**Notes on Modeling: Session XI: Non-Zero-Sum Games**

Prepared by:

Joseph Malkevitch

Mathematics Department

York College (CUNY)

Jamaica, NY 11451

email:

__malkevitch@york.cuny.edu__

web page:

__http://york.cuny.edu/~malk/__

Many of the most important games that are played by two players (two countries; union management; pirates and ships; etc.) do not have zero-sum payoffs. Sometimes it is hard to determine exactly how to measure the payoffs in such games by cardinal utilities. How does one attach a utility to a political confrontation that results in a nuclear war or a union-management conflict that results in a strike? However, often one can give verbal descriptions of the possible outcomes and each player can assign an ordinal score to these outcomes. The usual way this is done for a 2x2 game, which has four outcomes, is to assign 4 to the most favored outcome, and 1 to the least favored outcome.

Almost certainly, the two best known non-zero-sum games are prisoner's dilemma and chicken. These games have "stories" that are used to describe the typical set up. Both of these games have "paradoxical" structure or create quandaries to playing them. They both are used to model extremely confrontational games. These is a huge literature about these games to which philosophers, economists, political scientists, biologists, and mathematicians have contributed. The research literature is concerned with many variants which depend on whether or not the game is played once or many times and if it is played many times whether or not it is known exactly how many times the game will be played. There are also issues of whether or not the players can cooperate or not.

Prisoner's Dilemma

Column I | Column II | |

Row 1 | (2,2) | (-30, 20) |

Row 2 | (20, -30) | (-5, -5) |

Chicken:

Column I | Column II | |

Row 1 | (0, 0) | (-1, 1) |

Row 2 | (1, -1) | (-10, -10) |

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