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1 Electonic Supplementy Mteil On the coevolution of socil esponsiveness nd behvioul consistency Mx Wolf, G Snde vn Doon & Fnz J Weissing Poc R Soc B 78, ; 0 Bsic set-up of the model Conside the model s descibed in the min text Individuls within n evolving popultion e engged in lge numbe of ounds of piwise intections In ech ound, individuls in the popultion e ndomly mtched in pis Within ny intection between focl individul nd its opponent, ech of the two individuls cn dopt one out of two ctions Pyoffs e obtined ccoding to the pyoff mtix ction ction ction ction () such tht ij epesents the pyoff to focl individul tht plyed ction i ginst n opponent tht plyed j Thoughout, we ssume tht nd () to ensue tht mixed-sttegy ESS exists fo the single-shot gme Individuls cn eithe dopt esponsive o n unesponsive sttegy The sttegy of unesponsive individuls is given by single numbe p, 0 p tht detemines the pobbility with which n individul chooses ction in ech of its intections Responsive individuls tke into ccount the lst intection of thei ptne nd choose thei behviou ccoding to simple evesdopping sttegy: choose ction if opponent chose ction in its lst intection, othewise choose ction Responsiveness is costly nd educes the pyoff of esponsive individuls pe intection by c

2 Pyoffs At ech point in time, ou popultion cn be chcteized by the tuple ( f, θ( p )), whee f gives the fequency of esponsive individuls in the popultion nd θ( p ) gives the fequency distibution of unesponsive sttegies in tht popultion Fo ny distibution θ( p ), we denote its expected vlue by E( p ) nd its vince by V( p ) The expected pyoff of n unesponsive individul with sttegy p in esident envionment fˆ, θˆ( p ˆ) is given by pe intection, whee, ˆ, ˆ ˆ ˆ ˆ W p f θ f w f w (3) ui, ˆ u u, u, uˆ w e the expected pyoffs to n unesponsive individul dependent on whethe it intects with esponsive o n unesponsive individul These pyoffs e given by: ˆ w p pˆ pˆ θ( pdp ˆ) ˆ p pˆ pˆ θˆ( pdp ˆ) ˆ uu, ˆ 0 0 ( ˆ) ( ˆ) ( ˆ) ( ˆ) p E p E p p E p E p w p p p p p p u, ˆ The expected pyoff of esponsive individul in the esident envionment fˆ, θˆ( p ˆ) is given by: pe intection, whee ˆ, ˆ ˆ ˆ ˆ,, uˆ W f θ f w f w c (5) w e the expected pyoffs to n unesponsive individul i, ˆ dependent on whethe it intects with esponsive o n unesponsive individul These pyoffs e given by: w pˆ pˆ pˆ pˆ pˆ pˆ θˆ ( pdp ˆ) ˆ u, ˆ 0 ( ˆ) ( ˆ) ( ˆ) V( pˆ) E( pˆ) E( pˆ) E( pˆ) V( pˆ) E p E p V p w Φ Φ Φ Φ Φ Φ, ˆ (4) (6)

3 whee Φ is the popotion of esponsive individuls tht chose ction in thei lst ound, which chnges ccoding to the diffeence eqution: t ˆ t Φ f Φ fˆ E( pˆ ) (7) The unique stble equilibium solution of this ecuence eltion is given by: 3 Evolutiony Equilibium ( ) fˆ fˆ E pˆ Φ fˆ Responsive individuls invde popultion of unesponsive individuls Conside popultion of unesponsive individuls, tht is f ˆ 0 We ssume tht this popultion is t its evolutiony equilibium At tht point, the fequency distibution of p is constined by the fct tht both ctions chieve equl pyoff The expected pyoff to ction is given by the expected pyoff to ction by E( pˆ) ( E( pˆ)), (9) E( pˆ) ( E( pˆ)) (0) At evolutiony equilibium, the fequency distibution of p thus stisfies whee δ (8) E( pˆ ) δ () Conside the invsion pospects of e esponsive mutnt in such popultion With equtions (3) nd (5), the pyoff diffeence Δ u, ˆ between this mutnt nd the unesponsive esident is given by Δ u, ˆ u, ˆ uu ˆ, ˆ w w c () pe intection, which, with equtions (4) nd (6), simplifies to δ V( pˆ ) c (3) Responsive individuls cn thus invde popultion of unesponsive individuls wheneve the vition pesent in this popultion is lge enough, tht is c V( pˆ ) (4) δ 3

4 Responsive individuls do not go to fixtion Conside the invsion pospects of e unesponsive mutnt with sttegy p in popultion of esponsive individuls, tht is f ˆ With equtions (3) nd (5), the pyoff diffeence between this mutnt nd the esponsive esident is then given by: Δ ( p) w w c (5) u, ˆ u, ˆ ˆ, ˆ pe intection, which, with equtions (4), (6) nd (8) educes to Δ u, ˆ ( p) δ p p pc 4 Fom this follows diectly tht n unesponsive mutnt with p obtins highe pyoff thn the esident since (6) Δ () c 0, (7) u, ˆ which shows tht popultion of esponsive individuls is not evolutionily stble In fct, wheneve, ll unesponsive mutnts with p obtin highe pyoff thn the esident since Δ ( p ) is stictly convex function in p, tht is u, ˆ nd d dδ Δ dp u, ˆ dp u, ˆ p δ 0 (8) (9) Similly, wheneve, ll unesponsive mutnts with p obtin highe pyoff thn the esident Coexistence of unesponsive nd esponsive sttegies Fom the lst two sections follows diectly tht, t ny evolutiony equilibium, both esponsive nd unesponsive sttegies coexist, tht is * 0 f (0) 4

5 Unesponsive individuls dopt pue sttegies At n evolutiony equilibium, unesponsive individuls will lwys dopt one of the pue sttegies, tht is p 0 o p To see this, conside tht n intenl evolutiony equilibium, 0 p, is chcteized by vnishing selection gdient, dw dp 0 (no diectionl selection), nd negtive second deivtive dwu dp 0 ( p is locl ESS) t tht equilibium Fom this follow diectly tht n intenl p cn neve be locl ESS since u dw u dp fˆ δ 0 () Dimophisms e evolutionily unstble It follows fom the bove nlysis tht thee e thee cndidte evolutiony equilibi left: two dimophic equilibi t which esponsive individuls coexist with one unesponsive type ( p 0 o p ) nd timophic equilibium t which esponsive individuls coexist with two unesponsive types ( p 0 nd p ) Let f nd f denote the fequency of individuls tht lwys choose ction ( p ) nd ( p 0 ), espectively, nd f f f the fequency of esponsive individuls in the popultion Dependent on f, f, f the pyoffs to these thee behvioul types pe intection e given by whee W f f f W f f f W f f f w, () w Φ Φ Φ Φ (3), gives the expected pyoff of n esponsive individuls when encounteing nothe esponsive individul, nd Φ the popotion of esponsive individuls tht chose ction in thei lst ound, which is, with eqution (8), given by f f (4) Φ f 5

6 We now show tht, wheneve (5) neithe of the two dimophic equilibi is evolutionily stble The poof fo uns nlogously Conside fist dimophic cndidte equilibium with f 0 At tht cndidte, individuls tht dopt the pue sttegy obtin the pyoff W f f (6) Compe this with the pyoff to n unesponsive mutnt tht lwys dopts ction which is given by W f f (7) which, since is the mximl pyoff tht cn be obtined, is lwys highe thn tht of the unesponsive esident individul A dimophic cndidte with f 0 is thus neve evolutionily stble Conside next dimophic cndidte equilibium with f 0 At tht cndidte, individuls tht dopt the pue sttegy obtin the pyoff W f f (8) Compe this with the pyoff to n unesponsive mutnt tht lwys dopts ction which is given by W f f (9) The pyoff diffeence between the unesponsive mutnt nd the unesponsive esident is thus given by W W f ( ) (30) The unesponsive mutnt cn invde if nd only if this pyoff diffeence is positive, tht is, wheneve f (3) 6

7 t tht cndidte The unesponsive mutnt cn thus invde if nd only if the fequency of esponsive types t the cndidte is not too lge We now show tht condition (3) is in fct lwys tue t ou cndidte To simplify mttes, we mke use of the fct tht we cn lwys escle ou pyoff mtix () by multiplying ll elements with the sme numbe nd/o dding the sme numbe to ll elements without chnging the equilibium popeties () of ou gme In pticul we fist subtct fom ll elements nd then divide ll elements by Ou new pyoff mtix is thus given by ction ction ction 0 (3) whee nd ction with, in view of ou ssumptions () nd (5), nd (33) (34) 0 (35) (36) At the cndidte, the pyoff diffeences between the esponsive esident nd the unesponsive esident type is given by Δ u, W W f 0 (37) which is stictly decesing in the fequency f of esponsive individuls in the popultions since whee δδ δf u, 3 A f 3 f B f 0 (38) A 0 B 0 (39) 7

8 Notice tht the esponsive mutnt cn invde if nd only if f (40) At tht theshold the pyoff diffeence between the esponsive esident nd the unesponsive esident type is given by 3 4, Δu f (4) which, in view of (35) nd (36) is lwys negtive Since the pyoff diffeence is stictly decesing in the fequency f of esponsive individuls in the popultions nd ll esident types hve to chieve equl pyoff t n equilibium, the equilibium fequency of esponsive types t the cndidte equilibium must lwys stisfy (4) Fom this follows tht the unesponsive mutnt tht lwys dopts ction obtins highe pyoff thn the unesponsive esident (see eqution (30)) The cndidte is thus neve n evolutiony equilibium 4 Refeences Hofbue J & Sigmund K (998) Evolutiony Gmes nd Popultion Dynmics (Cmbidge Univ Pess, Cmbidge, UK) 8