on the model developed by J.F. Nash Jr (Nobel laureate in Economic Sciences, 1994).
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The players have to say not only if they prefer to win, draw or lose, but also to order according to their preferences every possible way to win, draw or lose. However, it is accepted that a player is indifferent to certain results, so it may be reasonable to assume that in many cases in chess there are just three different outcomes and that a player strictly prefers winning to drawing and drawing to losing. All the outcomes that lead to win, in this case, are considered equivalent.
To simplify, a chess player might state they are indifferent about the way they win, or draw, or lose, and assign 1 to a victory, 0 to a draw, −1 to a defeat.
The idea of a mixed strategy predates Nash’s contribution. Indeed, the beautiful theorem by von Neumann on the existence of optimal solutions in a zero-sum game is based precisely on the idea of mixed strategy. Nash’s theorem it is an extension of this. However, it is important to stress that Nash’s approach is totally new compared to von Neumann’s contributions, which defined optimal strategies in zero-sum games, using the idea of strategies that implement the conservative values of the game. The crux of his theorem is to show the coincidence of these conservative values, a very different approach from Nash’s, and not really meaningful if the game is not strictly competitive.
Since the game is fair, against a player who knows the optimal strategies the expected value is instead zero, as it is obvious, so it would be better for them to deny entry to those who know theory (and how to count).
Even if both players had agreed to answer “100 euros to the other player”, this agreement would be unreliable (unless we introduce the possibility of a retaliation, which, however, changes the nature of the game).
These are the games in which, if we consider any two outcomes, if one player improves their payoff by passing from one outcome to the other, then automatically the other player worsens it.
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Lucchetti, R. N for Nash. Lett Mat Int 5, 151–154 (2017). https://doi.org/10.1007/s40329-017-0176-2
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DOI: https://doi.org/10.1007/s40329-017-0176-2