Lecture 20 Nash EquilibriumLecture Outline 1 Nash Equilibrium 2 Relation between IEDS and Nash equilibrium 3 Examples 1. Cournot Duopoly 2. Design Crowdsourcing 3. The Commons Problem

Nash EquilibriumRelation between IEDS and Nash equilibrium

Examples

Lecture 20Nash 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 31, 2019

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http://engineering.purdue.edu/delp


Examples

Lecture Outline

1 Nash Equilibrium

2 Relation between IEDS and Nash equilibrium

3 Examples1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem

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

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Examples

Two-Player Game - Example 1

Prisoner’s Dilemma

1 / 2 Cooperate DefectCooperate −1,−1 −3, 0

Defect 0,−3 −2,−2

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Examples

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.

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Examples

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)

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Examples

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.

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Examples

Two-Player Game: Example 2

Battle of Sexes

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

Opera (O) 0, 0 1, 3

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Examples

Two-Player Game: Example 3

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

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Examples

Nash Equilibrium: Example 4

The Odd Couple: Felix and Oscar share an apartment. It takes 12 hours ofwork per week to make the apartment spotlessly clean, 9 hours to be livable,and anything less leaves the apartment in a state that would not standinspection by the local rodent police. Felix and Oscar each get a (gross)payoff of 2 from a livable apartment, but Felix assigns a payoff of 10 to aspotless apartment whereas Oscar gets a payoff of only 5. A filthy apartmentis worth -10 to Felix but only -5 to Oscar. Each person’s net payoff equals hisrespective gross payoffs minus his respective hours worked cleaning.

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

https://www.chegg.com/homework-help/questions-and-answers/consider-odd-couple-game-felix-oscar-share-apartment-state-cleanliness-public-good-takes-1-q1984393

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Examples

Nash Equilibrium: Example 5

Coordination game

1 / 2 Party HomeParty 2, 2 0, 0

Home 0, 0 1, 1

Note that one Nash Equilibrium is also Pareto Optimal!

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Examples

Motivation for Nash Equilibrium

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

1 Preplay communication2 Rational introspection3 Trial and error

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Examples

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.

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Examples

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

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Examples

1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem

Application 2: 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

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Examples


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

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Examples


How much should each firm produce?

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

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Examples


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

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Examples


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

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Examples


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

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Examples


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

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Examples


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.

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Examples


Example Application: Design Crowdsourcing

Crowdsourcing

The practice of outsourcing tasks, traditionally performed by employees orsuppliers, to a large group of people in the form of open tournaments.

School of Mechanical Engineering ! Purdue University 3

2000+ platforms!!!

www.crowdsourcing.org/directoryME 597: Fall 2019 Lecture 20 22 / 39


Examples


Crowdsoucing in Engineering Design

Design related tasksidea generation

problem solving

classification

evaluation of designs

Challengesquality control

spamming

filtering low-quality / irrelevantsolutions

individuals may not participate!

Airplane Bearing Bracket Challenge

Airbus Cargo Drone Challenge

Handrail Clamp Assembly Challenge

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Examples


Designing Crowdsourcing Initiatives: The Sponsor’s View

Design Alternatives:single stage vs. multistagetournament

open entry vs. restricted entry vs.entry fee

single competition vs. multiplesmaller competitions

Fixed prize vs. auction-styletournaments

Outcomes:solution quality

number of contributors

amount of effort

overall cost of running the contest

probability of getting a goodsolution

cost of filtering good solutions

Research Question

How does the design of a crowdsourcing tournament affect its outcomes inengineering design?

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Examples


Analysis Framework

… …

Contest Designer

Contestants

Objective: Maximize payoff (πD)

π1 π2 π3 πnπi

Objective: Maximize payoff

Example contest:• Design requirements

• Max. strength• Min. weight• Min. cost

• Prizes• $1500 first prize• $1000 second prize

• Deadline• Two weeks

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Contestant’s Payoff

Contestant i ’s expected payoff:

E(πi ) = Πi Pi − Ci

Prize amount Winning probability Cost

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Examples


1. Prize Amount


E(πi ) = Πi Pi − Ci

Fixed Prize ContestsΠi = Π

AuctionsΠi = bi

bi is the bidding price

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2. Winning Probability

Contestant i ’s winning probability (Pi ) depends on:Quality of i ’s submission (qi )Probability distribution on quality of other contestants, F (q−i )

Contestant’s expected payoff:

E(πi ) = Πi Pi (qi , q−i ) − Ci

AssumedContest Success Functions (CSFs)

Pi =

f (qi )

n∑j=1

f (qj )

ifn∑

j=1

f (qj ) > 0

1n

otherwise

Derived from F (q−i )

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

q

F(q

−i)

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3. Cost

The cost of a solution depends on the quality desired. Ci = Ci (qi )


E(πi ) = ΠiPi (qi , q−i )− Ci (qi )

Linear

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

qi

Ci

Diminishing Returns

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

qi

Ci

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Examples


Specific Instantiation (1): Fixed Prize Contests

Contestant’s expected payoff:

E(πi ) = Πi Pi (qi , q−i ) − Ci (qi )

Prize amountFixed Prize: Π

Winning probability

Pi =qm

in∑

j=1qm

j

Costqi = αei

Ci = cei =( cα

)qi

Expected Payoff for Two Players:

E(πi ) = ΠPi − Ci

= Π

(em

i

emi + em

−i

)− cei

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Examples


Specific Instantiation (1): Solution of Two-player Scenario

Strategy: Invest effort ei thatmaximizes E(πi )

maxei

E(πi )

Solution:Rational reaction (RRSi ):

em−1i em

−i

(emi + em

−i )2 −

cΠm

= 0

Unique Nash equilibrium:

ei = e−i =Πm4c

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

e1

e 2

RRS1

RRS2

Nash eqm.

Rational Reaction Sets form = 1,

Π

c= 1

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Examples


Fixed Prize Contest vs. Auctions

Fixed Prize Contest

Assume:

Πi = Π; Pi =

∫ qi

−∞F (q)dq; Ci = q2

i

Solution: Optimal Quality andPrize

qopt = E [Pi (qi )] = E[q2

i

Π

]Πopt = arg max

Π(qopt − Π)F (qopt )

Auction

Assume:

Πi = bi ; Pi =

∫ qi

−∞F (q)dq; Ci = q2

i

Solution: Optimal Quality andPrize

qopt = arg minq

(Ci

qi − πD

)bopt = Ci/C′i

Insight:

For the same quality, the prize in fixed prize contest is greater than the prizein auctionsa

Πopt > bopt

aChe and Gale 2003

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Examples


Insights from Analytical Models

Auction style tournaments reduce the sponsor’s expenditure.

Free and open entry is not optimal. Optimum number of contestants istwo.

Optimal strategy: allocate the entire prize to a single winner.

Bidding after quality revelation: fixed prize may cost lower.

Note

These models assume a single period innovation process. However,engineering design can involve sequential information acquisition.

(Fullerton et al. 2002)(Taylor 1995)(Fullerton and McAfee 1999)(Moldovanu and Sela 2001)(Schottner 2008)

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Application 3: The Commons Problem

Assumptions:

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)

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Examples


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

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Examples


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

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Examples


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?

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Examples


Summary

1 Nash Equilibrium

2 Relation between IEDS and Nash equilibrium

3 Examples1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem

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Examples


References

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

2 Panchal J. H. (2015). “Using Crowds in Engineering Design – Towards aHolistic Framework”, International Conference on Engineering Design(ICED 2015), Milan, Italy, July 27-30, 2015.

3 Chaudhari, A.M., Thekinen, J., Panchal, J.H. (2016). “Using Contests forEngineering Systems Design: A Study Auctions and Fixed PrizeTournaments,” DESIGN 2016, Cavtat, Croatia, May 16-19, 2016, pp.946-956.

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Documents

Lecture 20 Nash EquilibriumLecture Outline 1 Nash Equilibrium 2 Relation between IEDS and Nash equilibrium 3 Examples 1. Cournot Duopoly 2. Design Crowdsourcing 3. The Commons Problem