Min Max


Min Max,[1] this is a Min  

I am not benevolent, I am not benevolent

Min Max, this is a Min Max, you are not benevolent

Have you ever tried to maximize? Have you ever tried to minimize?

In a two zero-sum-game,[2] in a two-zero-sum-game, 

in a two- zero-sum-game, in a two zero-sum-game

In a two-zero-sum-game, this is the value of the game,[3] 

in a two-zero-sum-game, this is the value of the game

In a two zero-sum-game, this is the value of the game, 

in a two zero-sum-game, this is the value of the game

This is a saddle point,[4] this is a Nash equilibrium of the game[5]

Min Max this is, I am not benevolent, I am not benevolent, I am not benevolent.



*Text by Marco Marini. Music by Marco Marini, Elisa Pezzuto and Valter Sacripanti. All rights reserved.

Voices: Marco Marini, Elisa Pezzuto. Guitars: David Pieralisi, Marco Marini, Bass: David Pieralisi. Drum: Valter Sacripanti, 

Production: Valter Sacripanti. Mastering: Fabrizio De Carolis.



[1]The minmax (or minimax) is a seminal equilibrium concept due to John von Neumann (1928) and also extensively studied by John von Neumann and Oskar Morgenstern (1944) for zero-sum games, i.e. games where one player's gain is equal to the other player’s loss. The theorem on this concept states that in any finite, two-player zero-sum game, there is a (Nash) equilibrium where every player obtains a payoff equal to both their maxmin and minmax value: the minimum payoff that a player can guarantee for herself is equal to the maximum payoff the opponent can guarantee for herself.

[2] A zero-sum game is a game in which the players’ payoffs sum up to zero. In this case, the obvious choice of a player is to maximize their own payoff and minimize that of rivals.

[3] In a two-player zero-sum game the maximum expected payoff for the first player is equal to the minimum expected loss for the second player. This common value is called the value of the game

[4] In mathematics, a saddle point or minimax point is a point on the surface of a graph of a function where the slopes (derivatives) in orthogonal directions are all zero (critical point), but that is not a local extremum of the function. An example of a saddle point is when there is a critical point with a relative minimum along one axial direction (between peaks) and at a relative maximum along the crossing axis. However, a saddle point need not be in this form. For example, the function x2+y2 has a critical point at (0,0).

[5] In a two-player zero-sum game every max min strategy profile (or, equivalently, min max strategy profile) is a Nash equilibrium. Therefore, all Nash equilibria yield the same payoff for all players.         


REFERENCES

von Neumann, J. (1928): “Zur Theorie der Gesellschaftsspiele.” Mathematische Annalen, 100, 1928, pp. 295–320.

von Neumann, John & Oskar Morgenstern (1944) Theory of Game and Economic Behavior, Princeton, Princeton University Press.