Intermediate
Microeconomics
Professor
Yongmin Chen
Topic 10. Game Theory and Strategic Behavior
Game
Theory: An Introduction
In
oligopoly markets firms interact with each other when making decisions. The study of strategic interactions between
firms is a subject of game theory. A
major development of microeconomics in the last decade is in game theory and
its application to understanding the strategic interactions on the oligopoly
markets.
A
game is a decision situation where more than one decision agent is
involved. A participant of the game (a
firm, a consumer, government, etc) is called a player. Each player may only undertake certain
actions. Depending on the actions of
all players, each player receives a payoff from the game (that is why each
player needs to think about the actions of other players before he chooses an
action).
Example
10-1. A bidding game. (To be explained in class. All of you and I are players)
To
simplify analysis, we will focus on situations where there are only two
players. Let's call these two players A
and B. A strategy of a player is a plan
by this player on what actions to take.
A pair of strategies is said to be a Nash equilibrium if each player's
strategy is optimal given the strategy of the other player. In other words, a Nash equilibrium is a
situation where no player can unilaterally improve his (her) payoff.
Example
10-2. The Prisoner's Dilemma. Two suspects are arrested and charged with a
crime. The police lack sufficient
evidence to convict the suspects unless at least one confesses. The two suspects are held at two separate
cells and can choose either to confess or to deny. If one confesses but the other dos not, then the one confessed
will be set free immediately (payoff = 0)
and the one not confessed will receive
5-year sentence (payoff = -10); if both confess, then both will be sentenced to
4 years in prison (each has payoff = -4); if neither confesses, both will be
convicted a minor offense and receive 3-month sentence (payoff = -1). What are the strategies of each
suspect? What are the Nash equilibria
of this game?
|
B confess
deny confess
(-4,-4) (0,-10) A deny (-10,0)
(-1,-1)
|
The
strategies of each player are: confess, and deny. The only Nash equilibrium is (confess, confess). To see this pair of strategies constitute a
Nash equilibrium, we need to check that given the other player is confessing, a
player does best by confessing too. To
see that there are no other Nash equilibrium, we need to check that at any
other pair of strategies, at least one player can benefit by deviating to
confessing.
|
H boxing
movie boxing (1,2)
(0,0) W movie (0,0)
(2,1)
|
Example
10-3. The battle of the sexes.
A young couple need to decide what to do for an evening. They both want to spend the evening
together, but the husband prefers to watch a boxing game, while the wife
prefers to watch a movie. The game is
described as follows:
What are the strategies of the husband
and the wife? What are the Nash
equilibria of the game?
Each
has two strategies: boxing, and movie.
There are two Nash equilibria of the game: (boxing, boxing); (movie,
movie). This game is an example of
coordination games.
Not
all games have a Nash equilibrium, however.
|
B Heads Tails Heads (-1,1)
(1,-1) A Tails (1,-1)
(-1,1)
|
Example
10-4. Matching pennies game:
Each player's strategies include heads or
tails, but this game has no Nash equilibrium in pure strategies.
A player's dominant strategy is one that is
best for the player, regardless of the strategies of the other players. In some games, there exist a dominant
strategy for a player. For instance, in
the Prisoner's Dilemma game, the dominant strategy for each player is to
confess. There are also games where
there is no dominant strategy for a player, such as the battle-of-sexes
game. In general, it is not easy to see
what a player should do in a game, even if a Nash equilibrium exists (recall
the battle of sexes). But if a player
has a dominant strategy, then clearly the player should play the dominant
strategy. A situation where each player
is choosing the dominant strategy is clearly a Nash equilibrium, but it is not
necessary that a player is choosing the dominant strategy at a Nash
equilibrium. Sometimes there is no
dominant strategy for a player, but a Nash equilibrium may still exist.
Example
10-5. Price Wars. Suppose two firms are competing on a
market. Each firm can either charge
high price or low price. The game is
described as follows:
|
B Hp Lp Hp (5,5) (0,8) A Lp (8,0) (3,3)
|
The only Nash equilibrium for the game is Lp-Lp: both firms charge low
prices. This is a situation where price
war occurs. In fact, this is a game
where each player has a dominant strategy: Lp. Although both firms will be better off if they can collude
(charge high prices), but if one firm charges high price, the other firm has an
incentive to undercut and benefit more.
This game is another version of the Prisoner's Dilemma.
Example
10-6. The incentives for collusion.
When
firms choose their strategies non-coorperatively, the equilibrium outcome often
does not maximize the industry's profit.
This can be clearly seen from the Nash equilibrium at a Cournot model. In example 10-6, each firm's equilibriu
output is 33. Thus the industry's total
output is 66, and the market price will be 34.
Each firm's profit is (34 - 1) x 33 = 1089, and the combined profit of
two firms is 2178.
Now
if the two firms make decisions jointly like a monopoly, then the total output
is given by: 100 - 2Q = 1. Tha is, Q = 99/2 = 49.5, and the market
price will be p = 50.5. The combined
frofit of two firms is (50.5 - 1) x
49.5 = 2450.25. Thus firms can
have higher profit if they collude on their prices or output. This is why there will be cartels. But cartels may not be stable, because each
member has incentives to cheat.
Market Signaling
In many market situations, there is asymmetry in information. For example, a producer has better information about the quality of its product; a job applicant has better information about his own ability; an incumbent firm has better information about cost and demand conditions than a potential entry; etc. Sometimes, such information asymmetry can be resolved through signaling by the informed party.
--Job-market signaling. Suppose a firm needs to hire a worker. There are two types of possible applicants: those with high ability and those with low ability. An applicant knows whether he has high or low ability, but the firm does not know it before hiring, and it wants to hire a worker with high ability. It has been suggested that, to certain extent, education can be a signal. If education is more costly to a low-ability person than to a high-ability person, then a high-ability person may choose to have more education than a low-ability person. Thus, even if education does not increase a person's productivity, it may still be acquired to signal a person's ability. In reality, education can often increase a person's productivity, but it may also have a signaling effect.
--High price to signal high quality. But what prevents a low-quality producer from also charging a high price? One possibility is: Suppose some consumers are informed and some are not. If the proportion of informed consumers is large enough, then high price can indeed signal high quality. This is because if a low-quality producer tries to imitate the high-quality producer by charging high prices, it will lose business from the informed consumers. Such losses can be more than what it can benefit from fooling the uninformed customers with high prices.
International Competition and the Role of Government
Increasingly, firms are now competing in the global market. An important feature of international competition is that not only it involves profits of firms, it also affects the welfare of different nations. Some economists have argued that in international competition, certain government intervention can increase a country's welfare, although often in the expense of other countries. An example that has often been used is as follows:
Both Boeing and Airbus are considering to produce an aircraft. Without government intervention, the game is:
|
Airbus Produce Not Produce (-25,-25) (300,0) Boeing Not
(0, 300) (0,0)
|
The game has two Nash equilibrium: (Produce, Not) and (Not, Produce). Now suppose Boeing can choose first (it has the first move advantage), then clearly it will choose Produce, and it will receive 300 while Airbus will receive 0.
Airbus, and the governments of European countries, are not happy with this outcome. They want to change the rule of the game. Suppose the European governments decide to intervene, providing a subsidy of 30 to Airbus if it produces. The new game would then be:
|
Airbus Produce Not Produce (-25,5) (300,0) Boeing Not (0,330) (0,0)
|
Now the only Nash equilibrium is for Airbus to produce and for Boeing not to produce, with the payoffs being (0,330). Thus by providing 30 subsidies, the European countries have increased their welfare by 300, and U.S. has lost 300. This example illustrates that in international competition, government intervention can sometimes increase a country's welfare.
There are, however, serious problems to implement such a theory in practice. First, the gains of a country are often the losses of another country under such policies, and this is likely to cause retaliation from the country that loses. And as a result, both countries can be worse off. Second, the government may not have the proper information to decide when to provide subsidies and when not to, and how much if subsidy is needed. Finally, this may opens the door for various interest groups to influence government decisions.
The Inspection Game: Mixed Strategies
In the previous lectures, we have studied several games where each player takes some action with certainty, and in equilibrium, each player correctly anticipates the action its opponent will take. In some situations, however, it might be best for a player to randomize over some actions so that his action will be unpredictable. This is called using mixed strategies. Let me explain this through the Inspection Game.
Suppose there are two players: an agent and a principal. The agent is employed by the principal. The agent can either work (W) or not work (NW). Working costs the agent 10 and would produce output of value 20. The principal can either inspect (I) or not inspect (NI). An inspection costs 5, but would provide evidence if the agent did not work. The principle pays the agent a wage 14. However, if the agent is caught not-working, he will receive 0. The game is illustrated below:
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Principal I NI W 4, 1 4, 6 Agent NW 0,-5 14,-14
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First, it is easy to check that this game has no pure-strategy equilibrium. If the agent knows that the principal would inspect, he would choose to work. But if the agent is choosing to work, the principal should choose not to inspect. However, if the principal would not inspect, the agent should choose not to work. Any pair of pure strategies cannot be a Nash equilibrium.
But this game has a mixed-strategy Nash equilibrium. That is, an equilibrium in which the principal inspects only with some probability, and the agent avoids working with some probability. Let us see how these probabilities can be determined. The basic idea is that these probabilities have to be such that the principal will be indifferent between inspection and not, and the agent will be indifferent between work and not work. Suppose the probability that the principal inspects is g, and the probability that the agent does not work is h. Then g has to be such that
4 = 14 (1 -g)
and h has to be such that
-5h + 1(1-h) = -14h + 6(1-h)
Therefore, g = 10/14 = 5/7, and h = 5/14.
There are many other applications of mixed strategies. Some examples: pricing strategies in duopoly; war of attrition.
Bargaining
Almost everyone has some experience of bargaining. When two parties bargain over some outcomes, the situation can be described by a bargaining game. Intuitively, the bargaining outcome depends on the bargaining strength of each party. But what are the factors that determine one's bargaining strength?
--Alternative opportunities. When you try to negotiate prices with a car salesman, it certainly helps if you have some idea of what kind of price you can get from another car dealer. A firm often tries to maintain relationships with several suppliers. An employee would most likely to get the pay raise he wants if there is an outside offer. The better your alternative opportunities are, the stronger your bargaining position is.
--Relative costs of delaying an agreement. In a labor dispute, unions can benefit from establishing a striking fund, or if demand for the product is very high. The management can improve its bargaining position in case of a strike by holding inventories of finished goods.
--The advantage of being able to commit.
Example. Suppose two people are bargaining over how to divide $100. If an agreement is not reached, then no one receives anything. Now suppose one person is able to make an offer and then never return to bargaining, what will be his strategy?
Example. It is a common practice for a semiconductor firm that has developed a new, proprietary chip to license another firm to produce the chip in competition with itself. User firms must make a large investment in retooling in order to be able to use the new chip. They may worry how reliable the supply of the new chips will be. Licensing allows the innovating firm to commit to a reliable supply of chips.