Solution to Tutorial 6

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1 Solutio to Tutorial /2013 Semester I MA4264 Game Theory Tutor: Xiag Su October 12, Review Static game of icomplete iformatio The ormal-form represetatio of a -player static Bayesia game: {A 1,..., A ; T 1,..., T ; P 1,..., P ; u 1,..., u }. A strategy for player i is a fuctio s i : T i A i. Player i s expected payoff whe her/his type is t i : E t i u i s it i ), a i ; t i ) = Pt i t i )u i s it i ), a i ; t i ). t i T i Bayesia Nash equilibrium: I the static Bayesia game G = {A 1,..., A ; T 1,..., T ; P 1,..., P ; u 1,..., u }, the strategy profile s = s 1,..., s ) is a pure-strategy) Bayesia Nash equilibrium if for each player i ad for each of i s type t i i T i, s i t i) maximizes Player i s expected payoff. A static game of icomplete iformatio ca be trasferred to a game of complete iformatio. 2 Tutorial Exercise 1. Cosider the followig asymmetric-iformatio model of Bertrad duopoly with differetiated products. Demad for firm i is q i p i, p j ) = a p i + b i p j. Costs are zero for both firms. The sesitivity of firm i s demad to firm j s price is either high or low. That is, b i is either b H or b L, where b H > b L > 0. For each firm, b i = b H with probability θ ad b i = b L with probability 1 θ, idepedet of the realizatio of b j. Each firm kows its ow b i but ot its competitor s. All of this is commo kowledge. What are the actio spaces, type spaces, beliefs, ad utility fuctios i this game? What are the strategy spaces? Assume that θb H + 1 θ)b L < 2. Fid the pure-strategy Bayesia Nash equilibrium of this game. xiagsu@us.edu.sg. Suggestio ad commets are always welcome. 1

2 MA4264 Game Theory 2/8 Solutio to Tutorial 6 Solutio. Firm i s actio space: A i = {p: p 0}. Firm i s type space: T i = {H, L}. Firm i s beliefs: θh + 1 θ)l. Firm i s strategy space: S i = {p ih, p il ): p ih, p il A i }. Firm i s utility fuctio for type t): [a p it + b t θp jh + 1 θ)p jl )]p it. For type t = H, L, Firm i s maximizatio problem: By the first order coditio, we have That is, for i = 1, 2, Let b = θb H + 1 θ)b L. The we have max p it [a p it + b t θp jh + 1 θ)p jl )]p it. a 2p it + b t θp jh + 1 θ)p jl ) = 0. p ih = a 2 + b Hθp jh + 1 θ)p jl ), 2 p il = a 2 + b Lθp jh + 1 θ)p jl ). 2 p ih = a 2 + ab H 4 + bb H p il = a 2 + ab L 4 + bb L θp ih + 1 θ)p il, 4 θp ih + 1 θ)p il. 4 Therefore, for i = 1, 2, [ 1 1 p ih = b2 2 a b H) + 1 θ ] 8 abb H b L ), [ 1 1 p il = b2 2 a b L) + θ ] 8 abb H b L ). Exercise 2. Fid all the pure-strategy Bayesia Nash equilibria i the followig static Bayesia game: i) Nature determies whether the payoffs are as i Game 1 or as i Game 2, each game beig equally likely. T 1, 1 0, 0 B 0, 0 0, 0 Game 1 T 0, 0 0, 0 B 0, 0 2, 2 Game 2 ii) Player 1 lears whether ature has draw Game 1 or Game 2, but Player 2 does ot.

3 MA4264 Game Theory 3/8 Solutio to Tutorial 6 iii) Player 1 chooses either T or B; Player 2 simultaeously chooses either L or R. iv) Payoffs are give by the game draw by ature. Solutio. There are two players: Player 1 ad Player 2; Type spaces: T 1 = {1, 2}, ad T 2 = {{1, 2}}; Believes: Player 1 s belief o Player 2 s type is 1 o {T, B}, ad Player 2 s belief o Player 1 s types is 1/2 o T ad 1/2 o B; Actio spaces: A 1 = {T, B}, ad A 2 = {L, R}; Strategy spaces: S 1 = {T T, T B, BT, BB}, ad S 2 = {L, R}. Now we will fid the best-respose correspodece for each player ad each associated type: let a 1, a 2 be Player 1 s actios i Game 1 ad Game 2, respectively, b Player 2 s actio. If Game 1 is draw by Nature, the Player 1 s best-respose correspodece is { a {T }, if b = L; 1b) = {T, B}, if b = R. If Game 2 is draw by Nature, the Player 1 s best-respose correspodece is { a {T, B}, if b = L; 2b) = {B}, if b = R. Sice Player 2 does ot kow which game is beig draw, he will choose b to maximize his expected payoff. The followig table is Player 2 s expected payoff table: T T 1/2 0 T B 1/2 1 BT 0 0 BB 0 1 Thus we get Player 2 s best-respose correspodece: {L}, if a 1 a 2 = T T ; b {R}, if a 1 a 2 = T B; a 1, a 2 ) = {L, R}, if a 1 a 2 = BT ; {R}, if a 1 a 2 = BB. Therefore, by defiitio, we will get all the Bayesia Nash equilibria: T T, L), T B, R) ad BB, R). The reaso is as follows: If Player 2 plays L, the Player 1 must play L i Game 1 ad Player 1 is idifferet betwee T ad B i Game 2). Note that, if Player 1 plays B i Game 2, the Player 2 must play R. So, give that Player 2 plays L, the oly possible pure-strategy Bayesia Nash equilibrium is T T, L) i this case.

4 MA4264 Game Theory 4/8 Solutio to Tutorial 6 If Player 2 plays R, the Player 1 must play B i Game 2 ad Player 1 is idifferet betwee T ad B i Game 1). Note that, R is Player 2 s best respose for T B ad BB. So, give that Player 2 plays R, there are two pure-strategy Bayesia Nash equilibria: T B, R) ad BB, R). Exercise 3. The worker has a outside opportuity v kow by himself. The firm believes that v = 6 ad v = 10 with probabilities 2/3 ad 1/3 respectively. A wage w = 8 is preset by the uio. The firm ad the worker simultaeously aouce whether to accept or reject the wage. The worker will be employed by the firm if ad oly if both of them accept the wage. If the firm accepts the wage, its payoff is 3 if the worker is employed ad 1 otherwise. If the firm rejects the wage, the its payoff is 0 regardless the worker s actio. The worker s payoff is w if he is employed ad v otherwise. Fid the Bayesia Nash equilibria. Depict the extesive-form represetatio i which Nature draws the outside opportuity for the worker. Solutio. Let Game 1 ad Game 2 be as follows: Firm A R A 8, 3 6, 0 Worker R 6, 1 6, 0 Game 1, v = 6 Firm A R A 8, 3 10, 0 Worker R 10, 1 10, 0 Game 2, v = 10 There are two players: firm ad worker; Type spaces: T f = {{1, 2}}, ad T w = {1, 2}; Believes: work s belief o firm s type is 1 o {1, 2}, ad firm s belief o work s types is 2/3 o 1 ad 1/3 o 2; Actio spaces: A w = A f = {A, R}; Strategy spaces: S f = {A, R} ad S w = {AA, AR, RA, RR}. Now we will fid the best-respose correspodece for each player ad each associated type: let a 1 ad a 2 be worker s actios i Game 1 ad Game 2, respectively, b firm s actio. If Game 1 is draw by Nature, the worker s best-respose correspodece is { a {A}, if b = A; 1b) = {A, R}, if b = R. If Game 2 is draw by Nature, the worker s best-respose correspodece is { a {R}, if b = A; 2b) = {A, R}, if b = R.

5 MA4264 Game Theory 5/8 Solutio to Tutorial 6 Firm A R AA 3 0 Worker AR 5/3 0 RA 1/3 0 RR 1 0 Sice firm does ot kow which game is beig draw, it will choose b to maximize its expected payoff. The followig table is firm s expected payoff table: Thus we get firm s best-respose correspodece is {A}, if a 1 a 2 = AA; b {A}, if a 1 a 2 = AR; a 1, a 2 ) = {A}, if a 1 a 2 = RA; {R}, if a 1 a 2 = RR. Therefore, by defiitio, we will get all the Bayesia Nash equilibria: AR, A) ad RR, R). The reaso is as follows: If firm chooses A, the worker should choose A ad R i Game 1 ad Game 2, respectively. Note that, if worker chooses AR, the firm should choose A. So, give that firm chooses A, the oly possible pure-strategy Bayesia Nash equilibrium is AR, A). If firm chooses R, the worker ca choose ay strategy i each game. Note that, oly whe worker chooses RR, R is firm s best respose. So, give that firm chooses R, the oly possible pure-strategy Bayesia Nash equilibrium is RR, R). Exercise 4. Cosider the followig static Bayesia game. Nature selects Game 1 with probability 1/3, Game 2 with probability 1/3 ad Game 3 with probability 1/3. Player I lears whether Nature has selected Game 1 or ot; Player II lears whether Nature has selected Game 2 or ot. Players I ad II simultaeously choose their actios: Player I either T or B, ad Player II either L or R. Payoffs are give by the game selected by Nature. T 0, 0 6, 1 B 1, 6 4, 4 Game 1 T 1, 3 0, 0 B 0, 0 3, 1 Game 2 T 2, 2 2, 2 B 2, 2 2, 2 Game 3 All of this is commo kowledge. Fid all the pure-strategy Bayesia Nash equilibria. Solutio. Leave as Questio 2 of Assigmet 3.

6 MA4264 Game Theory 6/8 Solutio to Tutorial 6 Exercise 5. Cosider a first-price, sealed-bid auctio i which the bidders valuatios are idepedetly ad uiformly distributed o [0, 1]. Show that if there are bidders, the the strategy of biddig 1)/ times oe s valuatio is a symmetric Bayesia Nash equilibrium of this auctio. Proof. There are players; Type spaces: T i = [0, 1], that is, each t i T i is a valuatio; Actio spaces: A i = [0, 1], that is, each a i A i is a bid; Strategy spaces: S i = {s i : T i A i }; Payoff: t i a i, if a i > a j, j i; t u i a i, a i, t i ) = i a i k, if a i is oe of the k largest bids; 0, otherwise. where Aim: show that s 1, s 2,..., s ) is a Bayesia Nash equilibrium, where s i t i) = 1 t i. It suffices to show that for each Player i ad each associated type t i, s i t i) solves E t i u i s it i ), a i ; t i ) = By computatio, we have max E t i u i s it i ), a i ; t i ), a i A i t i T i P i t i t i ) u i s it i ), a i ; t i ) = t i a i ) Proba i > s jt j ), j i) t i a i + Proba i is oe of the k largest bids) k k=2 Proba i is oe of the k largest bids) ProbPlayer i shares the wier of the auctio with aother player, say Player j) = Probs jt j ) = a i ) = Probt j = a i 1 ) = 0 Note that here we use the fact Probt j = l) = 0 for ay l [0, 1] sice t j is uiformly distributed o [0, 1]. Moreover, we have Proba i > s jt j ), j i) = Prob a i > 1 ) t j, j i = Π j i Prob a i > 1 ) t j = Π j i Prob t j < ) 1 a i defiitio of s jt j ) idepedece

7 MA4264 Game Theory 7/8 Solutio to Tutorial 6 Whe a i 1, Π j i Prob t j < ad hece the maximizer is 1.1 Whe a i 1, Π j i Prob t j < ) = Π j i 1 a i = ) 1 a i ) 1 a i = 1, so Player i s expected payoff is t i a i, 1 a i Therefore the expected payoff of Player i is ) 1 a 1 i t i a i ), 1 ) 1 uiform distributio ad the uique maximizer is 1 t i = s i t i). Therefore, the global maximizer is 1 t i = s i t i), ad every Player i s strategy s i t i) = t i costitutes a symmetric) Bayesia Nash equilibrium. 1 Exercise 6. There are 2 players who were at the scee where a crime was committed. But either player kows whether she has bee the oly witess to the crime, or whether there was aother witess as well. Let π be the probability with which each player believes the other player is a witess. Each player, if she is a witess, ca call the police or ot. The payoff to Player i is 2/3 if she calls the police, 1 if someoe else calls the police, ad 0 if obody calls. i) Write dow each player s types ad strategies. ii) For each value of π [0, 1], fid the Bayesia Nash equilibria. Solutio. i) Sice each player kows that he is i the crime scee, each oe has oly oe type: Player 1 s type is Player 1 is a witess, ad Player 2 s type is Player 2 s type is a witess. There is o possibility that they are ot i the crime scee. 2 However, they do t kow whether the other perso is also i the crime scee or ot. Hece, what they are ucertai about is the other player s type. Each Player i has oe types: t i = o the scee. For π [0, 1], each Player i has two strategies C call) ad N ot call). ii) Each Player i thiks that he is playig the followig games: Game 1: if Player i thiks that Player j is also o the spot probability π). The Player i s payoff table is as follows: Player j C N C 2/3 2/3 Player i N 1 0 Game 1: Player j is o the scee Game 2: if Player i thiks that Player j is ot o the spot probability 1 π). The Player i thik that he will get 2/3 if he chooses C, ad 0 otherwise, o matter what Player j chooses. 1 Thaks for Mr. Yusheg Luo for poitig out this issue. 2 Aother acceptable solutio is: Player i s type space is {Player i is a witess, Player i is ot a witess}. While there is o available actio whe the type is Player i is ot a witess.

8 MA4264 Game Theory 8/8 Solutio to Tutorial 6 Player j C N C 2/3 2/3 Player i N 0 0 Game 2: Player j is ot o the scee Therefore, Player i s expected payoff is i the payoff table G 1, ad the game i fact ca be represeted by the payoff table G 2. Player j C N C 2/3 2/3 Player i N π 0 G 1 Player j C N C 2/3, 2/3 2/3, π Player i N π, 2/3 0, 0 G 2 Thus the Bayesia Nash equilibria are as follows: If 2/3 > π 0, the there is oly oe Bayesia Nash equilibrium C, C); If π = 2/3, the there are three Bayesia Nash equilibria C, C), C, N) ad N, C); If 1 π > 2/3, the there are two Bayesia Nash equilibria C, N) ad N, C). Ed of Solutio to Tutorial 6

Notes on Expected Revenue from Auctions

Notes on Expected Revenue from Auctions Notes o Epected Reveue from Auctios Professor Bergstrom These otes spell out some of the mathematical details about first ad secod price sealed bid auctios that were discussed i Thursday s lecture You