Game Theory
Section 2: Externalities
Agenda
• Key terms and definitions
• Complementarity and cross-partial derivatives)
– Partnership game
– Cournot game
• Mixed strategy equilibria
– Conceptualization
– How to solve
– Application: Iran and IAEA
Key Terms and Definitions
• Set of rationalizable strategies
– Those that survive iterated dominance
– Motivated by common knowledge assumptions
• Congruity
– Weakly congruous strategies
• Each strategy is a best response to others’ strategy
– Best response complete
• Set includes i’s best response to all others’ strategies
Quick Congruity Quiz
Player 2
Player 1
Left
Center
Right
Top
3,4
2,3
9,15
Middle
2,5
-1,8
4,0
Down
8,2
4,4
5,0
Pareto
Optima
Nash Eqm
(TR & DC)
strategy profiles
Set of Rationalizable Strategies: {T,D} x {C,R}
Set of Weakly Congruent Strategies: {T,D} x {C,R}
Set of Best Response Complete: {T,D} x {C, R} or {T,D} x {L,C,R}
Set of Congruent Strategies: {T,D} x {C, R}
Externalities: The Dirty Little Secret
When One Person’s Actions Affect Others’ Welfare
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Classic Examples
Pollution
Lightening rods
Security
Grades and curving
LaTeX and presentations
Cooperation and
competition
Contextual Strategic Issues
• Complementarity
– Agents share the benefits of
each other’s actions
• Substitutability
– Agents share the costs of
each other’s actions
• Pareto efficiency
Partnership Game: Setting
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2 Partners in business
Complete information about the situation
Simultaneous moves (no verifiable contract)
Two players: i, j
– Each expends effort: i, j
– Payoffs are:
πi = 2(i + j + cij) - i2
πj = 2(i + j + cij) - j2
• Interpret these equations
Partnership Game: Analysis
• Payoffs are:
• πi = 2(i + j + cij) - i2
• πj = 2(i + j + cij) - j2
• Best response functions:
– d (πi)/di = 2+2cj - 2i
– set equal to zero: 2i = 2 + 2cj
• i*=1+cj … This is our BR function for agent i
– symmetry requires that j*=1+ci
• If each is responding to the BR of the other
• j*=1+c(1+cj*), j*=1/(1-c) = i* also by symmetry
• πi,j= 2[1/(1-c) + 1/(1-c) + c(1/(1-c))2] - (1/(1-c))2
• πi,j= (3-2c)/(1-c)2
Partnership Game: Pareto-Efficiency
In Strategic Setting, Equilibrium was i*,j*=1/(1-c), πi,j= (3-2c)/(1-c)2
• Now we leave the strategic setting (S.S.) and
maximize total profits (pareto efficiency)
πF = 4(i + j + cij) - i2- j2
• d (πF)/di = 4 +4cj - 2i
• d (πF)/dj = 4 +4ci - 2j
Set equal to zero: 2i = 4 + 4cj
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i* = 2 + 2cj, j* = 2 + 2ci, i* = 2 + 2c(2 + 2ci*)
i* = 2/(1-2c) = j*, πF = 8/(1-2c),
Profit per partner, πi.j= 4/(1-2c) > (3-2c)/(1-c)2 in S.S.
Note that output is also lower in the strategic setting
Partnership Game: Explanation
Positive Cross-Partial Derivatives = Complementarity
πi = 2(i + j + cij) - i2- j2
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d (πi)/di = 2 +2cj - 2i
d ((πi)/di)/dj = 2c > 0
Why? What does this mean?
Agent j’s effort has positive effect on agent i’s productivity
– They complement each other
– All benefits are shared, but costs are private
» In the strategic setting, contributing more than i* would
contribute to total output (through an indirect effect via j)
» But would not contribute to i’s welfare
• Therefore this contribution is not made
– Outcome is not Pareto Efficient (there is deadweight loss)
Cournot Oligopoly: The Setting
• 2 Companies in competition: Firm 1 and Firm 2
• Each sets quantity (q1 , q2), which in turn effects price
p = 1000 - q1 - q2
• When nothing is produced, p = 1000
• Price = 0 when 1000 units are produced
• Every additional unit produced lowers price by $1
• Quantity and price are inversely related
• Firms are profit-maximizing
– Cost (to a company) of each item sold: $100
π1 = (1000 - q1 - q2)q1 - 100 q1
π2 = (1000 - q1 - q2)q2 - 100 q2
Cournot Oligopoly: Analysis
• Profit functions for firms 1 and 2
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π1 = (1000 - q1 - q2)q1 - 100q1
π2 = (1000 - q1 - q2)q2 - 100q2
d (π1)/dq1 = 1000 - 2q1 - q2 - 100
Set equal to zero, so... 2q1 = 1000 - q2 - 100
q1* = 450 - (q2)/2
q2* = 450 - (q1)/2
q1* = 450 - (450 - (q1)/2)/2 = 225 + q1/4
(3/4)q1* = 225
225(4/3) = q1* = $300 = q2* by symmetry
• Price is 1000 - q1 - q2 = $400, so π1,2 = $120,000
Cournot Oligopoly:Pareto Problemo
Recall: in strategic setting, q1,2* = $300, π1,2* = $120,000
• Now we leave the strategic setting and
maximize total welfare (pareto efficiency)
πF = (1000 - Q)Q - 100Q
• d (πF)/dQ = 1000 - 2Q - 100
• Set equal to zero…
– 2Q* = 1000 -100 so Q* = 450
– Firms split 450 equally, so q1,2* = $225
– Price = 1000 - 450 = $550, so π1,2 = $123,750
• Profits in pareto world > Profits in strategic world
• Why?
Cournot Oligopoly: Explanation
Positive Cross-Partial Derivatives = Complementarity
π1 = (1000 - q1 - q2)q1 - 100q1
d (π1)/dq1 = 1000 - 2q1 - q2 - 100
d ((π1)/dq1)/dq2 = -1 < 0
• Why? What does this mean?
• Firm 2’s production negatively effects firm 1’s production
– They substitute for each other goods
– All benefits are private, but costs are shared
» In the strategic setting, contributing less than i* would add to
total profit (by raising prices across all goods sold)
» But individual firms produce until marginal revenue = $100
• Therefore firms overproduce, outcome is not efficient
Mixed Strategy Equilibria
Mixed Strategy Nash Equilibrium: mixed-strategy profile such that
no player can increase his payoff by switching strategies, given the
other players’ strategies
Mixed Strategy Nash Equilibrium put positive probability (support)
only on pure strategies that are themselves best responses
Player 1
Up
Down
Player 2
Left
a,b
e,f
Right
c,d
g,h
Assume Player 2 follows a mixed strategy supporting Left and Right.
Assume Player 1 plays a mixed strategy, and the two are in a MSNE.
What are Player 1’s payoffs to playing Up vs. Down in any game?
What are Player 2’s payoffs to playing Left vs. Right in any game?
Mixed Strategies: Conceptualization
• Many situations: more than one pure
strategy equilibrium
• Agents try to influence others’ actions
– This can be done by keeping others’ off balance
by selecting one’s own actions probabilistically
– Mixed strategies force rivals to thin out
capabilities by not being able to concentrate
them in response to a single strategic action
• Another way: proportions in a population
Mixed Strategies: Logic
The Questions that Motivate the Analysis
• What mixed strategy would make my rival indifferent between
some mix of his pure strategies?
• If he’s indifferent, he may find he also have an incentive to mix.
If so, what is the equilibrium?
Prob (q) Prob (1-q)
Prob (p)
Prob (1-p)
Player 1
Up
Down
Player 2
Left
a,b
e,f
Right
c,d
g,h
Player 1: (p, 1-p) to make Player 2
indifferent between Left and Right
Player 2: (q, 1-q) to make Player 1
indifferent between Up and Down
(p)(b)+(f)(1-p) = (p)(d) + (h)(1-p)
(q)(a)+(1-q)(c) = (q)(e)+(1-q)(g)
p* = (h - f)/[(b+h)-(f+d)]
q* = (g - c)/[(a+g)-(c+e)]
Mixed Strategies: Application
Iran and the International Atomic Energy Agency
• Story
– Iran is eager to build nuclear weapons
– IAEA wants to prohibit nuclear proliferation
Players have rival preferences
IAEA
Iran
Make nukes
(p)
No nukes
Inspect
Don’t inspect
(q)
(1-q)
-10,5
8,-10
3,-1
0,0
(1-p)
Consider effect of IAEA’s power
• Iran wants to eliminate IAEA’s
incentive to inspect
• IAEA wants to eliminate Iran’s
incentive to make nukes
(p)(5) + (1-p)(-1) = (p)(-10) + 0
(q)(-10) + (1-q)(8) = (q)(3) + 0
Mixed Strategy N.E. {(1/16),(8/21)}
Mixed Strategy: Dénouement
For make nukes, Iran gets
[(8/21)*-10 + (13/21)*8] = 24/21
IAEA
Inspect
Don’t inspect
For no nukes, Iran gets
[(8/21)*3 + (13/21)*0] = 24/21
Iran
(8/21) (13/21)
Make nukes
-10,5
8,-10
For inspect, IAEA gets
[(1/16)*5 + (15/16)*-1] = -10/16
3,-1
0,0
For don’t inspect, IAEA gets
[(1/16)*-10 + (15/16)*0] = -10/16
(1/16)
No nukes
(15/16)