Zero-Sum Games

A zero-sum game is one in which no wealth is created or destroyed. So, in a two-player zero-sum game, whatever one player wins, the other loses. Therefore, the player share no common interests. There are two general types of zero-sum games: those with perfect information and those without.

In a game with perfect information, every player knows the results of all previous moves. Such games include chess, tic-tac-toe, and Nim. In games of perfect information, there is at least one "best" way to play for each player. This best strategy does not necessarily allow him to win but will minimize his losses. For instance, in tic-tac-toe, there is a strategy that will allow you to never lose, but there is no strategy that will allow you to always win. Even though there is an optimal strategy, it is not always possible for players to find it. For instance, chess is a zero-sum game with perfect information, but the number of possible strategies is so large that it is not possible for our computers to determine the best strategy.

In games with imperfect information, the players do not know all of the previous moves. Often, this occurs because the players play simulataneously. Here are some examples of such games:

Rock, Paper, Scissors

Game 1: Suppose two people are playing a simple game with nickels and quarters. At the same time, they each put out either a nickel or a quarter. If at least one player plays a nickel, player 1 gets both coins. Otherwise, player 2 gets both. Naturally, both players wish to gain as much money as possible. How should they play in order to do this?

Game 2: Suppose two people are playing a similar game with nickels and quarters. Now, if player 1 plays a nickel, player 2 gives him 5 cents. If player 2 plays a nickel and player 1 plays a quarter, player 1 gets 25 cents. If both players play quarters, player 2 gets 25 cents.

Game 3: We still have two people playing a game with nickels and quarters. Now, if both players play the same coin, player 2 gives player 1 the average value of the coins; otherwise, player 1 gives player 2 the average value of the coins.

The three games are implemented below, with the computer being player 1 and you being player 2. The computer is using the best strategy available to it.

Although the three games seem similar, the methods used to find the best strategies in each are very different. Game 1 is solved by eliminating dominant strategies, game 2's solution is known as a saddle point, and game 3 requires a mixed strategy.

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