1

Static Games of Complete Information: Equilibrium Concepts

APEC 8205: Applied Game Theory

2

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

3

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.

– …

4

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 theory’s 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.

5

What makes a good solution concept?

• Existence

• Uniqueness

• Logical Consistency

• Predictive Performance– In Equilibrium

– Out of Equilibrium

6

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 = S1S2…Sn is the set of all possible pure strategy outcomes. – si is a specific strategy for player i (si Si)

– s~i is a specific strategy for everyone but player i (s~i S~i = S1… Si-1Si+1 …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.

7

Dominant Strategy Equilibrium

• Definitions– Strategy si weakly dominates strategy ti if gi(si,s~i) ≥ gi(ti,s~i) for all s~i S~i.

– Strategy si dominates strategy ti if gi(si,s~i) ≥ gi(ti,s~i) for all s~i S~i and gi(si,s~i) > gi(ti, s~i) for some s~i S~i.

– Strategy si strictly dominates strategy ti if gi(si,s~i) > gi(ti,s~i) for all s~i S~i.

– 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 strategydoes not depend on your opponents’ strategy choices!

Note: An equilibrium is defined by the strategy profile thatmeets the definition of the equilibrium!

8

Example: Prisoners’ Dilemma

• Player 1– Choose Defect if Player 2 Cooperates (3 > 2).

– Choose Defect if Player 2 Defects (1 > 0).

– Defect is a Dominant Strategy!

• Player 2– Choose Defect if Player 1 Cooperates (3 > 2).

– Choose Defect if Player 1 Defects (1 > 0).

– Defect is a Dominant Strategy!

• (Defect, Defect) is a strictly dominant strategy equilibrium.

Player 2 Cooperate Defect

Player 1

Cooperate 2

2 3

0

Defect 0

3 1

1

9

• 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

– h~i: highest bid value of all players not including i

– gi = vi – h~i if bi > h~i and 0 otherwise.

• What is a players strategy? – bi ≥ 0 (e.g. bid value)

Example: Second Price Auction

10

Claim: bi* = vi for all i is a weakly dominant strategy equilibrium!

• Suppose bi > vi:

– If h~i ≥ bi, gi = 0 (Same as if bi = vi).

– If h~i < vi, gi = vi - h~i (Same as if bi = vi).

– If bi > h~i ≥ vi, gi = vi - h~i ≤ 0 (0 if bi = vi).

– Does not improve payoff under any circumstances and may reduce payoff!

• Suppose vi > bi:

– If h~i ≥ vi, gi = 0 (Same as if bi = vi).

– If h~i < bi, gi = vi - h~i (Same as if bi = vi).

– If vi > h~i ≥ bi, gi = 0 (vi - h~i ≥ 0 if bi = vi).

– Does not improve payoff under any circumstances and may reduce payoff!

11

Good & Bad Of Dominance Equilibrium

• Good– Tends to predict behavior pretty well!

• Bad– Often does not exist!

12

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!

13

Player 2 L C R

U 3

4 1

5 2

6 Player 1

M 1

2 4

8 6

3

D 0

3 6

9 8

2

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.

14

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.

15

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 i’s minimum possible payoff:

– s* is a maxi-min equilibrium if si* is a maxi-min strategy for all i.

iis ssg

iiii SsSsi ~* ,minmax

~~

16

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 1’s 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 2’s minimum possible reward.

• (D, R) is the Maxi-Min equilibrium strategy.

Player 2 L R

Player 1

U 0

0 1

3

D 3

4 2

1

17

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.

18

Pure Strategy Nash Equilibrium

• Definition:– s* S is a pure strategy Nash equilibrium if for all players i N, gi(si*,s~i*) ≥

gi(si,s~i*) for all si Si (there are no profitable unilateral deviations).

• Alternative Definition:– Best Response Function: bri(s) = {si Si: gi(si,s~i) ≥ gi(si’,s~i) 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).

19

Player 2 Cooperate Defect

Player 1

Cooperate 2

2 3

0

Defect 0

3 1

1

Example: Prisoners’ Dilemma

• Player 1– Defect is the best response to Cooperate (3 > 2).

– Defect is the best response to Defect (1 > 0).

• Player 2– Defect is the best response to Cooperate (3 > 2).

– Defect is the best response to Defect (1 > 0).

• (Defect, Defect) is a pure strategy Nash equilibrium.– Same as dominant strategy equilibrium!

* *

*

*

20

Welfare & Nash

• First Fundamental Welfare Theorem:– A competitive equilibrium is Pareto efficient.

• A Nash equilibrium need not be Pareto efficient!

g2

g1

(0,3)

(2,2)

(3,0)

(1,1) Nash

Pareto Efficient

Pareto Efficient

Pareto Efficient

21

Player 2 L C R

U 3

4 1

5 2

6 Player 1

M 1

2 4

8 6

3

D 0

3 6

9 8

2

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!

*

*

**

*

*

22

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!

Player 2 L R

Player 1

U 0

0 1

3

D 3

4 2

1

*

**

*

23

How can we choose between these two equilibria?

• Motivation for equilibrium refinements!

• What may make sense for this game?– Pareto Dominance!

Player 2 L R

Player 1

U 0

0 1

3

D 3

4 2

1

*

**

*

24

Player 2 L R

Player 1

U 9

9 8

0

D 0

8 7

7

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)?

*

*

*

*

25

Player 2 Football Ballet

Player 1

Football 1

2 0

0

Ballet 0

0 2

1

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!

*

*

*

*

26

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, economistsare relying more and more on people to show them how the games will be played.

27

Spencer Heads Tails

Mason

Heads -1

1 1

-1

Tails 1

-1 -1

1

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!

*

**

*

28

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.

ii iiSs Ss

iiiiiii ssgss~~

~~~ ,

29

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( *).

30

Spencer Heads Tails

Mason

Heads -1

1 1

-1

Tails 1

-1 -1

1

What is Mason’s 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) : Mason’s expected payoff from choosing Heads.

M(T) = -S + (1 – S) : Mason’s expected payoff from choosing Tails.

M(H) >/=/< M(T) for S >/=/< ½

•

2/1for,1,0

2/1for,1,1,0

2/1for,0,1

S

SMM

S

Mbr

31

Spencer Heads Tails

Mason

Heads -1

1 1

-1

Tails 1

-1 -1

1

What is Spencer’s best response for Matching Pennies?

S(H) = -M + (1 – M) : Spencer’s expected payoff from choosing Heads.

S(T) = M – (1 – M) : Spencer’s expected payoff from choosing Tails.

S(H) >/=/< S(T) for ½ >/=/< M

•

2/1for,0,1

2/1for,1,1,0

2/1for,1,0

M

MSS

M

Sbr

32

Do we have a mixed strategy equilibrium?

S

M1

1

½

½

brM()

brS()

Nash Equilibrium:{(½, ½), (½, ½)}

33

Player 2 Football Ballet

Player 1

Football 1

2 0

0

Ballet 0

0 2

1

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 1’s Optimization Problem

• Player 2’s Optimization Problem

21212121212101

3111110102max1

21212121212101

322211210101max2

34

Solving for Player 1

212101

31max1

121211 131 L

031 121

1 L

011

1 L

01

Lagrangian

First Order Conditions

Implications

3/1for,1,0

3/1for,1,1,0

3/1for,0,1

2

211

2

1

br

01 011

1 L

01 11

1 L

35

Solving for Player 2

212101

3222max2

221212 13222 L

032 212

2 L

022

2 L

02

Lagrangian

First Order Conditions

Implications

3/2for,1,0

3/2for,1,1,0

3/2for,0,1

1

122

1

2

br

02 022

2 L

01 22

2 L

36

Do we have a mixed strategy equilibrium?

2

11

1

2/3

1/3

br2()

br1()

Nash Equilibrium:{(2/3, 1/3), (1/3, 2/3)}

Is that all?Nash Equilibrium:

{(1, 0), (1, 0)}

Nash Equilibrium:{(0, 1), (0, 1)}

37

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!

38

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 other’s 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

39

Nash Equilibrium for Cournot Duopoly

• Find each firm’s best response function:– FOC for interior: a – 2qi – qj – c = 0

– SOC: –2 < 0 is satisfied

– Solve for qi:

• Find a Mutual Best Response:

iiji

q

cqqqqai

max0

2

jji

qcaqq

q1

q2

a – c2

a – c

a – ca – c2

q1*

q2*

3,

3**, 21

cacaqq

q1(q2)

q2(q1)

40

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

• Find each firm’s best response function:– FOC for interior: a – 2qi – q~i – c = 0

– SOC: –2 < 0 is satisfied

– Solve for qi:

• Find a Mutual Best Response:– qi* = a – c – Q* where Q* = qi* + q~i*

– Sum over i:

– Solve for Q*:

– Solve for qi*:

ii

n

jj

q

cqqqai

10

max

2

~~

iii

qcaqq

**or**11

QcanQQcaqn

i

n

ii

n

canQ

1

*

n

caqi

1*

41

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!

42

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, q~i) = 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

43

Nash Equilibrium for Common Property Resource

• Find each ranchers’s best response function:– FOC for interior: p(a – 2qi – q~i) – c = 0

– SOC: –2p < 0 is satisfied

– Solve for qi:

• Find a Mutual Best Response:– pqi* = pa – c – pQ*

– Sum over i:

– Solve for Q*:

– Solve for qi*:

ii

q

cqQ

qQaQp

i

2

0

max

p

pqcpaqq i

ii 2~

~

**or**11

pQcpanpQpQcpapqn

i

n

ii

pn

cpanQ

1

*

pn

cpaqi

1*

44

Implications as n Gets Large

• Individual rancher equilibrium stocking decreases.

• Equilibrium industry stocking approaches a – c/p.

• Individual rancher’s payoff approaches zero.

• Stocking rate becomes increasingly inefficient!

45

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?

Firm

Comply (F) Don’t Comply (1-F) Regulator

Audit (R)

BF - CF BR – CA

BF – S S – CA

Don’t Audit (1-R)

BF - CF BR

BF 0

46

Assuming S > CF & S > CA, what is the Nash equilibrium for this game?

• Regulator R(Audit) = F(BR – CA) + (1 – F)(S – CA)

R(Don’t Audit) = FBR + (1 – F)0

R(Audit) >/=/< R(Don’t Audit) for 1 – CA/S >/=/< F

• Firm F(Comply) = R(BF – CF) + (1 – R)(BF – CF)

F(Don’t Comply) = R(BF – S) + (1 – R)BF

F(Comply) >/=/< F(Don’t Comply) for R >/=/< CF/S

Firm

Comply (F) Don’t Comply (1-F) Regulator

Audit (R)

BF - CF BR – CA

BF – S S – CA

Don’t Audit (1-R)

BF - CF BR

BF 0

47

What is the equilibrium?We know we have at least one!

F

R1

1brF()

brR()

Nash Equilibrium:{(CF/S, 1 - CF/S), (1 – CA/S, CA/S)}

1 – CA/S

CF/S

48

What are the implications of this equilibrium?

• Equilibrium Audit Probability:– Increasing in the Firm’s cost of compliance!

– Decreasing in the Regulator’s sanction!

• Equilibrium Compliance Probability:– Decreasing in the Regulator’s cost of Auditing!

– Increasing in the Regulators sanction!

Shoot Jaywalkers with Zero Probability!