Online Appendix to. Mandating Behavioral Conformity in Social Groups with Conformist Members

Transcription

1 Onlne Appendx to Mndtng Behvorl Conformty n Socl Groups wth Conformst Members Peter Grzl Andrze Bnk (Correspondng uthor) Deprtment of Economcs, The Wllms School, Wshngton nd Lee Unversty, Lexngton, 4450 VA, USA. Eml: grzlp@wlu.edu Deprtment of Economcs, Centrl Europen Unversty, Ndor u. 9, 1051 Budpest, Hungry. Eml: bnk@ceu.hu

2 1. Summry Ths Onlne Appendx contns seres of mthemtcl dervtons nd results supplementng the mterl n the mn body of, nd the Appendx to, the pper. The presented mterl s orgnzed s follows. Secton explores our set-up under full nformton. Secton 3 relxes the ssumpton tht the men of θ 's s zero. The mn conclusons bsed on the nlyss presented heren re: Frst, the comprson of group welfre under dfferent scenros the centrl m of our pper s n the most generl cse lgebrclly untrctble under both the full nformton cse nd when relxng the ssumpton tht the men of θ 's s zero. In these cses, the comprson of group welfre under dfferent scenros s trctble only under strong ssumptons regrdng the extent of the group's homogenety. Second, the scenro when the plnner mndtes behvor (but not behvorl conformty) s mthemtclly smlr under both the full nformton cse nd the ncomplete nformton scenro n tht fully chrcterzng the problem's soluton for the most generl cse s nlytclly untrctble. We do, however, stte set of suffcent condtons under whch we cn nlytclly fully chrcterze the soluton to the plnner's problem of mndtng behvor (but not behvorl conformty); under those condtons, the soluton to the plnner's problem of mndtng behvor (but not behvorl conformty) n the ncomplete nformton cse concdes wth the soluton to the plnner's problem of mndtng behvorl conformty.. Full Informton Key ssumpton:θ 1,,θ n re common knowledge. Two defntons, whch we use n nlyss below: Def n 1 (Lmted homogenety): ω =ω, = for ll. Def n (Full homogenety): ω =ω, =, θ =θ for ll..1 Non-Coopertve Equlbrum (N) Indvduls ply sttc gme wth full nformton. The equlbrum concept s Nsh equlbrum. To fnd recton functon of ndvdul, we mxmze U (expresson (1) n the pper) choosng, whch mples the followng frst-order condton:

3 (S1) U 1 = ( θ ) (1 )( ) = 0. n 1 After re-rrngng terms n (S1), we obtn (S) = θ. The system of equtons (S) for =1,,n hs unque soluton ( 1 N,, n N ) becuse the mtrx of coeffcents s domnnt dgonl mtrx, nd hence non-sngulr. (The mtrx of coeffcents s dentcl to the one fetured n expresson (A5) of the pper.) Thus, under full nformton, we hve unque Nsh equlbrum n the non-coopertve scenro. From (S), however, t s cler tht ( 1 N,, n N ) depends n complcted wy on 's nd θ 's, whch, s we demonstrte below (see Secton.4), unfortuntely renders welfre nlyss n the generl cse untrctble. (S3) Under 'lmted homogenety' the system (S) for =1,,n becomes = θ for =1,,n. Addng up equtons (S3) for =1,,n nd smplfyng we obtn (S4) = θ. Next, multply LHS nd RHS of (S4) by ()/(n 1) to obtn (S5) 1 + = n 1 n 1 n 1 θ. Addng up (S3) nd (S5) nd solvng for gves (S6) ( ) = θ + θ. n n N Under 'full homogenety', we thus obtn (usng (S6)) N =θ.. Mndtng Behvorl Conformty (MBC) The socl plnner, knowng ll ω 's, 's, nd θ 's, chooses the group-wde common cton to mxmze ωu = ω ( θ ). Obtnng the correspondng frst-order condton nd re-rrngng terms, we hve MBC ω (S7) θ. = ω 3

4 Then, under 'lmted homogenety', we hve MBC =θ. MBC 1 = θ θ nd under 'full homogenety', n.3 Mndtng Behvor (but not Behvorl Conformty) (MB) We defne 'mndtng behvor' s scenro where the socl plnner, knowng ll ω 's, 's, nd θ 's, chooses vector of ctons ( 1,, n ) to mxmze group welfre W ω U, where U s defned n expresson (1) n the pper. Note tht ths s the frst-best soluton from group welfre pont of vew. Hvng n understndng of the lgebr of ths problem, however, s nstructve for our nlyss n Secton 3 of ths Onlne Appendx. The frst-order condtons for 'mndtng behvor' re: W U U (S8) = ω + ω 0, 1,..., n = =, where U (S9) = ( θ ) (1 )( ), (S10) U = ( ). n 1 Pluggng (S9) nd (S10) nto (S8) gves (S11) ω[ ( θ ) (1 )( )] + ω ( ) = 0, whch, fter smplfyng, yelds: ω + ωθ + ω (S1) ω(1 ) ω (1 ) 0. k k = n 1 ( n 1) We cn re-wrte the lst prt of LHS of (S1) s 1 ( ) ω (1 ) = k k (S13) 1 1 ω (1 ) (1 ). ω k k ( n 1) + ( n 1) k k 4

5 Multplyng (S1) by ( 1) nd usng (S13) then gves (S14) 1 ω + ω (1 ) ( ) 1 ω ω ω (1 ). k + k = ωθ n 1 n 1 ( n 1) k k To ese the exposton, let's defne: 1 (S15) b = ω + (1 ) ( n 1) ω nd 1 (S16) b = ω + ω ω (1 ) k k n 1 n 1 ( n 1) k k so tht (S14) cn be wrtten s (S17) b+ b = ωθ. From (S15), observe tht b s strctly postve. The sgn of b, on the other hnd, s mbguous (see (S16)). We would lke to chrcterze the soluton to the system (S17) for =1,,n. In prtculr, we would lke to know f the soluton exsts, nd, f t exsts, whether t s unque. To do so, we would lke to verfy or refute non-sngulrty of the mtrx of coeffcents of system (S17) for =1,,n. Let us cll ths mtrx B=[b ] n n. To check non-sngulrty of B, we try to verfy f the mtrx of coeffcents s domnnt dgonl, whch s true f b > b for ll. As t turns out, the mtrx B s not domnnt dgonl for ll possble prmeter vlues. To see ths, tke the exmple of n=3, 1 = =0.9, 3 =0.1, ω 1 =ω =0.1, ω 3 =0.8, n whch cse b 11 b 1 b 13 = 0.5<0. We lso know tht f mtrx s not domnnt dgonl, t cn stll be non-sngulr. (For the exmple of n=3, 1 = =0.9, 3 =0.1, ω 1 =ω =0.1, ω 3 =0.8, the mtrx B s ndeed non-sngulr: usng Mthemtc, we verfed tht the determnnt of the mtrx equls ) 5

6 We hve been unble to nlytclly verfy or refute non-sngulrty of the mtrx B n the most generl cse. (Workng wth the cse for n=3, we tred to check whether the determnnt of B s ever zero for ny of the permssble prmeter vlues {( 1,, 3,ω 1,ω,ω 3 ): (0,1), ω >0 for ll G}. A numercl explorton tht mde use of Mthemtc's commnds such s FndMnmum nd NMnmze dd not ndcte tht the determnnt of B s equl to zero for ny of the permssble prmeter vlues; the determnnt of B seemed to lwys be postve. Ths, however, s clerly not n nlytclly-grounded sttement nd cnnot be generlzed to the most generl cse for rbtrry n.) As result, we chrcterze the soluton to the system (S17) for =1,,n by fndng suffcent condtons such tht mtrx B s domnnt dgonl nd, hence, non-sngulr. In prtculr, we hve the followng result: Result: The mtrx of the coeffcents of the system defned by (S17) for =1,,n s domnnt dgonl, nd, hence, non-sngulr, f b <0 for ll nd, whch n turn holds f = nd ω =ω for ll,. Proof: To prove ths Result, suppose tht b, defned n (S16), s negtve for ll nd. Then, 1 b = [ ω + ω ω (1 )] k k n 1 n 1 ( n 1) k k 1 n = ω(1 ) + ω(1 )[ ] n 1 ( n 1) 1 = ω (1 ) + (1 ), ( n 1) ω whch s smller thn b =b defned n (S15). Hence, the resultng mtrx of coeffcents of the system defned by (S17) for =1,,n s domnnt dgonl, therefore non-sngulr, nd, thus, the soluton to the system defned by (S17) for =1,,n, whch we denote s ( MB 1,, MB n ), s unque. Note tht ω =ω nd = for ll mples tht b = [ω()n]/(n 1) <0, whch proves the lst prt of Result. Two remrks re n plce. Frst, b n generl cn be postve. For exmple, f nd re close to 1 nd ll other 's re close to 0 (tht s, the socl group cn be descrbe s 6

7 heterogeneous wth respect to the strength of members' nnte conformst tendences), then, from (S16), b >0. Second, when t s unque, the closed-form soluton to the problem of 'mndtng behvor' s evdently lgebrclly very messy. In contrst, we re ble to obtn more trctble closed-form solutons to the problem of 'mndtng behvor' usng Defntons 1 nd. Under 'lmted homogenety', (S14) becomes n (1 ) n (S19) = θ. n 1 ( n 1) Summng up expressons (S19) for =1,,n, we obtn (S0) = θ. Multplyng LHS nd RHS of (S0) by ()n/(n) nd ddng up the resultng expresson nd expresson (S19) gves (S1) + ( n 1) ( n 1) n n (1 ) n = θ + θ Expresson (S1) cn thus be used to solve for the optml vlue of : MB 1 (S) = ( n 1) θ (1 ) n θ n n + +. From (S), t follows tht under 'full homogenety' MB =θ..4 Welfre Anlyss. We know tht 'mndtng behvor' concdes wth the frst-best soluton from the group welfre pont of vew. It s therefore sensble only to ttempt to compre group welfre under 'mndtng behvorl conformty' nd under the non-coopertve equlbrum. From (S), t s cler tht ( N 1,, N n ) depends n complcted wy on 's nd θ 's. As result, the comprson of group welfre under the non-coopertve scenro wth tht under mndted behvorl conformty s n generl lgebrclly untrctble. In fct, the lgebr of group welfre comprson under the two scenros s untrctble even under 'lmted homogenety', n whch cse group welfre under mndted behvorl conformty equls (see (S7)) (S3) W MBC = ( ), ω θ θ nd group welfre under the non-coopertve scenro cn be shown (usng (S6)) to equl 7

8 (S4) W N (1 )( ) = ω θ θ n n ( ) ω(1 ) θ θ n n Evdently, comprson of (S3) nd (S4) s nlytclly untrctble. In contrst, under 'full homogenety', N = MBC =θ, nd, s result, W N = W MBC trvlly. 3. Relxng the ssumpton Eθ =0 for ll Key ssumpton: Eθ =µ. We ntroduce nother two defntons, whch we use n the nlyss below: Def n 3 (Lmted ex-nte homogenety): µ =µ for ll Def n 4 (Full ex-nte homogenety): ω =ω, =, µ =µ for ll. 3.1 Non-Coopertve Equlbrum (N) Group members ply Byesn-Nsh gme. To fnd the non-coopertve equlbrum, follow the steps outlned n Proof of Lemm n the Appendx of the pper. Under the ssumpton tht Eθ =µ, expresson (A4) becomes (S5) 1 E ( θ) (1 )( n 1) E ( θ) = µ. Upon comprson of (S5) wth (A4), t s cler tht they dffer only n terms of the RHS. Thus, the mtrx of the coeffcents mpled by system (S5) for =1,,n s domnnt dgonl nd, thus, non-sngulr. Therefore, there exsts unque vector (E * 1 (θ 1 ),,E * n (θ n )), whch solves the system (S5) for =1,,n nd depends n complcted wy on 's nd µ 's. As result, N, s mpled by (A3), no longer equls θ, but rther equls 1 * (S6) ( θ) = θ + (1 )( n 1) E ( θ), n expresson contnng { } nd µ 's. Expresson (S6) s not sgnfcntly smplfed even f we ssume 'lmted ex-nte homogenety', n whch cse (S5) becomes (S7) 1 E ( θ) (1 )( n 1) E ( θ) = µ. 8

9 Agn, t s cler tht the vector (E * 1 (θ 1 ),,E * n (θ n )), nd thus equlbrum N 's (see (A3)), depend n complcted wy on 's nd µ. In contrst, under 'full ex-nte homogenety', (S7) smplfes to (S8) 1 E ( θ) (1 )( n 1) E ( θ ) = µ. Observe tht the system (S8) for =1,,n s lgebrclly dentcl to the system (S3) for =1,,n f n (S3) we replce wth E (θ ), wth E (θ ), nd θ wth µ. Hence, usng steps nlogous to (S4)-(S6), t follows tht E (θ )=µ nd, usng (S6), we re ble to obtn trctble closed-form soluton for N equl to (S9) N = θ +()µ. 3. Mndtng Behvorl Conformty (MBC) The group plnner chooses to mxmze condton mples ω ( µ ) = 0, whch n turn mples (S30) MBC ω µ ω =. E[ ω( θ ) ]. The resultng frst-order From (S30), t follows tht under both 'lmted ex-nte homogenety' nd 'full ex-nte homogenety', therefore, MBC =µ. 3.3 Mndtng Behvor (but not Behvorl Conformty) (MB) The socl plnner, knowng ll ω 's nd 's, but not knowng exct relztons of θ 's, chooses vector of ctons ( 1,, n ) to mxmze expected group welfre EW = E ω U, where U s W defned n expresson (1) n the pper. The frst-order condtons re E = 0for =1,,n. Usng the sme steps s n Secton.3 of ths Onlne Appendx nd tkng expectton, we obtn: W (S31) E = ω[ ( µ ) (1 )( ) + ω ( ) = 0. n 1 Observe tht equton (S31) s lgebrclly dentcl to equton (S11) f n (S11) we replce θ by µ. Therefore, the nlyss of the system of equtons (S31) for =1,,n s lgebrclly 9

10 dentcl to tht of the system (S11) for =1,,n. We cn therefore mmedtely stte the crucl expresson, whch s n lgebrc equvlent of expresson (S14) wth θ replced by µ : (S3) 1 ω + ω (1 ) ( ) 1 ω ω ω (1 ). k + k = ωµ n 1 n 1 ( n 1) k k From the nlyss n Secton.3 n ths Onlne Appendx, we know tht f 1 (S33) ω + ω (1 ) 0 ωk k > n 1 n 1 ( n 1) k k for ll nd, then the mtrx of coeffcents of the system (S3) for =1,,n s domnnt dgonl nd hence non-sngulr, nd the system hs unque lbet lgebrclly messy soluton. Note n prtculr tht f µ =0 for ll, s ssumed n the pper, nd (S33) holds, then the unque soluton s =0 for ll, tht s, mndtng behvor (but not conformty) mples 'mndtng behvorl conformty'. Note tht the system (S3) for =1,,n s not much smpler even f we ssume 'lmted ex-nte homogenety' when µ0 (n the sense tht t does not llow for trctble closed-form soluton). On the other hnd, 'full ex-nte homogenety' mples, usng nlogous steps s those n Secton.3 of ths Onlne Appendx, tht MB =µ. 3.4 Welfre Anlyss Gven nlyss n Sectons of ths Onlne Appendx, t s cler tht whle the model cn stll be solved when relxng the ssumpton Eθ =0 for ll, the welfre nlyss the centrl m of ths pper becomes untrctble. Algebrclly, the reson s tht n the most generl cse, ssumng ether Eθ =µ or Eθ =µ results n non-coopertve equlbrum cton, whch depends on 's nd µ 's (or µ) nd smlrly untrctble closed-form soluton n the cses of 'mndtng behvor' (but not behvorl conformty) nd 'mndtng behvorl conformty'. Welfre nlyss s n fct untrctble even f we ssume 'lmted ex-nte homogenety' (Defnton 3). In contrst, 'full ex-nte homogenety' does llow for trctblty. Mndtng 10

11 behvor (MB) nd mndtng behvorl conformty (MBC) leds to MB = MBC =µ. Therefore, the expected group welfre under the two regmes equls (S34) ω θ µ ω. MB MBC M EW = EW EW = E( ) = n In contrst, under the non-coopertve scenro, we hve (see (S9)) N =θ +()µ. Therefore, evluted t ( 1 N,, n N ), E( θ ) (1 ) = nd n 1 E U = (1 ) + nd therefore N n 1 (S35) EW = ωe U = nω(1 ) +. Thus, to compre EW M nd EW N, clculte (S36) EW EW N M n 1+ = (1 ). n E( ) = so tht We next check whether the RHS of (S36) s greter or smller thn 1: n 1 + ( n ) (S37) (1 ) 1 =. n 1 n 1 It s strghtforwrd to see tht (S37) s lwys negtve for ny n nd (0,1): If n=, then (S37) equls <0. If n 3, then the numertor of (S37) s qudrtc functon of wth roots 0 nd n, mplyng tht for (0,1) ths functon s negtve. From (S36), we thus hve EW N /EW M <1, whch, becuse EW M <0 (see (S34)), mples tht EW N >EW M. 11