Download Game Theory - Mr. P. Ronan

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 policymakers – 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.
Betty (B)
Actions
Don’t Confess
Confess
Art (A)
Don’t Confess
A:1 B: 1
A: 10 B: 0
Confess
A:0 B: 10
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 B
Actions
Low Output
High Output
Firm A
Low Output
A:6 B: 6
A: 2 B: 9
High Output
A:9 B: 2
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 B
Actions
Advertise
Don’t advertise
Firm A
Advertise
A:100 B: 20
A: 0
B: 10
Don’t advertise
A:50 B: 70
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 B
Actions
Price = $20
Price = $15
Firm A
Price = $20
A:40 B: 37
A:49 B: 30
Price = $15
A:35 B: 39
A:38 B: 35
(a) Firms A and B are members of an oligopoly. Explain the interdependence that
exists in oligopolies using the payoff matrix facing the two firms.
(b) Assuming that the two firms cooperate, what is the solution to the problem facing
the firms?
(c) Given your answer to b, explain why cooperation would be mutually beneficial
(d) Given your answer to c, explain why one of the firms might cheat on a cooperative
agreement.
3: Consider trade relations between the United States and Mexico. Assume that the
leaders of the two countries believe the payoffs to alternative trade policies are as
follows (all figures are in US$ billions):
Mexico’s decision
Actions
Low tariffs
High tariffs
United States’ decision
Low tariffs
High tariffs
A:25 B: 25
A: 30 B: 10
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?