Write out a pay off matrix when two players are offered $100 bills. If one bids $2 and the other bids $1 they pay $3, and the higher bidder gets the money leaving him with net gain of $98 while the other with a net loss of $1. The possible are $0, $1, $2. Also, if they both bid the same amount, they split the $100.
We know that this payoff matrix will be 9 cells, and will be a 3x3 matrix because each player has three choices. They can either bid 0, 1, or 2 dollars. Since both players have 3 options, we know that their are nine possible outcomes. It is common practice to show the Row player's payoff first, and the column player's payoff second. With this in mind, we can create the matrix, and start to populate the different payoff cells.
If they both bid $0, then neither player loses money, and they split the $100 earning $50 a piece. If one bids $1, and the other $0, then the player who bid $1 receives $99 and the other zero. Using this same sort of logic, you can populate the cells of the payoff matrix, and you should get a result similar to the one below:
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Column Player
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$0
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$1
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$2
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Row Player
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$0
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$50/$50
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$0/$99
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$0/$98
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$1
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$99/$0
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$49/$49
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$-1/$98
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$2
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$98/$0
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$98/$-1
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$48/$48
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This is an interesting problem because it should be fairly
obvious that both players will choose to bid $2 in order to get the $48
payoff. However, if they were to
cooperate, they could both receive $50, but then there is an incentive to cheat
in order to move to the $99 payoff leaving the other person with nothing. Because both players have a dominant strategy
to bid $2, the Nash equilibrium is going to be the payoff cell at the bottom
right.
