Population Protocols that Correspond to Symmetric Games

P opulation Proto cols that Corresp ond to Symmetric Games ✩ Olivier Bournez a , J´ er ´ emie Chalopin b , Johanne Cohen c , Xavie r Ko egler d a Ec ole Po lyte chnique & L ab or atoir e d’Informatique (LIX), 91128 Palaise au Ce dex, F r anc e b CNRS & L ab or atoir e d’Informa t ique F ondamentale de Marseil le, CNRS & Ai x -Marseil le Un iversit ´ e, 39 rue J oliot Curie, 13453 Marseil le Ce dex 13, F r anc e c CNRS & PRiSM, 45 Avenue des Etats Unis, 78000 V ersail les, F ra n c e d ´ Ec ole Normale Sup´ eri eu re & Universit ´ e Paris Dider ot - Paris 7, Case 7014, 75205 Paris Ce dex 13, F r anc e Abstract P opulation proto cols ha ve b een in tro duced b y Angluin et al. as a mo del of net w orks consisting of v ery limited mobile a gen ts that in teract in pairs but with no c o n trol o v er their ow n mo veme nt. A collection of anon ymous a gen ts, mo deled by finite automata, inte ra ct pairwise according to some rules tha t up date their stat es. The mo del has b een conside red as a computational mo del in se veral pa - p ers. Input v alues are initially distributed among the agen ts, and the agen ts m ust ev en tually con v erge to the the correct output. Predicates on the init ia l configurations that can b e computed by suc h pro t o cols hav e b een character- ized under v arious h yp otheses. The mo del has initially b een motiv ated by sensor-net w orks, but it can b e seen more generally as a mo del of net w o r ks of anon ymous agen ts in teracting pa irwise. This includes sensor net w o r ks, ad-ho c net works , or models from c hemistry . In an orthogonal w ay , sev eral distributed systems hav e b een termed in literature as b eing realizations of games in the sense of game theory . In this pap er, we in ves tig ate under whic h conditions p opulation proto cols, or ✩ This work and all authors w ere partly supp orted by A NR Pro ject SOGEA and b y ANR Pro ject SHAMAN, Xavier Ko eg ler was par tly suppo rted by COST Action 295 DYNAMO and ANR Pro ject ALADDIN Pr eprint submitte d to Elsevier Novemb er 10, 2021 more generally pairwise in teraction rules, can b e considered as the result o f a symmetric game. W e prov e that not all rules can b e considered as symmetric games.W e characterize the computational p ow er of symmetric games. W e pro ve that they hav e very limited p ow er: they can coun t un til 2 , compute ma jority , but they can not eve n coun t un til 3. As a side effect of our study , w e also prov e that any p opulatio n proto col can b e simulated b y a symmetric one (but not necess arily a game). Key wor ds : P opulation Proto cols, Computation Theory, Distributed Computing, Algorithmic G ame Theory 1. In tro duction The computational p ow er of netw orks of anon ymous resource-limited mo- bile agents ha s b een in ves tig ated rece ntly . In particular, Angluin et al. prop osed in [2] a mo del of dis t r ibuted compu- tations. In this mo del, called p opulation pr oto c ol s , finitely man y finite-state agen ts in teract in pairs c hosen b y an adv ersary . Eac h interaction has the effect of updat ing the state of the t w o agents according to a joint transition function. A proto col is said to (stably) c ompute a predicate on the initial states of the agen ts if , in an y fair execution, af t er finitely many in teractions, all agen t s reac h a common output that corresponds t o the v alue of the predicate. The mo del w as orig ina lly prop osed to mo del computations realized b y sensor net works in whic h passiv e ag ents are carried a long b y other en tities. The canonical example of [2] corresp onds to sensors attac hed to a flo c k of birds and that mus t b e programmed to che ck some global prop erties, lik e determining whether more than 5% of the p opulation has elev ated temp er- ature. Motiv ating scenarios a lso include mo dels of the propagation of trust [9]. Muc h of the work so far on p opulation proto cols has concen trated on c haracterizing whic h predicates on the initial states can be computed in dif- feren t v ariants of the mo del and under v ario us assumptions. In particular, the predicates computable by the unrestricted p opulation proto cols fr o m [2] ha v e b een characterize d as b eing precisely the semi-linear pr edicates, that is those predicates on counts of input agen ts definable in first- order Presburger arithmetic [20]. Semi-linearity w as sho wn to b e sufficien t in [2] and necessary in [3]. 2 V ariants considered so far include restriction to one-w ay comm unications, restriction to particular in t eraction graphs, to random in teractions, with p o s- sibly v arious kind of failures of agen ts. Solutions to classical pro blems of distributed algo r ithmics hav e also b een considered in this mo del. R efer to [4] for a surv ey and a complete discus sion. The p opulatio n proto col mo del shares man y features with other mo dels already consid ered in the literature. In part icular, mo dels of pairwise in terac- tions ha ve b een used to study the propagation of diseases [14], or rumors [8]. In che mistry , the c hemical master equation has b een justified using (sto c has- tic) pairwise in teractions b et we en the finitely man y mo lecules [17, 13]. In that sense , the mo del of p opulation prot o cols may b e considered as funda- men tal in sev eral fields of study , since it app ears as so on as anony mous agen ts in teract pairwise. In an orthogonal wa y , pairwise in teractions b et w een finite-state agen ts are sometimes motiv ated b y the study of the dynamics of particular t wo-pla y er games from game theory . F or example, pap er [10] considers the dynamics of the so-called P AV LO V b eha vior in the iterated prisoner lemma. Sev eral results ab out the time of con v ergence of this particular dynamics to w ards the stable state can be found in [10], and [11], for rings, and complete graphs. The purp ose of this article is to b etter understand whether and when pairwise in teractions, and hence p o pulation proto cols, can b e considered as the result of a game. W e wan t to understand if restricting to rules t ha t come from a symmetric game is a limita tion, and in particular whether restricting to rules that can b e termed P AV LO V in the spirit of [10 ] is a limitation. W e do so by giving solutions to sev eral basic problems using rules of interactions asso ciated to a symmetric ga me, and by c hara cterizing they pow er: W e pro v e that they can coun t until 2, they can compute M AJ O RI T Y , but t hey can not ev en coun t until 3. As suc h proto cols m ust also b e symmetric, w e are also discussing whether restricting to symmetric rules in p opulation prot o cols is a limitation. W e pro ve that any p o pula t io n proto col can b e sim ulated by a symmetric one (but not necessarily a game). In Section 2, w e briefly recall p opulation proto cols. In Section 3, w e r ecall some basics from g ame t heory . In Section 4, w e discuss ho w a game can be turned into a dynamics, and in tro duce the notion of Pa vlovian p opulation proto col. In Section 5 w e prov e that any symmetric deterministic 2- states p opulation prot o col is P a vlov ian, and that the pro blem of computing the OR, AND, as w ell as the leader election and ma jority problem admit P a vlovian 3 solutions. W e then characterize there p ow er by pro ving that they can coun t un til 2, but they can not count un til 3 in Section 6. W e prov e that symmet- ric p opulation proto cols, unlik e the restricted class of Pa vlovian p opulation proto cols can compute all semi-linear predicate in Section 7. R elate d work. P o pulation proto cols ha ve b een in tro duced in [2], and pro v ed to compute all semi-linear predicates. They hav e b een pro v ed not to b e able to compute more in [3]. V arious restrictions on the initial mo del ha ve b een considered up to no w. An (almost) up to date surv ey can b e found in [4]. V ariants include discussions ab out the influence o f remo ving the assump- tion of tw o-wa y in teraction: One-w ay interaction mo dels include v a rian ts where agen ts comm unicate by anon ymous message-passing, with immediate deliv ery or dela ye d deliv ery , or where agen ts can record it has sen t a mes sage, or queue incoming messages [1]. Ho w ev er, as far as w e kno w, the constrain t of restricting to symme tr ic rules has not b een ye t explicitly considered, nor restricting to rules that corr espo nd to games in the p o pula t io n proto col lit- erature. More generally , p opulation proto cols arise a s soon as p opulations of anon y- mous agents interact in pairs. Our origina l motiv ation w a s to consider rules corresp onding to t w o- pla y ers games, and p o pula t ion proto cols aro se quite inciden tally . The main adv antage of the [2] settings is that it pro vides a clear understanding o f what is called a computation b y the mo del. Many distributed systems ha v e b een described as the result of games, but as far as w e kno w there has not b een attempts to c hara cterize what can b e computed b y g ames in the spirit of this computationa l mo del. In this pap er, we turn tw o play ers games into dynamics ov er agen ts, by considering P AV LO V b ehav io r . This is inspired by [10, 11, 16 ] that con- sider the dynamics of a pa rticular set of rules termed the P AV LO V b ehav - ior in the iterated prisoner lemma. The P AV LO V b ehavior is sometimes also termed WIN-ST A Y, LOSE-SHIFT [18, 5]. Notice , t hat we extended it from t wo-strategies t w o- pla ye rs games to n- strategies t wo-pla y ers games, whereas ab o ve references only talk ab out tw o- stra t egies tw o- pla y ers games, and mostly of the iterated prisoner lemma. This is clearly not the o nly w a y to asso ciate a dynamic to a ga me. They are sev eral famous classical approac hes: The first consists in repeating games: see for example [19, 6]. The sec ond in using mo dels from ev olutiona ry game theory: refer to [15 , 21 ] for a presen tation of this latter approac h. The ap- proac h considered here falls in metho d that consider dynamics obtained by 4 selecting at eac h step some play ers and let them play a fixed game. Alter- nativ es to P AV LO V b ehavior could include M Y O P I C dynamics (at each step eac h pla y er c ho oses the b est resp onse to previously pla y ed strategy by its a dvers a ry), or the w ell-known and studied F I C T I O U S − P LAY E R dy- namics (at each step eac h pla y er c ho oses the b est resp onse to the statistics of the past history o f strategies play ed b y its adve rsary). W e refer to [12, 6] for a presen ta t io n of results kno wn ab out the pro p erties of the obtained dy- namics according to the prop erties of the underlying game. T his is clearly non-exhaustiv e, and w e refer to [5] fo r an incredible zo ology of p ossible b e- ha viors for the particular iterated prisoner lemma game, with discussions o f their compared merits in experimental tournaments . W e obtain a c hara cterization of the p ow er o f P avlo vian p opulation pro to- cols in terms of closure prop erties that sho w t ha t they can count until 2 , but not un til 3. Notice that sev eral v aria n ts of (one-wa y) p opulation proto cols ha v e b een ch a racterized in [1 ] in a C O U N T k hierarc h y . The class obtained here seems close to the C O U N T 2 lev el of this latter hierarc hy [1]. Ho w ev er, on one hand, this is not exactly this class (f or example M AJ O RI T Y is com- putable but not in the C O U N T 2 lev el), and on the o ther hand, as no class there is for ma lly pro v ed to corresp ond to C O U N T 2 , this shows that the class considered here is differen t, and no t reducible to the v a rian ts of [1]. Notice that a preliminary v ersion of this article has b een presen ted in Complexity of Simple Pr o gr a ms CSP’08 . Compared to this conference v er- sion, w e simplifie d some constructions, we added a few proto cols, w e extend ed deeply related w ork discussions, and mainly , we solv ed the statemen t s con- jectured there: w e prov ide here a c haracterization of the p ow er o f Pa vlovian p opulation proto cols, whereas it was op en a t the time of the presen tation of this preliminary vers io n. 2. P opulation Proto cols A proto col [2, 4 ] is giv en b y ( Q, Σ , ι, ω , δ ) with the f ollo wing comp onents. Q is a finite set of states . Σ is a finite set of i n put symb ols . ι : Σ → Q is the initial state mapping, and ω : Q → { 0 , 1 } is the individual output function. δ ⊆ Q 4 is a join t transition r elat io n tha t describes ho w pairs of agen ts can in teract. Relation δ is sometimes described b y listing all p ossi- ble interactions using t he notat io n ( q 1 , q 2 ) → ( q ′ 1 , q ′ 2 ), or ev en the notation q 1 q 2 → q ′ 1 q ′ 2 , for ( q 1 , q 2 , q ′ 1 , q ′ 2 ) ∈ δ (with the con v en t io n that ( q 1 , q 2 ) → ( q 1 , q 2 ) when no rule is sp ecified with ( q 1 , q 2 ) in the left-ha nd side). The proto col 5 is termed de terministic if for all pairs ( q 1 , q 2 ) there is only one pair ( q ′ 1 , q ′ 2 ) with ( q 1 , q 2 ) → ( q ′ 1 , q ′ 2 ). In that case, we write δ 1 ( q 1 , q 2 ) for the unique q ′ 1 and δ 2 ( q 1 , q 2 ) for the unique q ′ 2 . Notice tha t , in general, rules can b e non-symmetric: if ( q 1 , q 2 ) → ( q ′ 1 , q ′ 2 ), it do es no t neces sarily follow that ( q 2 , q 1 ) → ( q ′ 2 , q ′ 1 ). Computations of a proto col pro ceed in t he follo wing wa y . The computa- tion t ak es place among n agents , where n ≥ 2. A c onfigur ation of the system can b e described by a v ector of all the agen ts’ states. The state of eac h agen t is an elemen t of Q . Because agen ts with the same states are indistin- guishable, eac h configuration can b e summarized as an unor dered m ultiset of states, and henc e of elemen ts of Q . Eac h agent is giv en initially some input v alue from Σ: Eac h agen t’s initial state is determined b y applying ι to its input v a lue. This determines the initial configuration o f the p opulation. An execution of a proto col pro ceeds from the init ia l configura tion by in teractions b et wee n pairs of agents. Suppo se that t wo agents in state q 1 and q 2 meet and hav e a n in teraction. They can c hange in to state q ′ 1 and q ′ 2 if ( q 1 , q 2 , q ′ 1 , q ′ 2 ) is in the transition relation δ . If C and C ′ are t w o configurations, w e write C → C ′ if C ′ can b e obtained from C by a single in teraction of tw o agen ts: this means that C con tains tw o stat es q 1 and q 2 and C ′ is obtained b y replacing q 1 and q 2 b y q ′ 1 and q ′ 2 in C , where ( q 1 , q 2 , q ′ 1 , q ′ 2 ) ∈ δ . An exe cution of the proto col is an infinite sequen ce of configurations C 0 , C 1 , C 2 , · · · , where C 0 is an initial configuration and C i → C i +1 for all i ≥ 0. An execution is fair if for all configurations C that app ear infinitely often in the execution, if C → C ′ for some configuration C ′ , then C ′ app ears infinitely of ten in the execution. A t any p oin t during an exec utio n, eac h agen t’s state determines its out- put a t t hat time. If the a gen t is in state q , its output v alue is ω ( q ). The configuration output is 0 (respectiv ely 1) if all the individual o utputs are 0 (resp ectiv ely 1 ). If the individual outputs are mixed 0s a nd 1 s then the output of the configuration is undefined. Let p b e a predicate o ver multise ts of elemen ts of Σ. Predicate p can b e considered a s a function whose range is { 0 , 1 } and whose do main is the collection of these multis ets. The predicate is said to b e computed b y the proto col if, for ev ery mu lt iset I , and ev ery fair execution tha t starts from the initial configuration corresp onding to I , the output v alue o f ev ery agen t ev en tually stabilizes to p ( I ). Multisets o f eleme nts of Σ are in clear bijection with elemen ts of N | Σ | : a 6 m ultiset o v er Σ can be iden tified by a ve ctor of | Σ | comp onen ts, where each comp onen t represen ts the multiplicit y of the corresp onding elemen t of Σ in this m ultiset. It follo ws that predicates can also b e considered as functions whose range is { 0 , 1 } a nd whose domain is N | Σ | . The follow ing w as t hen pro v ed in [2, 3]. Theorem 1 ([2, 3]) . A pr e dic ate is c omputable in the p opulation pr oto c ol mo d e l if and o n ly i f it is semiline ar. Recall that semilinear sets are kno wn to corresp ond to predicates on coun ts o f input ag en ts definable in first-order Presburger arithmetic [20]. W e will use the following notation as in [1]: the set of all functions from a set X to a set Y is denoted b y Y X . Let Σ be a finite non-empt y set. F o r all f , g ∈ R E , w e define the usual v ector space op erations ( f + g )( σ ) = f ( σ ) + g ( σ ) for all σ ∈ Σ ( f − g )( σ ) = f ( σ ) − g ( σ ) for all σ ∈ Σ ( cf )( σ ) = cf ( σ ) for all σ ∈ Σ , c ∈ R ( f .g )( σ ) = P σ f ( σ ) g ( σ ) . Abusing not a tion as in [1], w e will write σ for the f unction σ ( σ ′ ) = [ σ = σ ′ ], for all σ ′ ∈ Σ, where [ co ndition ] is 1 if condition is t r ue, 0 otherwise. 3. Game Theory W e now recall the simplest concepts from Game Theory . W e fo cus o n non-co op erat ive games, with complete information, in extensiv e form. The simplest game is made up of tw o play ers, called I and I I , with a finite set of options, called pur e str ate gies , S tr at ( I ) a nd S tr at ( I I ). Denote b y A i,j (resp ectiv ely: B i,j ) the score for play er I (resp. I I ) when I uses strategy i ∈ S tr at ( I ) a nd I I uses strategy j ∈ S tr at ( I I ). The scores are g iven b y n × m matrices A and B , where n and m are the cardinalit y of S tr at ( I ) and S tr at ( I I ). The game is termed symmetric if A is t he transp o se of B : t his implies that n = m , and w e can assume without loss of generality that S tr at ( I ) = S tr at ( I I ). In this paper, we will restrict to symmetric games. Example 1 (Prisoner’s dilemma) . The c ase wher e A and B ar e the fol lowing matric es 7 A =  R S T P  , B =  R T S P  with T > R > P > S and 2 R > T + S , is c al le d the prisoner’s dilemma . We denote by C (for c o op e r ation) the first pur e str ate gy, an d by D (for de fe ction) the se c ond pur e str a te gy of e ach player. As the game is symmetric, matrix A and B c an also b e denote d by: Opp onent C D Player C R S D T P A strategy x ∈ S tr at ( I ) is said to b e a b est r esp onse to strategy y ∈ S tr at ( I I ), denoted b y x ∈ B R ( y ) if A z ,y ≤ A x,y (1) for all strategies z ∈ S tr at ( I ). A pair ( x, y ) is a (pur e) Nash e quilibrium if x ∈ B R ( y ) and y ∈ B R ( x ). A pure Nash equilibrium do es not alw a ys exist. In other words, tw o strategies ( x, y ) form a Nash equilibrium if in that state neither o f the pla yers has a unilateral in terest to deviate from it. Example 2. On the example of the prisoner’s dilemma, B R ( y ) = D for al l y , and B R ( x ) = D for al l x . So ( D , D ) is the unique Nash e quilibrium, and it is pur e. In it, e ach pla yer has s c or e P . The wel l-kno wn p ar adox is that if they had playe d ( C , C ) (c o o p er ation) they would ha ve had sc or e R , that is mor e. The so cial optimum ( C , C ) , is differ ent fr om the e quilibrium that is r e ache d by r ational pla yers ( D , D ) , sinc e in an y other state, e a c h player fe ars that the adversary plays C . W e will also intro duce the following definition: G iv en some strategy x ′ ∈ S tr at ( I ), a strategy x ∈ S tr at ( I ) is said to b e a b est resp onse to strategy y ∈ S tr at ( I I ) among those differen t from x ′ , denoted by x ∈ B R 6 = x ′ ( y ) if A z ,y ≤ A x,y (2) for all strategy z ∈ S tr at ( I ) , z 6 = x ′ . Of course, the ro les of I I and I can b e inv erted in the previous definition. 8 There are t w o main approac hes to discuss dynamics of games. The first consists in rep eating games [19, 6]. The second in using mo dels from ev o- lutionary game theory . Refer to [15, 21] for a presen tation of this latter approac h. R ep e ating Game s. Rep eating k times a game, is equiv alen t to extending the space of c hoices into S tr at ( I ) k and S tr at ( I I ) k : play er I (resp ectiv ely I I ) c ho oses his or her action x ( t ) ∈ S tr at ( I ), (resp. y ( t ) ∈ S tr at ( I I )) at time t for t = 1 , 2 , · · · , k . Henc e, this is equiv alen t to a t wo-pla y er g ame with resp ectiv ely n k and m k c hoices for pla y ers. T o av oid confusion, w e will call actions the c ho ices x ( t ) , y ( t ) of eac h pla ye r at a given t ime, and str ate gies the sequence s X = x (1) , · · · , x ( k ) and Y = y ( 1) , · · · , y ( k ), that is to sa y t he strategies for the global game. If the game is rep eated an infinite n umber of times, a strategy b ecomes a function from in tegers to the se t of actions, and the game is still equiv alen t to a tw o- pla ye r game 1 . Behaviors. In practice, pla y er I (resp ectiv ely I I ) has to solve the follow ing problem at each time t : giv en the history of the game up to now, that is to sa y X t − 1 = x (1) , · · · , x ( t − 1 ) and Y t − 1 = y (1) , · · · , y ( t − 1) what should I (resp. I I) play at time t ? In other w ords, ho w to c ho ose x ( t ) ∈ S tr at ( I )? (resp. y ( t ) ∈ S tr at ( I I )?) Is is nat ura l to supp ose tha t this is giv en by some b eha vior rules: x ( t ) = f ( X t − 1 , Y t − 1 ) , y ( t ) = g ( X t − 1 , Y t − 1 ) for some par t icular functions f and g . 1 but whos e matrices ar e infinite. 9 The Sp e cific Case of the Prisoner’s L emma. The question of the b est b eha v- ior rule to use fo r the prisoner lemma gav e birth to an imp ortant literature. In pa r t icular, after the b o ok [5], that describ es the results of tournaments of b eha vior rules f or the iterated prisoner lemma, a nd tha t argues that there exists a b est b eha vior rule called T I T − F O R − T AT . This consists in co- op erating at the first step, and then do the same thing as the advers a ry at subseque nt times. A lot of other b eha viors, most of them with v ery picturesque na mes ha v e b een prop o sed and studied: see f or example [5]. Among p ossible b eha viors there is P AV LO V b eha vior: in the iterated prisoner lemma, a pla yer co op erates if a nd only if b oth play ers opted for the same alternative in the previous mov e. This name [5, 16, 18] stems from the fact that this strategy em b o dies an a lmo st reflex-lik e resp o nse to the pa y off: it rep eats its f ormer mo ve if it w as rew a rded by R or T p oin ts, but switc hes b eha vior if it was punished by receiving only P o r S p oints. Refer to [1 8] for some study of this strategy in the spirit of Axelro d’s to urnamen ts. The P AV LO V b eha vior can also b e termed WIN-ST A Y, LOSE- S HIFT since if t he pla y on the previous round results in a success, then the agent pla ys the same strategy on the next round. Alternativ ely , if the pla y resulted in a f ailure the agent switc hes to ano ther action [5, 18]. Going F r o m 2 Players to N Players. P AV LOV b ehav io r is Mark ovian: a b eha vior f is Markovian , if f ( X t − 1 , Y t − 1 ) dep ends only on x ( t − 1) and y ( t − 1). F rom suc h a b eha vior, it is easy to obta in a distributed dynamic. F o r example, let’s follow [10], for the prisoner’s dilemma. Supp ose that we hav e a connected graph G = ( V , E ), with N v ertices. The v ertices corresp o nd to pla y ers. An instantaneous configuration of the system is given by an elemen t of { C , D } N , tha t is to sa y b y the state C or D of eac h v ertex. Hence, there are 2 N configurations. A t eac h time t , one c ho oses randomly and unifor mly one edge ( i, j ) o f the graph. At t his momen t, play ers i and j play the prisoner dilemma with the P AV LO V b ehav ior . It is easy to see that this corresp onds to executing the follo wing r ules:        C C → C C C D → D D D C → D D D D → C C . (3) 10 What is the final state reac hed by the system? The underlying mo del is a ve r y large Mark o v chain with 2 N states. The state E ∗ = { C } N is absorbing. If the gra ph G do es not ha ve a ny isolated v ertex, this is the unique absorbing state, and there exists a sequence of tr a nsformations that transforms an y state E in to this state E ∗ . As a consequenc e, from we ll- kno wn classical results in Mark ov c hain theory , whateve r t he initial configuration is, with probability 1, the system will ev en tually b e in state E ∗ [7]. The system is self-stabiliz ing . Sev eral results ab out the time of con v ergence tow ards this stable state can b e fo und in [10], and [11], for rings, and complete graphs. What is in teresting in this example is that it show s ho w to go from a game, and a b eha vior to a distributed dynamic on a graph, and in pa r t icular to a p o pulation proto col when the graph is a complete graph. 4. F rom Games T o Population Pr o t o cols In the spirit of the previous discuss ion, to an y symmetric game, w e can asso ciate a populatio n pro to col as follows. Definition 1 ( Associat ing a Proto col to a G a me) . Assume a symmetric two-player game is given. L et ∆ b e some thr eshold. The pr oto c ol asso ciate d to the game is a p opulation pr oto c ol whose set of states is Q , wh e r e Q = S tr at ( I ) = S tr at ( I I ) is the set of str ate gies of the game, and whose tr ansition rules δ ar e given as fol lows: ( q 1 , q 2 , q ′ 1 , q ′ 2 ) ∈ δ wher e • q ′ 1 = q 1 when M q 1 ,q 2 ≥ ∆ • q ′ 1 ∈ B R 6 = q 1 ( q 2 ) when M q 1 ,q 2 < ∆ and • q ′ 2 = q 2 when M q 2 ,q 1 ≥ ∆ • q ′ 2 ∈ B R 6 = q 2 ( q 1 ) when M q 2 ,q 1 < ∆ , wher e M is the matrix of the game. 11 Remark 1. By subtr acting ∆ to e ach en try of the matrix M , we c an assume without loss of gener ality that ∆ = 0 . We wil l do so fr om now on. Definition 2 (P avlo vian P opulation Proto col) . A p opulation pr oto c ol is Pa vlo- vian if it c an b e obtaine d fr om a game as ab ove. Remark 2. Cle arly a Pavlovian p o pulation pr oto c ol must b e symmetric : in- de e d, whenever ( q 1 , q 2 , q ′ 1 , q ′ 2 ) ∈ δ , one has ( q 2 , q 1 , q ′ 2 , q ′ 1 ) ∈ δ . 5. Some Sp ecific Pa vlovian Prot o cols W e now discuss whether assuming proto cols P av lovian is a restriction. W e start b y an easy consideration. Theorem 2. Any symmetric deterministic 2 -states p opulation pr oto c ol is Pavlovian. Pr o of. Consider a deterministic symmetric 2-states p opulation proto col. Note Q = { + , −} its set of states. Its transition function can b e written as follo ws:        ++ → α ++ α ++ + − → α + − α − + − + → α − + α + − −− → α −− α −− (4) for some α ++ , α + − , α − + , α −− . This corresp onds to the symmetric game given by the following pay-off matrix M Opp onen t + - Pla y er + β ++ β + − - β − + β −− where for all q 1 , q 2 ∈ { + , −} , • β q 1 q 2 = 1 if α q 1 q 2 = q 1 , • β q 1 q 2 = − 1 otherwise . 12 Unfortunately , not all rules corresp ond to a game. Prop osition 1. Some symmetric p opulation pr oto c ols ar e not Pavlovian. Pr o of. Consider for example a deterministic 3-stat es p opulation proto col with set of stat es Q = { q 0 , q 1 , q 2 } and a join t transition function δ suc h that δ 1 ( q 0 , q 0 ) = q 1 , δ 1 ( q 1 , q 0 ) = q 2 , δ 1 ( q 2 , q 0 ) = q 0 . Assume b y con tradiction that there exists a 2-play er game corresp onding to this 3-states po pula t ion proto col. Consider its pa y o ff matrix M . Let M ( q 0 , q 0 ) = β 0 , M ( q 1 , q 0 ) = β 1 , M ( q 2 , q 0 ) = β 2 . W e mus t hav e β 0 ≥ ∆ = 0 , β 1 ≥ ∆ = 0 since all agen ts that in teract with a n agen t in state q 0 m ust c hange their stat e. No w, since q 0 c hanges to q 1 , q 1 m ust b e a strictly b etter resp onse to q 0 than q 2 : hence, w e m ust ha v e β 1 > β 2 . In a similar w a y , since q 1 c hanges to q 2 , w e m ust hav e β 2 > β 0 , and since q 2 c hanges to q 0 , w e must ha v e β 0 > β 1 . F rom β 1 > β 2 > β 0 w e reac h a con t r a diction. This indeed motiv ates the follo wing study , where w e discuss whic h prob- lems admit a P a vlov ia n solution. 5.1. Basic Pr oto c ols Prop osition 2. Ther e is a Pavlo vian p r oto c ol that c omputes the lo gic al O R (r esp. AN D ) of input bits. Pr o of. Consider the following proto col to compute O R ,        01 → 11 10 → 11 00 → 00 11 → 11 (5) and the fo llowing proto col to compute AN D ,        01 → 00 10 → 00 00 → 00 11 → 11 (6) Since they are b oth deterministic 2-states p opulatio n pro to cols, they are P av lovian. 13 Remark 3. Notic e that O R (r esp e ctively AN D ) pr oto c ol c orr esp o n ds to the pr e dic ate [ x. 0 ≥ 1] (r esp. [ x. 0 = 0] ), wher e x is the i n put. A simple change of notation yield s a pr oto c ol to c ompute [ x.σ ≥ 1] and [ x.σ = 0] for any input symb ol σ . Remark 4. Al l pr e v i o us pr oto c ols a r e “natur al ly br o adc asting” i.e., eventu- al ly al l age nts agr e e on some (the c orr e c t) value. With pr evious defin i tion s (which ar e the classic al ones for p opulation pr oto c ols), the fol lowing pr oto c ol do es not c ompute the X O R or inp ut bits, or e quivalently do es not c ompute pr e dic ate [ x. 1 ≡ 1 ( mod 2)] .        01 → 01 10 → 10 00 → 00 11 → 00 (7) Inde e d, the answer is not eventual ly kn o wn by al l the agents. It c omputes the X O R in a we aker form i.e. , eventual ly, al l agents w i l l b e in state 0 , if the X O R of input bits is 0 , or eventual ly only one agent wil l b e in state 1 , if the X O R of input bits is 1 . Prop osition 3. T h er e is a Pav l o vian pr oto c ol that c o m putes the thr eshold pr e dic ate [ x.σ ≥ 2] , which is true when ther e ar e at le ast 2 o c curr enc es of input symb ol σ in the input x . Pr o of. The follow ing proto col is a solution taking • Σ = { 0 , σ } , Q = { 0 , σ, 2 } , • ω (0 ) = ω ( σ ) = 0, • ω (2 ) = 1.                            00 → 00 0 σ → 0 σ σ 0 → σ 0 02 → 22 20 → 22 σ σ → 22 σ 2 → 2 2 2 σ → 22 22 → 22 (8) 14 Indeed, if there is at least tw o σ , then b y fairness and by the rule n umber 6, they will ultimately b e changed in to t w o 2 s. Then 2s will turn all other agen ts in to 2s. Now, this is the only w a y to create a 2. This is a P a vlovian proto col since it corresp onds t o the fo llowing pa y off matrix. Opp onen t 0 σ 2 Pla y er 0 0 0 − 1 σ 0 − 1 − 1 2 1 1 1 Hence, Pa vlo vian p opulation proto cols can coun t until 2. W e will pro ve later on that they can not count un til 3. 5.2. L e ader Ele c tion The classical solution [2] to the leader election problem (starting from a configuration with ≥ 1 leaders, ev en tually exactly o ne leader surviv es) is the follo wing:        LL → LN LN → LN N L → N L N N → N N (9) Notice that w e use the terminology “leader election” as in [2] fo r this proto col, but that it ma y b e considered more as a “mutual exclusion” pro t o col. Unfortunately , this proto col is non- symmetric, and hence non-P avlo vian. Remark 5. A ctual ly, the pr oblem is with the first rule, sinc e one wants two le ad e rs to b e c ome only one. If the two le aders ar e i d entic al, this is cle arly pr oble matic with symm e tric rules. Ho w eve r, the leader election problem can actually b e solv ed b y a P av lo - vian proto col, a t the price of a less trivial proto col. 15 Prop osition 4. T he fol lowing Pavlovian pr oto c ol solv e s the le ader ele ction pr oble m, as so on as the p opulation is of size ≥ 3 .                            L 1 L 2 → L 1 N L 1 N → N L 2 L 2 N → N L 1 N N → N N L 2 L 1 → N L 1 N L 1 → L 2 N N L 2 → L 1 N L 1 L 1 → L 2 L 2 L 2 L 2 → L 1 L 1 (10) Pr o of. Indeed, starting from a configuration containing no t only N s, ev en tu- ally after some time configurations will hav e exactly one leader, tha t is one agen t in state L 1 or L 2 . Indeed, the first rule and the fifth rule decrease strictly the num b er of leaders whenev er there are more than tw o leaders. Now t he other rules, pre- serv e the num b er of leaders, a nd are made suc h that an L 1 can a lw a ys b e transformed in to an L 2 and vice-v ersa, and hence a re made such that a con- figuration where first or fifth rule applies can alw ays b e reac hed whenev er there are more than tw o leaders. The fact tha t it solv es the leader elec- tion problem then fo llows fro m the h yp o thesis of fairness in the definition of computations. This is a P a vlov ia n proto col, since it corresp onds to the follow ing pa y off matrix. Opp onen t L 1 L 2 N Pla y er L 1 − 3 0 − 3 L 2 − 1 − 3 − 3 N − 2 − 3 0 5.3. Majority Prop osition 5. The majority pr oble m (given some p opulation of input sym- b ols σ and σ ′ , determine whether ther e ar e mor e σ than σ ′ ) c an b e so l v e d by a Pavlovian p opulation pr o to c ol. 16 Remark 6. If on e pr efers, the pr e dic ate [ x.σ ≥ x.σ ′ ] , wher e σ and σ ′ ar e two input symb ols, and x is the input, is c omputable b y a Pavlovian p opulation pr oto c ol. Pr o of. W e claim tha t t he following proto col outputs 1 if there are mo r e σ than σ ′ in the initial configuration and 0 o therwise,                        N Y → Y Y Y N → Y Y N σ → Y σ σ N → σ Y Y σ ′ → N σ ′ σ ′ Y → σ ′ N σ σ ′ → N Y σ ′ σ → Y N (11) taking • Σ = { σ , σ ′ } , Q = { σ, σ ′ , Y , N } , • ω ( σ ) = ω ( Y ) = 1, • ω ( σ ′ ) = ω ( N ) = 0. In this proto col, the states Y and N ar e “neutral” elemen ts for our pred- icate but they should b e understoo d as Y es and No . They are the “answ ers” to the question: a r e there more 0s than 1s. This proto col is made suc h that the n um b ers of σ and σ ′ are preserv ed except when a σ meets a σ ′ . In that lat ter case , the t wo agen ts are deleted and transformed into a Y and a N . If there are initially strictly more σ tha n σ ′ , from the fairness condition, eac h σ ′ will b e paired with a σ and at some p oint no σ ′ will left. By fa ir ness and since there is still at least a σ , a configuration con taining only σ and Y s will b e reached. Since in suc h a configuration, no rule can mo dify the state of any a gen t, and since the output is defined and equals to 1 in suc h a configuration, the proto col is correct in this case By symme try , one can sho w that the proto col outputs 0 if there are initially strictly mor e σ ′ than σ . Supp ose now that initially , there ar e exactly the same n um b er of σ and σ ′ . By fa irness, there exists a step when no more agen ts in the stat e σ or σ ′ 17 left. Note that a t the momen t where the last σ is matc hed with the last σ ′ , a Y is created. Since this Y can b e “bro a dcasted” o v er the N s, in the final configuration all a g en ts are in the state Y a nd th us the o utput is correct. This proto col is P av lovian, since it corresp onds to the follo wing pay off matrix. Opp onen t N Y σ σ ′ N 1 − 1 − 1 1 Pla y er Y 0 1 1 − 1 σ 0 0 0 − 1 σ ′ 0 0 − 1 0 6. Bounds on the Po wer of P a vlovian P opulation Proto cols W e prov ed that predic at es [ x.σ = 0], [ x.σ ≥ 1], [ x.σ ≥ 2] can b e computed b y some P a vlov ia n populatio n proto cols, as well as [ x.σ ≥ x.σ ′ ]. It is clear that the subset o f the predicates computable b y P a vlov ia n p opulation proto cols is closed b y negation: just switc h the v alue of the indi- vidual output function of a proto col computing a predicate to get a proto col computing its negation. Notice t hat, unlik e what happ ens for general p o pulation prot o cols, com- p osing Pa vlo vian p opulation proto cols into a P av lovian populatio n proto col is not easy . It is not clear whether P avlo vian computable predicates are closed b y conjunctions: classical constructions for general p opulation prot o cols can not b e used directly . The p ow er of P avlo vian p opulation proto cols is actually rather limited as they can count up to 2, but not 3. Theorem 3. Ther e is no Pavlovi a n pr oto c ol that c omputes the thr eshold pr e dic ate [ x.σ ≥ 3] , which is true when ther e ar e at le ast 3 o c curr enc es of input symb ol σ in the input x . Pr o of. W e will pro v e this b y con tradiction. Assume there exists suc h a P avlo- vian proto col. Without loss of generalit y w e ma y assume that Σ = { 0 , σ } is a subset o f the set of states Q . As the pro to col is P a vlovian, and hence symmetric, an y rule q q → q ′ q ′′ , is suc h that q ′ = q ′′ , that is to sa y of the form q q → q ′ q ′ for all q ∈ Q . 18 Let then consider the sequence o f r ules suc h that σ σ → q 1 q 1 → q 2 q 2 → · · · → q k q k → . . . where σ, q 1 q 2 , q 3 , . . . , q k ∈ Q . Since Q is finite, there exist tw o distinct in t egers k and ℓ suc h t ha t q k = q ℓ and k < ℓ . The case k + 1 = ℓ is not p ossible. Indeed, w e w ould ha v e the rule q k q k → q k q k . Consider the inputs x 3 and x 4 suc h that x 3 = { σ, σ } and x 4 = { σ, σ, σ, σ } . x 4 m ust b e accepted. F ro m x 4 there is a deriv a tion x 4 → { q 1 , q 1 , σ, σ } → { q 1 , q 1 , q 1 q 1 } → ∗ { q k , q k , q k , q k } . This latter configurat io n is terminal fr o m the ab ov e rule. Since x 4 m ust b e accepted, w e m ust ha ve ω ( q k ) = 1. How ev er, fro m x 3 there is a deriv ation x 3 → { q 1 , q 1 } → ∗ { q k , q k } , where the last configura t io n is also terminal. W e reac h a contradiction, since its output would b e ω ( q k ) = 1, whereas x 3 m ust b e rejected. Hence, k + 1 < ℓ , and q k q k → q k +1 q k +1 → · · · → q ℓ q ℓ → q k q k . Let T b e then the set of states T = { q i : k ≤ i ≤ ℓ } . Since q i q i → q i +1 q i +1 is among the rules, since the proto col is P a vlovian with a matrix M , and by definition of Pa vlo v b ehav io r, w e must hav e q i +1 = B R 6 = q i ( q i ) (with the con v ention that q ℓ +1 is q k ). So, B R ( q i ) can b e q i +1 or q i . Let then discu ss the rules q i q j → q ′ i q ′ j (12) for q i , q j ∈ T . There are three p ossibilities for the v alue o f q ′ i : 1. q ′ i = q i if M q i q j ≥ ∆ 2. q ′ i = q j , if M q i q j < ∆ and if q j = B R 6 = q i ( q j ) 3. q ′ i = q j +1 if M q i q j < ∆ and if q j +1 = B R 6 = q i ( q j ) In any case, w e see that the v alue of q ′ i is in T . Symmetrically , we hav e t hree p ossibilities for q ′ j , all of them in T . Hence, all rules of the form (12) preserv e T : w e ha ve q ′ i , q ′ j ∈ T , as so on as q i , q j ∈ T . Consider still then t he inputs x 3 and x 4 suc h that x 3 = { σ, σ } and x 4 = { σ, σ, σ, σ } . F rom x 4 there is a deriv ation x 4 → { q 1 , q 1 , σ, σ } → { q 1 , q 1 , q 1 q 1 } → ∗ { q k , q k , q k , q k } . F rom this last configuration, b y ab o ve re- mark, the state of all agen ts will b e in T . As x 4 m ust b e accepted, ultimately all agen ts will b e in states that b elong to T whose image by ω is 1. Consider no w x 3 . F rom x 3 there is a deriv ation x 3 → { q 1 , q 1 } → ∗ { q k , q k } that then will go trough all configuratio ns { q i q i } , for the q i ∈ T in turn. This can 19 not ev en tually stabilize to elemen ts whose image by ω is 0, as some of the elemen ts of T ha v e image 1 by ω , and hence x 3 is no t accepted. This yields a con tra diction, and hence suc h a P avlo vian proto col can not exist. 7. The Po w er of Symmetric Popula t ion Proto cols P av lovian P opula t io n proto cols a re symmetric. W e just pro ved that they ha v e a v ery limited computational p o w er. How ev er, assuming p opulation proto cols symmetric ( no t-necessarily P av lovian) is not truly a restriction. Prop osition 6. Any p opulation pr oto c ol c an b e simulate d by a symmetric p opulation pr oto c ol, as so on as the p opulation is of size ≥ 3 . Before prov ing this prop osition, w e state the (immediate) main conse- quence. Corollary 1. A p r e dic ate is c omputable by a symmetric p opulation pr oto c ol if and only if it i s semiline ar. Pr o of. T o a p opulation proto col ( Q, Σ , ι, ω , δ ), with Q = { q 1 , · · · , q n } asso- ciate p opulation pro t o col ( Q ∪ Q ′ , Σ , ι, ω , δ ′ ) with Q ′ = { q ′ 1 , · · · , q ′ n } , ω ( q ′ ) = ω ( q ) for all q ∈ Q , and for all rules q q → α β in δ , the following rules in δ ′ :                        q q ′ → αβ q ′ q → β α q q → q ′ q ′ q ′ q ′ → q q q γ → q ′ γ q ′ γ → q γ γ q → γ q ′ γ q ′ → γ q for all γ ∈ Q ∪ Q ′ , γ 6 = q , γ 6 = q ′ , and f o r all pairs of rules  q r → αβ r q → δ ǫ 20 with q , r ∈ Q , the follo wing rules in δ ′ :        q r ′ → αβ r ′ q → β α r q ′ → δ ǫ q ′ r → ǫδ . The obtained p opulatio n pro t o col is clearly symmetric. 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