Static Games of Complete Information: Equilibrium Concepts

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 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.

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 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.

7
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!
8
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.

9
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)

10
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!

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
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 is minimum possible payoff
• s is a maxi-min equilibrium if si is a maxi-min
  strategy for all i.

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 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.

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,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).

19
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!

20
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
21
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!

23
How can we choose between these two equilibria?

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

24
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
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,
economists are relying more and more on people to
show them how the games will be played.
27
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.
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
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

31
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

32
Do we have a mixed strategy equilibrium?
?S
brS(?)
1
brM(?)
Nash Equilibrium (½, ½), (½, ½)
½
?M
1
½
33
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

34
Solving for Player 1
Lagrangian
First Order Conditions
Implications
35
Solving for Player 2
Lagrangian
First Order Conditions
Implications
36
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
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 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

39
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
40
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

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, 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

43
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

44
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!

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?

46
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

47
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
48
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!