Notes on Strategic Substitutes and Complements in Global Games

Transcription

1 Note on Strategic Subtitute an Complement in Global Game Stephen Morri Cowle Founation, Yale Univerity, POBox , New Haven CT 06520, U S A tephenmorri@yaleeu Hyun Song Shin Lonon School of Economic, Houghton Street, Lonon WC2A 2AE U K hhin@leacuk January 10, A Private Value Global Game A continuum of player chooe action 0 or action 1 Each player i ha a payoff parameter x i The x i are normally itribute in the population with mean θ an preciion β; the mean i unknown to the player, an i itelf normally itribute with mean y an preciion α The payoff to action 1 i x i Thepayoff to action 0 i cl, wherec i a poitive or negative contant an l i the proportion of player chooing action 0 We analyze eentially thi game in Morri an Shin (2002), in the pecial cae where c =1 Inpire by Guenerie (2004) icuion of euctive tability with incomplete information, we olve here for the cae where c can be negative 2 Summary of Reult We will how: Propoition 1 Thi game i ominance olvable (ie, ha an eentially unique trategy profile urviving iterate eletion of trictly ominate trategie) if an

2 only if 2π c 2π α 2 β Thi follow from corollorie 7 an 11 below Notice that in the pecial cae where α 0, thi conition become r π β c But a α,thiconitionbecome r r 2π 2π β c β Thi nicely illutrate an important point in Guenerie (2004): aing aggregate uncertainty (ecreaing α for a fixe β) ten to make ominance olvability eaier to atify in the cae of trategic complementaritie, but harer to atify in the cae of trategic ubtitute Propoition 2 If c< 2π, thi game ha no threhol equilibrium If 2π c 2π, α 2 β there i a unique threhol equilibrium If c> 2π, there are three threhol equilibria α 2 β Thi follow from corollary 5 an propoition 8 below Interetingly, a imilar obervation to Guenerie hol for equilibrium a well: a α 0, thereialway exactly one threhol equilibrium uner trategic ubtitute, but there may be multiple equilibria uner trategic complementaritie q But a α,therei 4π no threhol equilibrium for c<, but there i alway a unique threhol β equilibrium uner trategic complementaritie 2

3 3 Key Expreion We introuce the key function to analyze thi game Oberve that player i believe that any other player private ignal x j i itribute normally with mean αy + βx i an preciion β () α +2β Thu the probability that any opponent oberve a ignal le than x i à β () Φ x αy + βx! i α +2β Note that thi i alo the expecte proportion oberving a ignal le than x The bx-threhol trategy i ½ 1,ifx bx (x) = 0, ifx<bx Now uppoe that all player follow the x -threhol trategy Then the expecte payoff to action 1 i x i ; the expecte payoff to action 0 i à β () cφ x αy + βx! i α +2β Thu the expecte gain to chooing action 1 i à u (x i,x β (),y)=x i cφ x αy + βx! i α +2β Propoition 3 There i a unique value of x olving u (x, x, y) =0(for all y) if an only if c 2π 3

4 PROOF u (x, x, y) =x cφ Ã! β () α (x y) α +2β β () u (x, x, y) = 1 c x α +2β = 1 c φ Ã α β () φ α +2β Ã β () α +2β α α! (x y) (x y) The expreion on the right han ie i minimize when x = y Sinceφ (0) = 1 2π, we have u (x, x, y) βα = 1 c 2 x y=x φ (0) = 1 c 1 2π Thi etablihe the ufficiency of c 2π for uniquene (for any y) Now uppoe that c> 2π Now if y = x = c,wehaveu (x, x, y) =0an 2 x u (x, x, y) y=x= c 2 =1 c 1 2π o there are other olution to u (x, x, y) =0 4 < 0,!

5 4 Strategic Complementaritie (c 0) Aume throughout thi ection that c 0 The reult in thi ection are minor variant of our reult elewhere (eg, in Morri an Shin (2002)), an exploit the fact that u (x i,x,y) i increaing in x i 41 Equilibrium Propoition 4 There i a bx-threhol equilibrium if an only if u (bx, bx, y) =0 Corollary 5 There i a unique threhol equilibrium (for all y) ifanonlyif c 2π 42 Dominance Solvability Let x (y) an x (y) be the mallet an larget olution to the equation u (x, x, y) = 0 Propoition 6 A trategy urvive iterate eletion of trictly ominate trategie if an only if x<x(y) (x) =0an x>x (y) (x) =1 Corollary 7 The game i ominance olvable (for all y) ifanonlyif c 2π 5 Strategic Subtitute (c <0) Aume throughout thi ection that c<0 In thi cae, we have that u (x i,x,y) i trictly increaing in x Note that we o not necearily have u increaing in x i 5

6 51 Equilibrium Propoition 8 If c< 2π, then thi game ha no threhol equilibrium (for ome y) If 2π c 0, then, for all y, there i a unique threhol equilibrium with cutoff equal to the unique olution to u (x, x, y) =0 PROOF A neceary conition for an bx threhol equilibrium i clearly that u (bx, bx, y) =0 If c 0, then (by Propoition 3) there i at mot one threhol equilibrium Now a ufficient conition for thi to be an equilibrium woul be that u (x i,x,y) 0 x i Now à u (x i,x β () β β (),y) = 1+c φ x αy + βx! i x i α +2β α +2β 1 1+c 2π So if c 2π, we have exitence But uppoe thi conition fail, an we have c< 2π Suppoe that y = c The unique value of x olving u (x, x, y) =0i then cbut 2 2 ³ c u x i 2, c 2, c 1 =1+c < 0 2 2π Thi contraict the exitence of the threhol equilibrium 6

7 52 Dominance Solvability Now conier the value of x olving the equation u (x, x,y)=0 Any olution mutlieinthecompactinterval[ 1, 0] Sobycontinuityofu, thereexitx (x,y) an x (x,y), the larget an mallet olution to the equation u (x, x,y)=0 Since lim u (x, x x,y)= an lim u (x, x,y)=, oberve (by continuity) x that u (x, x,y) > 0 for all x>x (x,y) an u (x, x,y) < 0 for all x<x(x,y) Now oberve that if x x,thenu(x, x,y) >u(x, x,y) > 0 for all x> x (x,y) an u (x, x,y) <u(x, x,y) < 0 for all x<x(x,y), implying that x (x,y) x (x,y) an x (x,y) x (x,y) Thubothx (x,y) an x (x,y) are ecreaing in x Now efine z k an z k inuctively by z 0 =, z 0 =, z k+1 = x z k,y an z k+1 = x z k,y Since z 0 > z 1 >z 1 >z 0, we have that by inuction that z k i an increaing equence an z k iaecreaingequencewithz k z k for all k Let z (y) = limz k an z (y) = lim z k k k We will how (by inuction) that trategy urvive k roun of eletion of trictly ominate trategie if an only if x<z k (x) =0an x>z k (x) =1 Vacuouly true for k =0 Suppoe it i true for k Now the payoff gain to chooing action 1 for a player oberving x if hi opponent i following a trategy urviving k roun i at mot u x, z k,y So if x<x z k,y = z k+1,thenaction1 cannot be a bet repone Thu any trategy urviving k +1roun ha x<z k+1 (x) =0 Alo the payoff gain to chooing action 1 for a player oberving x if hi opponent i following a trategy urviving k roun i at leat u x, z k,y So if x<x z k,y = z k+1,thenaction0 cannot be a bet repone Thu any trategy urviving k +1roun ha x>z k+1 (x) =1 Finally, oberve that if x z k+1, z k+1 Thu we have: Propoition 9 A trategy urvive iterate eletion of trictly ominate trategie if an only if x<z (y) (x) =0an x>z (y) (x) =1 Oberve that by contruction z (y) an z (y) are the unique pair of number atifying the following propertie: 1 z (y) z (y); 2 u (z (y),z (y),y)=0; 7

8 3 u (z (y), z (y),y)=0; 4 if (z, z) atify (i) z z, (ii) u (z,z,y)=0, an (iii) u (z, z, y) =0,then z (y) z z z (y) Propoition 10 z (y) =z (y) for all y if an only if β c 2π PROOF Do there exit z,z an y uch that (1) z>z;(2)u (z, z,y)=0;an (3) u (z, z,y) =0?Thuwerequire z > z z = cφ z = cφ Ã β () α +2β Ã β () α +2β Now carrying out the change of variable w = an = β () α +2β z z z! αy + βz! αy + βz (z z), αy + βz the firt equation become > 0 an, ubtracting the thir equation from the econ equation, we have = c [Φ (w + ) Φ (w)] But if there exit > 0 an w atifying the oberve equation, then we can clearly chooe z,z an y o that (1), (2) an (3) above are atifie Now oberve that Φ (w + ) Φ (w) < 2π 8

9 So a neceary conition to olve the above equation i that 2π< c But if thi conition hol, then etting w =0, < c [Φ (w + ) Φ (w)] for ufficiently mall > 0, > c [Φ (w + ) Φ (w)] for ufficently large > 0, obycontinuitythereexit > 0 olving = c [Φ (w + ) Φ (w)] Thu we have: Corollary 11 The game i ominance olvable (for all y) ifanonlyif 6 Common Value c 2π The above analyi all concerne a "private value global game" Conier exactly the ame game, except that the payoff to action 1 i θ intea of x i Thi i the game firttuiebycarlonanvandammeaninmuchoftheapplie literature The correponing ominance olvability conition for thi game can (by imilar metho) be hown to be: β 2π c 2π α 2 () But now a α 0, wehave r π β c ; but a α, the conition i never atifie for any c 6= 0 9

10 Reference [1] Guenerie, R (2004) "When trategic ubtitutabilitie ominate trategic complementaritie: towar a tanar theory for expectational coorination?" [2] Morri, S an H Shin (2002) "Heterogeneity an Uniquene in Interaction Game," forthcoming in The Economy a an Evolving Complex Sytem III, eite by L Blume an S Durlauf Santa Fe Intitute Stuie in the Science of Complexity New York: Oxfor Univerity Pre,