Prisoners Dilemma and Cheating on the Cartel

Prisoner's Dilemma:

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This game is a symmetric. Although the pair would be best off by both Denying guilt, Confess is a dominant strategy for each. Consequently the dominant strategy equilibrium is the upper-left element of the game matrix. Yet if the two were innocent!

Cheating on the Cartel:

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Suppose Coke and Pepsi consider to charge a high price for soda. If the game matrix were with payoffs in millions of dollars of profit per week, each firm has a dominant strategy of cheating on the cartel by charging the low price. The dominant strategy equilibrium is the lower-right element: the cartel falls apart despite the joint advantage of charging the high price.

With larger strategy spaces a few strategies can be dominated without leaving a dominant strategy. In such cases we eradicate dominated strategies iteratively. That is, some strategies that did not appear to be dominated do appear to be dominated once other dominated strategies are eliminated. That is true in the following example:

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First, Jill does not have a dominant strategy. Her best reply depends on whether Jack plays (Left or Middle) or Right. Second however would jack ever play Right? No, Middle dominates Right (no matter what Jill does). So Right is a dominated strategy, and is irrelevant. Therefore, we have:

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Third, having eliminated Jack playing Right, we now see that Jill playing Down can be eliminated: for Jill, the payoff to Up exceeds the payoff to Down no matter what Jack does.

Since Jill’s dominant strategy is Up, the dominant strategy equilibrium to this game is the upper-center element of the original game matrix.

To pull things together, our method for finding a solution to a one-shot game is first to eliminate dominated strategies and second to associate the solution with any dominant strategy that remains, if one exists.

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