AMS 335/ECO 355 Game Theory
Fall 2010
Exercise of 1.1 Normal Form Game
Zhen Xu
Page
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Exercise 1.1.1
the Prisoners’ Dilemma
An especially simple description used by Aumann (1987) is the game in which each
player can simply announce to a referee: "Give me $1,000," or "Give the other player
$3,000." Note that the monetary payments come from a third party, not from either of
the players; the Prisoner's Dilemma is a
variablesum game.
The players can discuss the game in advance but the actual decisions must be
independent. The Cooperate strategy is for each person to announce the $3,000 gift,
while the Defect strategy is to take the $1,000 (and run!). Table 1 below depicts the
payoff matrix to the Aumann version of the Prisoner's Dilemma, where the units of
the payoff are thousands of dollars.
Table 1 The Prisoner's Dilemma
Player 2
Cooperate
Defect
Player
1
Cooperate
3, 3
0, 4
Defect
4, 0
1, 1
We will discuss this game in more detail below, but we should point out the
"dilemma" before proceeding. The problem is that each party has an incentive to
defect, regardless of what he or she believes the other party will do. If I believe that
the other person will cooperate and give me a $3,000 gift, then I will get $4,000 in
total by defecting. On the other hand, if I believe that the other person will defect and
just take the $1,000, then I do better by taking $1,000 for myself.
The “dilemma” is caused by the existence of dominated and dominant strategy. Note:
Cooperate is strictly dominated by Defect. There is no credibility that any players in
the game will play Cooperate, or to say, the probability that a player plays Cooperate
is 0. Therefore the game can never reach the pareto optimal since Defect is always a
better choice for each player.
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 Fall '10
 Xu
 Game Theory, player, Normal Form Game, Zhen Xu

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