EC941 - Game Theory

Francesco SquintaniEmail:

f.squintani@warwick.ac.uk

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Lecture 1

Syllabus1. Games in Strategic Form

Definition and Solution ConceptsApplicationsReadings: Chapter 2, 3, 12

2. Mixed StrategiesNash Equilibrium and RationalizabilityCorrelated EquilibriumReadings: Chapter 4

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3. Bayesian Games DefinitionInformation and Bayesian GamesCournot Duopoly and Public Good Provision Readings: Sections 9.1 to 9.6

4.Bayesian Game ApplicationsJuries and Information AggregationAuctions with Private InformationReadings: Sections 9.7 to 9.8

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5. Extensive-Form GamesDefinitionSubgame Perfection and Backward InductionApplicationsReadings: Chapters 5, 6 and 7

6. Extensive-Form Games with Imperfect Information Definition Spence Signalling Game Crawford and Sobel Cheap Talk Readings: Chapter 10 4

7. Repeated GamesInfinitely Repeated GamesNash and Subgame-Perfect EquilibriumFinitely Repeated GamesReadings: Chapter 14 and 15

8. BargainingUltimatum Game and Hold Up ProblemRubinstein Alternating Offer BargainingNash Axiomatic BargainingReadings: Section 6.2 and Chapter 16

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9. Review SessionSolution of Past Exam Questions

Reference: An Introduction to Game Theory Martin J. Osborne, Oxford University Press, 2003.

Assessment: Final Exam (100% of the grade)

Office Hours: Wednesday 9:00-11:00 – Room 1.123

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Structure of the Lecture

Definition of Games in Strategic Form.

Solution Concepts Nash Equilibrium, Dominance and Rationalizability.

Applications Cournot Oligopoly, Bertrand Duopoly, Downsian Electoral Competition, Vickrey Second Price Auction.

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What is Game Theory?

Game Theory is the formal study of strategic interactions.

A strategic interaction involves two or more agents. They maximize their payoffs and are aware that their opponents maximize payoffs.

Applications range from economics to politics, to biology and computer science.

Games in Strategic Form

A game in strategic form is

a set of players: {1, 2, …, I} for each player i, a set Si of strategies si

for each player i, preferences over the set of strategy profiles S={(s1 , …, sI )}, represented by u : S RI

(a strategy profile includes one strategy for each player).

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Solution Concepts

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A solution concept is a mathematical rule to find the solution of a game.

It allows the modeler to formulate a prediction on the play of the interaction she modeled as a game.

Today, we will study 3 solution concepts:• Nash Equilibrium• Dominance• Rationalizability

Nash Equilibrium

A (pure-strategy) Nash equilibrium is a strategy profile

s∗ with the property that no player i can do

better by choosing a strategy different from si

∗, given that every

other player j adheres to sj∗.

Definition The strategy profile s∗ is a Nash equilibrium

if, for every player i, ui(s∗) ≥ ui(s’i, s−i∗) for

every strategy s’i

of player i.

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There are two main justifications for Nash Equilibrium:

Self-Enforcing Contract. The players meet and agree before playing on the course of actions s∗. The contract s∗

is self-enforcing if no player has reasons to deviate if the others do not.

Learning Equilibrium Play.The play s∗ is in equilibrium if no player i would deviate, were she to learn the opponents’ play s-i

∗, because of communication, observation, or repetition.

Dominance

A player’s strategy strictly dominates another one if it

gives a higher payoff, no matter of what other players do.

Definition Player i ’s strategy si strictly dominates strategy s’i

if ui (si , s−i) > ui (s’i , s−i) for every profile s−i of opponents’

strategies.

Theorem A strictly dominated strategy si is never part of any

Nash equilibrium s∗.

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A player’s strategy weakly dominates another strategy if it is always at least as good, and sometimes better.

Definition Player i ’s strategy si weakly dominates her strategy s’i if • ui (si , s−i) ≥ ui(s’i , s−i) for every profile s−i of opponents’ strategies

• ui(si , s−i) > ui(s’i , s−i) for some profile s−i of opponents’ strategies.

Note There exist games with Nash Equilibria s∗ that include weakly dominated strategies si for some player i. 14

There are two Nash Equilibria: (A,C) and (B,D).

The Nash Equilibrium (B,D) is weakly dominated.

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Player 2

C D Player 1

A

B

1, 1

0, 0 0, 0

0, 0

Rationalizability is defined via iterated deletion of strictly

dominated strategies.

Consider a finite game G = (I, S, u).

For each player i, and round t = 1, . . . , T, iteratively

define the set Xit of strategies of player i as

follows.

Xi1 = Si (start with the set of all possible

strategies).

Rationalizability

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For each t = 0, . . . , T − 1, Xit+1 is a subset

of Xit such that every strategy of player i in

Xit that is not in Xi

t+1 is strictly dominated in the game where the set of strategy of each player j is reduced to Xj

t

(in each round, delete all strictly dominated strategies).

The final index T is such that no strategy in Xi

T is strictly dominated in the game where the set of strategies of each player j is reduced to Xj

T

(proceed until no strategy is strictly dominated).

The set XiT is the set of rationalizable

strategies of player i.

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Rationalizability is justified by common knowledge of rationality.

Each player is rational: She does not play strictly dominated strategies.

Each player knows that every player is rational: She can reduce the game by deleting all players’ strictly dominated strategies from her model of the interaction (the game).

Each player knows that every player knows that every player is rational: She deletes all strictly dominated strategies in the reduced game.

The procedure is iterated until it stops.

Best Response Correspondences

The best response correspondence Bi of player i assigns

to each profile s-i of opponents’ strategies, the set of

player i ’s strategies that maximizes her payoff.

Definition The best response correspondence Bi of player i is:

Bi (s−i) = {si in Si : ui(si, s−i) ≥ ui(s’i, s−i) for all s’i in Si}.

Proposition The strategy profile s∗ is a Nash equilibrium of a

game G=(I, S, u) if and only if every player’s strategy is a best

response to the other players’ strategies: s∗i

belongs to Bi(s∗−i ) for

every player i.

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Prisoner’s Dilemma

Two prisoners are separately interviewed. By accusing the

other suspect, one’s prison term is reduced. But if they

both stayed quiet, they would not be incarcerated.

Players: The two suspects.

Strategies: Each player’s set of strategy is {Quiet, Fink}.

Preferences: Suspect 1’s ordering of the strategy profiles, from best to worst is (F, Q), (Q, Q), (F, F), (Q, F). Suspect 2’s ordering is (Q, F), (Q, Q), (F, F), (F, Q).

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Suspect 2

Quiet Fink Suspect 1

Quiet

Fink

2, 2 0, 3

3, 0 1, 1

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Solutions of Prisoner’s Dilemma

Quiet Fink

Quiet

Fink

2, 2 0, 3

3, 0 1, 1

Fink is the best response of each player, regardless of what the other player does. Fink is the strictly dominant and rationalizable strategy. (Fink, Fink) is the Nash Equilibrium.

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Bach or StravinskyTwo daters would rather be together than

separate, but dater 1 prefers Bach and dater 2 prefers

Stravinsky.

Players: The two daters.

Strategies: Each dater’s strategy set is {Bach, Stravinsky}.

Preferences: Dater 1’s ordering of the strategy profiles, from best to worst is (B, B), (S, S), (B, S) = (S, B).Dater 2’s ordering is (S, S), (B, B), (S, B) = (B, S).

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Dater 2

BachStravinsky

Dater 1

Bach

Stravinsky

2, 1 0, 0

0, 0 1, 2

If they can coordinate, either the two daters go to Bach’s concert or to Stravinsky’s concert.

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Solutions of Bach or Stravinsky

Bach

Stravinsky

2, 1 0, 0

0, 0 1, 2

For each player, B is the best response to B, and S is the best response to S. There are two Nash Equilibria, (B, B) and (S, S). All strategies are rationalizable, and none is dominated.

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Bach Stravinsky

Matching PenniesPlayer 1 wins if the coins are matched. Player 2 wins if they are not matched.

Players: The two players.

Strategies: Each player’s set of actions is {Head, Tail}.

Preferences: Player 1’s ordering of the strategy profiles, from best to worst, is (H, T) = (T, H), (H, H) = (T, T).Player 2’s ordering is (H, H) = (T, T), (H, T) = (T, H).

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Player 2

Head Tail Player 1

Head

Tail

-1, 1

1, -1

There is no sure way to win for either of the players.

-1, 1

1, -1

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Solutions of Matching Pennies

For player 1, H is the best response to T and viceversa, for player 2, H is the best response to H and T is the best response to T. All strategies are rationalizable and none is dominated. There are no Nash Equilibria.

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Head Tail Head

Tail

-1, 1

1, -1 -1, 1

1, -1

Cournot Oligopoly A good is produced by n firms.

Firm i’s cost of producing qi units is Ci(qi).Ci is an increasing function.

The firms' total output is Q = q1 + … + qn.

The market price is P(Q). P is the inverse demand function,

decreasing if positive.29

Firm i’s revenue is qi P(q1 + … + qn).

Firm i’s profit is revenue minus cost: pi(q1 + … + qn) = qi P(q1 + … + qn) - Ci (qi).

Ci (qi) = cqi, i=1, …, n.

P (Q) = a - Q if a > Q, P(Q) = 0 if a < Q.

pi(q1, …, qn) = qi [ a – (q1 + … + qn)] - cqi.

Linear Costs and Demand

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To find the Best Response Functions, differentiate pi with respect to qi, set it equal to zero, and obtain:dpi (q1, …, qn)/dqi = a – qi – (q1 + … + qn) – c = 0.

Best Response functions: bi (qi) = [ a – (q1 + … + qi-1 + qi+1 +…+ q n) – c]/2.

To find the Nash equilibria, we solve the system of best-response functions.

Because this system is linear and symmetric, we equalize qi* across i = 1,…,n:

qi* = bi (qi*) = [ a – (n-1) qi* – c]/2.

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Solving the above equation, we find that the Nash equilibrium quantity is:

qi* = [ a – c]/(n+1).

Substituting in the formula for the price, we find that the Nash equilibrium price is:

pi (qi*) = a – Q* = a – n[ a – c]/(n+1)

= [ a + nc]/(n+1).

The Nash equilibrium profits are:pi (qi*) = qi*[ a – Q*] – cqi*

= [ a – c ]2/(n-1)2.

[a – c]/3

[a – c]/3 q1

q2

b1(q2)

b2(q1)

(q1*, q2*) qi* = [ a – c]/3, i = 1,2.

With n = 2, b1(q2) = [a – q2– c]/2.b2(q1) = [a – q1– c]/2.

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Bertrand Competition

Unlike Cournot competition, firms compete in prices.

The demand function is denoted by D, if the good is available at the price p, then the total amount demanded is D(p).

The firm setting the lowest price sells to all the market.

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Linear Costs and Demand

Ci(qi) = cqi, i=1, …, n.

D(p) = a – p if a > p, D(p) = 0 if a < p.

Let pi = min {pj, j different from i}.

The profit is:pi(p1, …, pn) = (pi – c)(a - pi) if pi < pi, pi(p) = (pi – c)(a - pi)/|{k : pk = pi}| if pi = pi,pi(p) = 0 if pi > pi.

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Best-Response Correspondence

Suppose that there only two firms, so that pi= pj.

If pj < c, then pi(p) < 0 for pi < pj,

pi(p) = 0 for pi > pj : bi(p) = {pi : pi > pj}.pi

pi

pj

c pm

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If pj = c, then pi(p) < 0 for pi < pj,

pi(p) = 0 for pi > pj : bi(p) = {pi : pi > pj}. If pj > pm, then bi(p) = {pm}.

pi

pj

c pm pi

pi

pj c pm pi

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If c < pj < pm then pi(p) increases in pj, but discontinuously drops at pi = pj. So, bi(p) = f.The best response correspondence is empty.pi

pi

pj c

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In sum, the best-response correspondence is:

bi(p) = {pi : pi > pj}, if pj < c, bi(p) = {pi : pi > pj}, if pj = c, bi(p) = , f if c < pj < pm, bi(p) = {pm}, if pj > pm.

The Nash equilibrium is pi = c, for all i = 1,…,n.

Intuitively, selling at any price pi < c yields negative profit.

If the lowest industry price were pj > c, then firm i sells to

the whole industry at any price pi with c < pi < pj.

In equilibrium, pi = c, for all i.

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Downsian Electoral Competition

The players are 2 candidates in an election.

A strategy is a real number x, representing a policy on the left-right political ideology spectrum.

After the candidates choose policies, each citizen votes for the candidate with the policy she prefers.

The candidate who obtains the most votes wins. Candidates care only about winning.

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The voters are a continuum with diverse ideologies, with cumulative distribution F.

For any k, a voter with ideology y is indifferent between the policies y - k and y + k.

The median m is such that 1/2 of voters’ has ideologies y > m, and 1/2 has ideologies y < m. So, F (m) = 1/2.

Fix the policy x2 of candidate 2 and consider 1’s choice.

Suppose that x2 < m, the case for x2 > m

is symmetric.

If candidate 1 chooses x1 < x2 then she wins the votes of citizens with ideology y < ½ ( x1 + x2 ).

Because ½ ( x1 + x2 ) < x2 < m, it follows that

F(½ ( x1 + x2 ) ) < ½, so that candidate 1 wins less than ½ of the votes, and loses the election.

Best Response Functions

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If x1 > x2, then candidate 1 wins the votes of citizens with ideology y > ½ ( x1 + x2 ).

She wins more or less than ½ of the votes if and only if 1 – F(½ ( x1 + x2 )) > ½.

In this case, she wins the election.

This is equivalent to ½ ( x1 + x2 ) < m, i.e. x1 < 2m - x2.

So, b1 (x2) = {x1 : x2 < x1 < 2m - x2 } for x2 < m.

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For x2 > m, b1 (x2) = {x1 : 2m - x2 < x1 < x2}.

If x2 = m, then player 1 loses the election unless she plays x1 = m. So b1 (m) = {m}.

By using the best response correspondences the unique Nash Equilibrium is (m, m). The candidates’ political platforms converge to the median policy.

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Intuitively, consider any pair of platforms (x1, x2) other than (m, m). One candidate can win the election by deviating and locating e closer to m than x2.

Hence (x1, x2) is not a Nash Equilibrium.

If instead x1=m, then candidate 2 loses the election for sure unless she plays x2 = m.Hence (m, m) is a Nash Equilibrium.

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Vickrey Second -Price Auctions

In an “English” auction, n bidders submit increasing bids for a good, until only one is left, who wins the auction.

The price paid by the last bidder is her last bid.

Suppose each bidder’s valuation of the good is independent of the other bidders’ values. For example, Vickrey’s model applies when the good is a work of art, but not when it is a oil field.

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The English auction is equivalent to a sealed-bid auction, in which each bidder decides, before bidding begins, the most she is willing to bid.

To win, the bidder with the highest valuation needs to bid slightly more than the second highest maximal bid.

If the bidding increment is small, the price the winner pays equals the second highest maximal bid.

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Second-Price Auction Game

Players: n bidders. Bidder i’s valuation is vi, we order v1> … > vn > 0, without loss of generality.

Strategies: bidder i’s maximal bid is bi.

Let bi = max {bj : j different from i}.

Payoffs: ui(b1, … ,bn) = vi - bi if bi > bi

0 if bi < bi 48

1. (b*1,… , b*

n) = (v1, …, vn).

Bidder 1 wins the object, payoff: v1 – b*2 = v1

– v2 > 0.

If bidding b1 < v2, she loses the object, the payoff is 0.

If bidding b1 > v2, her payoff is v1 – v2 > 0.

The payoff of bidders i = 2, …, n is 0. If bidding bi > v1, the payoff is vi – b1 = vi – v1

< 0.If bidding bi < v1, she loses the object, the

payoff is 0.

Nash Equilibria

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2. (b*1,… , b*

n) = (v1, 0,… , 0)

Bidder 1 wins the object, her payoff is v1.

The payoff of bidders i = 2, …, n is 0. If bidding bi > v1, the payoff is vi – b1 = vi

– v1 < 0.

If bidding bi < v1, she loses the object, the payoff is 0.

3. (b*1,… , b*

n) = (v2, v1 , 0,… , 0)

Bidder 2 wins the object, payoff v2 – b1 = v2 – v2 = 0.

To win, bidder i = 1, 3, …, n must bid bi > b2 = v1,

so the payoff is vi – b1 < vi – v1 < 0.

Any of these bidders’ payoff is at least as good if losing the good.

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Weakly Dominant Solution

The Nash Equilibrium (b*1,… , b*

n) = (v1, …, vn) is the

unique weakly dominant solution.

bi < vi

bi < bi or

bi = bi & i wins

bi < bi < vi or

bi=bi & i loses

bi > vi

vi - bi

vi - bi vi - bi

0 0

0bi = vi

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bi > vi

bi < vi

vi < bi < bi or

bi =bi & i wins vi - bi

vi - bi vi – bi (< 0)

0 0

0

bi > bi or

bi =bi & i loses

In sum, bidding bi = vi yields at least as high a payoff as bidding bi > vi or bi < vi for any opponents’ bids.

bi = vi

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Definition of Games in Strategic Form.

Definition of Solution Concepts Nash Equilibrium, Dominance and Rationalizability.

Applications Cournot Oligopoly, Bertrand Duopoly, Downsian Electoral Competition, Vickrey Second Price Auction.

Summary of the Lecture

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Preview Next Lecture

Mixed Strategies.

Nash Equilibrium and Rationalizability.

Correlated Equilibrium.

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