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Static Games of Complete Information Equilibrium

Concepts

- APEC 8205 Applied Game Theory

Objectives

- Understand Common Solution Concepts for Static

Games of Complete Information - Dominant Strategy Equilibrium
- Iterated Dominance
- Maxi-Min Equilibrium
- Pure Strategy Nash Equilibrium
- Mixed Strategy Nash Equilibrium

Introductory Comments On Assumptions

- Knowledge
- I know the rules of the game.
- I know you know the rules of the game.
- I know you know I know the rules of the game.
- Rationality
- I am individually rational.
- I believe you are individually rational.
- I believe that you believe I am individually

rational.

Normative Versus Positive Theory

- Normative
- How should people play games?
- What should they be trying to accomplish?
- Positive
- How do people play games?
- What do they accomplish?
- What are obstacles to the theorys predictive

performance? - Players do not always fully understand the rules

of the game. - Players may not be individually rational.
- Players may poorly anticipate the choices of

others. - Players are not always playing the games we think

they are.

What makes a good solution concept?

- Existence
- Uniqueness
- Logical Consistency
- Predictive Performance
- In Equilibrium
- Out of Equilibrium

Normal Form Games Notation

- A set N 1,2,,n of players.
- A finite set of pure strategies Si for each i ? N

where S S1?S2??Sn is the set of all possible

pure strategy outcomes. - si is a specific strategy for player i (si ? Si)
- si is a specific strategy for everyone but

player i

(si ? Si S1?? Si-1?Si1 ??Sn). - s is a specific strategy for each and every

player (e.g. a strategy profile s ? S) - A payoff function gi S ?? for each i ? N.

Dominant Strategy Equilibrium

- Definitions
- Strategy si weakly dominates strategy ti if

gi(si,si) gi(ti,si) for all si ? Si. - Strategy si dominates strategy ti if gi(si,si)

gi(ti,si) for all si ? Si and gi(si,si) gt

gi(ti, si) for some si ? Si. - Strategy si strictly dominates strategy ti if

gi(si,si) gt gi(ti,si) for all si ? Si. - Strategy profile s ? S is a weakly/strictly

dominant strategy equilibrium if for all i ? N

and all ti ? S, si weakly/strictly dominates ti.

Note In a dominant strategy equilibrium, your

best strategy does not depend on your opponents

strategy choices!

Note An equilibrium is defined by the strategy

profile that meets the definition of the

equilibrium!

Example Prisoners Dilemma

- Player 1
- Choose Defect if Player 2 Cooperates (3 gt 2).
- Choose Defect if Player 2 Defects (1 gt 0).
- Defect is a Dominant Strategy!
- Player 2
- Choose Defect if Player 1 Cooperates (3 gt 2).
- Choose Defect if Player 1 Defects (1 gt 0).
- Defect is a Dominant Strategy!
- (Defect, Defect) is a strictly dominant strategy

equilibrium.

Example Second Price Auction

- Who are the players?
- Bidders i 1, , n who value and are competing

for the same object. - Who can do what when?
- Players submit bids simultaneously.
- Who knows what when?
- Players know their value of the object before

submitting their bid. They do not know the value

of others. - How are players rewarded based on what they do?
- vi value to i of winning Auction
- hi highest bid value of all players not

including i - gi vi hi if bi gt hi and 0 otherwise.
- What is a players strategy?
- bi 0 (e.g. bid value)

Claim bi vi for all i is a weakly dominant

strategy equilibrium!

- Suppose bi gt vi
- If hi bi, gi 0 (Same as if bi vi).
- If hi lt vi, gi vi - hi (Same as if bi vi).
- If bi gt hi vi, gi vi - hi 0 (0 if bi

vi). - Does not improve payoff under any circumstances

and may reduce payoff! - Suppose vi gt bi
- If hi vi, gi 0 (Same as if bi vi).
- If hi lt bi, gi vi - hi (Same as if bi vi).
- If vi gt hi bi, gi 0 (vi - hi 0 if bi

vi). - Does not improve payoff under any circumstances

and may reduce payoff!

Good Bad Of Dominance Equilibrium

- Good
- Tends to predict behavior pretty well!
- Bad
- Often does not exist!

Iterative Dominance

- Definition
- Messy
- Not Very Instructive
- Intuition
- Easy! No rational player will ever choose a

dominated strategy. - Repeatedly eliminate dominated strategies for

each player until no dominated strategies remain!

Example Iterative Dominance

- R strictly dominates C, so C is gone.
- U strictly dominates M, so M is gone.
- U strictly dominates D, so D is gone.
- L strictly dominates R, so R is gone.
- (U,L) is the iterative dominant strategy.

Good Bad Of Iterative Dominance

- Good
- May be able to use it when there is not dominant

strategy equilibrium! - Bad
- Does not predict as well. Particularly, if lots

of iterations are involved.

Maxi-Min Equilibrium

- Motivation
- How should we play if we want to be particularly

cautious? - Definition
- Strategy si is a maxi-min strategy if it

maximizes is minimum possible payoff - s is a maxi-min equilibrium if si is a maxi-min

strategy for all i.

Example MaxiMin Equilibrium

- Player 1
- The minimum possible reward from choosing U is 0.
- The minimum possible reward from choosing D is 1.
- D maximizes Player 1s minimum possible reward.
- Player 2
- The minimum possible reward from choosing L is 0.
- The minimum possible reward from choosing R is 1.
- R maximizes Player 2s minimum possible reward.
- (D, R) is the Maxi-Min equilibrium strategy.

Comments on Maxi-Min Equilibrium

- Popular Solution Concept for Zero Sum Games
- Your gain is your opponents loss, so they are out

to get you and it makes sense to be cautious. - Game theorist version of the precautionary

principle.

Pure Strategy Nash Equilibrium

- Definition
- s ? S is a pure strategy Nash equilibrium if for

all players i ? N, gi(si,si) gi(si,si) for

all si ? Si (there are no profitable unilateral

deviations). - Alternative Definition
- Best Response Function bri(s) si ? Si

gi(si,si) gi(si,si) for all si ? Si. - Best Response Correspondence br(s) br1(s) ?

br2(s) ?? brn(s). - s ? S is a pure strategy Nash equilibrium if s

? br(s) (s

is a best response to itself).

Example Prisoners Dilemma

- Player 1
- Defect is the best response to Cooperate (3 gt 2).
- Defect is the best response to Defect (1 gt 0).
- Player 2
- Defect is the best response to Cooperate (3 gt 2).
- Defect is the best response to Defect (1 gt 0).
- (Defect, Defect) is a pure strategy Nash

equilibrium. - Same as dominant strategy equilibrium!

Welfare Nash

- First Fundamental Welfare Theorem
- A competitive equilibrium is Pareto efficient.
- A Nash equilibrium need not be Pareto efficient!

g2

(0,3)

Pareto Efficient

(2,2)

Pareto Efficient

(1,1)

Nash

(3,0)

Pareto Efficient

g1

Iterative Dominance Example Revisited

- Player 1
- U is a best response to L.
- D is a best response to C.
- U is a best response to R.
- Player 2
- L is a best response to U.
- R is a best response to M.
- R is a best response to D.
- (U, L) is a pure strategy Nash equilibrium.
- Same as the iterative dominant equilibrium!

Maxi-Min Example Revisited

- Player 1
- D is a best response to L.
- U is a best response to R.
- Player 2
- R is a best response to U.
- L is a best response to D.
- Pure strategy Nash equilibria
- (U, R)
- (D, L)
- Multiple Nash!
- Neither is the Maxi-Min equilibrium!

How can we choose between these two equilibria?

- Motivation for equilibrium refinements!
- What may make sense for this game?
- Pareto Dominance!

Is Pareto dominance always a good strategy?

- Player 1
- U is a best response to L.
- D is a best response to R.
- Player 2
- L is a best response to U.
- R is a best response to D.
- Pure strategy Nash equilibria
- (U, L) Pareto Dominant
- (D, R)
- Is (U, L) really more compelling than (D, R)?

Example Battle of the Sexes

- Player 1
- Football is the best response to Football.
- Ballet is the best response to Ballet.
- Player 2
- Football is the best response to Football.
- Ballet is the best response to Ballet.
- Pure Strategy Nash Equilibria
- (Football, Football)
- (Ballet, Ballet)
- Neither strategy is Pareto dominant!

Focal Points (Schelling)

- Suppose you and a friend go to the Mall of

America to shop. As you leave the car in the

parking garage, you agree to go separate ways and

meet back up at 4 pm. The problem is you forget

to specify where to meet. - Question Where do you go to meet back up with

your friend?

Historically, equilibrium refinement relied much

on introspection. With the emergence and

increasing popularity of experimental methods,

economists are relying more and more on people to

show them how the games will be played.

Matching Pennies Revisited

- Mason
- Heads is the best response to Heads.
- Tails is the best response to Tails.
- Spencer
- Tails is the best response to Heads.
- Heads is the best response to Tails.
- There is no pure strategy Nash equilibrium!

Mixed Strategy Nash Equilibrium

- Definitions
- ?i(si) probability player i will play pure

strategy si. - ?i mixed strategy (a probability distribution

over all possible pure strategies). - ?i set of all possible mixed strategies for

player i (?i ? ?i). - ? ? 1, ? 2, , ? n mixed strategy profile.
- ? ? 1 ? ? 2 ?? ? n set of all possible mixed

strategy profiles (? ? ?). - Gi(?i,?i)
- ? ? ? is a mixed strategy Nash equilibrium if

for all players i ? N, Gi(?i,?i) Gi(si,?i)

for all si ? Si.

Note Dominant strategy equilibrium and iterative

dominant strategy equilibrium can also

be defined in mixed strategies.

Mixed Strategy Nash Equilibrium

- Another Definition
- Best Response Function

bri(?) ?i ? ?i Gi(?i, ?i) Gi(si,

?i) for all si ? Si. - Best Response Correspondence br(?) br1(?) ?

br2(?) ?? brn(?). - ? ? ? is a pure strategy Nash equilibrium if ?

? br(? ).

What is Masons best response for Matching

Pennies?

- (?S ,1 ?S) mixed strategy for Spencer where 1

?S 0 is the probability of Heads. - (?M,1 ?M) mixed strategy for Mason where 1

?M 0 is the probability of Heads. - ?M(H) ?S (1 ?S) Masons expected payoff

from choosing Heads. - ?M(T) -?S (1 ?S) Masons expected payoff

from choosing Tails. - ?M(H) gt//lt ?M(T) for ?S gt//lt ½

What is Spencers best response for Matching

Pennies?

- ?S(H) -?M (1 ?M) Spencers expected

payoff from choosing Heads. - ?S(T) ?M (1 ?M) Spencers expected payoff

from choosing Tails. - ?S(H) gt//lt ?S(T) for ½ gt//lt ?M

Do we have a mixed strategy equilibrium?

?S

brS(?)

1

brM(?)

Nash Equilibrium (½, ½), (½, ½)

½

?M

1

½

Battle of the Sexes Example Revisited

- (?1 ,1 ?1) mixed strategy for Player 1 where 1

?1 0 is the probability of Football. - (?2,1 ?2) mixed strategy for Player 2 where 1

?2 0 is the probability of Football. - Player 1s Optimization Problem
- Player 2s Optimization Problem

Solving for Player 1

Lagrangian

First Order Conditions

Implications

Solving for Player 2

Lagrangian

First Order Conditions

Implications

Do we have a mixed strategy equilibrium?

Is that all?

Nash Equilibrium (1, 0), (1, 0)

?2

br2(?)

1

br1(?)

Nash Equilibrium (2/3, 1/3), (1/3, 2/3)

1/3

Nash Equilibrium (0, 1), (0, 1)

?1

1

2/3

Why do we care about mixed strategy equilibrium?

- Seems sensible in many games
- Matching Pennies
- Rock/Paper/Scissors
- Tennis
- Baseball
- Prelim Exams
- If we allow mixed strategies, we are guaranteed

to find at least one Nash in finite games (Nash,

1950)! - Games with continuous strategies also have at

least one Nash under usual conditions (Debreu,

1952 Glicksburg, 1952 and Fan, 1952). - Actually, finding a Nash is usually not a

problem. The problem is usually the multiplicity

of Nash!

Application Cournot Duopoly

- Who are the players?
- Two firms denoted by i 1, 2.
- Who can do what when?
- Firms choose output simultaneously.
- Who knows what when?
- Neither firm knows the others output before

choosing its own. . - How are firms rewarded based on what they do?
- gi(qi, qj) (a qi qj)qi cqi for i ? j.
- Question What is a strategy for firm i?
- qi 0

Nash Equilibrium for Cournot Duopoly

- Find each firms best response function
- FOC for interior a 2qi qj c 0
- SOC 2 lt 0 is satisfied
- Solve for qi
- Find a Mutual Best Response

q2

a c

q1(q2)

a c 2

q2

q2(q1)

q1

a c

a c 2

q1

But what if we have n firms instead of just 2?

- Find each firms best response function
- FOC for interior a 2qi qi c 0
- SOC 2 lt 0 is satisfied
- Solve for qi
- Find a Mutual Best Response
- qi a c Q where Q qi qi
- Sum over i
- Solve for Q
- Solve for qi

Implications as n Gets Large

- Individual firm equilibrium output decreases.
- Equilibrium industry output approaches a c.
- Equilibrium price approaches marginal cost c.
- We approach an efficient competitive equilibrium!

Application Common Property Resource

- Who are the players?
- Ranchers denoted by i 1, 2, , n.
- Who can do what when?
- Each rancher can put steers on open range to

graze simultaneously. - Who knows what when?
- No rancher knows how many steers other ranchers

will graze before choosing how many he will

graze. - How are ranchers rewarded based on what they do?
- gi(qi, qi) p(aQ Q2)qi/Q cqi where Q is the

total number of steers grazing the range land. - Question What is a strategy for rancher i?
- qi 0

Nash Equilibrium for Common Property Resource

- Find each rancherss best response function
- FOC for interior p(a 2qi qi) c 0
- SOC 2p lt 0 is satisfied
- Solve for qi
- Find a Mutual Best Response
- pqi pa c pQ
- Sum over i
- Solve for Q
- Solve for qi

Implications as n Gets Large

- Individual rancher equilibrium stocking

decreases. - Equilibrium industry stocking approaches a c/p.
- Individual ranchers payoff approaches zero.
- Stocking rate becomes increasingly inefficient!

Application Compliance Game

- Who are the players?
- Regulator Firm
- Who does what when?
- Regulator chooses whether to Audit Firm Firm

chooses whether to Comply. - Choices are simultaneous.
- Who knows what when?
- Regulator Firm do not know each others choices

when making their own. - How are the Regulator and Firm rewarded based on

what they do?

Assuming S gt CF S gt CA, what is the Nash

equilibrium for this game?

- Regulator
- ?R(Audit) ?F(BR CA) (1 ?F)(S CA)
- ?R(Dont Audit) ?FBR (1 ?F)0
- ?R(Audit) gt//lt ?R(Dont Audit) for 1 CA/S

gt//lt ?F - Firm
- ?F(Comply) ?R(BF CF) (1 ?R)(BF CF)
- ?F(Dont Comply) ?R(BF S) (1 ?R)BF
- ?F(Comply) gt//lt ?F(Dont Comply) for ?R gt//lt

CF/S

What is the equilibrium?We know we have at least

one!

?F

brF(?)

1

Nash Equilibrium (CF/S, 1 - CF/S), (1 CA/S,

CA/S)

1 CA/S

brR(?)

?R

1

CF/S

What are the implications of this equilibrium?

- Equilibrium Audit Probability
- Increasing in the Firms cost of compliance!
- Decreasing in the Regulators sanction!
- Equilibrium Compliance Probability
- Decreasing in the Regulators cost of Auditing!
- Increasing in the Regulators sanction!

Shoot Jaywalkers with Zero Probability!

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