1Lecture 5-6 Applications of Nash equilibrium Rationalizablity & Backwards Induction14.12 Game Theory Muhamet YildizRoad Map1. Cournot (quantity) Competition1. Nash Equilibrium in Cournot duopoly2. Nash Equilibrium in Cournot oligopoly3. Rationalizability in Cournot duopoly2. Bertrand (price) Competition3. Commons Problem4. Quiz5. Mixed-strategy Nash equilibrium6. Backwards induction2Cournot Oligopoly• N = {1,2,…,n} firms;• Simultaneously, each firm i produces qiunits of a good at marginal cost c,• and sells the good at priceP = max{0,1-Q}where Q = q1+…+qn.• Game = (S1,…,Sn; π1,…,πn) where Si= [0,∞∞),),11QPπi(q1,…,qn) = qi[1-(q1+…+qn)-c] if q1+…+qn < 1, -qicotherwise.Cournot Duopoly -- profit0 1-0.201-qj-c(1-qj-c)/2-cqiqi(1- qi-qj-c)Profitqj=0.2c=0.23C-D – best responses•qiB(qj) = max{(1-qj-c)/2,0};• Nash Equilibrium q*:q1* = (1-q2*-c)/2;q2* = (1-q1*-c)/2;•q1* = q2* = (1-c)/3q1q2q2=q2B(q1)q1=q1B(q2)q*21 c−1-cCournot Oligopoly --Equilibrium • q>1-c is strictly dominated, so q ≤ 1-c. • πi(q1,…,qn) = qi[1-(q1+…+qn)-c] for each i.•FOC:• That is, • Therefore, q1*=…=qn*=(1-c)/(n+1)..0)1()]1([),,(***111**=−−−−−=∂−−−−∂=∂∂==inqqiniqqiniqcqqqcqqqqqqLLKπcnqqqcqqqcqqqnnn−=+++−=+++−=+++11212**2*1**2*1**2*1LMLL4Cournot oligopoly – comparative statics11QPn = 1n = 2cn = 3n = 4Rationalizability in Cournot Duopolyq1q221 c−1-c21 c−1-cAssume that players are rational.5Players are rational:q1q221 c−1-c21 c−1-cAssume that players know this.Players are rational and know that players are rationalq1q221 c−1-c21 c−1-cAssume that players know this.6Players are rational; players know that players are rational; players know that players know that players are rationalq1q221 c−1-c21 c−1-cAssume that players know this.Rationalizability in Cournot duopoly• If i knows that qj≤ q, then qi≥ (1-c-q)/2.• If i knows that qj≥ q, then qi≤ (1-c-q)/2.• We know that qj≥ q0 = 0.• Then, qi≤ q1= (1-c-q0)/2 = (1-c)/2 for each i;• Then, qi≥ q2= (1-c-q1)/2 = (1-c)(1-1/2)/2 for each i;•…• Then, qn≤ qi≤ qn+1or qn+1≤ qi≤ qnwhere qn+1 = (1-c-qn)/2 = (1-c)(1-1/2+1/4-…+(-1/2)n)/2.•As n→∞,qn→ (1-c)/3.7Bertrand (price) competition• N = {1,2} firms.• Simultaneously, each firm i sets a price pi;• If pi <pj, firm i sells Q = max{1 – pi,0} unit at price pi; the other firm gets 0.• If p1 = p2, each firm sells Q/2 units at price p1, where Q = max{1 – p1,0}.• The marginal cost is 0.()otherwise. if if02/)1()1(,21211111211pppppppppp =<−−=πBertrand duopoly -- EquilibriumTheorem: The only Nash equilibrium in the “Bertrand game” is p* = (0,0).Proof:1. p*=(0,0) is an equilibrium. 2. If p = (p1,p2) is an equilibrium, then p = p*.1. If p = (p1,p2) is an equilibrium, then p1 = p2... 2. Given any equilibrium p = (p1,p2) with p1 = p2, p = p*.8Commons Problem• N = {1,2,…,n} players, each with unlimited money;• Simultaneously, each player i contributes xi≥ 0 to produce y = x1+…xnunit of some public good, yielding payoff Ui(xi,y) = y1/2–xi.Stag Hunt(5,5)(0,4)(4,0)(2,2)9Equilibrium in Mixed StrategiesWhat is a strategy?– A complete contingent-plan of a player.– What the others think the player might do under various contingency.What do we mean by a mixed strategy?– The player is randomly choosing his pure strategies.– The other players are not certain about what he will do.Stag Hunt(5,5)(0,4)(4,0)(2,2)10Mixed-strategy equilibrium in Stag-Hunt game• Assume: Player 2 thinks that, with probability p, Player 1 targets for Rabbit. What is the best probability q she wants to play Rabbit?• His payoff from targeting Rabbit: U2(R;p) = 2p + 4(1-p) = 4-2p.•From Stag:U2(S;p) = 5(1-p) • She is indifferent iff4-2p = 5(1-p) iff p = 1/3.()[]1/3p if1/3p if1/3p if11,00>=<∈= qpqBR0 0.2 0.4 0.6 0.8 100.511.522.533.544.55 4 - 2p5(1-p)Best responses in Stag-Hunt game1/3p1/3q11Bertrand Competition with costly search• N = {F1,F2,B}; F1, F2 are firms; B is buyer• B needs 1 unit of good, worth 6;• Firms sell the good; Marginal cost = 0.• Possible prices P = {1,5}.• Buyer can check the prices with a small cost c > 0.Game:1. Each firm i chooses price pi;2. B decides whether to check the prices;3. (Given) If he checks the prices, and p1≠p2, he buys the cheaper one; otherwise, he buys from any of the firm with probability ½. Bertrand Competition with costly searchF1F2HighLowHigh LowF1F2HighLowHigh LowCheckDon’t Check12Mixed-strategy equilibrium• Symmetric equilibrium: Each firm charges “High” with probability q;• Buyer Checks with probability r.• U(check;q) = q21 + (1-q2)5 – c = 5 - 4 q2–c;• U(Don’t;q) = q1 + (1-q)5 = 5 - 4 q;• Indifference: 4q(1-q) = c; i.e.,• U(high;q,r) = 0.5(1-r(1-q))5;• U(low;q,r) = qr1 + 0.5(1-qr) • Indifference = r = 4/(5-4q).Dynamic Games of Perfect Information & Backward Induction13DefinitionsPerfect-Information game is a game in which all the information sets are singleton.Sequential Rationality: A player is sequentially rational iff, at each node he is to move, he maximizes his expected utility conditional on that he is at the node – even if this node is precluded by his own strategy. In a finite game of perfect information, the “common knowledge” of sequential rationality gives “Backward Induction” outcome.A centipede game12ADαδ(4,4) (5,2)(1,-5)ad(3,3)114Backward InductionTake any pen-terminal nodePick one of the payoff vectors (moves) that gives ‘the mover’ at the node the highest payoffAssign this payoff to the node at the hand;Eliminate all the moves and the terminal nodes following the nodeAny non-terminalnodeYesNoThe picked movesBattle of The Sexes with perfect information2 12 TB L LRR(2,1) (0,0) (0,0) (1,2)15Note• There are Nash equilibria that are different from the Backward Induction outcome.• Backward Induction always yields a Nash Equilibrium.• That is, Sequential rationality is stronger than rationality.Matching Pennies (wpi)122HeadTailhead tailhead tail(-1,1) (1,-1)(1,-1) (-1,1)16Stackelberg DuopolyGame:N = {1,2} firms w MC = 0;1. Firm 1 produces q1units 2. Observing q1, Firm 2 produces q2units3. Each sells the good at priceP = max{0,1-(q1+q2)}.11QPπi(q1, q2) = qi[1-(q1+q2)] if q1+ q2< 1, 0otherwise.“Stackelberg equilibrium”• If q1 > 1, q2*(q1) = 0. • If q1 ≤ 1, q2*(q1) = (1-q1)/2.• Given the function q2*, if q1 ≤