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7 - More Game Theory

Published online by Cambridge University Press:  11 March 2021

David C. Vella
Affiliation:
Skidmore College, New York
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Summary

We continue the study of constant-sum games by illustrating how to solve them if the payoff matrix is larger than 2 x 2.We derive the method of equalizing expectation to solve such games, Williams's method of oddments, and finally, we show how to solve any constant-sum game using linear programming. This provides us with a full proof of the minimax theorem. Also, using linear programming, we can prove the square subgame theorem, which states that the solution to any constant-sum game is the same as a solution to one of its subgames that has a square payoff matrix. We then illustrate how to use Microsoft Excel or Wolfram Mathematica to solve such games. In the final section of the chapter, we study variable-sum games and introduce the notions of payoff polygon and Pareto efficiency of an outcome. We show that not every such game has a universally accepted solution, so there is no analog of the minimax theorem for such games. In the 2 x 2 case, we show how to find a Nash equilibrium using mixed strategies if necessary (Nash proved that any game has one). However, the equlibrium point so obtained may not be Pareto efficient so may not be a good "solution" to the game.

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Publisher: Cambridge University Press
Print publication year: 2021

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  • More Game Theory
  • David C. Vella, Skidmore College, New York
  • Book: Invitation to Linear Programming and Game Theory
  • Online publication: 11 March 2021
  • Chapter DOI: https://doi.org/10.1017/9781108568166.009
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  • More Game Theory
  • David C. Vella, Skidmore College, New York
  • Book: Invitation to Linear Programming and Game Theory
  • Online publication: 11 March 2021
  • Chapter DOI: https://doi.org/10.1017/9781108568166.009
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • More Game Theory
  • David C. Vella, Skidmore College, New York
  • Book: Invitation to Linear Programming and Game Theory
  • Online publication: 11 March 2021
  • Chapter DOI: https://doi.org/10.1017/9781108568166.009
Available formats
×