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Game Theory

The term ‘game theory’ does not mean that we are really going to be playing games, but it is appropriate because each ‘game’ involves players, strategies and payoffs. To play a game, each player – different firms, labour unions, management or policy-makers – must consider the costs and benefits of alternative strategies as well as the possible strategies that might be adopted by other players. The purpose of each game is to win the payoff – market share, wages, profits, achievement of policy goals, or whatever. To be successful, a player must adopt a strategy that correctly anticipates the response of its opponent. For example, if a firm in an oligopoly industry is considering the introduction of anew product, it must consider not only the costs of product development and the likely response of consumers, but also whether its rival firms will also introduce new products. If only one firm introduces a new product, it may be able to capture a large market share and pay for development costs, but if all firms introduce new products, the development costs may exceed the increased sales revenue.

The Prisoner’s Dilemma

Suppose that two criminals, Art and Betty, are held as suspects in a bank robbery. The evidence is convincing, but without a confession, the most that the police can pin on each of them is a one-year jail sentence for a known previous petty crime. If they both confess, each will get a five-year jail term. Thus the best strategy is for both suspects to hold out and spend only a year in jail – but the police want a confession to the bank robbery. To coax a confession out of the prisoners, the police can use a simple application of game theory. Put Art and Betty in separate rooms so they cannot communicate, and offer each a suspended sentence (zero years) for confessing and naming the other as an accomplice. The accomplice will then go to jail for 10 years. This offer is made to each suspect. Betty knows that if she and Art both clam up, they get only a year in jail, but if Art confesses and she does not confess, she will go to jail for 10 years. Art knows the same thing. What should the suspects do?

The payoff matrix in the following table illustrates the dilemma faced by the two prisoners.

| |Art (A) |

| |Actions |Don’t Confess |Confess |

|Betty (B) | | | |

| |Don’t Confess |A:1 B: 1 |A:0 B: 10 |

| |Confess |A: 10 B: 0 |A:5 B: 5 |

Interpreting the payoff matrix is straightforward. The entry (A:1, B:1) in the northwest corner shows what happens if both Art and Betty hold out – both go to jail for one year. The entry (A:0, B:10) in the northeast corner shows what happens if Art confesses and Betty holds out – Art gets the suspended sentence and Betty goes to jail for 10 years.

What is the most likely outcome of the game? To make the most advantageous decision, each player needs to consider the action of the other. Consider the situation from Art’s standpoint. Suppose that Betty confesses. If Art also confesses, he gets 5 years; if he holds out, he will get 10 years. The best strategy is to confess. But what if Betty holds out? If Art also holds out, he will get 1 year. If he confesses, he will get a suspended sentence. Again the right choice is to confess. You get the same situation if you look at it from Betty’s standpoint.

Confessing is the dominant strategy because it gives each player the best payoff regardless of the strategy chosen by the other player. A dominant strategy is the only likely outcome of a prisoner’s dilemma game. When both players adopt their dominant strategies, the game rests in dominant strategy equilibrium. However, it should be noted that in the above example, the equilibrium is not the most beneficial equilibrium to both players.

Cartel Behaviour

The prisoner’s dilemma has been applied to several areas in economics, most notably, oligopoly behaviour. One can apply it to the classic case of cartel cheating.

Suppose that two firms share a market and must decide whether to produce high quantity (H) or low quantity (L). If the firms form a cartel and agree to restrict production, they can charge high prices and earn $6 million in profits each. This is represented in the northwest corner (A:6, B:6) of the table below :

| |Firm A |

| |Actions |Low Output |High Output |

|Firm B | | | |

| |Low Output |A:6 B: 6 |A:9 B: 2 |

| |High Output |A: 2 B: 9 |A:3 B: 3 |

If there is no cartel agreement and both firms produce high output, price will fall and bring profits down to $3 million per firm. This is represented by the (A:3, B:3) entry in the southeast corner of the table. The other entries in the table show what happens if one firm cheats on the cartel while the other maintains low production. The cheater will increase sales at the expense of the rival, and profits rise to $9 million for the cheater and fall to $2 million for the rival.

What is the most likely outcome to this game? Look at this situation from the perspective of Firm A. If Firm B keeps to the cartel agreement, then Firm A can increase its profits from $6 million to $9 million by cheating. And if Firm B cheats, Firm A should still cheat; otherwise its profits will fall to $2 million. Firm B faces the same choices, so the dominant strategy for both firms is to cheat on the cartel.

Questions

1: The payoff matrix below shows the profit two firms earn if both advertise, neither advertise, or one advertises and the other does not. Profits are reported in millions of dollars. Does either firm have a dominant strategy?

| |Firm A |

| |Actions |Advertise |Don’t advertise |

|Firm B | | | |

| |Advertise |A:100 B: 20 |A:50 B: 70 |

| |Don’t advertise |A: 0 B: 10 |A:20 B: 60 |

2: Using the payoff matrix below, answer the following questions. The payoff matrix indicates the profit outcome that corresponds to each firm’s pricing strategy.

| |Firm A |

| |Actions |Price = $20 |Price = $15 |

|Firm B | | | |

| |Price = $20 |A:40 B: 37 |A:35 B: 39 |

| |Price = $15 |A:49 B: 30 |A:38 B: 35 |

(a) Firms A and B are members of an oligopoly. Explain the interdependence that exists in oligopolies using the payoff matrix facing the two firms.

(b) Assuming that the two firms cooperate, what is the solution to the problem facing the firms?

(c) Given your answer to b, explain why cooperation would be mutually beneficial

(d) Given your answer to c, explain why one of the firms might cheat on a cooperative agreement.

3: Consider trade relations between the United States and Mexico. Assume that the leaders of the two countries believe the payoffs to alternative trade policies are as follows (all figures are in US$ billions):

| |United States’ decision |

| |Actions |Low tariffs |High tariffs |

|Mexico’s decision | | | |

| |Low tariffs |A:25 B: 25 |A: 30 B: 10 |

| |High tariffs |A:10 B: 30 |A: 20 B: 20 |

(a) What is the dominant strategy for the United States? For Mexico? Explain

(b) What is the Nash Equilibrium for trade policy?

(c) In 1993, the US Congress ratified the North American Free Trade Agreement (NAFTA) in which the United States and Mexico agreed to reduce trade barriers simultaneously. Do the perceived payoffs as shown here justify this approach to trade policy?

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