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Applied Microeconomics. Game Theory I: Strategic-Form Games. Outline. Strategic-form games Dominance and dominance solvable games Common knowledge Nash equilibrium Mixed strategies Mixed Nash Equilibrium . Readings. Kreps: Chapter 21 Perloff: Chapter 13 Zandt: Chapter 9. Introduction.

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Applied Microeconomics

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Applied microeconomics

Applied Microeconomics

Game Theory I: Strategic-Form Games


Outline

Outline

  • Strategic-form games

  • Dominance and dominance solvable games

  • Common knowledge

  • Nash equilibrium

  • Mixed strategies

  • Mixed Nash Equilibrium


Readings

Readings

  • Kreps: Chapter 21

  • Perloff: Chapter 13

  • Zandt: Chapter 9


Introduction

Introduction

  • So far we assumed that the firm maximizes profit assuming its competitors remain passive

  • This is a reasonable assumption if the firm is one of many small firms as in a competitive market or if the firm is a monopoly producing of a good with no close substitutes


Introduction1

Introduction

  • However, in an oligopoly, a market with a few firms that have some market power, we need to relax this assumption and allow for strategic interaction

  • To model this situation we need a new tool called non-cooperative game theory


Example

Example

  • Alpha has a local monopoly in the market for a good and is making a profit of 2 million Euros

  • Beta is considering entering the market

  • If Beta enters the market, Alpha could either fight by setting a low price or accommodate, by setting a high price:

    • If Alpha fights, then both firms get a profit of –1 million Euros

    • If Alpha accommodates, then both firms get a profit of 1 million Euros


Example1

Beta

Not

Enter

Alpha

0,2

Fight

Acc.

1,1

-1,-1

Example

  • We could model this strategic interaction in the following ways:

Extensive-form game

Strategic-form game

Beta

Alpha


Game theory

Game Theory

  • Game theory:

    • Cooperative

    • Non-cooperative

  • Non-cooperative game theory:

    • Sequential/extensive-form games

    • Strategic-form/normal-form games

  • Strategic-form games:

    • Infinite action

    • Finite action


Strategic form games

Strategic-Form Games

  • Static analysis

  • An extensive-form game can be represented in strategic form

  • Infinite action or finite action games

  • Infinite and finite games


Examples of finite action games

Prisoner’s Dilemma

Cooperate

Fink

Cooperate

1,1

-1,2

Fink

2,-1

0,0

Cooperation

Movie

Theater

Movie

2,2

-5,0

Theater

0,-5

1,1

Examples of Finite Action Games


Example of an infinite action game

Example of an Infinite Action Game

  • Three firms, Alpha, Beta, and Gamma, simultaneously and independently decides an amount xi to spend on advertisement, where xi is non-negative

  • Total sales in the market is 1 and each firm sells in proportion to its share of total ad expenditure

  • Alpha’s profit is xAlpha/(xAlpha+ xBeta+ xGamma)-xAlpha and Beta’s is xBeta/(xAlpha+ xBeta+ xGamma)-xBeta, and Gamma’s is xGamma/(xAlpha+ xBeta+ xGamma)-xGamma


Concepts

Concepts

  • A strategy si is a complete contingent plan for a player in the game

    • Example 1: si=fink

    • Example 2: si=2/9

  • A strategy profile s is a vector of strategies, one for each player

    • Example 1: s=(fink,fink)

    • Example 2: s=(xAlpha=2/9, xBeta=2/9, xGamma=2/9)


Concepts1

Concepts

  • A payoff function ui(si,s-i) gives the payoff or utility to player i for any of her strategies si and any of her opponents’ strategies s-i

    • Example 1: U1(fink,fink)=0, U1(fink,cooperate)=2, U1(cooperate,fink)=-1,U1(cooperate,cooperate)=1

    • Example 2: UAlpha(xAlpha, xBeta, xGamma)= xAlpha/(xAlpha+ xBeta+ xGamma)-xAlpha

  • Payoffs are ordinal (we could f.i. multiply all of them by a positive constant without changing the game), and the players are generally assumed to be expected utility maximizers


Strategic form games1

Strategic Form Games

  • A strategic-form game is:

    • A set of players: N={1,2,…,n}

    • A set of strategies for each player i: Si={si1,…,siJ}

    • A payoff function for each player i: ui(si,s-i)


Predicting play dominance

Predicting Play: Dominance

  • A strategy si is dominated for player i if there is some other strategy s’i that gives a weakly better payoff no matter what strategies the other players use, and strictly better payoff for some s-i: si is dominated if there is s’i≠ si such that ui(si,s-i)≤ui(s’i,s-i) for any s-i and ui(si,s-i)<ui(s’i,s-i) for some s-i

  • A strategy si is strictly dominated for player i if there is some other strategy s’i that gives a strictly better payoff no matter what strategies the other players use: si is strictly dominated if there is s’i≠ si such that ui(si,s-i)<ui(s’i,s-i) for any s-i


Example2

Example

  • If we assume that no player would ever play a strictly dominated strategy, we can sometimes make predict the strategies that will be played

  • Is any strategy strictly dominated in any of these games?


Iterated elimination of dominated of strategies

Iterated Elimination of Dominated of Strategies

  • Eliminating strictly dominated strategy rarely gives precise predictions, but the procedure can be extended

  • Suppose we delete all the strictly dominated strategies from the strategy set S1 of the game G1

  • Once we have done this we get a new game G2 with strategy set S2 from which we once again can delete the strictly dominated strategies

  • We can keep on deleting strategies in this way until we reach a game Gt with strategy set St such that no more strategies can be deleted


Example3

Example

  • Find the set of strategies that survives iterated elimination of strictly dominated strategies in the following game


Iterated elimination of dominated of strategies1

Iterated Elimination of Dominated of Strategies

  • This procedure is called iterated elimination of strictly dominated strategies

  • In finite games (and most infinite games), the set of strategies that survives iterated elimination of strictly dominated strategies does not depend on the order in which we eliminate strategies!

  • If only one strategy for each player survives the procedure, then the game is said to be dominance solvable


What assumptions would result in iesds

What Assumptions Would Result in IESDS?

  • Deduction: If all players know their payoff function and never play a strictly dominated strategy, know that their opponent knows his own payoff function and never plays a strictly dominated strategy, know that their opponent knows that they know their payoff function and never play a strictly dominated strategy etc.

  • Evolutionary selection: Alternatively, we can assume that players recurrently are drawn from a large population of individuals to play the game, and that individuals that perform worse disappear or change strategy


Experimental evidence

Experimental Evidence

  • Few subjects play dominated strategies

  • More subjects play iteratively dominated strategies

  • The more iterations are needed to solve game, the worse predictive power of IESDS


Iterated elimination of weakly dominated strategies

Iterated Elimination of Weakly Dominated Strategies

  • The set of strategies that survive iterated elimination of weakly dominated strategies in finite action games may depend on the order of elimination (compare eliminating b and B)


Nash equilibrium

Nash Equilibrium

  • Often many strategies iterative elimination of strictly dominated strategies

  • We want to be able to make predictions also in this case

  • The most famous concept for doing this is the Nash Equilibrium


Nash equilibrium1

Nash Equilibrium

  • A Nash equilibrium is a strategy profile such that no player could gain by deviating unilaterally and play a different strategy

  • Formally, s* is a NE if for all i in N, and all si in Si, ui(si,s-i*)≤ui(si*,s-i*)

  • If the inequality is strict, then s* is said to be a strict NE


Examples

Examples


Finding nash equilibria

Finding Nash Equilibria

  • In order to find the Nash equilibria of a game it is useful to find each player’s best reply to any strategy played by the opponent(s) – the best reply correspondence

  • A NE is a strategyprofile such that allplayers are playinga best reply


What assumptions would result in ne

What Assumptions Would Result in NE?

  • Deduction

  • Evolutionary selection

  • Learning


Nash equilibrium and iterated elimination

Nash Equilibrium and Iterated Elimination

  • A strategy that is eliminated by iterated elimination of strictly dominated strategies cannot be part of a NE

  • If the game is dominance solvable, the surviving strategy profile is the unique NE of the game

  • If the game is dominance solvable using iterated elimination of weakly dominated strategies, the resulting strategy profile is a NE, but there may be more NE


Example without nash equilibrium in pure strategies

Example Without Nash Equilibrium in Pure Strategies

  • Sometimes, there does not exist a NE is pure strategies

  • To deal with this we define mixed strategies pi as probability distributions over the set of pure strategies in Si

  • Motivation:

    • Each player is randomizing

    • Game played against a randomly drawn individual from a large population where a share pi1 play strategy si1, a share pi2 play strategy si2 etc.


Example4

Example

  • Suppose two players are playing the following game

  • A mixed strategyfor player onecould be p1=(0.3,0.7); to play A with prob. 0.3 and B with prob. 0.7


Nash equilibrium in mixed strategies

Nash Equilibrium in Mixed Strategies

  • Let ∆Si set of all mixed strategies over Si, and let p be a mixed strategy profile, p=(p1,p2,…,p3)

  • Formally, p* is a NE if for all i in N, and all pi in ∆Si, ui(pi,p-i*)≤ui(pi*,p-i*)

  • Nash (1950):Any finite game has at least one Nash equilibrium in mixed strategies

  • Not that pure strategies are just mixed strategies that play a particular pure strategy with probability one


Example5

Example

  • Find a NE in the above game

  • If there is a NE such that player 1 is randomizing when 2 is playing a with prob. p and b with prob. 1-p, then player 1 must get the same expected payoff from both of his strategies:

    • U(A)=-2p+1(1-p)=1-3p

    • U(B)=1p-1(1-p) =2p-1

  • Hence, 1-3p=2p-1 or p=2/5=0.4


Example6

Example

  • Likewise, if there is a NE such that player 2 is randomizing when player 1 is playing A with prob. q and B with prob.1-q, then player 2 must get the same expected payoff from both of his strategies:

    • U(a)=1q-1(1-q)=2q-1

    • U(b)=-1q+1(1-q)=1-2q

  • Hence, 2q-1=1-2q or q=2/4=0.5

  • The NE is thus given by ((qA=0.5,qB=0.5),((pA=0.4,pB=0.6))


The best reply correspondence

The Best-Reply Correspondence

  • Let G be a strategic-form game with players i=1,…,n, strategy sets S1,…,Sn, and payoff functions u1,…,un

  • For each player, define player i’s best-reply correspondence Bi(s-i) that maps any strategy choice s-i by the opponent(s) to the most profitable strategy for player i

  • Mathematically: Bi(s-i)={si:ui(si,s-i)≥ui(si’,s-i)} for all si’ in Si


Nash equilibrium and the best reply correspondence

Nash Equilibrium and the Best Reply Correspondence

  • We can define a NE using the best-reply correspondence: it is a strategy profile s* such that all players are playing a best reply

  • Mathematically: s* is an NE if si*∈Bi(s-i*) for i=1,…,n

  • Hence, the NE can be found using the best-reply correspondences of all players


Example finite game

a

b

A

2,3

-5,0

B

0,-5

0,0

Example: Finite Game

  • Suppose the column player plays a with prob. Pa

  • The row player’s expected utility of A is then U(A)=2Pa-5(1- Pa)=-5+7Pa

  • The row player’s expected utility of B is then U(B)=0Pa+0(1- Pa)=0

  • Hence, A is a best reply for Pa≥5/7 and B for Pa≤5/7 and any randomization over the two for Pa=5/7


Example finite game1

PA

1

5/8

B1(Pa)

5/7

1

Pa

0

Example: Finite Game


Example finite game2

Example: Finite Game

  • Suppose the row player plays A with prob. PA

  • The column player’s expected utility of a is then U(a)=3PA-5(1- Pa)=-5+8PA

  • The row player’s expected utility of b is then U(b)=0PA+0(1- PA)=0

  • Hence, a is a best reply for PA≥5/8 and B for PA≤5/8 and any randomization over the two for PA=5/8


Example finite game3

PA

1

B2(PA)

5/8

B1(Pa)

5/7

1

Pa

0

Example: Finite Game


Example finite game4

Example: Finite Game

  • The NE are given by the points where B1(Pa)=PA and B2(PA)=Pa

  • Graphically this is where the two best-reply correspondences intersect

  • Hence, the NE of the game are (P*A=0,P*a=0), (P**A=1,P**a=1), and (P***A=5/8,P***a=5/7)


Example quantity competition

Example: Quantity Competition

  • Suppose two firms, i=Alpha, Beta, are competing in quantities and simultaneously and independently decide a non-negative amount xi to produce (Cournot competition)

  • Each firm has per unit cost c<1 and the market inverse demand function is given by P(xi,x-i)=1-(xi+x-i) for xi+x-i<1 and 0 otherwise

  • Payoffs are given by uAlpha(xAlpha,xBeta)= xAlpha(1-(xAlpha+xBeta)-c)uBeta(xBeta,xAlpha)= xBeta(1-(xAlpha+xBeta)-c)


Example quantity competition1

Example: Quantity Competition

  • To find the best-reply correspondence for firm i we solve: Maxxi≥0 xi(1-(xi+x-i)-c)with the first-order condition 1-(x-i+xi)-c-xi=0

  • Solving for xi gives Bi(x-i)=(1-c-x-i)/2for 1-c≥x-i and 0 for x-i>1-c

  • If we plot the best-reply function for both firms in a diagram, we get the following picture


Example quantity competition2

xBeta

BAlpha(xBeta)

1-c

Nash Equilibrium

(1-c)/2

Bbeta(xAlpha)

(1-c)/2

1-c

xAlpha

Example: Quantity Competition


Example quantity competition3

Example: Quantity Competition

  • The best-reply correspondences are down-ward sloping since each firm wants to produce less if the competitor is producing more: quantities are strategic substitutes

  • In order to find the NE we solve:xAlpha=(1-xBeta-c)/2xBeta =(1-xAlpha-c)/2

  • This gives the NE: xAlpha= xBeta=(1-c)/3


Example quantity competition4

Example: Quantity Competition

  • This game is symmetric sinceuAlpha(xAlpha,xBeta)=uBeta(xBeta,xAlpha), implying identical best-reply correspondences for both firms

  • Moreover, from the first-order condition we see that both firms must produce equal amounts in a NE since xi=x-i=1-(x-i+xi)-c

  • We can use this fact to solve for an NE in a convenient way: xi=1-2xi-c gives xi=(1-c)/3, implying a price of p(x)=(1+2c)/3

  • Compare this to the monopoly price of p=(1+c)/2 and the competitive price of p=c


Example quantity competition5

Example: Quantity Competition

  • Using this technique we can easily solve for the NE in a market with n identical firms of the same type

  • Denoting total quantity by x, we can write the first-order condition for firm i’s profit maximization: xi=1-x-c

  • Since all firms will produce equal amounts in a NE this means xi=1-nxi-c or xi=(1-c)/(n+1) for all i=1,…,n

  • This gives a price of p(x)=(1+nc)/(n+1), that converges to c as n goes to infinity


Conclusion

Conclusion

  • Game theory is a tool for modeling strategic interaction

  • A strategic-form game consists of the players, the players’ strategies, and their payoff functions

  • Ways of predicting play:

    • Iterated elimination of dominated strategies

    • Nash equilibrium, strict Nash equilibrium

    • Nash equilibrium in mixed strategies

  • The Nash equilibria can be calculated using the best-reply correspondences


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