Lecture 12 Nash Equilibrium - Purdue University...Felix / Oscar 3 hours 6 hours 9 hours 3 hours 13;...

Nash EquilibriumRelation between IEDS and Nash equilibrium

Application: Cournot DuopolyApplication: The Commons Problem

Lecture 12Nash Equilibrium

Jitesh H. Panchal

ME 597: Decision Making for Engineering Systems Design

Design Engineering Lab @ Purdue (DELP)School of Mechanical Engineering

Purdue University, West Lafayette, INhttp://engineering.purdue.edu/delp

October 30, 2014

c©Jitesh H. Panchal Lecture 12 1 / 27

http://engineering.purdue.edu/delp



Lecture Outline

1 Nash Equilibrium

2 Relation between IEDS and Nash equilibrium

3 Application: Cournot Duopoly

4 Application: The Commons Problem

Dutta, P.K. (1999). Strategies and Games: Theory and Practice. Cambridge, MA, The MIT Press. Chapters 5 and 6.




Best Response

Best Response

A strategy s∗i is a best response to a strategy vector s∗

−i of the other players if

πi (s∗i , s

∗−i ) ≥ πi (si , s∗

−i ), for all si

s∗i is a “dominant strategy” in a very weak sense.




Nash Equilibrium

Nash Equilibrium

The strategy vector s∗ = s∗1 , s

∗2 , . . . , s

∗N is a Nash equilibrium if

πi (s∗i , s

∗−i ) ≥ πi (si , s∗

−i ), for all si and all i

Requirements of Nash equilibrium:

Each player must be playing a best response against a conjecture.

The conjectures must be correct.




Nash Equilibrium - Example of a two-player scenario

Two players: 1 and 2

Strategies:1 {a1 and a2} for player 12 {b1 and b2} for player 2

(a2, b1) is a Nash equilibrium if and only if

π1(a2, b1) ≥ π1(a1, b1)

π2(a2, b1) ≥ π2(a2, b2)




Nash Equilibrium - Example 1

Battle of Sexes

Husband / Wife Football (F) OperaFootball (F) 3, 1 0, 0

Opera (O) 0, 0 1, 3





Prisoner’s Dilemma

1 / 2 Confess Not ConfessConfess 0, 0 7,−2

Not Confess −2, 7 5, 5





Bertrand Pricing

Firm 1 / Firm 2 High (H) Medium (M) Low (L)High (H) 6, 6 0, 10 0, 8

Medium (M) 10, 0 5, 5 0, 8Low (L) 8, 0 8, 0 4, 4





The Odd Couple

Felix / Oscar 3 hours 6 hours 9 hours3 hours −13,−8 −1,−4 7,−46 hours −4,−1 4,−1 4,−49 hours 1, 2 1,−1 1,−4





Coordination game

1 / 2 7:30 10:307:30 1, 1 0, 0

10:30 0, 0 0, 0




Motivation for Nash Equilibrium

Scenarios under which the two requirements (players playing best responseagainst conjectures, and conjectures being correct) may be appropriate:

1 Play prescription2 Preplay communication3 Rational introspection4 Focal point5 Trial and error




Relationship between IEDS and Nash equilibrium

Proposition

Consider any game in which there is an outcome to IEDS. It must be the casethat this outcome is a Nash equilibrium.

However, not every Nash equilibrium can be obtained as the outcome toIEDS.




Relationship between IEDS and Nash equilibrium - Illustration

Consider a 2-player 3-strategy game.

A / B b1 b2 b3

a1

a2

a3

Suppose that {a2, b3} is the IEDS solution.

For Nash equilibrium, need to show that:

a2 � a1, a2 � a3 when played against b3

b3 � b1, b3 � b2 when played against a2




Application: Cournot Duopoly

Extreme types of markets:

Monopoly (single firm)

Perfect competition (infinitely many firms)

Few firms in a given market:

Automobile market : 3 domestic and 10 foreign manufacturers

Aircraft manufacturers: 1 domestic manufacturer and 1 foreignmanufacturer

World oil market : 10 manufacturing nations account for 80% of oilproduction




The Basic Cournot Duopoly Model

Firms compete in a market for ahomogenous product (single demandcurve).

Q = α− βP

where α > 0, β > 0, and Q = Q1 + Q2

Cost: C1 = cQ1 and C2 = cQ2

Question

How much should each firm produce?

Quantity (Q)

Price (P)

0 10

10

Figure: 6.1 on Page 77




How much should each firm produce?

1 Make a conjecture about other firm’s production.2 Determine the quantity to produce.




Cournot Nash Equilibrium

Step 2

Make a conjecture about other firm’s production (say Q2).

The price is:

P =α

β− Q1 + Q2

β

P = a− b(Q1 + Q2)

where a =α

β, b =

1β

Step 2

Determine the quantity to produce.

MaxQ1 [a− b(Q1 + Q2)− c]Q1




Cournot Nash Equilibrium: Best Response Functions

R1(Q2) =

a− c − bQ2

2bif Q2 ≤

a− cb

;

0 if Q2 >a− c

b.

R2(Q1) =

a− c − bQ1

2bif Q1 ≤

a− cb

;

0 if Q1 >a− c

b.

Solving these equations for Q1 and Q2, we get:

Q1 = Q2 = Q∗ =a− c

3b

P∗ =a + 2c

3




Cournot Duopoly: Cartel Solution

If the two firms operate as a cartel, they coordinate their production decisionsto maximize their joint profits.

MaxQ1,Q2 [a− b(Q1 + Q2)− c][Q1 + Q2]

Solution:Q1 = Q2 = Q =

a− c4b

P =a + c

2

Per-Firm Profit =(a− c)2

8b




Generalization: Cournot Oligopoly

For N firms,Q = Q1 + Q2 + · · ·+ QN

Best reply correspondence:

R1¯(Qi ) =

a− c − (N − 1)bQi

2b

Nash equilibrium quantities:

Q∗i =

a− c(N + 1)b

Price:P∗ =

aN + 1

+Nc

N + 1




Generalization: Stackelberg Model

Suppose Firm 1 decides on its quantity before Firm 2 (i.e., Firm 2 knows Firm1’s quantity).

1 Firm 1 knows that Firm 2 will play a best response2 So, Firm 1 should choose Q1 knowing that Q2 = R2(Q1)

MaxQ1{a− b[Q1 + R2(Q1)]− c}Q1

Solving this we, get:

Q1 =a− c

2b

Q2 =a− c

4bNote: Firm 1’s profits are higher in the Stackelberg solution than in the Nashequilibrium.




The Commons Problem

Suppose that there is a common property resource of size y > 0

Each player can consume a non-negative amount, c1 or c2

Consider two time periods:1 Decide how much to consume in the first period2 Decide how much to consume from the available quantity: y − (c1 + c2)




The Commons Problem: Nash equilibrium

Assume: Utility from consumption: log(c1) and log(c2)

Player 1’s best response problem:

Maxc1 log(c1) + logy − (c1 + c2)

2

Solving this,

R1(c2) =y − c2

2Similarly,

R2(c1) =y − c1

2

The Nash equilibrium is: c∗1 = c∗

2 =y3




The Commons Problem: Social Optimality

Definition (Social optimality)

A pattern of comsumption, c1, c2 is socially optimal if it maximizes the sum ofthe two players’ utility, that is, if it solves the following problem:

Maxc1,c2 = log(c1) + log(c2) + 2 logy − (c1 + c2)

2

Socially optimal solution:c1 = c2 =

y4




The Commons Problem: For a Large Population

Nash equilibrium:c1 = c2 = · · · =

yN + 1

Socially optimal solution:

c1 = c2 = · · · =y

2N

Key Question

How can we balance the private desire for utility or profits against the socialimperative of sustainable resource use?




Summary

1 Nash Equilibrium

2 Relation between IEDS and Nash equilibrium

3 Application: Cournot Duopoly

4 Application: The Commons Problem




References

1 Dutta, P.K. (1999). Strategies and Games: Theory and Practice.Cambridge, MA, The MIT Press. Chapters 5 and 6.

2 Keeney, R. L. and H. Raiffa (1993). Decisions with Multiple Objectives:Preferences and Value Tradeoffs. Cambridge, UK, Cambridge UniversityPress.

3 Clemen, R. T. (1996). Making Hard Decisions: An Introduction toDecision Analysis. Belmont, CA, Wadsworth Publishing Company.

4 Simon, H.A., A Behavioral Model of Rational Choice, Vol. 69, No. 1, pp.99-118.

5 Stanovich, K.E. (1999) Who is Rational? Studies of IndividualDifferences in Reasoning, Psychology Press, Mahwah, NJ.


THANK YOU!


Lecture 12 Nash Equilibrium - Purdue University...Felix / Oscar 3 hours 6 hours 9 hours 3 hours 13;...

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