Lectures Dynamic Games with Incomplete Information Game Theory Muhamet Yildiz

Transcription

1 Lectures Dynamic Games with Incomplete Information 4.2 Game Theory Muhamet Yildiz

2 Road Map. Sequential Rationality 2. Sequential Equilibrium 3. Economic Applications. Sequential Bargaining with incomplete information 2. Reputation 2

3 What is wrong with this equilibrium? x (2,6) T B L R L R (0,) (3,2) (-,3) (,5) 3

4 Beliefs Beliefs of an agent at a given information set is a probability distribution on the information set. x.---(2,6) For each information set, we must specify the beliefs of 2 /. _~._.. _l~~_ the agent who moves at that information set. L / R L R (0,) (3,2) (-,3) (,5) 4

5 Assessment An assessment is a pair (0,-) where a is a strategy profile and. is a belief system:.(,/) is a probability distribution on for each information set I. 5

6 Sequential Rationality An assessment (0,J) is sequentially rational if 0i is a best reply to O_i given IJ(.I/) for each player i, at each information set player i moves. That is, 0i maximizes his expected payoff given IJ(.I/) and given others stick to to their strategies in the continuation game. 6

7 Sequential Rationality implies x (2,6) B I-f.! 2 L " R L ' R (0,) (3,2) (-,3) (,5) 7

8 Example L (0,0) (3,2) (-,3) R (,5) 8

9 Consistency (a,lj) is consistent ifthere is a sequence (am,lj m) - (a,lj) where am is "completely mixed" and IJm is computed from am by Bayes rule 9

10 Example T B L R L R (0,0) (3,2) (-,3) (,5) 0

11 Example 2~ -/ B 3 f 0. L.-.-.-~ ~-.-? o 2 2 o o o R

12 Sequential Equilibrium An assessment: (0,J) where is a strategy profile and IJ is a belief system, lj(h)e~(h) for each h. An assessment (0,J) is sequentially rational if at each hj, OJ is a best reply to O_j given lj(h). (0,J) is consistent if there is a sequence (om,lj m) ~(0,J) where om is "completely mixed" and IJm is computed from om by Bayes rule An assessment (0,J) is a sequential equilibrium if it is sequentially rational and consistent. 2

13 Example T 2~ B 2 o o L R o 2 o 3

14 Beer - Quiche I) 0 :-:... ('/ ~~c beer quiche 2,\ i {. } dol} 't 3 C.o~, 0 tw 0 0 ~('/ beer {.9} 0 ts ~ OV 0 quiche 3,\ C.o~ dol} 't 2 4

15 Beer - Quiche, An equilibrium I) 0 :-:., ('/ ~~c beer quiche 2,\ i. {. } dol} 't 3 C.o~. 0 tw 0 0 ~('/ 0 ts ~ OV 0.9 beer {.9} quiche 0 3,\ C.o~ dol} 't 2 5

16 Beer - Quiche, Another equilibriu 0 :-:.. 0,,('/ ~~c beer quiche 2, \ {. } 3 C.o~ 0 tw 0 0 ts o beer {.9} quiche 3,\ C.o~ dol} 't 2 6

17 Beer - Quiche, revised I) -2 :-:... ('/ ~~c beer quiche o,\ i {. } dol} 't 3 C.o~, 0 tw 0 0 ~('/ 3,\ C.o~ beer {.9} 0 ts ~ OV 0 quiche dol} 't 2 7

18 Beer - Quiche, Revised I) -2 :-:., ('/ ~~c beer quiche o,\ i 0 {. } dol} 't 3 C.o~, 0 tw 0 0 ~('/ 3,\ C.o~ 0 ts ~ OV 0 beer {.9} quiche 0 dol} 't 2 8

19 Beer - Quiche in weakland I) 0 :-:... ('/ ~~c beer quiche 2,\ i {.8} dol} 't 3 C.o~, 0 tw 0 0 ~('/ beer {.2} 0 ts ~ OV 0 quiche 3,\ C.o~ dol} 't 2 9

20 Beer - Quiche in weakland Unique PBE 0 I~.5 ('/ beer quiche :-:... ~~c 2.5,\ i.5 4 {.8} 3/4 dol} 't 3 C.o~, 0 tw ~('/.5,\ C.o~ 0 ts ~ OV 0.5 beer {.2} quiche 0 dol} 't 2 20

21 .9 Example. (4,4) 2 \ \ \ \ \ \ \ \ \ (5,2) '. \ 2 '. (3,3) (,-5) r ,----- (0,-5) (-,4 ) (0,2) (-,3 ) 2

22 .9 Example - solved a=7/9 2 P=/2 ;or--=----:~ ~ (,-5) \ '. f.l=7/8 \ \ \ \ \ \ (4,4) (5,2) '. (3,3). \ 2 '. (0,-5) (-,4 ) (0,2) (-,3 ) 22

23 Sequential Bargaining. -period bargaining - 2 types 2. 2-period bargaining - 2 types 3. -period bargaining - continuum 4. 2-period bargaining - continuum 23

24 Sequential bargaining -p A seller S with valuation 0 A buyer B with valuation v; B knows v, S does not v = 2 with probability TC = with probability -TC S sets a price p ~ 0; B either buys, yielding (p,v-p) or does not, yielding (0,0). P 2-p s H o p o I-p L o o 24

25 Solution. B buys iff v > p;. If P ::;;, both types buy: S gets p. 2. If < P ::;; 2, only H-type buys: S gets np. 3. If P > 2, no one buys. 2. S offers if C < ~, 2 if C > ~ p 2 25

26 Sequential bargaining 2-perio( A seller S with valuation Po ~ 0; 0 2. B either A buyer B with valuation v, '. At t = 0, S sets a ~ rice buys, yielding (Po,v-Po) or does not, then B knows v, S does not 3. At t =, S sets another v = 2 with probability l price P ~ 0; = with probability -l 4. B either buys, yielding (i:ipj,i:i(v-p)) or does not, yielding (0,0) 26

27 Solution, 2-period. Let J.l = Pr(v = 2history at t=). 2. At t =, buy iff v ;::: P; 3. If J.l > Yz, P = 2 4. If J.l < Yz, P =. 5. If J.l = Yz, mix between and B with v= buys at t=o if Po <. 7. If Po >, J.l = Pr(v = 2 Po,t= ) ::; n. 27

28 Solution, cont. t </2 I. f..t = Pr(v = 2IPo,t=) < n </2. 2. Att =, buy iff v ~ P; 3. P =. 4. B with v=2 buys at t=o if (2-po) > 0(2-) = 0 ~ Po ::; Po = : n(2-0) + (l-n)o = 2n(-0) + 0 < -0+0 =. 28

29 Solution, cont. t >/2 If v=2 is buying at Po > 2-0, then Il = Pr(v = 2Po > 2-8,t=) = 0; P = ; v = 2 should not buy at Po > 2-8. If v=2 is not buying at 2> Po > 2-0, then Il = Pr(v = 2Po > 2-8,t=) = : > 2; P = 2; v = 2 should buy at 2 > Po > 2-8. No pure-strategy equilibrium. 29

30 Mixed-strategy equilibrium, t > 2. For Po > 2-0, f.l(po) = %; 2. ~(Po) = - Pr(v=2 buys at Po) j3(po)jr - Jr f. = = - <;::> j3(po)jr = -Jr <;::> j3(po) =-. j3(po)jr+(-jr) 2 Jr 3. V = 2 is indifferent towards buying at Po: 2- Po = OY(Po) :> Y(Po) = (2- Po)/o where Y(Po) = Pr(P=Ipo) 30

31 Sequential bargaining -period A seller S with valuation 0 A buyer B with valuation v; B knows v, S does not v is uniformly distributed on [0,] S sets a price p 2 0; B either buys, yielding (p,v-p) or does not, yielding (0,0). 3

32 Sequential bargaining, v in [O,a] period: B buys at p iff v ~ p; S gets U(p) = p Pr(v ~ p); v in [O,a] => U(p) = p(a-p)/a; p = a/2. 32

33 Sequential bargaining 2- periods If B does not buy at t = 0, then at t= S sets a price P ~ 0; B either buys, yielding (8P' 8(V-P)) or does not, yielding (0,0). 33

34 Sequential bargaining, v in [0,] 2 periods: (Pa'P) At t = 0, B buys at Po iff v ~ a(po); P = a(po)/2; Type a(po) is indifferent: S gets a(po) - Po = 8(a(po) - P ) = 8a(po)/2 ~ a(po) = Po/(-8/2) 34

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