Matthew Lew

Section 13.e

Game Theory and Prisoner’s Dilemma

Game theory is the study of rational behavior in situations involving interdependence. It is a formal way to analyze interaction among a group of rational individuals who behave strategically. A rational individual takes action that is consist with goals; not consistently making the same mistake. Interdependence implies that any player is affected by what others do; actions must depend upon prediction of others’ responses. In order for an individual to decide what to do, they must determine how are others are going to act (or react). This requires knowledge of other’s aims as well as the options available to them. There are many implications of game theory outside of economics:

 Buyers and sellers negotiating a price

 Employer and employee interaction

 Firm and its competitors

 Auctioneer and bidders

Game theory has many applications outside of economics and that why it continues to be one of the most studied topics. Outside examples include:

 Presidential candidates

 Congress and the President

 Opposing generals at War

The Prisoner’s Dilemma

Cooperation in game theory is usually analyzed by means of a non-zero-sum game called the “Prisoner’s Dilemma.” The two players in the game can choose between two moves, either “cooperate” or “defect.” The idea is that each player gains when both cooperate, but if only one of them cooperates, the other one who defects, will gain more. If both defect, both lose or gain very little, but not as much as the “cheated” cooperator whose cooperation is not returned.

Figure 13.e.1: The Classic Prisoner’s Dilemma

Action of A / Action of B	Cooperate	Defect
Cooperate	Fairly Good (+5)	Bad (-10)
Defect	Good (+10)	Mediocre (0)

Figure 13.e.1 shows typical payoffs for a prisoner’s dilemma game. Prisoner’s dilemma games are considered zero-sum in that there is no mutual cooperation: either each gets 0 when both defect, or when one of them cooperates, the defector gets (+10), and the cooperator (-10), a total of 0. On the other hand, if both cooperate, there will be a positive gain: each of them gets 5, a total of 10. The gain for mutual cooperation (5) in the prisoner’s dilemma is kept smaller than the gain for one-sided defection (10), so that there would always be a temptation to defect. The prisoner’s dilemma is meant to study short-term decision-making where the actors do not have any specific expectations about future interactions or collaborations (as in the case in the original situation of the jailed criminals).

Figure 13.e.2: Contribution Towards a Bridge

Player 1 / Player 2	Contribute	Do Not Contribute
Contribute	32, 32	28, 35
Do Not Contribute	35, 28	30, 30

Figure 13.e.2 shows another example of a prisoner’s dilemma type game. The situation involves construction of a bridge by two different individuals. If they both contribute to the building of this bridge, then they both receive a utility of 32. However, if they both fail to contribute, they are left with a utility of just 30. If one player contributes and the other one does not, then the player who does not contribute is a ‘free rider’ and will receive a utility of 35. The contributing player is left with 28.

Pareto Efficiency

Pareto efficiency is a term that can be used when analyzing prisoner dilemma games. An outcome (of the game) is said to be pareto efficient if there is no other outcome in which some other individual is better off and no individual is worst off. In figure 13.2.e, there are three pareto efficient outcomes: (contribute, contribute), (do not contribute, contribute), and (contribute, do not contribute). At these three payoffs, there is no other spot to choose that doesn’t make at least one player worst off. At (35, 28), if you move to any other space, one player will be worst off. However, if you are at (do not contribute, do not contribute), each player will get a payoff of 30. This outcome is said to be pareto dominated by (contribute, contribute) because you can move to that outcome and make both players better off.

Would the payoff (38,1) be pareto efficient?
(38,1) would be pareto efficient because you could not move to any other spot within the game and not make at least one of the players worst off.
Would the payoff (31,31) be pareto efficient?
(31,31) would NOT be pareto efficient because it is pareto dominated by (32,32). You could play the (contribute, contribute) strategy and make both players better off.

Sequential Move Games

The Prisoner’s Dilemma game is a game in which both players move simultaneously. There is another type of game where players make their moves in specific sequences. This is a different type of game, but it can be put in the normal form (box diagram) as well.

F
Accommodate

(1,1)
igure 13.e.3

Enter

Price War

B

A

(-1,-1)


Stay Out

(0,3)

Player A is a new entrant into a market. He has two choices: he can either enter the market or stay out. Player B is already in the market and if player A enters, he has the choice either to accommodate his competitor and not raise prices, or to start a price war. The logical move for player A would be to enter the market because he knows that if he does enter, player B will choose ‘accommodate’ and receive a payoff of (1), rather than try to start a price war and get (-1). For player A, a payoff of (1) is better than (0), the payoff for staying out of the market.

Nash Equilibrium

Nash equilibrium occurs when a player plays optimally and correctly guesses what the other player will do. In other words, each player plays a best-response strategy by assuming the other player’s moves. Nash equilibrium occurs where the best response functions cross.

Duopoly:

Assume there is a duopoly where p=a-bx, where a=17, c=1, b=1. The competitive solution is 16 units of output and the monopoly solution is 8 units of output.

Profits i = [17-(xi + x-i)]xi – xi
The best response function will be xi = 8 – (x-i)/2
Figure 13.e.4

X2

16


X1

16

8

8

X1=8 – (X2)/2

X2=8 – (X1)/2

16/3

16/3

Figure 13.e.3 shows that the best response strategy for each firm is where the best response functions cross at 16/3.

Coordination game

The Nash equilibrium in a coordination game is determined by each player playing a best response what he/she thinks the other player might do.

Figure 13.e.5

	Left	Right
Up	-1, -1	2, 0
Down	1, 1	1, 1

Figure 13.e.4 shows a typical coordination game where player one’s choices are (up) or (down). Player two’s choices are (left) or (right). To find a Nash equilibrium in a coordination game, you must analyze each player’s best response (in this case, best response refers to the choice that will give the player the highest payoff):

Player 1: If Player 1 assumes Player 2 will choose Left, Player 1 will choose Down

If Player 1 assumes Player 2 will choose Right, Player 1 will choose Up

Player 2: If Player 2 assumes Player 1 will choose Up, Player 2 will choose Right

If Player 2 assumes Player 1 will choose Down, Player 2 will be indifferent between Left or Right

Figure 13.e.6

	Left	Right
Up	-1, -1	2*, 0*
Down	1*, 1*	1, 1*

Note: * denotes a best-response payoff for a player.
By analyzing each player’s best response, it is easy to determine that this game has two Nash equilibriums at (Down, Left) and (Up, Right). Both of these strategies contain payoffs that are both best responses for each player.