Game Theory: An Introduction

Text: An Introduction to Game Theory by Martin J. Osborne

Strategic Games

• A strategic game (with ordinal preferences) consists of:

• a set of players• for each player, a set of actions• for each player, preferences over the set of

action profiles.

• A very wide range of situations may be modeled as strategic games. For example, the players may be firms, the actions prices, and the preferences a reflection of the firms’ profits.

• Or the players may be candidates for political office, the actionscampaign expenditures, and the preferences a reflection of the candidates’ probabilities of winning.

The famous example: Prisonner’s Dilemma

• One of the most well-known strategic games is the Prisoner’s Dilemma. Its name comes from a story involving suspects in a crime; its importance comes from the huge variety of situations in which the participants face incentives similar to those faced by the suspects in the story.

Prisonner’s Dilemma

• Two suspects in a major crime are held in separate cells. There is enough evidence to convict each of them of a minor offense, but not enough evidence to convict either of them of the major crime unless one of them acts as an informer against the other (finks). If they both stay quiet, each will be convicted of the minor offense and spend 1 year in prison. If one and only one of them finks, she will be freed and used as a witness against the other, who will spend 4 years in prison. If they both fink, each will spend 3 years in prison

• This situation may be modeled as a strategic game:

• Players:The two suspects.• Actions: Each player’s set of actions is {Quiet,

Fink}.• Preferences: Suspect1:(Fink, Quiet) (Quiet,

Quiet) (she gets one year in prison), (Fink, Fink) (Quiet, Fink) Suspect 2:(Quiet, Fink), (Quiet, Quiet), (Fink, Fink), (Fink, Quiet).

Guess what happens?

Suspect 1

Suspect 2

Quiet Fink

Quiet (1,1) (4,0)

Fink (0,4) (3,3)

Duopoly• In a simple model of a duopoly, two firms produce the

same good, for which each firm charges either a low price or a high price.

• Each firm wants to achieve the highest possible profit. If both firms choose High then each earns a profit of $1000. If one firm chooses High and the other chooses Low then the firm choosing High obtains no customers and makes a loss of $200, whereas the firm choosing Low earns a profit of $1200 (its unit profit is low, but its volume is high). If both firms choose Low then each earns a profit of $600. Each firm cares only about its profit.

Firm 1

Firm 2

High Low

High (1000,1000) (-200,1200)

Low (1200,-200) (600,600)

Football or Ballet?

• In the Prisoner’s Dilemma themain issue iswhether or not the playerswill cooperate (choose Quiet). In the following game the players agree that it is better to cooperate than not to cooperate, but disagree about the best outcome.

The Battle of the Sexes

• A couple wish to go out together. Two choices are available: a football game and a ballet show. Husband prefers the footbal game and the wife prefers ballet. The important thing to remember here is that they want to do this together.

Wife

Husband

Ballet Football

Ballet (2,1) (0,0)

Football (0,0) (1,2)

Dominated actions

• In any game, a player’s action “strictly dominates” another action if it is superior, no matter what the other players do.

• In the Prisoner’s Dilemma, for example, the action Fink strictly dominates the action Quiet.

• In ballet vs football, on the other hand, neither action strictly dominates the other: Ballet is better than Football if the other player chooses Ballet, but is worse than Football if the other player chooses Football.

• If an action strictly dominates the action ai, we say that ai is strictly dominated.

• A strictly dominated action is not a best response to any actions of the other players: whatever the other players do, some other action is better. Since a player’s Nash equilibrium action is a best response to the other players’ Nash equilibrium actions,

• a strictly dominated action is not used in any Nash equilibrium.

Weak Domination

• In any game, a player’s action “weakly dominates” another action if the first action is at least as good as the second action, no matter what the other players do, and is better than the second action for some actions of the other players.

• Can an action be weakly dominated in a nonstrict Nash equilibrium?

Cournot’s Model of Oligopoly

• How does the outcome of competition among the firms in an industry depend on the characteristics of the demand for the firms’ output, the nature of the firms’ cost functions, and the number of firms?

• Will the benefits of technological improvements be passed on to consumers?

• Will a reduction in the number of firms generate a less desirable outcome?

General Model

• A single good is produced by n firms.• The cost to firm i of producing qi units of the

good is Ci(qi), where Ci is an increasing function (more output is more costly to produce).

• All the output is sold at a single price, determined by the demand for the good and the firms’ total output.

• Obviously the profit function of the firm takes the form:

• πi(q1, . . . , qn) = qiP(q1 + · · · + qn) − Ci(qi)

The model

• Players: The firms.• Actions: Each firm’s set of actions is the set of

its possible outputs (nonnegative numbers).• Preferences: Each firm’s preferences are

represented by its profit.• Assumptions: 2 firms. Firms decide

simultaneously (static game). Quantity competition.

Bertrand’s model of oligopoly

• In Cournot’s game, each firm chooses an output; the price is determined by the demand for the good in relation to the total output produced. In an alternative model of oligopoly, associated with a review of Cournot’s book by Bertrand (1883), each firm chooses a price, and produces enough output to meet the demand it faces, given the prices chosen by all the firms. The model is designed to shed light on the same questions that Cournot’s game addresses; as we shall see, some of the answers it gives are different.

• The economic setting for the model is similar to that for Cournot’s game. A single good is produced by n firms; each firm can produce qi units of the good at a cost of Ci(qi). It is convenient to specify demand by giving a “demand function” D, rather than an inverse demand function as we did for Cournot’s game. The interpretation of D is that if the good is available at the price p then the total amount demanded is D(p).

The model

• Players: The firms.• Actions: Each firm’s set of actions is the set of

possible prices (nonnegative numbers).• Preferences: Firm i’s preferences are

represented by its profit• Assumptions: 2 firms. Static game. Price

competition.

Next Week

• Dynamic imperfect competition: Stackelberg Model