Exercise sheets micro-economy game theory – KönigsteinExercise sheet 1 Exercise 1.1 Give one example of a game-like situation from your everyday life. Identify the players, the nature of interaction, the strategies available, and the objectives that each player is trying to achieve. Teamwork: Group: Interaction: Strategic: Rational: Students jointly preparing a case study If I put little effort in the project someone else must work harder to get the work done Estimating the likelihood of free-riders in the group Comparison of the benefits of extra work (better grade) against the costs of extra work (less leisure time) R&D expenditures Group: e.g., pharmaceutical companies Interaction: The first developer of a drug makes the most pro fits (thanks to the associated patent) Strategic & rational: Choosing how much to spend involves thinking about competitors’ commitments and possible reactions to own decision Exercise 1.2 By carefully examining the components of a game situation – group, interaction, strategy, and rationality – discuss whether the following situations qualify as a game. a) Purchasing a house b) A trial by a jury a) Purchasing a house group: seller, buyers different situation of buying a house Many buyers achieve, objectives: buyer: buy a house; buyers wants to minimize price seller: objective of the seller is to get the best price interact in a strategic sense? Individual bargain? buying a house is usually speci fic, small, thin market → Strategic game problem → do i want to influence the price? Are they other buyers? b) A trial by a jury -Players: the jury members -strategy of each jury member: penalty or range to penalty (1 or 2 or 3 years in jail) → you might as well say: this is a suggested decision → yes or no → specific penalty or just choice between yes or no → there are different types -goal of a jury member: come up with a justice decision → neutral, unbiased decision -nature of interaction – more decision of individual choice or more an anticipate what other will choose → interaction is possible – decision making in a group → both activities can be modelled to remember: 4 categories: a) players b) players choice sets c) goals d) nature of strategic action Exercise 1.3 Consider two persons. Person 1 has “altruistic” preferences. This means, she does not care only for her own income but also for the income of person 2. Precisely, she attaches to each unit of own income the same value as for two units of person 2’s income. For example, she is indifferent between two actions a and b when action a generates an income of 1 for her and 0 for person 2 and action b generates an income of 0 for her and 2 for person 2. y) → positive linear transformation – with b > 0 → this transformation preserves the order. a) Are the preferences of this DM also represented by the payoff function v for which v(a)=1.5 Determine whether each of the games differs from the Prisoner’s Dilemma as introduced in the lecture only in the names of the actions or whether it differs also in one or both of the players’ preferences. d. v(b)=0. LEFT → PD-Game has certain solution aspects: individual rationality. b) How about the function w for which w(a)=w(b)=0 and w(c)=8? u(a) = 0 u(b) = 1 u(c) = 4 a) v(a) = -1.4) = 1 + ½ * 4 = 3 u (x. collective rationality.1) → preference sign → not equal ~ but indifferent <. it depends on a positive b Exercise 1. >.4 A decision maker’s preferences over the action set A={a.y) = a + b u(x. where in each case the first number indicates her own income and the second number is person 2’s income? b) Give a payoff function consistent with these preferences. v(c) = 2 → yes. ~ are preference operators b) they are many payoff functions payoff vector: (x. dominant strategies.0) > (2. because we have the same payoffs for a and b Exercise 1.a) How do person 1’s preferences order the actions c.0) respectively. but what constitutes a PD GAME? → choice options? → Preference → certain preference → For a game to be represented as a PD game it must be a 2 by 2 game with a specific order of .0) > (2.b. and (3.1) → (1.5 u(1. and u(c)=4. (2. because it preserves the order → this order represents the same references b) no.y) payoff x = payoff from p1 payoff y = payoff from p2 → utility function = u(x. 1.y) = x + ½y u(3.0) = 3 + ½ * 0 = 3 u(2.c} are represented by the payoff function u for which u(a)=0.4). a) order the actions: → (3. e associated with outcomes (1.y) = (x + ½y)2 = [u(x. and v(c)=2? 2.1) = 2 + ½ * 1 = 2. u(b)=1.1).4) ~ (3.y)]2 → monotonic transformation does not matter W(x. v(b) = 1. it will be a PD game SOLUTION => left game is not a PD game. dass p1 y nimmt. wenn er auf x ausweicht? Answer: NO! → nash solution is y.y → no one can benefit from deviation → the suggested solution is the nash equilibrium! Exercise sheet 2 Exercise 2.1 → preference of p1 is: according strategy vector is: (b1. dass p2 y nimmt.b2) > (a1.b2) > (a2.a2) > (b1.0 b2 0.a2) > (b1.2 3.b2) > (a1. wie bei einem klassischen PD-Game => right is a PD game since preference ordering is as in PD-game-way for both players => right is not symmetric. The expected distribution of the viewers is given in the following game table: a) Determine the Nash equilibrium! b) Is it reasonable that TV-Beta considers the broadcasting of sports events? a) two players: TV-Alpha + TV-Beta two options: sport or show Payoff vectors: each channel will maximizes the percentage of viewers it is not symmetric: ansosnten müssten gleiche Zahlenwerte bei show. but however it is nevertheless a PD Game → notion of individual rationality is important fpr PD game Nash equilibrium analysis: What is the best choice given some choice by the other player → suggested solution → we have to check.1: The inhabitants of the city Hope can only choose between two TV channels: TVAlpha and TV-Beta. since preference ordering differs → die unteren beiden haben für p2 beide den value 1 und deshalb ist der dann indifferent und es gibt nicht die gleich Anordnung der Präferenzen.payoffs: example for PD: p1 p2 a1 b1 a2 2. kann er dann benefit. than p1 takes sports → arrow → suppose p2 chooses show2. where no player can benefit from deviating → suppose p2 chooses sports2. bloß halt andersrum. wenn er abweicht von y auf x? Answer: NO! Wenn p2 proposes. than p1 takes sports → arrow → sport1 and show1 = Sp1 and Sh1 Sp1 = BR1 (best reply choice1) (Sp2) Best reply function of p1 Sp1 = BR1 (Sh2) → horizontal arrow for p2 → vertical arrow for p1 Sh2 = BR2 (Sp1) Best replied function of p2 . Each channel has to decide whether to send a sports or a show program.a2) > (a1. kann p1 dann benefit.b2) → preference of p2 is: (a1. → we search the vector. sport und bei sport und show stehen.3 1.b3) if the preferences are like this. wether one part can benefit from deviation Wenn p1 annimmt. but no PD – Game → efficiency: all action profiles are efficient. ranking the outcomes according to the other person’s comfort. but.a2) Є [(sit1.x2) = α1x1 + α2x2 → social welfare function → Evaluate the ending of the game . Model the situation as a strategic game. a2) Є A with A = A1 x A2 → three scenarios → three number to characterize the preferences for a Player Player 2 Player 1 Sit1 Stand1 Sit2 2.) u1(a1.3 Stand2 3. prefers to stand than to sit if the other person stands.a2).2: Two people enter a bus. Two adjacent cramped seats are free.Sh2 = BR2 (sh1) → (Sp1.). sit2) . Model the situation as a strategic game. sit) → but the introduction of altruism isn't good W (x1. u2(.1 → it is a dominant solvable game. Each person must decide whether to sit or stand. P2 A1: sit1. out of politeness. (sit1. (stand1.1 → PD – Game utility numbers → The social outcome is here (sit. each element is a action vector or action profile → set of feasible action profiles! (Еa1. u2(a1. stand1 A2: sit2. stand2). because is not part of any nash equilibrium → sp2 is never part of a best replied choice Exercise 2. Sh2) → unique nash equilibrium of the game! b) no. Sitting alone is more comfortable than sitting next to the other person. stand2) → efficiency is a collective viewpoint → but just sit.1 1.2 3. which is more comfortable than standing. Is this game a Prisoner’s Dilemma? Find the Nash equilibrium (equilbria?). bis auf (stand1. b) Suppose that each person is altruistic. stand2)] → Set with four elements. sit is the equilibrium b) Player 2 Player 1 Sit1 Stand1 Sit2 2. c) Compare the people’s comfort in the equilibria of the two games! a) Players: action profile set utility functions: P1. sit2). (stand1. Is this game a Prisoner’s Dilemma? Find the Nash equilibrium (equilibria?). a) Suppose that each person cares only about his own comfort. utility2 u1(.3 1. stand2 Utility1.2 1.a2) 4 possible action profiles (a1.0 Stand2 0. fink2) 1+1*1=2 1+1*1=2 Player 2 Player 1 quiet1 fink1 quiet2 4.f2 f1. sit2 => Welfare => 2 * 2 + 2 = 6 sit 1.q2 quiet1 fink1 Player 2 quiet2 2+2α . 1+α u2 2+α*2 3+α*0 0+ α*3 . each action pair results in the players' receiving amounts of money equal to the numbers corresponding to that action pair in the game table below. j is the other player. and α is a given nonnegative number.2 → unique nash equilibrium → dominant strategy → but no PD. a) This game is an Example of generalization: On top of the monetary values we can form into utility functions → what is the utility for example for quiet1 and quiet2 ? utility1(i) utility2(j) (quiet1. fink2) 0+1*3=3 3+1*0=3 (fink1.3 2.3 fink2 3. for example. is 2 + 2α. quiet and fink. For values of α for which the game is not the Prisoner's Dilemma. 3 1+α. because its end up with a efficient solution b) Player 1 q1.= 2 * u 1 + u2 → attach a weights! Attach depends on taste sit1. sit2 => 2 * 1 + 3 = 5 stand 1 . 3α u1 2+α*2 0+ α*3 3+ α*0 fink2 3α . where mi(a) is the amount of money received by player i when the action profile is a.4 3. The players are not “selfish”.3: Each of two players has two possible actions. a) Formulate a strategic game that models this situation in the case α = 1. stand 2 => 2 * 3 + 1 = 7 stand 1. quiet). stand2 => 2 * 1 + 1 = 3 → WHAT SHOULD HAPPENED IN THE SOCIETY? c) comfort is actually higher. Is this game the Prisoner's Dilemma? b) Find the range of values of α for which the resulting game is the Prisoner's Dilemma. Player 1's payoff to the action pair (quiet. find its Nash equilibria. 2+2α 3 . quiet2) 2 + 1* 2 = 4 2+1*2=4 (quiet1. when they both individual or egoistic Exercise 2.q2 q1. the preferences of each player i are represented by the payoff function mi(a) + αmj(a). quiet2) 3+1*0=3 0+1*3=3 (fink1. rather. that this is a PD-GAME b) new start: We have to start new. quiet” to “fink. fink” => 2 + 2α > 1+ α <=> 2 (1 + α) > 1 + α → conditions (i).f1. because of the simultaneously → When you want to model a social interaction.f2 1+ α*1 1+ α*1 → For what values of α do we get a PD game? → to have q1 as individually rational choice by p1 the following must hold. (iii) must hold simultaneously → what constitutes a game depends on the structure of the game. if p2 chooses q2 2 + 2α ≥ 3 <=> 2α ≥ 1 <=> α ≥ ½ => for α > ½ q1 is BR (q2) → to have q1 as BR (f2). when the other Player chooses “quiet”: => 3 > 2 + 2α <=> 3 > 2 (1 +α) (ii) A Player prefers “fink” when the other player chooses “fink”: => 1 + α > 3α (iii) each player prefers “quiet. there are many ways to do this → few parameters or less parameters? That is on ones choice → possibilities for solving this 3 equations: a) infinite numbers of solutions b) one solution c) no solution (iii) → it is always fulfilled (redundant) => 2 > 1 (i) 3 > 2 (1 +α) => ½ > α (ii) 1 + α > 3α => ½ > α → For α < ½ the game is a PD – GAME Suppose alternatively. to get a PD-game: conditions: (i) A player prefers “fink” to “quiet”. (ii). and the payoffs from each player → two strategies in the PD-Game → Wether there is a equilibrium interests just in the solution #noCONDITION → we can formulate conditions for each player separately → because the game is symmetrical. the following must hold 3α ≥ 1 + α <=> 2α ≥ 1 <=> α > ½ → By symmetry we get q2 is BR(q1) and q2 is BR (f1) if α > ½ => if α > ½. than q1 and q2 is nash equilibrium → dominant strategy qi → unique nash equilibrium → inefficient equilibrium → but it is not inefficient! → What constitutes a PD-GAME FAZIT: we should not say. (i)' α<½ (ii)' α < 1/3 . we can just prove one player → 6 conditions which boiled down to 3. cause this contains the other ones! If α = ½ → put in → Table from 2. stag2) 2 (stag1. Philosopher Jean-Jacques Rousseau describes an example of a group of hunters who wish to catch a stag.2! → two strict nash equilibria → is the action weakly dominated? No! → Pareto → collective 2. There are group interests and individual interests 2 type of hunters hunters action profiles: ai € Ai. 3/2 fink2 3/2.1: (Stag Hunt Game). harei} A = Ai x aj = A1 x A2 (stag1. find the Nash equilibria. The hunters will succeed if they all remain sufficiently attentive.2 .1 1.2 1. Ai = {stagi. a) Consider the group of hunters comprising N = 2 persons and assume that each hunter prefers one half of the stag to a hare.1 → one allocation is efficiency. What problem do you see in modelling the situation as a strategic game? Despite this problem. Describe the situation as a strategic game and find the Nash equilibria.(iii)' α < ¼ ??? → You have to pick the strongest conditions. but each is tempted to leave her post and catch a hare. Which action profiles are collectively rational (efficient)? b) Consider the group of hunters comprising N = 5 persons and assume that each hunter prefers 1/5 of the stag to a hare.0 hare2 0. f2 is Nash quilibrium in the original table => no PD-GAM If α > ½ Player 2 quiet2 Player 1 fink2 quiet1 fink1 → only q1. stag2) 1 (hare1.3 Player 1 fink1 3. hare2) 0 (hare1.3. hare2) 1 Payoff1 2 1 0 1 Payoff2 Player 2 Player 1 stag1 hare2 stag2 2.q2 is nash equilibria => no PD Exercise Sheet 3 Exercise 3. 3/2 → All action profiles are nash equilibria → Only f1. 3 3/2. when at least one player gets more without hunting → 2.b) Player 2 quiet2 quiet1 3. 5 (hare1.. since a single or multiple players would benefit von deviating the proposed action profile (all asymmetric profiles) are not nash equilibria.5) = hare → equilibiriums payoff = ui# b) ui (stagi. stagj Vj ≠ i) c) Conclusion: Since the hunter prefers 1/5 of stag to 1 hare. Vi.payoff vs. harej Vj ≠ I) =0 → deviation payoff =~ui c) conclusion: since ui# > ~ui the action profile harei Vi = 1..stag5) = (stagi) Vi = 1. Problem: Many action profiles (because we have 5 players) a profile is a 5-dimensional vector → 25 different possible profiles 2.…. stagi) = ui# = 0 → define: ~ui is i's utility from deviating to harei ~ui = harei → Player i would benefit from deviating → Thus.…. Problem: We can not draw a 5 dimensional matrix However: we can distinguish different types of action profiles and analyse these (stag1.. 2.5. stag i Vi = 1. 5 → two symmetric action profiles (2 types of action profiles) and: There are asymmetric action profiles where some players i choose stagi and some playersi choose harei → consider (i) stagi Vi = 1. stag2. with 2 ≤ m .….…5 nash equilibrium it is beneficial for one of the i-group or one for the j-group to deviate → no? Than it is a Nash equilibrium Asymmetric profiles a) action profiles:(harej.2.hare5) = (harei) Vi = 1. ….5 → Is this a Nash-equilibrium? a) What is playeri's payoff given the proposed equilibrium profile (What is the proposed equilibrium payoff) → payoff according to the proposed solution (or candidate solution) => ui (stagj Vj = 1.2. i≠j) ui (harej.…..2: Consider variants of the n-hunter stag hunt game in which only m hunters.….j = 1. hare2.2. strategie payoff vectors → collective strategy domination → individual b) 1.2.5 → Is this a nash equilibrium? a) ui (harej Vj = 1. stagi..….2…5) is Nash equilibrium (Deviation is not beneficial) → consider (ii) harei Vi = 1.2.2.5) = 1/5 of the stag b) What is the playeri's payoff from deviating => ui(harei. Graphic example: All othersj stag j harej playeri stagi 2 0 harei 1 1 → stag: 10 hare: 1 nothing = 0 b) uj (harej.2... stagei) = uj# = hare ~uj = Exercise 3. < n.b) symmetric: (i) stagi Vi => ui# = 1/k stag => ~ui = hare since n > k. n))))) Deviation payoff? ~ui = hare → Since ui# is > than ~ui. find the Nash equilibria of the according strategic game and find out what would be socially optimal if the caught stag and hares would be divided by all hunters: a) Each hunter prefers the fraction 1/n of the stag to a hare.2.2. or ui# is prefer to ~ui .a) that only the 2 symmetric action profiles are Nash eq'a of the game → social optimal: m stag hunters and n-m hare hunters! → No equilibrium is the social optimal solution! 3. b) Each hunter prefers the fraction 1/k of the stag to a hare. Under each of the following assumptions on the hunters’ preferences. A captured stag is shared only by those hunters who catch it. we conclude playeri prefers 1/k stag to hare => ui# > ~ui → playeri (staghunter) can not benefit from deviation. the candidate action profile Is a Nash equilibrium (ii) suppose (harei) Vi is the candidate solution => ui# = hare Deviation payoff? ~ui = 0 → (harei) Vi is Nash equilibrium (iii) suppose k players with k < m hunt stag while the rest n-k players hunt hare: → Suppose playeri is one of the k stag hunters ui# = 0 ~ui = hare => playeri can benefit from deviation => candidate profile is not a Nash equilibrium (iv) suppose k players hunt stag with m ≤ k < n while the rest n-k players hunt hare a) suppose playeri is one the k stag hunters ui# = 1/k stag ~ui = hare since 1/n stag is preferred to hare and since 1/k stag is > 1/n stag . 3. …. we conclude for 3. (1/n) stag < (1/k) stag => playeri prefers have (~ui > ui#) => stagi Vi is not Nash eq'a (ii) harei for all i => ui# = hare // ~ui = 0 asymmetric . where k is an integer with m ≤ k ≤ n. b) Check hare hunters → Suppose playeri is one of the hare hunters ui# = hare ~ui = 1/(k+1) stag → since 1/n stag is preferred to hare and since 1/k+1 stag ≥ 1/n stag we conclude that playeri benefits from deviation => all asymmetric profiles with stag hunters and m ≤ k < n are not a nash eq'a Overall.a) (i) suppose (stagi) Vi is the candidate solution => ui# = 1/n stag (((((→ ui (stagj Vj .2. need to pursue the stag in order to catch it (assume there is only a single stag). but prefers a hare to any smaller fraction of the stag. j = 1. but any smaller fraction of stag is worse than hare. agg2). Formulate the situation as a strategic game and find its Nash equilibria. agg2) . The candidate profile is not a nash equilibrium since a stag hunter benefits from deviating.. harej Vj = m+1. and harej Vj = k+a…n) → playeri : ui# = (1/k)stag. pass2) > (pass1. if opponent is passive (ii) u2 (pass1.. ~ui = hare ui# > ~ui => playeri does not benefit from deviation # → playerj : uj = hare...stagi Vi i = 1… k-1 harei Vj j = k … n . Each prefers to be aggressive if its opponent is passive. (agg1. ~ui = hare => We know that 1/L stag > hare . each animal prefers the outcome in which its opponent is passive to that in which its opponent is aggressive. Players: p1.3: (Hawk-Dove Game). Thus 1/k of stag is worse than hare => playeri benefits from deviation => candidate profile is not a nash equilibrium Collecting results of our analysis 3 types of equilibria: . Each can be passive (“dove”) or aggressive (“hawk”).(iii) suppose L players going for stag hunting L < m → no stag hunting → they get = 0 → this is no nash equilibrium → Let L hunters hunt stag and L smaller m.stagi Vi i = 1 … k harej Vj j = k+1 … n Exercise 3. (pass1. Furthermore. (iv) suppose L players going for stag hunting L = m (stagi Vi = 1. ~uj = (1/(k+1)) stag → We know (1/k)stag is strictly preferred to hare. (pass1. Two animals are fighting over some prey. Therefore (1/(k+1)stag is worse than hare.n) – playeri = ui# = (1/m)stag ~ui = hare (1/m) stag > 1/k (stag) > hare 1/m of a stag is preferred to 1/k of stag which is preferred to hare => i does not benefit from deviation # – Player j: uj = hare ~uj = (1/m+1) stag → We know that player i prefers 1/k to hare => (1/(m+1))stag is strictly preferred to ≥ 1/k(stag) iv a) (1/(m+1))stag > 1/k stag if m+1 < k → no nash equilibrium iv b) (1/(m+1))stag = (1/k)stag of m+1 = k → nash equilibrium (v) suppose k stag hunters (stagi Vi = 1… k. but any smaller fraction of stay is worse than hare.pass2) → prefers to be aggressive. agg2) > (agg1.m.pass2) preferences: (i) u1 (agg1. pass2). and passive if its opponent is aggressive.harei Vi i = n . which means player j does not benefit from deviation => this profile is a nash equilibrium (vi) suppose L > k hunters an l < n → suppose playeri is a stag hunter → playeri → ui# = 1/L stag. agg2). p2 action profiles: (agg1. Formulate this situation as a strategic geme and find its Nash equilibria. – Is there an equilibrium in which more than k people contribute? One in which k peaple contribute? One in which fewer than k people contribute? Players: n people/ persons actions/ action profiles: ai = {contribute. => There exist no Nash equilibrium with m > k contributors Exercise sheet 4 Exercise 4.0 3. Each person ranks outcomes from best to worst as follows: (i) any outcome in which the good is provided and she does not contribute.2 => two strict nash equilibria Exercise 3. (4) m > k → a contributori would benefit from deviation since the public good would still be provided and i prefers not to contribute then. where 2 ≤ k ≤ n. agg2) > u2 (agg1.an) Preferences for each possible action profile as described from (i) to (iv) (1) Suppose number of 0 < m < k with m being the number of contributors → consider playeri contributing. agg2) 3 2 1 0 agg2 pass2 agg1 0. Is it the best reply? Yes. pass2) > u2 (Agg1. …. agg2) → prefers the outcome in which the opponent is passive to that in which its opponent is aggressive => complete ordering of the action profiles: u1 (agg1. contributions are not refunded.no . if it is not provided.3 2. (ii) any outcome in which the good is provided and she contributes (iii) any outcome in which the good is not provided and she does not contribute. (iv) any outcome in which the good is not provided and she contributes.no…no} → not contributing is best-reply for playeri => nash equilibrium (3) Suppose m = k Playeri: contributing: is this a best-reply? Yes! Because if player i deviates (no contribution) the public good is not provided → Playeri is not contribute.pass2) > u1 (pass1. Each of n people chooses whether to contribute a fixed amount toward the provision of a public good. Is this a last-reply? No! Playeri would benefit from deviation => any strategic profile with m < k contributors is not a Nash equilibrium (2) Suppose no contributors: m = 0 action profile {no . a2.1 pass1 1.1: Consider a two-player strategic game in which each player’s set of actions is the set . because the public good is provided anyway => Any action profile with m = k contribute and n-k non contribute is a Nash equilibrium.(iii) → prefers to be passive.4: (Public Good Game).no . The good is provided if and only if at least k people contribute. pass2) > u1 (pass1. not contribute} Vi = 1…n Action profile: a = (a1. if the opponent is aggressive u1 (agg1. agg2) and u1 (pass1. pass2) > u1 (pass1. b) Find the Nash equilibria. e) Determine the firms’ benefits from forming a cartel according to the percentage increase in profits. d) Suppose the two firms form a cartel: They coordinate on individual outputs and share the sum of profits. a2) → best response function player2 a2 derivative du1(a1. c) Dermine the Nash equilibria.a2) / da1 = 0 <=> a2 .a2#) = 1/5 (2/5 – 1/5) = 1/25 u2# = u(a1#. a2) = a1(a2 – a1) Vai € Ro u2 (a1. compute the percentage changes in price and quantity due to the cartel.2 Max u2 (a1.of the players’ payoff functions are u1(a1. a2) = a1(1 – a1 – a2) i = 1. market price and firms’ profits. Let firm i’s production cost per unit of output be equal to 1. Suppose each firm’s goal is to maximize profit πi. Determine the Nash equilibria of the cartel. a2#) = (1/5.2: (Cournot Duopoly Game).a2)/ da2 = 0 <=> 1. In additon compute industry output.) / da22 = -2 < 0 → â2 is a maximal point (best reply) b) => Condition (1) and (2) must hold simultaneously => (1) into (2): a2 = ½ – ½ * (½a2) <=> a2 = ½ – ¼ a2 <=> a2 # ¼a2 = ½ <=> a2 (1+ ¼ ) = ½ <=> a2 * 5/4 = ½ <=> a2# = 4/5 * ½ = 2/5 a2# into (1) : a1# = ½ a2# = ½ * 2/5 = 1/5 Nash equilibrium = (a1#. market price and firms’ profits.2a1 = 0 <=> â1 = ½ a2 → Best response generic on a2 → Best response function 2 d u1(. a2) = a1(a2 – a1) and a) Find the best response functions.a1 = 2a2 <=> â2 = ½ – ½ a1 → Best response function of player2 d2u2 (. 2 represent individual quantities and let the market price p per unit of output sold be determined according to the following function (“inverse demand function”): p = 36 – q with q representing the sum of both firms quantities (“industry output”). Also.a2#) = 2/5 (1 – 1/5 – 2/5) = 4/25 Exercise 4.) / da21 = -2 < 0 → â1 is a maximal point (best reply) du2(a1. In additon compute industry output. What does this mean for consumers? . a) u1 (a1. b) Determine and draw the best-response functions.2/5) Equilibrium payoff: u1# = u(a1#. Let qi with i = 1. Two firms compete in a market by producing and selling homogenous goods. a) Model the given situation as a strategic game. q2 -1 = 0 <=> 35 -q2 = 2q1 <=> <=> ^q1 = 35/2 – ½q2 → best response function player1 maximize ~u2 → dπ2 (.a) Players: action/ action profiles: preferences/utilities: 2 firms q1 € Ro+ .q = 35q – q2 dπ/dq = 0 <=> 35 – 2q = 0 <=> q## = 35/2 → total output that maximizes cartel profit π .q2) action profile profit profit = Revenue minus cost Cost = 1 * quantitiy → unitcost π1 = p * q1 – 1 * q1 π2 = p * q2 – 1 * q2 with p = 36 –q and q = q1 + q 2 (1) (2) use in (1) and (2) π1 = (36 -q1 – q2) * q1 – q1 = 36 q1 – q2 – q2q1 – q1 π2 = (36 -q1 – q2) * q2 – q2 = 36 q2 – q2 – q1q2 – q2 b) max π1 → dπ1/dq1 = 0 <=> 36 – 2q1 .1 π1# = π2# d) π = π1 + π 2 = p * q – q = (36) – q) * q . (q1.q1# = 38/3 * 35/3 – 35/3 = 1225/9 = ungefähr 136.) / dq2 = 0 <=> 46 – q1 – 2q2 – 1 = 0 <=> 35 – q1 = 2q2 <=> ^q2 = 35/2 – ½ q1 → Best response functions player2 q2 35 ^q1: ^q1 = 0 if q2 = 35 q1 = 35/2 if q2 = 0 ^q2: if q1 = 0 → ^q2 = 35/2 if q2 = 0 => q1 = 35 35/2 35/2 35 q2 c) solve (3) and (4) by plugin (3) in (4) => q2 = 35/2 – ½ (35/2 – ½ * q2) => q2 –1/4 q2 = 35/4 => q2 3/4 = 35/3 = 4/3 => q2# = 35/3 => q2# into (3) q1 = 25/2 – ½ * 35/3 = 35/2 – 35/ 6 <=> q1 = 2 * 35 / 6 = 35 /3 # total output q = q1# + q2# = 70 /3 market price: p# = 36 – q# = 38/3 profits π1# = p# * q1# . q2 € R0+ (actions). if player2 mixes with (1/3.5 qi2 with the parameter ai representing fixed cost. 1/3).p## = 36 – q## = 36 – 35/2 = 37/2 π# = 37/2 * 25/2 – 35/2 = 1225/4 = 206. Exercise 4.2/3) check p1: p1 can rationally mix between actions “Bach” and “Stravinsky” only if her expected payoff from choosing “Bach” is the same as the expected payoff from choosing “Stravinsky” => EU1 (Bach1. nonnegative numbers and u2(a1. Str2) EU2 (α1. Generalize the Cournot game described in 4. 2/3)) is a mixed strategy Nash equilibrium of the Bach or Stravinsky game presented below: → We need to check whether it is rational for player 1 to mix with (2/3. b) Assume the same model as in 4. Is (2/3.125 → Percentage increase = 12.5% % increase price: p## . a) Dermine the Nash equilibria. Game Theory – Exercise Sheet 5 Exercise 5.2 to the case of 3 firms and dermine the Nash equilibria.1 = 1.3: (Generalized Cournot Duopoly Game).2/3) and vice versa.4: (Cournot Game With 3 Firms). 1/3) the BR-choice given the choice of p2 (1/3. b) Show that with increasing N the market price converges to the level of marginal cost (the idealized result that should be achieved in perfectly competitive markets).1 % % increase quantitiy is about .2 to the case of N firms.1/3) is a best-response (any mixture is a best response including the proposed mixture (2/3. a2) = a2(1 – a1 – a2). α2) → This must hold! EU1 (Bach1.p# / p# * 100 = 37/2 – 38/3 / 38/3 = 46. Bach2) = EU2 (α1.125 e) Percentage increase in individual profit due to cartel → π1##/π1# = 153.2 but now assume a quadratic cost function: ci(qi) = ai + qi + 0. α2) = EU1 (Str1. α2) = 1/3 * 2 + 2/3 * 0 = 2/3 EU1 (Str1. Based on 4.1/ 136.25% Exercise 4.5: (Cournot Game With N Firms).1/3). At what level of fixed cost will firm i quit staying in business? Exercise 4. Str2) = 2/3 * 0 + 1/3 * 2 = 2/3 => Together the arguments established the propsed equilibrium . (1/3.1: Show that the strategy pair ((2/3.2 generalize the Cournot duopoly model and solve for Nash equilibria by proceeding as follows: a) Assume a general linear inverse demand function. Determine also relevant restrictions for the parameters of the model.1/3)) chek p2: Mixing rationally requires => EU2 (α1. Bach2) = 2/3 * 1 + 1/3 * 0 = 2/3 EU2 (α1. α2) = 1/3 * 0 + 2/3 * 1 = 2/3 => condition (#) is fullfilled => p1 can rationally mix => Thus α1 = (2/3. Generalize the Cournot game described in 4.25 => Each firm sells qi = ½ q# = 35/4 π#1 = ½ π# = 1225/8 = 153. q) = EU1 (3.0 2 0.0 3 0.….=> two pure equilibria (“arrows) and one equilibria in mixed strategies Exercise 5. 1) =¼*1+¼*0+¼*0+¼*0 =¼ EU2 (p. ¼.p2 . q) =… =-¼ => We have established that p1 can mix rationally between actions 1 to 4 and p = (¼. K is a best .q3 .0 0.0 … -1.1 0. ¼. then P1 pays 1€ to P2.0 -1.0 0.0 0. c) Assume a generalization such that every player may choose a positive integer up to K. ¼ ) → It is the BR for p1!!! → check best-reply condition für p2 EU2 (p. 2) =¼*0+¼*1+¼*0+¼*0 =¼ EU2 (p.1 0.0 0.1 … 0. Show that the game has a mixed strategy Nash equilibrium in which each player chooses each positive integer up to K with probability 1/K.p2 .0 1 -1.p3 . 3) = EU2 (p.… .0 … 0. ¼. 4) =… =¼ => This establishes the proposed nash-equilibrium (in mixed strategies) c) player1 player2 1 2 … K 1 -1.1 0.1 0. If the players choose the same number.q4) = q p1 mixed strategy: (p1 . q) = 0*¼–1*¼+0*¼+0*¼ =-¼ EU1 (3. ….2.1 → player1 chooses (p1 .qK) = q with qj = 1/K Vj = 1…K Check best-reply condition für player1: EU1 (a1. Each player’s preferences are represented by the expected monetary payoff.0 0.0 … … … … … 4 0. q) EU1 (1.0 0. q) = -1 * ¼ + 0 * ¼ + 0 * ¼ + 0 * ¼ = .2.K) => Player1 can rationally mix (1) and thus p1 0 1/K V I = 1.¼ EU1 (2. b) Show that the game has a mixed strategy Nash equilibrium in which each player chooses numbers 1 to 4 with probability 1/4.p4) = p → check best-reply condition für p1 EU1 (1.q2 .0 4 0. q) =… =-¼ EU1 (4.1 p2 mixed strategy: (q1 . ¼. otherwise no payment is made. 3) =… =¼ EU2 (p. q) = EU1 (4.0 -1.….0 -1.0 -1. q) = EU1 (2. 2) = EU2 (p. ¼) given q (¼. a) player2 b) 1 2 3 4 player1 2 0. 4) EU2 (p. q) = -1 * 1/K + 0 * 1/K = -1/K Va1 € A1 (Va1 = 1. a) Represent the game in game matrix.pK) = p with pi = 1/K Vi = 1…K → Player2 chooses (q1 .q2 .2: Players 1 and 2 each choose a positive integer up to 4. 1) = EU2 (p.0 0. 4 C 3. 2 -. And this mixture is a Best-reply) => Player1 can mix rationally (1) and the proposed mixture is (also) a bestreply ((. 1/3.-. 2/3) a Nash-equilibrium? (P1 → M has probability = 0 ) Check Player1: EU1 (T.3 0.…. (0. It does not destroy our candidate solution.2. the best-reply will not help you to conclude which action you have to use → PARETO → Some nash-equilibria are social better than other nash-equilibrium. q) = 0 * ? + 3 * 1/3 + 1 * 2/3 = 5/3 EU1 (M. C) =3*¾+1*¼ = 10/4 EU2 (p. q) = 5 * 1/3 + 0 * 2/3 = 5/3 (T and B must be equal and the expected utility has to be smaller or equal to T and B. Additional exercise: PLAYER1 PLAYER2 T M B L -. one part looeses → constant sum games → election → non constant sum game → coordination games: multiple nash-equilibria.K is a best reply (2) → Analysis gives two answers.¼).5. q) = 0 * 1/3 + 2 * 2/3 = 4/3 EU1 (B.0. L) = 2 * ¾ + 4 * 1/4 = 10/4 (ii) holds EU2 (p.reply (2) Check best-reply condition for player2: EU2 (p. Prediction what rational players will do → Nash-equilibrium → but if you have multiple nash-equilibrium.))) .L) ≤ EU2 (p.K) => Player2 can mix rationally (1) and thus qj = 1/K V i= 1.0.1 R 1. Some of them are preferred by some players.C) EU2 (p.2.7 Is α = ((¾.…. R) =1*¾+7*¼ = 10/4 (i) holds => proposed solution is a Nash-equilibrium (((→ zero sum games → Chess – one part wins. two conclusions → (1)+(2) => Thus the proposed strategy profile is a Nash equilibrium in mixed strategies. a2) = 1 * 1/K = 1/K Va2 € A2 (Va1 = 1.C) = EU2 (p.because any other mixture between T and B will also be a Bestreply)) Check Player2: → What exactly do we have to check? (i) EU2 (p.R) (ii) EU2 (p. because player one can rationally mix between B and T.1 2. if p > 1 – c B2 (p.0) probability “no effort” → both choose no effort (0.. i. Let the game be represented as below with c being a parameter for cost (disutility) of effort with 0 < c < 1: a) Determine both player best-response correspondences.Q) => p1 always chooses effort1 . für player 2 (symmetry): best-reply-correspondence player 2: (1. they are better off than if nobody exerts effort.0) . 1-p) = (q.) Two people can perform a task if. (1. Q) = EU1 (effort1. The worst outcome for a person is reached if she exerts effort while the other person doesn’t.1. If they do so.Exercise 5. 1-q) if p = 1 – c (0.e.1) (0.p-1) (0.Q) < EU1 (effort1. and only if..0). . the both exert effort.0) (p. b) How do the equilibrium choices vary with c? → two pure strat.1) if p > 1 – c B1(q. with probability = 1 or 1 – p = 1 or p = 0 best-reply-choice p = 0 If q > 1 – c <=> 0 > 1 – q – c <=> EU1(no effort1.Q) > EU1 (effort.Q) => Player1 chooses no effort with p = 1 best reply choice i p = 1 → best reply correspondence for player1 1is 0 if q < 1 – c p= p € [0.1] if q = 1 – c 1 if q > 1 – c simi.1) .: q 1-q p 1-p = prob (no effort2) = prob (effort2) = prob (no effort1) = prob (effort1) player1 mixes rationally if EU1 (no effort1.1-q) = (1. show them in a graph and determine all mixed strategy Nash equilibria.15 if q < 1-c <=> 0 < 1 – q – c <=> EU1 (no effort1.3: (A coordination game. equilibria Def. chooses effort1.1) if q < 1 – c if q = 1 – c if q > 1 – c 3 equilibria (2 are indifferent pure nash equilibria (1 is mixed) b) formally nash equilibria (1. Q) <=> 0 * q + (-c) * (1-q) = -c * q + (1-c) * (1-q) <=> 0 = 1 – q – c => Player1 mixes rationally if q = to 1 – c 12. 2.(1-c.c) (1-c.c) (B1) (B2) → Pure Equilibrium dont ??? with c.b) subcu)e : m+1 = k . just mixed strategy Correction EX 3.