Effects of dimers on cooperation in the spatial prisoners dilemma game

Effects of dimers on co op eration in the spatial prisoner’s dilemma game ∗ Haihong Li(), 1 , † Hongy an Cheng(), 1, 2 Qionglin Dai(), 1 Ping Ju(), 1 Mei Zhang(), 2 and Junzhong Y ang() 1 1 Scho ol of Scienc e, Beijing University of Posts and T el e c ommunic ations, Beijing, 100876, Pe ople’s R epublic of China 2 Dep artment of Physics, B eijing Normal University, Beijing, 100875, Pe ople’s R epublic of China Abstract W e in v estigate the ev olutionary prisoner’s dilemma game in structured p opu lati ons by in tro- ducing dimers, whic h are defin ed as that t w o pla y ers in eac h dimer alwa ys hold a same strategy . W e find that influ en ce s of d im er s on co op eration dep end on th e t yp e of d im er s and the p opu la- tion structur e. F or those dimers in whic h p lay ers interac t with eac h other, the co op eration lev el increases w ith the num b er of dimers though the co op eration impro v emen t leve l dep end s on the t yp e of net w ork structur es. On the other hand, the dimers, in whic h there are not m utual inter- actions, will not d o any go o d to the co op eratio n lev el in a single comm unity , but inte restingly , will impro v e the co op eration lev el in a p opulation with t w o comm unities. W e explore the relation- ship b et we en dimers and self-in teractions and find that the effects of d im er s are similar to th at of self-in teractions. Also, we fi nd that the dimers, which are established o v er tw o comm unities in a m ulti-comm unit y netw ork, act as one t yp e of int eraction throu gh which information b et wee n comm unities is comm unicated b y the requiremen t that t w o pla y ers in a dimer hold a same strategy . Keyw ords: Prisoner’s dilemma ga me, Co op eration fr equ ency , Netw orks P ACS num b ers: 02 .50.Le, 87.23 .Kg, 89.75.Fb ∗ This w ork is suppor ted by the pro jects of National Natural Science F oundation of China under Gr a n t No . 10775 022, Gr an t No. 909 21015 a nd the F undamental Resear c h F unds for the Central Univ ersities. † Electronic address: ha iho ngli@bupt.edu.cn 1 I. INTR ODUCTIO N The sp on taneous emergence of co op eration in groups of selfish individuals is ubiquitous in h uman society and biological system s and the evolutionary g ame theory has b een considered as an imp ortant approach to in ves tigate the co op erativ e b eha vior in those systems. As one of the most in triguing games, the evolutionary prisoner’s dilemma game (PDG) has attracted m uch a t ten tion o v er the last few decades [1] for gaining understanding the emergence of co op eration. In a PDG, each individual c ho oses co op eration (C) or defection (D) as her comp eting strategy . When the po pulation is w ell- mixed, a PDG fails to sustain co operat ion, whic h is often at o dds with realit y where mutual co op eration may also b e the final outcome of the ga me [2– 4]. In Now ak a nd Ma y’s seminal w o rks [5, 6], the t w o-dimensional (2D) square lattice and the in teraction b et wee n nearest neigh b ors enable co op erators to protect themselv es against exploitatio n of defectors by for ming compact clusters on the lattice. F rom then on, the evolutionary PDG on a structured populat io n has been a hot sp ot. The influences on co op eration by differen t f a ctors o f mo dels hav e b een studied in tensiv ely . F or example, a la rge set o f strategies were used [7–9], differen t ev olutionary rules w ere in tro duced [10, 11], differen t types of ra ndomnes s were considered [12–16], and so on. Also, the influences of p opulation structure on co op eration ha v e attra cted muc h attention, f o r example, types of net w o rk structure [5, 17 – 26] a nd the top ological prop erties of structure suc h as av erage degree [27], degree-mixing patt erns [28] a nd clustering co efficien t [29, 30]. Generally , the o pinion that the div ersit y in p ersonalities o f individuals o r in p opulation structures could enhance co op eration in an ev olutio nary PD G [10, 31–37] has b een widely accepted. No w, t he in v estigation on an ev olutiona ry PDG has been extended to structured p opula- tions with communities . Lozano et al. studied t he PDG in tw o practical net w orks in which in ter-comm unit y structure and in tra-communit y structure w ere designed and f o und that co op eration dep ends strongly on b oth intra-comm unity heterogeneit y and inte r-communit y connectivit y [3 8, 39]. In [40], t he authors considered the p opulation with tw o comm unities and studied the effects o f in ter-connection on co op eration. They found that co op eration ma y displa y a resonance-like b eha vior with the v aria t ion of the n umber o f inter-connec tions. Consider that, in h uman so ciet y , there alw ays exist man y small coalitions such a s family mem b ers, colleag ues, friends, colla b orat o rs, and so on. The mem b ers within one coalit io n 2 alw ays hold the same b elief and b ehav e in a consisten t wa y . It is an in teresting question that ho w co op eration v aries when the small coalitions are in tro duced in to structured p opulations. In this w ork, w e consider the influence s of the factors on co op eration b y in tro ducing dimers to a n ev olutio nary PDG in structured p opulations. In each dimer, tw o play ers alw a ys hold a same strategy . The pap er is organized as follow s. In section 2, the mo del incorp orating dimers and the categories of dimers ar e in t r o duced. In section 3, w e firstly discuss the effects of different categories of dimers on coo peration in square lattices and ER netw orks [41]. And then, we compare the effects o f dimers a nd self-in teraction. F inally , w e expand t he study to p opulations with tw o comm unities [40] and more rich phenomen a are found. In the final section, we giv e some discussions and conclusions. I I. MODEL In a standard ev olutionary PD G, there are t w o steps in one g ene ration. In the first step, eac h pla y er follows co op eration s x = ( 1 0 ) or defection s x = ( 0 1 ). The pay off of a pla y er x accum ulating b y playing PDGs with her neigh b ors can b e expressed as P x = X y ∈ Ω x s + x Qs y , (1) where s + x denotes the transp ose o f the state vec tor s x . Ω x includes a ll of the neigh b ors of the pla y er x . F or simplicit y , but without loss o f g enerality , w e follow the previous work [12] and ado pt the re-scaled pa y off matrix Q dep ending on one single parameter r for PDG Q =    1 − r 1 + r 0    , 1 < 1 + r < 2 . (2) In this notation, 1 < 1 + r < 2 measures a defector’s temptation to exploit the neigh b oring co op erators a nd − r denotes the suc ker’s pay off f or a co op erator encoun tering a defector. Here, r denotes t he ratio of the costs of co op eration to the net b enefits of co op eration. In the second step, the play er x will adopt the strategy s y of a randomly c hosen neigh b or y with a probabilit y whic h is determined by the pa y off difference b et we en them [15]: W ( s x ← s y ) = 1 / [1 + exp [( P y − P x ) /K ]] , (3) 3 where the pa r a meter K , whic h is analogous to the temp erature in F ermi-D ir a c distribution in statistical phy sics, c haracterizes the sto c hastic uncertainties in making decisions for t he pla y er x [14, 15]. Throughout the work, w e set K = 0 . 1. In this w ork, the play ers in the p opulation are divided in to tw o groups: one with ordinary pla y ers who follow the standard ev olutiona ry PDG, a nd the other with dimers. The play ers in dimers behav e differen tly from the ordinary play ers only in the step of strategy updating : in eac h dimer, the one with higher pa y off up dates her strategy ordinarily a nd t he other just follows her partner. Based on whether the t w o play ers in a dimer play g ame with eac h other or not, dimers can b e classified in t o t w o categories: in t eracting ones (I-D imer) and non-inte racting ones (N-Dimer). Each category of dimers can b e sub divided into lo cal dimers (L-D imer) and distan t dimers (D-D imer) dep ending on whether the play ers in a dimer a re neigh b ors or not on the g iven net w ork. T o b e noted, f o r ID-Dimer, the given net work structure is mo dified b y extra connections b et ween pla y ers in dimers and, f o r NL- Dimer, the g iven netw ork structure is mo dified b y cutting the connections b etw een play ers in dimers. In this work, w e do not consider NL-Dimer and just fo cus on the effects of the other three categories o f dimers on co opera t ion in differen t types of p opulation structures such as square lat tices with p erio dic b oundary conditions and degree z = 4, Erd¨ os-R ´ en yi (ER) [41] net works with mean degree z = 8 and structural p opulatio ns with t w o comm unities. In a t w o -comm unit y-structure p opulation where one communit y is a square la ttice with z = 4 and the other a square lattice with z = 8, dimers are established o ver these t w o comm unities, that is, tw o play ers in a dimer lo cate on differen t comm unities. Throughout this w ork, w e set the num b er of pla y ers in the p opulation to be N = 10 , 000. Initially , in the Monte Carlo s im ulations, m dimers are randomly assigned through the p opulation and play ers ta ke the strategy of C or D with equal pro ba bility . T o measure the co op eration lev el, the co op erator fr equency ρ c will b e monitored when the ev olution of strat- egy pattern reache s its steady state. All the f o llo wing data are obtained with sync hronous strategy up dating and each p oint is g ained b y av eraging 1000 generations after a transien t time of 6000 g eneratio ns and b y av eraging ov er 1 00 indep ende n t realizations. 4 0 . 0 0 . 1 0 . 2 0 . 3 0 2 0 0 0 4 0 0 0 m 0 . 0 0 . 1 0 . 2 0 . 3 0 2 0 0 0 4 0 0 0 0 . 0 0 0 . 0 2 0 . 0 4 0 2 0 0 0 4 0 0 0 ( f ) ( c ) ( b ) ( a ) 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 0 . 0 0 . 1 0 . 2 0 2 0 0 0 4 0 0 0 ( d ) ( e ) r m 0 . 0 0 . 1 0 . 2 0 2 0 0 0 4 0 0 0 r 0 . 0 0 . 1 0 . 2 0 2 0 0 0 4 0 0 0 r FIG. 1: (col or online) The contour graphs of the co op eration leve l ρ c for ev olutionary PDGs, as functions of r and the num b er of dimers m on square lattices with z = 4 (top panel) and ER net w orks with mean degree z = 8 (b ottom p anel). m increases from 0 to 4400 in all cases. (a, d) F or IL-Dimer. (b, e) F or I D-Dimer. (c, f ) F or ND-Dimer. I I I. SIMULA TION RESUL T S AND ANAL YSIS Generally , co op eration is main tained in an ev olutionary PDG on net w orks b y f orming co op erator clusters (C-clusters). The in teractions b et w een co op erators at the b oundaries of C-clusters and tho se inside C-clusters enable co op erators at t he b oundaries to ha v e high pa y o ffs to comp ete with surrounding defectors. In tuitiv ely , the same strategy held by the pla y ers in a dimer mak e it p ossible t ha t co op erator dimers (C-dimer) serv e as seeds for C-clusters and tend to accelerate the expansion of C-clusters. Therefore, it seems that the presence o f dimers in an ev olutio nary PDG w ould impro v e co op eration. How ev er, a s w e sho w b elo w, the effects of dimers on co op eration are not self-eviden t and whether co op eration is impro v ed or not dep ends on the t yp e of dimers and the structure of the underlying net w orks. Firstly , w e inv estigate the effects of dimers o n co op eration in p o pulations without multi- comm unity . The con tour graphs of co op erator frequency ρ c as functions of r a nd the n um b er of dimers m are presen ted in figure 1. The top panel sho ws the results for IL-Dimer, ID -Dimer 5 and ND-Dimer on square lattices, resp ectiv ely . F or the range of r that is not co v ered here, almost all of ρ c reac hes 0 or will descend to 0 a nd no mor e raise of ρ c app ears. Intere stingly , the presence o f interaction b et w een play ers in a dimer plays a decisiv e role on co op eration. Dimers enhance co op eration strongly f or both IL-Dimer and ID-Dimer. Esp ecially , in these t w o situations, the state that all play ers b ecome co op erators (All-C state) may b e r eac hed for a large rang e of r where co op erators die off in the absence of dimers. F urthermore, ID-Dimer sho ws a faster growth of ρ c with the n um b er of dimers than IL-Dimer in the range of r < 0 . 2 , whic h indicates a stronger co op eration enhancemen t fo r ID-Dimer than IL-Dimer. The stronger co op eration enhancemen t for ID -Dimer results from the presence of shortcuts established by the t w o pla y ers in each dimer. T og ether with a same strategy held b y play ers in a dimer, these shortcuts shorten the av erage distance b et w een a ny t wo pla ye rs, whic h sp eeds up the expansion of C-clusters. O n the other hand, the presence of ND-Dimer deteriorates co op eration, i.e., ρ c decreases with the n um b er of dimers. The deteriorat ion of co op eration f or ND-D imer could b e explained by t he b ehav ior of C-dimer. Since there are no in teractions b et we en pla y ers in these C-dimers, the adv antage in pay off f o r co op era t o rs at the b oundaries of C-clusters o v er defectors ma y b e w eak ened. Additionally , due to the absence of in teraction b et w een the pla y ers in these C-dimers, the sp eeding up of the expansion of C-clusters is lost and t he defectors will b enefit fro m these C-dimers prov ided that they are the neigh b ors of these C-dimers. Both of these suppress co op eration in ND-D imer and the suppression o f co operat ion increases with the num b er of dimers. The b ottom panel in figure 1 shows the results fo r IL-D imer, ID- Dimer and ND-Dimer on ER net works, resp ectiv ely . In comparison with the to p panel in figure 1 , t he improv e- men t of co op eration b y IL-Dimer and ID-Dimer is observ ed, though the impro v emen t is not prominen t. How ev er, the influence o f ND -Dimer on co op eration is quite differen t from that on square lattices: ρ c is insensitiv e to the presence of dimers in this situation. Consider t ha t, in ER net w orks where the mean distance b et w een a n y t w o pla y ers is short, the play ers in C-clusters are alw a ys exp osed to defectors and defectors alwa ys b enefit from co op erators in C-clusters in an or dina r y PDG. When ND-Dimer is intro duce d, defectors cannot g et more b enefits from C-dimer than those in the absence of ND-Dimer. Therefore, the deterioration of co op eration b y ND-Dimer on square latt ice s is missed on ER net w orks and co op eration is indep enden t of the n um b er of dimers. It is w ell kno wn that the inclusion of self-in teractio n in an ev olutionary PDG on net works 6 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 ( a ) c , d ( m ) c , s ( m ) r = 0 . 1 0 r = 0 . 2 4 ( b ) c , d ( m ) c , s ( m ) r = 0 . 1 0 r = 0 . 2 0 ( c) c r ( d ) c r FIG. 2: (color online) The relationships b et w een the co op erator frequencies ρ c,s for square lattices with self-interac tion and ρ c,d for those with dimers. (a) ρ c,d is for IL-Dimer. Closed squares are for r = 0 . 10 and op en circles for r = 0 . 24. (b) ρ c,d is for ID-Dimer. C losed s q u ares are f or r = 0 . 10 and op en circles for r = 0 . 20. (c) The closed sym b ols are for IL-Dimer with m = 2000 (squ ares) and m = 4400 (circles). The op en symb ols are for the case without d im er s and in whic h there are 4000 (squares) or 8800 (circles) play ers with self-in teraction. (d) The closed symb ols are for ID-Dimer with m = 2000 (squ ares) and m = 4400 (circles). Th e op en symbols are for the case without dimers and in whic h there are 4000 (squares) or 8800 (circles) p la y ers with self-in teraction. The structure of netw orks for the case with self-inte raction is same as th at in ID-Dimer, wh ic h is mo dified by the shortcuts b et ween p la y ers in dimers. [15] fa v ors co op eration. How ev er, how self-in teraction relat es to realit y is unknow n. As far as it is concerned that a same strategy is held b y tw o pla y ers in a dimer, w e find that the in teraction b et we en play ers in a dimer for IL-Dimer o r ID-Dimer acts as self-in teraction and either IL-D imer or ID-Dimer prov ides a wa y to realize self-in tera ctio n for play ers to some exten t. T o mak e it clear, w e compare the effects of IL- Dimer ( o r ID -Dimer) a nd self-in t era ctio n on co op eration. W e take square lat t ices with z = 4 as examples. It has to b e men tioned that the square lattices with the presence of ID-Dimer are mo dified b y the shortcuts b et ween play ers in dimers. Therefore, the net w orks with self-in teraction to b e studied a re ’square latt ices’ with the same shortcuts as those established b y dimers. T o 7 b e noted, not all pla y ers on net w orks hav e self-interaction and the num b er of pla y ers with self-in t era ctio n is t wice as man y as the n um b er of dimers. F or a giv en n um b er of dimers m , w e monitor the co op erator frequencies ρ c,s ( m ) for net w orks with self-in teraction and ρ c,d ( m ) for net works with dimers. As sho wn in the top panel in fig ur e 2 where the r elationships b et w een ρ c,s ( m ) and ρ c,d ( m ) are presen ted, ρ c,s ( m ) is p ositiv ely correlated with ρ c,d ( m ) and the slop e o f ρ c,d o v er ρ c,s is ro ug hly aro und 1, which means that IL-Dimer or ID-Dimer do es pro vide a w a y to realize self-in teraction. How ev er, some differences exist b et w een the systems with self-in teraction and those with IL-Dimer or ID-D imer. F or example, as sho wn in the b ottom panel in figure 2 where ρ c,s and ρ c,d against r for different m are presen ted, ρ c,s is alwa ys higher than ρ c,d for small r whereas ρ c,s b ecomes lo w er t ha n ρ c,d for large r . Additionally , the ab o v e mo del is inv estigated in square lattices and ER net w orks with some other mean degrees , suc h as z = 8 for square lattices and z = 4 f or ER net w orks. The analogous phenomena could b e obtained. Since that there is an explicit dep endence betw een the v a lue o f the K and the outcome of the prisoner’s dilemma follow ing the F ermi up date rule, w e test the w ork with different K and find t he robustness o f our results against the v ariation of K . Secondly , w e consider the effects of dimers on a p opulation with tw o commun ities: one is a square lattice with z = 4 and the other with z = 8. As stated in the model section, tw o pla y ers in each dimer b elong to differen t commun ities and are randomly selected. Clearly , dimers on this p opulation structure fall into the category of either ID-Dimer or ND-Dimer. The effects of dimers on co op eration for ID-D imer and ND-Dimer are illustrated in the top panel and the b ottom panel in figure 3, resp ectiv ely . The left and middle columns sho w the contour graphs of ρ c on the r - m space for the t w o commun ities resp ectiv ely and the righ t column is for the whole p opulation. Similar t o the p opulations without comm unity structure, co op eration is improv ed for ID-Dimer and a high lev el of co op eration may b e reac hed. A resonance-lik e b eha vior with the v aria tion of the num b er of dimers could b e observ ed when r > 0 . 1, whic h is similar to that in [40] whic h w as discussed part icularly . In con trast with the p opulations without comm unit y structure, ND-Dimer indeed enhances co op eration. F or ND -Dimer, thoug h co op eration is a little do wngraded in the comm unit y with z = 8 in whic h ρ c is hig her, co op eration, b oth in the whole p opulation and in the comm unity with z = 4 in whic h ρ c is low er, is enhanced in comparison with that in the absence of dimers. As men tioned ab ov e, the deterioration of co op eration in a square lattice 8 0 . 0 0 . 1 0 . 2 0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 ( a ) m 0 . 0 0 . 1 0 . 2 ( b ) 0 . 0 0 . 1 0 . 2 ( c) 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 0 . 0 0 0 . 0 4 0 . 0 8 0 . 1 2 0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 ( d ) r m 0 . 0 0 0 . 0 4 0 . 0 8 0 . 1 2 ( e ) r 0 . 0 0 0 . 0 4 0 . 0 8 0 . 1 2 ( f ) r 0 0 . 0 7 0 . 1 4 0 . 2 1 0 . 2 8 0 . 3 5 0 . 4 2 0 . 4 9 0 . 5 6 0 . 6 3 0 . 7 0 FIG. 3: (color online) Th e con tour graphs of th e co op eratio n leve l ρ c , as a function of r and th e n umber of d imers m in tw o comm unities with random dimer. The top panel is for ID-dimer and the b ottom one is for ND-dimer. (a, d) z = 4. (b, e) z = 8. (c, f ) F or the av eraged ρ c o ver t w o comm u nities. with ND -Dimer originates from t w o factors in volving C-dimers: pla y ers in C-dimers lo cating inside C-clusters can not supp ort their partners at the b oundaries of C-clusters, y et supp ort the defectors neigh b oring their partners indirectly . Ho w ev er, for ND- D imer defined ov er tw o comm unities, the deterioration of co op eration b y the first factor is lost and, consequen tly , the do wngrade of co op eration by ND-D imer in the comm unit y z = 8 with a hig h lev el of co op eration b ecomes w eak. On the other hand, the requiremen t that the play ers in dimers hold a same strategy ma y enhance co op eration if one of play ers is in a C-cluster. Com bining these together, w e o bserv e a w eak deterioration of co op eration in o ne comm unit y whereas the enhancemen t of co op eration in the other one and in the whole p opulation. An interes ting comparison can b e made b et we en the results in this w ork and those in [40]. Both w orks consider the ev o lution of co op eration in a p opulation with t w o in teracting comm unities, but the w ay s of in teraction are differen t. Either in tera ctio n through ID-Dimer or interaction through ND-Dimer ma y lead to t w o differen t ev olutions of co op eration in comparison with t hose in [40 ]. F or interaction with ty p e o f ID- Dimer, in teraction b et w een comm unities alw a ys impro v es co operat io n, whic h is indep enden t of the co op eration lev els in 9 isolated comm unities. Actually , ev en when co op eration is extinct in b oth isolated commu - nities, co op eration ma y still reac h a high lev el. F or in teraction with t yp e of ND -Dimer, the co op eration lev els in t w o comm unities may decrease sim ultaneously for w eak interaction, and a resonance-lik e b eha vior of co op eration against in teraction strength may a pp ear for that the co op eration lev els in tw o comm unities are no t in the in t erme diate range. IV. CONCLUSIO NS In conclusion, w e in tro duce dimers in whic h tw o play ers hold t he same strategy to an ev olutiona ry PDG on structured p opulations. The effects of dimers on co op eration are in- v estigated. W e find that influences of dimers on co op eration dep end on the t yp e of dimers and the p opulation structure. F o r example, ID-D imer alwa ys enhances co op eration and a high lev el of co op eration may b e reac hed with a large num b er of dimers. Ho w eve r, the influences of ND-Dimer on co op eration dep end on the p opulation structure strongly: dete- rioration of co op eration o n square lattices, lit t le v ariation of co op eration on ER net w orks, and enhancemen t of co op eration on the p opulation with t w o comm unities. So me in teresting discussions b et w een the results in t his w ork and previous studies are made. Fir stly , w e dis- cuss the relationship b et w een dimers and self-in teractions and find that the effect of dimers is similar to that of self-in teraction. Secondly , we compare the in teraction through inte r- connections b et w een comm unities, where interaction is realized by pla ying games b et w een pla y ers in differen t communities , with that through dimers, where in teraction is realized through holding a same strat egy b y tw o pla y ers in a dimer. F or the in teraction thro ugh dimers, rich phenomena a re observ ed. References [1] R. 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