7 - More Game Theory

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.