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Introduction to Game Theory (Deterministic Model) Non-cooperative Game Theory

with Complete Information

Prof. Dr. Jinxing Xie

Department of Mathematical Sciences

Tsinghua University, Beijing 100084, China

http://faculty.math.tsinghua.edu.cn/~jxie

Email: jxie@ math.tsinghua.edu.cn

Voice: (86-10)62787812 Fax: (86-10)62785847

Office: Rm. 1202, New Science Building

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What is Game Theory? “No man is an island”

Study of rational behavior in interactive or interdependent situations

Bad news:Knowing game theory does not guarantee

winning

Good news:Framework for thinking about strategic

interaction

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Games We Play Group projects free-riding, reputation Flat tire coordination GPA trap prisoner’s dilemma Tennis / Baseball mixed strategies Mean professors commitment Traffic congestion Dating information manipulation

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Games Businesses Play

Patent races game of chicken Drug testing mixed strategies FCC spectrum auctions Market entry commitment OPEC output choice collusion & enforcement Stock options compensation schemes Internet pricing market design

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Why Study Game Theory?Because the press tells us to…

“As for the firms that want to get their hands on a sliver of the airwaves, their best bet is to go out first and hire themselves a good game theorist.”

The Economist, July 23,1994 p. 70

“Game Theory, long an intellectual pastime, came into its own as a business tool.”

Forbes, July 3, 1995, p. 62. 

“Game theory is hot.” The Wall Street Journal, 13 February 1995, p. A14

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Why Study Game Theory?

Because we can formulate effective strategy…

Because we can predict the outcome of strategic situations…

Because we can select or design the best game for us to be playing…

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Why Study Game Theory?

McKinsey: John Stuckey & David White - Sydney

“To help predict competitor behavior and determine optimal strategy, our consulting teams use techniques such as pay-off matrices and competitive games.”

Tom Copeland - Director of Corporate Finance “Game theory can explain why oligopolies tend to be

unprofitable, the cycle of over capacity and overbuilding, and the tendency to execute real options earlier than optimal.”

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Outline -- Concepts

Recognizing the gameRules of the gameSimultaneous games

Anticipating rival’s movesSequential games

Looking forward – reasoning backMixed strategies

Sensibility of being unpredictableRepeated games

Cooperation and agreeing to agree

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Outline -- Applications

Winning the gameCommitment

Credibility, threats, and promisesInformation

Strategic use of informationBargaining

Gaining the upper hand in negotiationAuctions

Design and Participation

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Interactive Decision Theory

Decision theoryYou are self-interested and selfish

Game theorySo is everyone else

“If it’s true that we are here to help others,then what exactly are the others here for?”

- George Carlin

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The Golden Rule

COMMANDMENT

Never assume that your opponents’ behavior is fixed.

Predict their reaction to your behavior.

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The Matrix of Game Theory

Non-cooperative DynamicGames

Non-cooperative

Static

Games

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The Matrix of Non-cooperation Game

Complete (Full) Information

Incomplete

Information

Simultaneous Move (Static)

Nash Equilibrium

Bayesian Equilibrium

Sequential Move (Dynamic)

Subgame Perfect

(Nash)

Equilibrium

Subgame Perfect Bayesian Equilibrium

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Definition of a Game Must consider the strategic (Normal) environment

Who are the PLAYERS? (Decision makers) What STRATEGIES are available? (Feasible actions) What are the PAYOFFS? (Objectives)

Rules of the game What is the time-frame for decisions? What is the nature of the conflict? What is the nature of interaction? What information is available?

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The Assumptions

RationalityPlayers aim to maximize their payoffsPlayers are perfect calculators

Common knowledgeEach player knows the rules of the game Each player knows that each player knows the rules Each player knows that each player knows that

each player knows the rules Each player knows that each player knows that each player knows that

each player knows the rules Each player knows that each player knows that each player knows that each player knows that each player knows the rules

Etc. etc. etc.

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Nash (1950)

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Cournot’s Model of Oligopoly(Cournot, 1838)

Single good produced by n firms Cost to firm i of producing qi units: Ci(qi), where Ci

is nonnegative and increasing If firms’ total output is Q then market price is P(Q),

where P is nonincreasing Profit of firm i, as a function of all the firms’

outputs: qCqPqqqq

ii

n

jjini

121

...,,,

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Cournot’s Model of Oligopoly

Strategic (normal) form game: players: firms each firm’s set of actions: set of all possible

outputs each firm’s preferences are represented by its

profit

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Example: Duopoly

two firms Inverse demand:

• constant unit cost: Ci(qi) = cqi, where c <

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Example: Duopoly

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Example: Duopoly

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Example: Duopoly

Best response function is:

Same for firm 2: b2(q) = b1(q) for all q.

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Example: Duopoly

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Example: Duopoly

Nash equilibrium:

Pair (q*1, q*2) of outputs such that each firm’s action is a best response to the other firm’s action

or

q*1 = b1(q*2) and q*2 = b2(q*1)

Solution:

q1 = ( − c − q2)/2 and q2 = ( − c − q1)/2

q*1 = q*2 = ( − c)/3

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Example: Duopoly

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Example: Duopoly

Conclusion: Game has unique Nash equilibrium:

(q*1, q*2) = (( − c)/3, ( − c)/3)

At equilibrium, P* = ( + 2c)/3, each firm’s profit is

=( − c)2)/9

Total output 2( − c)/3 lies between monopoly output ( − c)/2 and competitive output − c.

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Cournot’s Model of Oligopoly: Notes

Dependence of Nash equilibrium on number of firms Comparison of Nash equilibrium with collusive outco

mes (monopoly):

If there are only one firm in the market:

Max (-q-c)q q* = (-c)/2 < 2(-c)/3

If P(Q) = -bQ: Cournot’s model gives

(q*1, q*2) = (( − c)/3b, ( − c)/3b)

P* = ( + 2c)/3 (unchanged)

=( − c)2)/9b

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Bertrand’s Model of Oligopoly (Bertrand, 1883)

Strategic variable price rather than output. Single good produced by n firms Cost to firm i of producing qi units: Ci(qi), where Ci

is nonnegative and increasing If price is p, demand is D(p) Consumers buy from firm with lowest price Firms produce what is demanded

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Bertrand’s Model of Oligopoly(Bertrand, 1883)

Strategic game: players: firms each firm’s set of actions: set of all possible prices each firm’s preferences are represented by its profit

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Example: Duopoly

2 firms Ci(qi) = c qi for i = 1, 2

D(p) = − p p = - D P in [0, ∞], or actually in [0, ]?

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Example: Duopoly

Nash Equilibrium:

(p1, p2) = (c, c)

total quantity produced = − c (?)

If each firm charges a price of c then the other firm can do no better than charge a price of c also (if it raises its price it sells no output, while if it lowers its price it makes a loss), so (c, c) is a Nash equilibrium.

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Example: Duopoly

No other pair (p1, p2) is a Nash equilibrium since

If pi < c then the firm whose price is lowest (or either firm, if the prices are the same) can increase its profit (to zero) by raising its price to c

If pi = c and pj > c then firm i is better off increasing its price slightly

if pi ≥ pj > c then firm i can increase its profit by lowering pi to some price between c and pj (e.g. to slightly below pj if D(pj) > 0 or to pmonop if pj > pmonop).

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Bertrand’s Model: Notes

If D(p) has the form: p = − bD:

Nash Equilibrium unchanged: (p1, p2) = (c, c)

total quantity produced = ( − c)/b (0?)

If the products produced by two firms are non-identical: Di(pi)= − pi + bpj (i=2-j)

pi*=( + c + bpj

*)

p1*=p2

*=( + c ) / (2-b)

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Hotelling’s Model of Electoral Competition

Several candidates run for political office Each candidate chooses a policy position Each citizen, who has preferences over policy

positions, votes for one of the candidates Candidate who obtains the most votes wins.

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Hotelling’s Model of Electoral Competition

Strategic game: Players: candidates Set of actions of each candidate: set of possible

positions Each candidate gets the votes of all citizens who

prefer her position to the other candidates’ positions; each candidate prefers to win than to tie than to lose.

Note: Citizens are not players in this game.

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Example

Two candidates Set of possible positions is a (one-dimensional)

interval. Each voter has a single favorite position, on

each side of which her distaste for other positions increases equally.

Unique median favorite position m among the voters: the favorite positions of half of the voters are at most m, and the favorite positions of the other half of the voters are at least m.

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Example

Direct argument for Nash equilibrium(m, m) is an equilibrium: if either candidate chooses

a different position she loses.No other pair of positions is a Nash equilibrium: If one candidate loses then she can do better by

moving to m (where she either wins or ties for first place)

If the candidates tie (because their positions are either the same or symmetric about m), then either candidate can do better by moving to m, where she wins.

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Sequential Game

“Life must be understood backward, but … it must be lived forward.”

- Soren Kierkegaard

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Games of Chicken A monopolist faces a potential entrant Monopolist can accommodate or fight Potential entrant can enter or stay out

Monopolist

Accommodate Fight

In 50 , 50 -50 , -50

Out 0 , 100 0 , 100

Pote

nti

al

En

trant

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Equilibrium Use best reply method

to find equilibria

Monopolist

Accommodate Fight

In 5050 , 5050 -50 , -50

Out 0 , 100100 00 , 100100

Pote

nti

al

En

trant

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Importance of Order Two equilibria exist

( In, Accommodate )( Out, Fight )

Only one makes temporal senseFight is a threat, but not credibleNot sequentially rational

Simultaneous outcomes may not make sense for sequential games.

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Sequential Games

E out

in M fight

acc

0 , 100

-50 , -50

50 , 50

The Extensive Form

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Looking Forward… Entrant makes the first move:

Must consider how monopolist will respond

If enter:

Monopolist accommodates

M fight

acc

-50 , -50

50 , 50

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Now consider entrant’s move

Only ( In, Accommodate ) is sequentially rational

… And Reasoning Back

E out

in M

0 , 100

50 , 50acc

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Sequential Rationality

COMMANDMENT

Look forward and reason back.

Anticipate what your rivals will do tomorrow

in response to your actions today

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Solving Sequential Games:Backward Induction

Start with the last move in the game Determine what that player will do Trim the tree

Eliminate the dominated strategies

This results in a simpler game Repeat the procedure

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Equilibrium

What is likely to happen when rational players interact in a game?

Type of equilibrium depends on the game Simultaneous or sequential Perfect or limited information

Concept always the same: Each player is playing the best response to other

players’ actions No unilateral motive to change Self-enforcing

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Summary

Recognizing that you are in a game Identifying players, strategies, payoffs Understanding the rules Manipulating the rules

Nash Equilibrium Subgame Perfect Equilibrium Best response strategy (function) Backward Induction Searching for possible outcomes