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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

- 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

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

- 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

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

- Static analysis
- An extensive-form game can be represented in strategic form
- Infinite action or finite action games
- Infinite and finite games

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 GamesExample 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

- 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)

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 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

- 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

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

- 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

Example

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

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?

- 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

- 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

- 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

- 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 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

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?

- Deduction
- Evolutionary selection
- Learning

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

- 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.

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

- 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

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

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

- 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

- 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

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 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 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

- 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 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 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 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 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

- 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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