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  1. Applied Microeconomics Game Theory I: Strategic-Form Games

  2. Outline • Strategic-form games • Dominance and dominance solvable games • Common knowledge • Nash equilibrium • Mixed strategies • Mixed Nash Equilibrium

  3. Readings • Kreps: Chapter 21 • Perloff: Chapter 13 • Zandt: Chapter 9

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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)

  13. 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

  14. 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)

  15. 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

  16. 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?

  17. 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

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

  19. 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

  20. 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

  21. 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

  22. 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)

  23. 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

  24. 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

  25. Examples

  26. 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

  27. What Assumptions Would Result in NE? • Deduction • Evolutionary selection • Learning

  28. 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

  29. 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.

  30. 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

  31. 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

  32. 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

  33. 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))

  34. 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

  35. 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

  36. 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

  37. PA 1 5/8 B1(Pa) 5/7 1 Pa 0 Example: Finite Game

  38. 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

  39. PA 1 B2(PA) 5/8 B1(Pa) 5/7 1 Pa 0 Example: Finite Game

  40. 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)

  41. 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)

  42. 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

  43. xBeta BAlpha(xBeta) 1-c Nash Equilibrium (1-c)/2 Bbeta(xAlpha) (1-c)/2 1-c xAlpha Example: Quantity Competition

  44. 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

  45. 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

  46. 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

  47. 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