Solution to Tutorial 8

Size: px

Start display at page:

Download "Solution to Tutorial 8"

Diane Holmes
6 years ago
Views:

1 Soltion to Ttorial 8 01/013 Semester I MA464 Game Theory Ttor: Xiang Sn October, 01 1 eview A perfect Bayesian eqilibrim consists of strategies an beliefs satisfying eqirements 1 throgh 4. eqirement 1: At each information set, the Player with the move mst have a belief abot which noe in the information set has been reache by the play of the game. eqirement : Given their beliefs, the players strategies mst be seqentially rational. eqirement 3: At information sets on the eqilibrim path, beliefs are etermine by Bayes rle an the players eqilibrim strategies. eqirement 4: At information sets off the eqilibrim path, beliefs are etermine by Bayes rle an the players eqilibrim strategies where possible. For the ynamic games, we have PBE SPE NE which gives s a stanar metho to fin perfect Bayesian eqilibrim. Ttorial Exercise 1. The following static game of complete information (Matching Pennies) has no pre-strategy Nash eqilibrim bt has one mixe-strategy Nash eqilibrim: each player plays H with probability 1/. Provie a pre-strategy Bayesian Nash eqilibrim of H T H 1, 1 1, 1 T 1, 1 1, 1 Game G xiangsn@ns.e.sg. Sggestion an comments are always welcome. 1

2 MA464 Game Theory /10 Soltion to Ttorial 8 a corresponing game of incomplete information sch that as incomplete information isappears, the players behavior in the Bayesian Nash eqilibrim approaches their behavior in the mixe-strategy Nash eqilibrim in the original game of complete information. Soltion. Consier the following game with incomplete information G(x): where H T H 1 + t 1, 1 1, 1 t T 1, 1 1, 1 Game G(x) Type spaces: T 1 = T = [0, x], t 1 an t are i.i.. ranom variables an niformly istribte on [0, x]. Action spaces: A 1 = A = H, T }. Strategy spaces: S 1 = S = s i is a fnction from [0, x] to H, T }}. Note that G(0) = G. In G(x), sppose (s 1, s ) is a Bayesian Nash eqilibrim, p = Probt 1 : s 1 (t 1) = H}, an q = Probt : s (t ) = H}. For Player 1, given his type t 1 an Player s strategy s, his expecte payoff is E[ 1 (a 1, s (1 + t 1 ) q 1 (1 q), a 1 = H; ) t 1 ] = 1 q + 1 (1 q), a 1 = T. Ths H is a best response if an only if (1 + t 1 ) q 1 (1 q) 1 q + 1 (1 q), that is, t 1 q 4. Hence, we have p = Probt 1 : s 1(t 1 ) = H} = 1 /q 4 x (1) For Player, given his type t an Player 1 s strategy s 1, his expecte payoff is E[ (a, s 1 p + 1 (1 p), a = H; 1) t ] = (1 t ) p + ( 1) (1 p), a = T. Ths H is a best response if an only if 1 p+1 (1 p) (1 t ) p+( 1) (1 p), that is, t 4 p. Hence, we have q = Probt : s (t ) = H} = 1 4 /p x () ewriting Eqations (1) an (), we will have p = 4 + (q 1)x, q = 4 + (1 p)x. As x 0, p, q 1, that is, the Bayesian Nash eqilibrim will converge to the mixestrategy Nash eqilibrim in G. Exercise. In the following extensive-form games, erive the normal-form game an fin all the pre-strategy Nash, sbgame-perfect, an perfect Bayesian eqilibria.

3 MA464 Game Theory 3/10 Soltion to Ttorial 8 1 M [p] [1 p], 4,1 3,0 0,1 Figre 1: Game (a) 1 M [p] M [1 p] M,4 1,3 1, 4,0 4,0 0, 3,3 Figre : Game (b) Soltion. (a) Game (a). Normal-form representation is as follows: S 1 =, M, }, S =, }. Payoff table: Player 1 Player 4, 1 0, 0 M 3, 0 0, 1,, (i) There are two pre-strategy Nash eqilibria (, ) an (, ). (ii) Since there is no sbgame, every Nash eqilibrim is sbgame-perfect, an hence (, ) an (, ) are all the sbgame-perfect Nash eqilibria. (iii) To check whether (, ) is a perfect Bayesian eqilibrim, we nee only to fin beliefs, satisfying eqirements 1,, 3 an 4. eqirement 1: For Player s information set, assign probability p on the left ecision noe, an 1 p on the right ecision noe. eqirement : To spport to be a best response for Player, we shol take p 1. eqirement 3: Since Player 1 chooses, by Bayes rle, Player s belief shol be (1, 0), that is, p = 1.

4 MA464 Game Theory 4/10 Soltion to Ttorial 8 eqirement 4: No information set is off the path, so eqirement 4 gives no restriction on p. Hence (, ) with p = 1 is a perfect Bayesian eqilibrim. To check whether (, ) is a perfect Bayesian eqilibrim, we nee only to fin beliefs, satisfying eqirements 1,, 3 an 4. eqirement 1: For Player s information set, assign probability p on the left ecision noe, an 1 p on the right ecision noe. eqirement : To spport to be a best response for Player, we shol take p 1. eqirement 3: No nontrivial information set is on the path, so eqirement 3 gives no restriction on p. eqirement 4: Since Player 1 chooses, Player s information set is off the path, so p col be arbitrary. Hence (, ) with p 1 is a perfect Bayesian eqilibrim. (b) Game (b). Normal-form representation is as follows: S 1 =, M, }, S =, M, }. Payoff table: Player 1 Player M 1, 3 1, 4, 0 M 4, 0 0, 3, 3, 4, 4, 4 (i) (, M ) is the niqe pre-strategy Nash eqilibrim. (ii) Since there is no sbgame, every Nash eqilibrim is sbgame-perfect, an hence (, M ) is the niqe sbgame-perfect Nash eqilibrim. (iii) To check whether (, M ) is a perfect Bayesian eqilibrim, we nee only to fin beliefs, satisfying eqirements 1,, 3 an 4. eqirement 1: For Player s information set, assign probability p on the left ecision noe, an 1 p on the right ecision noe. eqirement : To spport M to be a best response for Player, we shol take p [ 1 3, 3 ]. eqirement 3: No nontrivial information set is on the path, so eqirement 3 gives no restriction on p. eqirement 4: Since Player 1 chooses, Player s information set is off the path, so p col be arbitrary. Hence (, M ) with p [ 1 3, 3 ] is the niqe perfect Bayesian eqilibrim. Exercise 3. Consier the following game between three Players: Player 1 moves first. He has two actions: U an D. Action U gives the next move to Player, action D gives the next move to Player 3.

5 MA464 Game Theory 5/10 Soltion to Ttorial 8 If Player is given the move he also has two actions: T an B. Action T ens the game, action B gives the move to Player 3. If Player 3 is given the move he also has two actions: an. Both actions en the game. Player 3 oes not know whether the move was given to him by Player 1 or Player. The extensive form is given as follows: where the payoff vector (x, y, z) means that Player 1 U T D B 3,1,0,0,1 1,0, 3,0,4 0,,3 1 receives tility x, Player receives tility y an Player 3 receives tility z. (a) Fin all Nash eqilibria. (b) Fin all sbgame-perfect Nash eqilibria. (c) Fin all perfect Bayesian eqilibria. Soltion. (a) The normal-form representation is as follows: S 1 = D, U}, S = B, T }, S 3 =, }. Payoff table: Player 1 Player an Player 3 B B T T D, 0, 1 1, 0,, 0, 1 1, 0, U 3, 0, 4 0,, 3, 1, 0, 1, 0 There are two pre-strategy Nash eqilibria (D, B, ) an (U, T, ). (b) There is no sbgame, so (D, B, ) an (U, T, ) are all the sbgame perfect Nash eqilibria. (c) Assme Player 3 s belief is p on D an 1 p on B. To check whether (D, B, ) is a perfect Bayesian eqilibrim, we nee only to fin beliefs, satisfying eqirements 1,, 3 an 4. eqirement 1: For Player 3 s information set, assign probability p on the left ecision noe, an 1 p on the right ecision noe. eqirement : To spport to be a best response for Player 3, we shol take p 1. eqirement 3: Since Player 1 chooses D, by Bayes rle, Player 3 s belief shol be (1, 0), that is, p = 1.

6 MA464 Game Theory 6/10 Soltion to Ttorial 8 eqirement 4: No nontrivial information set is off the path, so eqirement 4 gives no restriction on p. Hence (D, B, ) with p = 1 is a perfect Bayesian eqilibrim. To check whether (U, T, ) is a perfect Bayesian eqilibrim, we nee only to fin beliefs, satisfying eqirements 1,, 3 an 4. eqirement 1: For Player 3 s information set, assign probability p on the left ecision noe, an 1 p on the right ecision noe. eqirement : To spport to be a best response for Player 3, we shol take p 1. eqirement 3: No nontrivial information set is on the path, so eqirement 3 gives no restriction on p. eqirement 4: Since Player 1 an Player choose U an T, respectively, Player 3 s belief col be arbitrary since his information set will not be reache. Hence (U, T, ) with p 1 is a perfect Bayesian eqilibrim. Exercise 4. Fin all perfect Bayesian eqilibria in the following signaling games. 1, [p] t 1 [q] 0,1,0 [ 1 ] 3,0 Natre [ 1 ] 1,0 3,1 [1 p] t [1 q], Figre 3: Game 1

7 MA464 Game Theory 7/10 Soltion to Ttorial 8 1,1 1,0 [p 1 ] t 1 [ 1 3 ] [q 1 ] 0,1,1 [p ] t [ 1 3 ] [q ] 1,1 1,0 1,1 [p 3 ] t 3 [ 1 3 ] [q 3 ],1 Figre 4: Game Soltion. (i) Game 1: The normal-form representation is as follows: T = t 1, t }, M =, }, A =, }. Payoff table: Sener 1/, 1 1/, 1 5/, 1/ 5/, 1/ 1, 1 3/, 3/, 0, 1 0, 1/ 3/, 0 3/, 1 3, 1/ 1/, 1/ 5/, 1 1/, 1/ 5/, 1 For example, U(, ) = Prob(t 1 )U(, t 1 ) + Prob(t )U(, t ) = 1 (0, 1) + 1 (3, 1) = (3/, 1) There is the niqe Nash eqilibrim (, ), which is also the niqe sbgame perfect Nash eqilibrim since there is no sbgame.

8 MA464 Game Theory 8/10 Soltion to Ttorial 8 To check whether (, ) is a perfect Bayesian eqilibrim, we nee only to fin beliefs, satisfying eqirements 1, S, an 3. eqirement 1: Assme the probability istribtions on left an right information set are (p, 1 p) an (q, 1 q), respectively, isplaye in the figre. eqirement S: Hols atomatically. (since (, ) is a Nash eqilibrim) eqirement : To spport ( when Sener chooses, when Sener chooses ) to be a best response for, we shol take p 1 3 an q 3. eqirement 3: Since Sener chooses, Bayes rle implies p col be arbitrary, an q = 1. Hence (, ) with p 1 3 an q = 1 is a perfect Bayesian eqilibrim. (ii) Game : The normal-form representation is as follows: T = t 1, t, t 3 }, M =, }, an A =, }. Payoff table: Sener 4/3, 1 4/3, 1 1/3, 0 1/3, 0 1, /3 5/3, 1 1/3, 0 1, 1/3 1, 1 1, /3 /3, 1/3 /3, 0 /3, /3 4/3, /3 /3, 1/3 4/3, 1/3 1, 1 1, /3 0, 1/3 0, 0 /3, /3 4/3, /3 0, 1/3 /3, 1/3 /3, 1 /3, 1/3 1/3, /3 1/3, 0 1/3, /3 1, 1/3 1/3, /3 1, 1/3 For example, U(, ) = Prob(t 1 )U(, t 1 ) + Prob(t )U(, t ) + Prob(t 3 )U(, t 3 ) = 1 3 (0, 1) (0, 0) + 1 (0, 0) = (0, 1/3) 3 There are two pre-strategy Nash eqilibria (, ) an (, ), which are also the sbgame perfect Nash eqilibria since there is no sbgame. To check whether (, ) is a perfect Bayesian eqilibrim, we nee only to fin beliefs, satisfying eqirements 1, S, an 3. eqirement 1: Assme the probability istribtions on left an right information set are (p 1, p, p 3 ) an (q 1, q, q 3 ), respectively, isplaye in the figre, where p 1 + p + p 3 = q 1 + q + q 3 = 1. eqirement S: Hols atomatically. (since (, ) is a Nash eqilibrim) eqirement S: It is obvios that is the best response for when Sener chooses. To spport to be a best response for when Sener chooses, we shol take q 3 1. eqirement 3: Since Sener chooses, Bayes rle implies p 1 = p = p 3 = 1 3 an q 1, q, q 3 col be arbitrary.

9 MA464 Game Theory 9/10 Soltion to Ttorial 8 Hence, (, ) with p 1 = p = p 3 = 1 3 an q 3 1. To check whether (, ) is a perfect Bayesian eqilibrim, we nee only to fin beliefs, satisfying eqirements 1, S, an 3. eqirement 1: Assme the probability istribtions on left an right information set are (p 1, p, p 3 ) an (q 1, q, q 3 ), respectively, isplaye in the figre, where p 1 + p + p 3 = q 1 + q + q 3 = 1. eqirement S: Hols atomatically. (since (, ) is a Nash eqilibrim) eqirement S: It is obvios that is the best response for when Sener chooses. To spport to be a best response for when Sener chooses, we shol take q 3 1. eqirement 3: Since Sener chooses, Bayes rle implies p 1 = p = 1, p 3 = 0 an q 1 = q = 0, q 3 = 1. Hence, (, ) with p 1 = p = 1, p 3 = 0 an q 1 = q = 0, q 3 = 1 is a perfect Bayesian eqilibrim. Exercise 5. Two partners mst issolve their partnership. Partner 1 crrently owns share s of the partnership, partner owns share 1 s. the partners agree to play the following game: partner 1 names a price, p, for the whole partnership, an partner then chooses either to by 1 s share for ps or to sell his or her share to 1 for p(1 s). Sppose it is common knowlege that the partners valations for owning the whole partnership are inepenently an niformly istribte on [0, 1], bt that each partner s valation is private information. What is the perfect Bayesian eqilibrim? Soltion. Figre 5 is the extensive-form representation. It is easy to see Partner s best response is s by, if v p; (p, v ) = sell, if v < p. Note that we assme Partner will by if v = p. This will not affect the or analysis of the game since the probability is zero for v = p. Given Partner s strategy s, Partner 1 s payoff is ps, if v p π 1 = v 1 (1 s)p, if v < p, an expecte payoff is E[π 1 ] = ps Probv : v p} + (v 1 (1 s)p) Probv : v < p} = ps(1 p) + (v 1 (1 s)p)p = (v 1 + s p)p By the first orer conition, we have p (v 1 ) = v 1+s. Each information set of Partner 1 is reache, so the belief on it shol be etermine by Bayes rle, an hence, Partner 1 s belief on each information set is a niform istribtion on [0, 1].

10 MA464 Game Theory 10/10 Soltion to Ttorial 8 Figre 5 Since v 1 [0, 1], p [ s, 1+s ], an hence Partner s beliefs shol be as follows: If p [ s, 1+s ], then the information set p is on the path, so Partner s belief abot Partner 1 s valation is a niform istribtion on [0, 1]; Otherwise, the information set p is off the path, so Partner s belief col be arbitrary. Therefore, the perfect Bayesian eqilibrim is: s 1(v 1 ) = p = v 1 + s, s (v p) = by, if v p sell, if v < p, Partner 1 s belief abot the Partner s valation is a niform istribtion on [0,1], an Partner s belief is given above. Exercise 6. A byer an a seller have valations v b an v s. It is common knowlege that there are gains from trae (i.e., that v b > v s ), bt the size of the gains is private information, as follows: the seller s valation is niformly istribte on [0, 1]; the byer s valation v b = k v s, where k > 1 is common knowlege; the seller knows v s (an hence v b ) bt the byer oes not know v b (or v s ). Sppose the byer makes a single offer, p, which the seller either accepts or rejects. What is the perfect Bayesian eqilibrim when k <? When k >? Soltion. The extensive-form representation of this game is as follows: Clearly, the byer has no incentive to offer p > 1, since the seller will accept p v s an v s is niformly istribte on [0, 1].

11 MA464 Game Theory 11/10 Soltion to Ttorial 8 Natre Byer v s p Seller Accept eject v b p,p 0,v s Figre 6 By backwars inction, the seller s best response is s accept, if v s p s(v s p) = reject, if v s > p. Note that we assme seller will accept if v s = p. This will not affect the or analysis of the game since the probability is zero for v s = p. The byer s maximization problem is: max E[v b p v s p]. 0 p 1 Since v b = kv s, the byer s maximization problem is: max 0 p 1 Therefore, the maximizer is p 0 (kv s p) v s = max 0 p 1 (k/ 1)p. p = 1, if k > 0, if k <. Each information set of byer is reache, so byer s belief is a niform istribtion on [0, 1]. To smmarize, the perfect Bayesian eqilibrim is: s b = 1, if k > p = 0, if k <, an for v s [0, 1], accept, s s(v s p) = accept or reject, reject, if v s < p if v s = p, if v s < p the byer s belief abot the seller s valation is a niform istribtion on [0, 1]. En of Soltion to Ttorial 8

SF2972 Game Theory Exam with Solutions March 19, 2015

SF2972 Game Theory Exam with Soltions March 9, 205 Part A Classical Game Theory Jörgen Weibll an Mark Voornevel. Consier the following finite two-player game G, where player chooses row an player 2 chooses