Static (or Simultaneous- Move) Games of Incomplete Information

Transcription

1 Stti or Simultneous- Move Gmes of Inomlete Informtion Introdution to Stti Byesin Gmes Teori dos Jogos - Filomen Gri

2 Tody s Agend Wht is stti gme of inomlete informtion? Prisoners dilemm of inomlete informtion Cournot duooly model of inomlete informtion Teori dos Jogos - Filomen Gri

3 Stti or simultneous-move gmes of omlete informtion A set of lyers t lest two lyers For eh lyer, set of strtegies/tions Pyoffs reeived by eh lyer for the ombintions of the strtegies, or for eh lyer, referenes over the ombintions of the strtegies All these re ommon knowledge mong ll the lyers. Teori dos Jogos - Filomen Gri 3

4 Stti or simultneous-move gmes of INCOMPETE informtion Pyoffs re no longer ommon knowledge Inomlete informtion mens tht At lest one lyer is unertin bout some other lyer s yoff funtion. Stti gmes of inomlete informtion re lso lled stti Byesin gmes Teori dos Jogos - Filomen Gri 4

5 Cournot duooly model of omlete informtion The norml-form reresenttion: Set of lyers: { Firm, Firm } Sets of strtegies: S [0,, S [0, Pyoff funtions: u, - -, u, - - All these informtion is ommon knowledge Teori dos Jogos - Filomen Gri 5

6 Cournot duooly model of inomlete informtion A homogeneous rodut is rodued by only two firms: firm nd firm. The untities re denoted by nd, resetively. They hoose their untities simultneously. The mrket rie: PQ-Q, where is onstnt number nd Q. Firm s ost funtion: C. All the bove re ommon knowledge Teori dos Jogos - Filomen Gri 6

7 Cournot duooly model of inomlete informtion ont d Firm s mrginl ost deends on some ftor e.g. tehnology tht only firm knows. Its mrginl ost n be IG: ost funtion: C. OW: ost funtion: C. Before rodution, firm n observe the ftor nd know extly whih level of mrginl ost is in. owever, firm nnot know extly firm s ost. Euivlently, it is unertin bout firm s yoff. Firm believes tht firm s ost funtion is C with robbility θ, nd C with robbility θ. All the bove re ommon knowledge Teori dos Jogos - Filomen Gri 7

8 Cournot duooly model of inomlete informtion ont d A solution for the Cournot duooly model of inomlete informtion Firm knows extly its mrginl ost is high or low. If its mrginl ost is high, i.e. C, then, for ny given, it will solve Mx s. t. [ 0 FOC: 0 is firm 's best resonse to, if its mrginl ost is high. Teori dos Jogos - Filomen Gri 8

9 Cournot duooly model of inomlete informtion ont d Firm knows extly its mrginl ost is high or low. If its mrginl ost is low, i.e. C, then, for ny given, it will solve Mx [ s. t. 0 FOC: 0 is firm 's best resonse to, if its mrginl ost is low. Teori dos Jogos - Filomen Gri 9

10 Cournot duooly model of inomlete informtion ont d Firm knows extly its ost funtion C. Firm does not know extly firm 's mrginl ost is high or low. But it believes tht firm 's ost funtion is C with robbility θ, nd C with robbility θ Euivlently, it knows tht the robbility tht firm 's untity is is θ, nd the robbility tht firm 's untity is is θ. So it solves Mx s. t. θ θ 0 [ [ Teori dos Jogos - Filomen Gri 0

11 Cournot duooly model of inomlete informtion ont d Firm 's roblem: 0.. [ [ s t Mx θ θ FOC: Teori dos Jogos - Filomen Gri FOC: 0 [ [ θ θ ene, [ [ θ θ is firm 's best resonse to the belief tht firm hooses with robbility θ, nd with robbility θ

12 Cournot duooly model of inomlete informtion ont d Now we hve Teori dos Jogos - Filomen Gri [ [ θ θ We hve three eutions nd three unknowns. Solving these gives us

13 Cournot duooly model of inomlete informtion ont d Firm hooses θ 3 6 θ 3 6 θ θ 3 Firm hooses if its mrginl ost is high, or if its mrginl ost is low. This n be written s,, One is the best resonse to the other A Nsh euilibrium solution lled Byesin Nsh euilibrium. Teori dos Jogos - Filomen Gri 3

14 Prisoners dilemm of omlete informtion Two susets held in serte ells re hrged with mjor rime. owever, there is not enough evidene. Both susets re told the following oliy: If neither onfesses then both will be onvited of minor offense nd sentened to one month in jil. If both onfess then both will be sentened to jil for six months. If one onfesses but the other does not, then the onfessor will be relesed but the other will be sentened to jil for nine months. Prisoner Prisoner Mum Confess Mum -, - -9, 0 Confess 0, -9-6, -6 Teori dos Jogos - Filomen Gri 4

15 Prisoners dilemm of inomlete informtion Prisoner is lwys rtionl selfish. Prisoner n be rtionl selfish or ltruisti, deending on whether he is hy or not. If he is ltruisti then he refers to mum nd he thinks tht onfess is euivlent to dditionl four months in jil. Prisoner n not know extly whether risoner is rtionl or ltruisti, but he believes tht risoner is rtionl with robbility 0.8, nd ltruisti with robbility 0.. Pyoffs if risoner is ltruisti Prisoner Mum Prisoner Confess Mum -, - -9, -4 Confess 0, -9-6, -0 Teori dos Jogos - Filomen Gri 5

16 Prisoners dilemm of inomlete informtion ont d Given risoner s belief on risoner, wht strtegy should rison hoose? Wht strtegy should risoner hoose if he is rtionl or ltruisti? Pyoffs if risoner is Prisoner rtionl Mum Confess Prisoner Mum -, - -9, 0 Confess 0, -9-6, -6 Pyoffs if risoner is ltruisti Prisoner Mum Prisoner Confess Mum -, - -9, -4 Confess 0, -9-6, -0 Teori dos Jogos - Filomen Gri 6

17 Prisoners dilemm of inomlete informtion ont d Solution: Prisoner hooses to onfess, given his belief on risoner Prisoner hooses to onfess if he is rtionl, nd mum if he is ltruisti This n be written s Confess, Confess if rtionl, Mum if ltruisti Confess is risoner s best resonse to risoner s hoie Confess if rtionl, Mum if ltruisti. Confess if rtionl, Mum if ltruisti is risoner s best resonse to risoner s Confess A Nsh euilibrium lled Byesin Nsh euilibrium Teori dos Jogos - Filomen Gri 7

18 Norml-form reresenttion of stti Byesin gmes The norml-form reresenttion of n n-lyer stti gme G of inomlete informtion seifies: A finite set of lyers {,,..., n}, lyers tion sets A, A, A3,..., An nd their yoff funtions more Remrk: lyer's yoff funtion deends on not only the n lyers' tions but lso her TYPE. T is lyer i 's tye set. i Exmle: T {, }, T {, } Teori dos Jogos - Filomen Gri 8

19 Norml-form reresenttion of stti Byesin gmes: yoffs Plyer i's yoff funtion is reresented s:.,,...,, for ;,...,, i i n n i n i T t A A A t u Exmle: [ ;, u [ ;, u Teori dos Jogos - Filomen Gri 9 Eh lyer knows her own tye. Euivlently, she knows her own yoff funtion. Eh lyer my be unertin bout other lyers' tyes. Euivlently, she is unertin bout other lyers' yoff funtions.

20 Norml-form reresenttion of stti Byesin gmes: beliefs robbilities Plyer i hs beliefs on other lyers' tyes, denoted by i t, t,..., ti, ti,..., tn ti for t T, t T,..., tn Tn. or i t i ti where ti t, t,..., ti, ti,..., tn, t T, t T,..., t Plyer i's beliefs re onditionl robbilities T n n. Exmle: Teori dos Jogos - Filomen Gri 0

21 Strtegy In stti Byesin gme, strtegy for lyer i is funtion s t for eh t T. i i i i s i t i seifies wht lyer i does for her eh tye ti Ti Exmle:, is strtegy for firm in the Cournot model of inomlete informtion version three. Teori dos Jogos - Filomen Gri

22 Byesin Nsh euilibrium: -lyer In stti Byesin -lyer gme { A, A ; T, T ;, ; u, u}, the strtegies s, s re ure strtegy Byesin Nsh euilibrium if for eh of lyer 's tyes t T, s t solves Mx A u, s t; t t t t T nd for eh of lyer 's tyes t T, s t solves Mx A u t T s t, ; t t t Teori dos Jogos - Filomen Gri

23 Byesin Nsh euilibrium: -lyer In stti Byesin -lyer gme { A, A ; T, T;, ; u, u}, the strtegies s, s re ure strtegy Byesin Nsh euilibrium if for eh i nd j, ssume T t, t,...}, T { t,,...} { t s t In the sense of exettion s t bsed on her belief s t s t t t M M j M s t i s t M n M s t n s t M M lyer s best resonse if her tye is t j lyer s best resonse if her tye is t i In the sense of exettion bsed on her belief Teori dos Jogos - Filomen Gri 3

24 Bttle of the sexes At the serte workles, Chris nd Pt must hoose to ttend either n oer or rize fight in the evening. Both Chris nd Pt know the following: Both would like to send the evening together. But Chris refers the oer. Pt refers the rize fight. Pt Chris Oer Prize Fight Oer, 0, 0 Prize Fight 0, 0, Teori dos Jogos - Filomen Gri 4

25 Bttle of the sexes with inomlete informtion version one Now Pt s referene deends on whether he is hy. If he is hy then his referene is the sme. If he is unhy then he refers to send the evening by himself nd his referene is shown in the following tble. Chris nnot figure out whether Pt is hy or not. But Chris believes tht Pt is hy with robbility 0.5 nd unhy with robbility 0.5 Pyoffs if Pt is unhy Chris Oer Pt Prize Fight Oer, 0 0, Prize Fight 0,, 0 Teori dos Jogos - Filomen Gri 5

26 Bttle of the sexes with inomlete informtion version one ont d ow to find solution? Pyoffs if Pt is hy with robbility 0.5 Chris Oer Pt Prize Fight Oer, 0, 0 Prize Fight 0, 0, Pyoffs if Pt is unhy with robbility 0.5 Chris Oer Pt Prize Fight Oer, 0 0, Prize Fight 0,, 0 Teori dos Jogos - Filomen Gri 6

27 Bttle of the sexes with inomlete informtion version one ont d Best resonse If Chris hooses oer then Pt s best resonse: oer if he is hy, nd rize fight if he is unhy Suose tht Pt hooses oer if he is hy, nd rize fight if he is unhy. Wht is Chris best resonse? If Chris hooses oer then she get yoff if Pt is hy, or 0 if Pt is unhy. er exeted yoff is If Chris hooses rize fight then she get yoff 0 if Pt is hy, or if Pt is unhy. er exeted yoff is Sine >0.5, Chris best resonse is oer A Byesin Nsh euilibrium: oer, oer if hy nd rize fight if unhy Teori dos Jogos - Filomen Gri 7

28 Bttle of the sexes with inomlete informtion version one ont d Best resonse If Chris hooses rize fight then Pt s best resonse: rize fight if he is hy, nd oer if he is unhy Suose tht Pt hooses rize fight if he is hy, nd oer if he is unhy. Wht is Chris best resonse? If Chris hooses oer then she get yoff 0 if Pt is hy, or if Pt is unhy. er exeted yoff is If Chris hooses rize fight then she get yoff if Pt is hy, or 0 if Pt is unhy. er exeted yoff is Sine >0.5, Chris best resonse is oer rize fight, rize fight if hy nd oer if unhy is not Byesin Nsh euilibrium. Teori dos Jogos - Filomen Gri 8

29 Cournot duooly model of inomlete informtion version three A homogeneous rodut is rodued by only two firms: firm nd firm. The untities re denoted by nd, resetively. They hoose their untities simultneously. The mrket rie: PQ-Q, where is onstnt number nd Q. All the bove re ommon knowledge Teori dos Jogos - Filomen Gri 9

30 Cournot duooly model of inomlete informtion version three ont d Firm s ost deends on some ftor e.g. tehnology tht only firm knows. Its ost n be IG: ost funtion: C. OW: ost funtion: C. Firm s ost lso deends on some other indeendent or deendent ftor tht only firm knows. Its ost n be IG: ost funtion: C. OW: ost funtion: C. Teori dos Jogos - Filomen Gri 30

31 Cournot duooly model of inomlete informtion version three ont d Firm 's untity deends on its ost. It hooses if its ost is IG if its ost is OW Firm 's untity lso deends on its ost. It hooses if its ost is IG if its ost is OW Teori dos Jogos - Filomen Gri 3

32 Cournot duooly model of inomlete informtion version three ont d Before rodution, firm knows extly its ost is IG or OW. owever, firm nnot know extly firm s ost. Euivlently, it is unertin bout firm s yoff. Firm believes tht if its ost is IG then firm s ost funtion is C with robbility, nd C with robbility. Firm believes tht if its ost is OW then firm s ost funtion is C with robbility, nd C with robbility. Exmle: θ θ s in version two. Teori dos Jogos - Filomen Gri 3

33 Cournot duooly model of inomlete informtion version three ont d Before rodution, firm knows extly its ost is IG or OW. owever, firm nnot know extly firm s ost. Euivlently, it is unertin bout firm s yoff. Firm believes tht if its ost is IG then firm s ost funtion is C with robbility, nd C C with robbility. Firm believes tht if its ost is OW then firm s ost funtion is C with robbility, nd C with robbility. Exmle: θ θ s in version two. Teori dos Jogos - Filomen Gri 33

34 Cournot duooly model of inomlete informtion version three ont d Firm knows extly its ost is high or low. If its ost is IG, i.e. C, then, given its belief on firm, it will solve 0.. [ [ s t Mx u, ; u, ; Teori dos Jogos - Filomen Gri 34 FOC: 0 [ [ ene, is firm 's best resonse to its belief robbility on firm 's, if firm 's ost is IG.

35 Cournot duooly model of inomlete informtion version three ont d Firm knows extly its ost is high or low. If its ost is OW, i.e. C, then, given its belief on firm, it will solve 0.. [ [ s t Mx FOC: u, ; u, ; Teori dos Jogos - Filomen Gri 35 FOC: 0 [ [ ene, is firm 's best resonse to its belief robbility on firm 's, if firm 's ost is OW.

36 Cournot duooly model of inomlete informtion version three ont d Firm knows extly its ost is high or low. If its ost is IG, i.e. C, then, given its belief on firm, it will solve 0.. [ [ s t Mx u, ; u, ; Teori dos Jogos - Filomen Gri 36 FOC: 0 [ [ ene, is firm 's best resonse to its belief robbility on firm 's, if firm 's ost is IG.

37 Cournot duooly model of inomlete informtion version three ont d Firm knows extly its ost is high or low. If its ost is OW, i.e. C, then, given its belief on firm, it will solve 0.. [ [ s t Mx u, ; u, ; Teori dos Jogos - Filomen Gri 37 FOC: 0 [ [ ene, is firm 's best resonse to its belief robbility on firm 's, if firm 's ost is OW.

38 Cournot duooly model of inomlete informtion version three ont d Now we hve four eutions with four unknowns. Teori dos Jogos - Filomen Gri 38 Solving these gives us the following Byesin Nsh euilibrium.,,

39 Cournot duooly model of inomlete informtion version three ont d The Byesin Nsh euilibrium:,,, If firm 's mrginl ost is IG then it hooses whih is its best resonse to firm 's, nd the robbility. If firm 's mrginl ost is OW then it hooses whih is its best resonse to firm 's, nd the robbility. If firm 's mrginl ost is IG then it hooses whih is its best resonse to firm 's, nd the robbility. If firm 's mrginl ost is OW then it hooses whih is its best resonse to firm 's, nd the robbility. Teori dos Jogos - Filomen Gri 39