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
ME 597: Fall 2019 Lecture 20 1 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
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.
ME 597: Fall 2019 Lecture 20 2 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
Two-Player Game - Example 1
Prisoner’s Dilemma
1 / 2 Cooperate DefectCooperate −1,−1 −3, 0
Defect 0,−3 −2,−2
ME 597: Fall 2019 Lecture 20 3 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
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.
ME 597: Fall 2019 Lecture 20 4 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
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)
ME 597: Fall 2019 Lecture 20 5 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
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.
ME 597: Fall 2019 Lecture 20 6 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
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
ME 597: Fall 2019 Lecture 20 7 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
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
ME 597: Fall 2019 Lecture 20 8 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
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
ME 597: Fall 2019 Lecture 20 9 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
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!
ME 597: Fall 2019 Lecture 20 10 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
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
ME 597: Fall 2019 Lecture 20 11 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
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.
ME 597: Fall 2019 Lecture 20 12 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
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
ME 597: Fall 2019 Lecture 20 13 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
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
ME 597: Fall 2019 Lecture 20 14 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
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
ME 597: Fall 2019 Lecture 20 15 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
How much should each firm produce?
1 Make a conjecture about other firm’s production.2 Determine the quantity to produce.
ME 597: Fall 2019 Lecture 20 16 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
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
ME 597: Fall 2019 Lecture 20 17 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
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
ME 597: Fall 2019 Lecture 20 18 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
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
ME 597: Fall 2019 Lecture 20 19 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
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
ME 597: Fall 2019 Lecture 20 20 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
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.
ME 597: Fall 2019 Lecture 20 21 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
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
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
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
ME 597: Fall 2019 Lecture 20 23 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
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?
ME 597: Fall 2019 Lecture 20 24 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
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
ME 597: Fall 2019 Lecture 20 25 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
Contestant’s Payoff
Contestant i ’s expected payoff:
E(πi ) = Πi Pi − Ci
Prize amount Winning probability Cost
ME 597: Fall 2019 Lecture 20 26 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
1. Prize Amount
Contestant i ’s expected payoff:
E(πi ) = Πi Pi − Ci
Fixed Prize ContestsΠi = Π
AuctionsΠi = bi
bi is the bidding price
ME 597: Fall 2019 Lecture 20 27 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
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)
ME 597: Fall 2019 Lecture 20 28 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
3. Cost
The cost of a solution depends on the quality desired. Ci = Ci (qi )
Contestant i ’s expected payoff:
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
ME 597: Fall 2019 Lecture 20 29 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
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
ME 597: Fall 2019 Lecture 20 30 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
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
ME 597: Fall 2019 Lecture 20 31 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
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
ME 597: Fall 2019 Lecture 20 32 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
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)
ME 597: Fall 2019 Lecture 20 33 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
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)
ME 597: Fall 2019 Lecture 20 34 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
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
ME 597: Fall 2019 Lecture 20 35 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
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
ME 597: Fall 2019 Lecture 20 36 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
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?
ME 597: Fall 2019 Lecture 20 37 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
Summary
1 Nash Equilibrium
2 Relation between IEDS and Nash equilibrium
3 Examples1. Cournot Duopoly2. Design Crowdsourcing3. The Commons Problem
ME 597: Fall 2019 Lecture 20 38 / 39
Nash EquilibriumRelation between IEDS and Nash equilibrium
Examples
1. Cournot Duopoly2. Design Crowdsourcing3. 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 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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