brief introduction to game theory - umass · 2020-04-24 · introduction game theory and games...

Brief Introduction to Game TheoryEcon 700 Tutorial

Weikai Chen

University of Massachusetts, Amherst

Sep 11, 2019

Introduction Game Theory and Games Solution Concepts Computing Equilibrium Summary

Map for Chapter 1

Weikai Chen (Umass Amherst) Brief Introduction to Game Theory Sep 11, 2019 2 / 27

Map for Chapter 1

Introduction

1 Introduction

2 Game Theory and GamesGame Theory: Definition and HistoryNormal Form Games

3 Solution ConceptsDominanceNash Equilibrium

4 Problem-Solving: How to find the Nash Equilibrium

5 Summary

Game Theory

What is it?Game Theory models situation in which multiple agents makestrategically interdependent decision.

classical & evolutionarynoncooperative & cooperativenormal form & extensive form (strategic & dynamic)

Game Theory

History of (Noncooperative) Game Theory

1 Cournot (1838), Bertrand (1883)2 John von Neumann & Oskar Morgenstern (1944)3 John Nash (1950)4 Selten (1965)5 Harsanyi (1967-68)

Normal Form Games

The game in normal (or strategic) form

Γ = (N, (Si)i∈N, (ui)i∈N)

has three elements:Players i ∈ N= {1, . . . ,N}. Denote all players other thanplayer i by −iPure-strategy space for each player i, Si.Payoff functions ui(s) : S→ R where s ∈ S = S1 × · · · × Sn

Normal Form Games

Finite Two-Player Games

We focus our attention on finite games, that is, games whereS is finite.Strategic forms for finite two-player games are often depictedas matrices.

Example (Prison Dilemma)

Player 2C D

Player 1C 1, 1 −1, 2D 2,−1 0,0

Then N= {1, 2}, S1 = S2 = {C,D}, and for (C,D) ∈ S1 × S2, we haveu1(C,D) = −1.

Finite Two-Player Games

We focus our attention on finite games, that is, games whereS is finite.Strategic forms for finite two-player games are often depictedas matrices.

Player 2C D

Player 1C 1, 1 −1, 2D 2,−1 0,0

Then N= {1, 2}, S1 = S2 = {C,D}, and for (C,D) ∈ S1 × S2, we haveu1(C,D) = −1.

Mixed Strategy: Definition

A mixed strategy σi is a probability distribution over purestrategies. Denote the set of mixed strategies by Σi and

Σ = Σ1 × · · · × ΣN

with element σ .The support of a mixed strategy σi is the set of pure strategiesto which σi assigns positive probability.Player i’s payoff to profile σ is∑

s∈S(ΠN

j=1σj(sj))ui(s)

Note that the set of mixed strategies contains the purestrategies.

Σ = Σ1 × · · · × ΣN

s∈S(ΠN

j=1σj(sj))ui(s)

Σ = Σ1 × · · · × ΣN

s∈S(ΠN

j=1σj(sj))ui(s)

Σ = Σ1 × · · · × ΣN

s∈S(ΠN

j=1σj(sj))ui(s)

Mixed Strategy: Example

Player 2C D

Player 1C 1, 1 −1, 2D 2,−1 0,0

Then N= {1, 2}, S1 = S2 = {C,D}, and suppose

σ1 = (12,12),σ2 = (1,0)

Thenu1(σ ) =

Player 2C D

Player 1C 1, 1 −1, 2D 2,−1 0,0

σ1 = (12,12),σ2 = (1,0)

Thenu1(σ ) =

Player 2C D

Player 1C 1, 1 −1, 2D 2,−1 0,0

σ1 = (12,12),σ2 = (1,0)

Thenu1(σ ) =

12(1 · 1 + 0 · (−1)) +

12(1 · 2 + 0 · 0) =

Solution Concepts

Solution concepts are the rules for predicting how a game willbe play.

DominanceNash equilibrium and its refinementsMaximin solutionRationalizability

Different assumptions on rationality and information are usedwith different solution concepts.Which one is better depends on the context.

Solution Concepts

Dominated Strategy

We would like to vary the strategy of a single player i whileholding the strategies of others fixed.

Let s−i ∈ S−i denote a strategy selection for all players but i,and write (s′i , s−i) for the profile

(s1, . . . , si−1, s′i , si+1, . . . , sN)

For the mixed strategies, we let

(σ ′i ,σ−i) = (σ1, . . . ,σi−1,σ′i ,σi+1, . . . ,σN)

Dominated Strategy

Pure strategy si is strictly dominated for player i if there existsσ ′i ∈ Σi such that

ui(σ ′i , s−i) > ui(si, s−i), ∀s−i ∈ S−i

It is weakly dominated if the inequality holds with weakinequality, and the inequality is strictly for at least one s−i.

Prison Dilemma: continued

No matter how player 2 plays, D gives player 1 a strictly higherpayoff than C does.

Player 2C D

Player 1C 1, 1 −1, 2D 2,−1 0,0

Then strategy C is said to be strictly dominated.

Dominated Strategy: Example

Example

Player 2L R

Player 1U 2,−1 −1,0M 0,0 0,0D −1,0 2,0

Strategy M strictly dominated and L is weakly dominated, why?

Dominated Strategy: Example

Example

Player 2L R

Player 1U 2,−1 −1,0M 0,0 0,0D −1,0 2,0

Strategy M strictly dominated and L is weakly dominated, why?

Nash Equilibrium

A mixed-strategy profile σ ∗ is a Nash equilibrium if, for all playersi,

ui(σ ∗i ,σ∗−i) ≥ ui(si,σ

∗−i) ∀si ∈ Si

A pure-strategy Nash equilibrium is a pure-strategy profile thatsatisfies the same conditions.If a player uses a mixed strategy, she must be indifferentbetween all pure strategies in the support of mixed strategy (towhich she assigns positive probability). (Why?)

Nash Equilibrium

∗−i) ∀si ∈ Si

Nash Equilibrium

∗−i) ∀si ∈ Si

Strict Nash Equilibrium

A pure-strategy profile s∗ is a strict Nash equilibrium if, for allplayers i,

ui(s∗i , s∗−i) > ui(si, s

∗−i) for all si , s∗i

Nash equilibria are “consistent" predictions of how the game willbe played, in the sense that if all players predict that a particular NEwill occur then no player has an incentive to play di�erently.

Strict Nash Equilibrium

A pure-strategy profile s∗ is a strict Nash equilibrium if, for allplayers i,

ui(s∗i , s∗−i) > ui(si, s

∗−i) for all si , s∗i

Nash equilibria are “consistent" predictions of how the game willbe played, in the sense that if all players predict that a particular NEwill occur then no player has an incentive to play di�erently.

Best Response Correspondence/Function

Player i’s best response correspondence, bi maps each strategyprofile σ to the set of mixed strategies that maximize player i’spayoff when her opponents play σ−i. That is

bi(σ ) = arg maxσ ′i ∈Σi

ui(σ ′i ,σ−i)

The profile σ ∗ such that σ ∗i ∈ bi(σ∗) for all player i is a Nash

equilibrium.Define the correspondence b : Σ→ Σ to be the Cartesianproduct of the bi. A fixed point of b is a σ ∗ such thatσ ∗ ∈ b(σ ∗). Thus a NE is a fixed point of b.Therefore, every finite strategy-form game has a mixed-strategyequilibrium, guaranteed by the so-called Kakutani’s fixed pointtheorem.

ui(σ ′i ,σ−i)

Best Response Correspondence: Example

Example (Matching Pennies)

Player 2H T

Player 1H 1,−1 −1, 1T −1, 1 1,−1

Example of the nonexistence of a pure-strategy equilibrium.The Monitor and Work Game (Problem 3 on p.503) has thesame structure.

Player 2H T

Player 1H 1,−1 −1, 1T −1, 1 1,−1

Suppose σ1 = (q, 1 − q),σ2 = (p, 1 − p), then

u1(H,σ2) = p · 1 + (1 − p) · (−1) = 2p − 1u1(T,σ2) = p · (−1) + (1 − p) · 1 = 1 − 2p

Player 2H T

Player 1H 1,−1 −1, 1T −1, 1 1,−1

Suppose σ1 = (q, 1 − q),σ2 = (p, 1 − p), then

u1(H,σ2) = p · 1 + (1 − p) · (−1) = 2p − 1u1(T,σ2) = p · (−1) + (1 − p) · 1 = 1 − 2p

Therefore, 1’s best response correspondence is

q = b1(σ2) =

1, p > 1

2[0, 1] p = 1

20, p < 1

Figure: Best response correspondences in the Matching Pennies Game

The Best Response Corres-pondence

q = b1(σ2) =

1, p > 1

2[0, 1] p = 1

20, p < 1

q = b1(p)

p = b2(q)

q = b1(σ2) =

1, p > 1

2[0, 1] p = 1

20, p < 1

q = b1(p)

p = b2(q)

q = b1(σ2) =

1, p > 1

2[0, 1] p = 1

20, p < 1

q = b1(p)

p = b2(q)

q = b1(σ2) =

1, p > 1

2[0, 1] p = 1

20, p < 1

q = b1(p)

p = b2(q)

Exercise

Find all the Nash equilibria.

Planting in Palanpur

Player 2Early Late

Player 1Early 4, 4 0, 3Late 3,0 2, 2

What would be the Maximin Solution?

Exercise

Find all the Nash equilibria.

Planting in Palanpur

Player 2Early Late

Player 1Early 4, 4 0, 3Late 3,0 2, 2

What would be the Maximin Solution?

Application: the household public goods problem

The action, ai, is a contribution of the public good, e.g.cleaning the house.Individual utility function ui = f ln(

∑i ai) − gai

FOC:∂ui∂ai=

f∑i ai− g = 0

Best Response:

aBRi =fg−∑j,i

Two individuals in a couple, i = 1, 2

aBR1 =fg− a2

aBR2 =fg− a1

∑i ai) − gai

FOC:∂ui∂ai=

f∑i ai− g = 0

Best Response:

aBRi =fg−∑j,i

aBR1 =fg− a2

aBR2 =fg− a1

∑i ai) − gai

FOC:∂ui∂ai=

f∑i ai− g = 0

Best Response:

aBRi =fg−∑j,i

aBR1 =fg− a2

aBR2 =fg− a1

∑i ai) − gai

FOC:∂ui∂ai=

f∑i ai− g = 0

Best Response:

aBRi =fg−∑j,i

aBR1 =fg− a2

aBR2 =fg− a1

The Nash Equilibria?

aBR2 =fg − a1

aBR1 =fg − a2

a1 + a2 = fg

Figure: Best response function in the Household Public Good Problem

aBR2 =fg − a1

aBR1 =fg − a2

a1 + a2 = fg

aBR2 =fg − a1

aBR1 =fg − a2

a1 + a2 = fg

aBR2 =fg − a1

aBR1 =fg − a2

a1 + a2 = fg

aBR2 =fg − a1

aBR1 =fg − a2

a1 + a2 = fg

Allocation and Distribution: First Mover Advantage inthe Household Public Good Problem

Suppose Player 1 is the first mover, what would be the outcome?

aBR2 =fg − a1

aBR1 =fg − a2

Figure: The first mover advantage in the Household Public GoodProblem

aBR2 =fg − a1

aBR1 =fg − a2

aBR2 =fg − a1

aBR1 =fg − a2

Summary

Strategic (or normal) Game: Γ = (N, (Si)i∈N, (ui)i∈N)Dominant Strategy Equilibrium and Nash Equilibrium

Find pure-strategic NE using payoff matrix

Find mixed NE in finite gameFind NE using best-response function with continuousstrategic space

Summary

Textbooks Recommendation

Classical Game Theory:Maschler, M., Zamir, S., & Solan, E. 2013. Game Theory.Cambridge University Press.Osborne, Martin J., and Ariel. Rubinstein. 1994. A Course inGame Theory. MIT Press.Myerson, Roger B. 1991. Game Theory : Analysis of Conflict.Harvard University Press.Fudenberg, Drew., and Jean. Tirole. 1991. Game Theory.MIT Press.

Textbooks Recommendation

Evolutionary Game Theory:Sandholm, William H. 2010. Population Games andEvolutionary Dynamics. MIT Press.Maynard Smith, John. 1982. Evolution and the Theory ofGames. Cambridge University Press.Weibull, Jorgen W. 1995. Evolutionary Game Theory. MITPress.

brief introduction to game theory - umass · 2020-04-24 · introduction game theory and games...

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