aa r X i v : . [ c s . G T ] J un T. Neary, D. Woods, A.K. Seda and N. Murphy (Eds.):The Complexity of Simple Programs 2008.EPTCS 1, 2009, pp. 3–15, doi:10.4204/EPTCS.1.1 c (cid:13)
O. Bournez, J. Chalopin, J. Cohen, X. Koegler
Playing With Population Protocols ∗ Olivier Bournez
Ecole Polytechnique & Laboratoire d’Informatique (LIX),91128 Palaiseau Cedex, France
J´er´emie Chalopin
CNRS & Laboratoire d’Informatique Fondamentale de Marseille, CNRS & Aix-Marseille Universit´e,39 rue Joliot Curie, 13453 Marseille Cedex 13, France
Johanne Cohen
CNRS & PRiSM,45 Avenue des Etats Unis, 78000 Versailles, France
Xavier Koegler
[email protected] ´Ecole Normale Sup´erieure & Universit´e Paris Diderot - Paris 7,Case 7014, 75205 Paris Cedex 13, France
Population protocols have been introduced as a model of sensor networks consisting of verylimited mobile agents with no control over their own movement: A collection of anonymousagents, modeled by finite automata, interact in pairs according to some rules.Predicates on the initial configurations that can be computed by such protocols havebeen characterized under several hypotheses.We discuss here whether and when the rules of interactions between agents can be seenas a game from game theory. We do so by discussing several basic protocols.
The computational power of networks of anonymous resource-limited mobile agents has beeninvestigated in several recent papers.In particular, Angluin et al. proposed in [1] a new model of distributed computations. In thismodel, called population protocols , finitely many finite-state agents interact in pairs chosen byan adversary. Each interaction has the effect of updating the state of the two agents accordingto a joint transition function.A protocol is said to (stably) compute a predicate on the initial states of the agents if, inany fair execution, after finitely many interactions, all agents reach a common output thatcorresponds to the value of the predicate.The model was originally proposed to model computations realized by sensor networks inwhich passive agents are carried along by other entities. The canonical example of [1] correspondsto sensors attached to a flock of birds and that must be programmed to check some global ∗ This work and all authors were partly supported by ANR Project SOGEA and by ANR Project SHAMAN,Xavier Koegler was partly supported by COST Action 295 DYNAMO and ANR Project ALADDIN
Playing With Population Protocols properties, like determining whether more than 5% of the population has elevated temperature.Motivating scenarios also include models of the propagation of trust [8].Much of the work so far on population protocols has concentrated on characterizing whichpredicates on the initial states can be computed in different variants of the model and undervarious assumptions. In particular, the predicates computable by the unrestricted populationprotocols from [1] have been characterized as being precisely the semi-linear predicates, that isto say those predicates on counts of input agents definable in first-order Presburger arithmetic[18]. Semilinearity was shown to be sufficient in [1] and necessary in [2].Variants considered so far include restriction to one-way communications, restriction to par-ticular interaction graphs, to random interactions, with possibly various kind of failures of agents.Solutions to classical problems of distributed algorithmics have also been considered in thismodel. Refer to survey [3] for a complete discussion.The population protocol model shares many features with other models already consideredin the literature. In particular, models of pairwise interactions have been used to study thepropagation of diseases [12], or rumors [7]. In chemistry the chemical master equation has beenjustified using (stochastic) pairwise interactions between the finitely many molecules present[16, 11]. In that sense, the model of population protocols may be considered as fundamental inseveral fields of study.Pairwise interactions between finite-state agents are sometimes motivated by the study ofthe dynamics of particular two-player games from game theory. For example, paper [9] considersthe dynamics of the so-called
PAV LOV behaviour in the iterated prisoner lemma. Several resultsabout the time of convergence of this particular dynamics towards the stable state can be foundin [9], and [10], for rings, and complete graphs.The purpose of the following discussion is to better understand whether and when pairwiseinteractions, and hence population protocols, can be considered as the result of a game. Wewant to understand if restricting to rules that come from a (symmetric) game is a limitation,and in particular whether restricting to rules that can be termed
PAV LOV in the spirit of [9] is alimitation. We do so by giving solutions to several basic problems using rules of interactions as-sociated to a symmetric game. As such protocols must also be symmetric, we are also discussingwhether restricting to symmetric rules in population protocols is a limitation.In Section 2, we briefly recall population protocols. In Section 3, we recall some basicsfrom game theory. In Section 4, we discuss how a game can be turned into a dynamics, andintroduce the notion of
Pavlovian population protocol. In Section 5 we prove that any symmetricdeterministic 2-states population protocol is Pavlovian, and that the problem of computing theOR, AND, as well as the leader election and majority problem admit Pavlovian solutions. Wethen discuss our results in Section 6.
A protocol is given by ( Q , S , i , w , d ) with the following components. Q is a finite set of states . S is a finite set of input symbols . i : S → Q is the initial state mapping, and w : Q → { , } isthe individual output function. d ⊆ Q is a joint transition relation that describes how pairs ofagents can interact. Relation d is sometimes described by listing all possible interactions usingthe notation ( q , q ) → ( q ′ , q ′ ), or even the notation q q → q ′ q ′ , for ( q , q , q ′ , q ′ ) ∈ d (with theconvention that ( q , q ) → ( q , q ) when no rule is specified with ( q , q ) in the left-hand side). . Bournez, J. Chalopin, J. Cohen, X. Koegler deterministic if for all pairs ( q , q ) there is only one pair ( q ′ , q ′ ) with( q , q ) → ( q ′ , q ′ ). In that case, we write d ( q , q ) for the unique q ′ and d ( q , q ) for the unique q ′ . Notice that, in general, rules can be non-symmetric: if ( q , q ) → ( q ′ , q ′ ), it does not neces-sarily follow that ( q , q ) → ( q ′ , q ′ ).Computations of a protocol proceed in the following way. The computation takes placeamong n agents , where n ≥ . A configuration of the system can be described by a vector of allthe agents’ states. The state of each agent is an element of Q . Because agents with the samestates are indistinguishable, each configuration can be summarized as an unordered multiset ofstates, and hence of elements of Q .Each agent is given initially some input value from S : Each agent’s initial state is determinedby applying i to its input value. This determines the initial configuration of the population.An execution of a protocol proceeds from the initial configuration by interactions betweenpairs of agents. Suppose that two agents in state q and q meet and have an interaction. Theycan change into state q ′ and q ′ if ( q , q , q ′ , q ′ ) is in the transition relation d . If C and C ′ aretwo configurations, we write C → C ′ if C ′ can be obtained from C by a single interaction of twoagents: this means that C contains two states q and q and C ′ is obtained by replacing q and q by q ′ and q ′ in C , where ( q , q , q ′ , q ′ ) ∈ d . An execution of the protocol is an infinite sequenceof configurations C , C , C , · · · , where C is an initial configuration and C i → C i + for all i ≥ .An execution is fair if for all configurations C that appear infinitely often in the execution, if C → C ′ for some configuration C ′ , then C ′ appears infinitely often in the execution.At any point during an execution, each agent’s state determines its output at that time. Ifthe agent is in state q , its output value is w ( q ). The configuration output is (respectively ) ifall the individual outputs are (respectively ). If the individual outputs are mixed s and s then the output of the configuration is undefined.Let p be a predicate over multisets of elements of S . Predicate p can be considered asa function whose range is { , } and whose domain is the collection of these multisets. Thepredicate is said to be computed by the protocol if, for every multiset I , and every fair executionthat starts from the initial configuration corresponding to I , the output value of every agenteventually stabilizes to p ( I ).The following was proved in [1, 2] Theorem 1 ([1, 2]) . A predicate is computable in the population protocol model if and only ifit is semilinear.
Recall that semilinear sets are known to correspond to predicates on counts of input agentsdefinable in first-order Presburger arithmetic [18].
We now recall the simplest concepts from Game Theory. We focus on non-cooperative games,with complete information, in extensive form.The simplest game is made up of two players, called I and II , with a finite set of options,called pure strategies , Strat ( I ) and Strat ( II ). Denote by A i , j (respectively: B i , j ) the score forplayer I (resp. II ) when I uses strategy i ∈ Strat ( I ) and II uses strategy j ∈ Strat ( II ).The scores are given by n × m matrices A and B , where n and m are the cardinality of Strat ( I )and Strat ( II ). The game is termed symmetric if A is the transpose of B : this implies that n = m , Playing With Population Protocols and we can assume without loss of generality that
Strat ( I ) = Strat ( II ). Example 1 (Prisoner’s dilemma) . The case where A and B are the following matrices A = (cid:18) R ST P (cid:19) , B = (cid:18) R TS P (cid:19) with T > R > P > S and R > T + S , is called the prisoner’s dilemma . We denote by C (forcooperation) the first pure strategy, and by D (for defection) the second pure strategy of eachplayer.As the game is symmetric, matrix A and B can also be denoted by:OpponentC DPlayer C R S D T P
A strategy x ∈ Strat ( I ) is said to be a best response to strategy y ∈ Strat ( II ), denoted by x ∈ BR ( y ) if A z , y ≤ A x , y (1)for all strategies z ∈ Strat ( I ).A pair ( x , y ) is a (pure) Nash equilibrium if x ∈ BR ( y ) and y ∈ BR ( x ). A pure Nash equilibriumdoes not always exist.In other words, two strategies ( x , y ) form a Nash equilibrium if in that state neither of theplayers has a unilateral interest to deviate from it. Example 2.
On the example of the prisoner’s dilemma, BR ( y ) = D for all y , and BR ( x ) = D forall x . So ( D , D ) is the unique Nash equilibrium, and it is pure. In it, each player has score P .The paradox is that if they had played ( C , C ) (cooperation) they would have had score R , that ismore. The social optimum ( C , C ) , is different from the equilibrium that is reached by rationalplayers ( D , D ) , since in any other state, each player fears that the adversary plays C . We will also introduce the following definition: Given some strategy x ′ ∈ Strat ( I ), a strategy x ∈ Strat ( I ) is said to be a best response to strategy y ∈ Strat ( II ) among those different from x ′ ,denoted by x ∈ BR = x ′ ( y ) if A z , y ≤ A x , y (2)for all strategy z ∈ Strat ( I ) , z = x ′ .Of course, the role of II and I can be inverted in the previous definition.There are two main approaches to discussing dynamics of games. The first consists inrepeating games. The second in using models from evolutionary game theory. Refer to [13, 19]for a presentation of this latter approach. Repeating Games.
Repeating k times a game, is equivalent to extending the space of choicesinto Strat ( I ) k and Strat ( II ) k : player I (respectively II ) chooses his or her action x ( t ) ∈ Strat ( I ),(resp. y ( t ) ∈ Strat ( II )) at time t for t = , , · · · , k . Hence, this is equivalent to a two-player gamewith respectively n k and m k choices for players.To avoid confusion, we will call actions the choices x ( t ) , y ( t ) of each player at a given time,and strategies the sequences X = x ( ) , · · · , x ( k ) and Y = y ( ) , · · · , y ( k ), that is to say the strategiesfor the global game. . Bournez, J. Chalopin, J. Cohen, X. Koegler . Behaviours.
In practice, player I (respectively II ) has to solve the following problem at eachtime t : given the history of the game up to now, that is to say X t − = x ( ) , · · · , x ( t − )and Y t − = y ( ) , · · · , y ( t − )what should I play at time t ? In other words, how to choose x ( t ) ∈ Strat ( I )? (resp. y ( t ) ∈ Strat ( II )?)Is is natural to suppose that this is given by some behaviour rules: x ( t ) = f ( X t − , Y t − ) , y ( t ) = g ( X t − , Y t − )for some particular functions f and g . The Specific Case of the Prisoner’s Lemma.
The question of the best behaviour rule touse for the prisoner lemma gave birth to an important literature. In particular, after the book[4], that describes the results of tournaments of behaviour rules for the iterated prisoner lemma,and that argues that there exists a best behaviour rule called
T IT − FOR − TAT . This consists incooperating at the first step, and then do the same thing as the adversary at subsequent times.A lot of other behaviours, most of them with very picturesque names have been proposedand studied: see for example [4], [5], [15].Among possible behaviours is
PAV LOV : in the iterated prisoner lemma, a player cooperatesif and only if both players opted for the same alternative in the previous move. This name[14, 17, 4] stems from the fact that this strategy embodies an almost reflex-like response to thepayoff: it repeats its former move if it was rewarded by R or T points, but switches behaviour ifit was punished by receiving only P or S points. Refer to [17] for some study of this strategy inthe spirit of Axelrod’s tournaments.The PAV LOV behaviour can also be termed
WIN-STAY, LOSE-SHIFT as if the play on theprevious round resulted in a success, then the agent plays the same strategy on the next round.Alternatively, if the play resulted in a failure the agent switches to another action [17, 4].
Going From Players to N Players.
PAV LOV behaviour is Markovian: a behaviour f is Markovian , if f ( X t − , Y t − ) depends only on x ( t − ) and y ( t − ).From such a behaviour, it is easy to obtain a distributed dynamic. For example, let’s follow[9], for the prisoner’s dilemma.Suppose that we have a connected graph G = ( V , E ), with N vertices. The vertices correspondto players. An instantaneous configuration of the system is given by an element of { C , D } N , thatis to say by the state C or D of each vertex. Hence, there are N configurations. but whose matrices are infinite. Playing With Population Protocols
At each time t , one chooses randomly and uniformly one edge ( i , j ) of the graph. At thismoment, players i and j play the prisoner dilemma with the PAV LOV behaviour. It is easy tosee that this corresponds to executing the following rules: CC → CCCD → DDDC → DDDD → CC . (3)What is the final state reached by the system? The underlying model is a very large Markovchain with N states. The state E ∗ = { C } N is absorbing. If the graph G does not have any isolatedvertex, this is the unique absorbing state, and there exists a sequence of transformations thattransforms any state E into this state E ∗ . As a consequence, from well-known classical resultsin Markov chain theory, whatever the initial configuration is, with probability , the system willeventually be in state E ∗ [6]. The system is self-stabilizing .Several results about the time of convergence towards this stable state can be found in [9],and [10], for rings, and complete graphs.What is interesting in this example is that it shows how to go from a game, and a behaviourto a distributed dynamics on a graph, and in particular to a population protocol when the graphis the complete graph. In the spirit of the previous discussion, to any symmetric game, we can associate a populationprotocol as follows.
Definition 1 (Associating a Protocol to a Game) . Assume a symmetric two-player game isgiven. Let D be some threshold.The protocol associated to the game is a population protocol whose set of states is Q , where Q = Strat ( I ) = Strat ( II ) is the set of strategies of the game, and whose transition rules d are givenas follows: ( q , q , q ′ , q ′ ) ∈ d where • q ′ = q when M q , q ≥ D • q ′ ∈ BR = q ( q ) when M q , q < D and • q ′ = q when M q , q ≥ D • q ′ ∈ BR = q ( q ) when M q , q < D ,where M is the matrix of the game. Definition 2 (Pavlovian Population Protocol) . A population protocol is
Pavlovian if it can beobtained from a game as above.
Remark 1.
Clearly a Pavlovian population protocol must be symmetric : indeed, whenever ( q , q , q ′ , q ′ ) ∈ d , one has ( q , q , q ′ , q ′ ) ∈ d . . Bournez, J. Chalopin, J. Cohen, X. Koegler We now discuss whether assuming protocols Pavlovian is a restriction.We start by an easy consideration.
Theorem 2.
Any symmetric deterministic -states population protocol is Pavlovian.Proof. Consider a deterministic symmetric -states population protocol. Note Q = { + , −} itsset of states. Its transition function can be written as follows: ++ → a ++ a ++ + − → a + − a − + − + → a − + a + − −− → a −− a −− (4)for some a ++ , a + − , a − + , a −− .This corresponds to the symmetric game given by the following pay-off matrix M Opponent + -
Player + b ++ b + − - b − + b −− taking threshold D = , where for all q , q ∈ { + , −} , • b q q = if a q q = q , • b q q = otherwise.Unfortunately, not all rules correspond to a game. Proposition 1.
Some symmetric population protocols are not Pavlovian.Proof.
Consider for example a deterministic -states population protocol with set of states Q = { q , q , q } and a joint transition function d such that d ( q , q ) = q , d ( q , q ) = q , d ( q , q ) = q . Assume by contradiction that there exists a -player game corresponding to this -states pop-ulation protocol. Consider its payoff matrix M . Let M ( q , q ) = b , M ( q , q ) = b , M ( q , q ) = b .We must have b ≥ D , b ≥ D since all agents that interact with an agent in state q must changetheir state. Now, since q changes to q , q must be a strictly better response to q than q :hence, we must have b > b . In a similar way, since q changes to q , we must have b > b , andsince q changes to q , we must have b > b . From b > b > b we reach a contradiction.This indeed motivates the following study, where we discuss which problems admit a Pavlo-vian solution.0 Playing With Population Protocols
Proposition 2.
There is a Pavlovian protocol that computes the logical OR (resp. AND ) ofinput bits.Proof.
Consider the following protocol to compute OR , → → → → (5)and the following protocol to compute AND , → → → → (6)Since they are both deterministic 2-states population protocols, they are Pavlovian. Remark 2.
Notice that OR (respectively AND ) protocol corresponds to the predicates on countsof input agents n ≥ (resp. n = ) where n , n are the number of input agents in state and respectively. Remark 3.
All previous protocols are “naturally broadcasting” i.e., eventually all agents agree onsome (the correct) value. With previous definitions (which are the classical ones for populationprotocols), the following protocol does not compute the
X OR or input bits, or equivalently doesnot compute predicate n ≡ ( mod ) . → → → → (7) Indeed, the answer is not eventually known by all the agents. It computes the
X OR in aweaker form i.e., eventually, all agents will be in state , if the X OR of input bits is , oreventually only one agent will be in state , if the X OR of input bits is . The classical solution [1] to the leader election problem (starting from a configuration with ≥ leaders, eventually exactly one leader survives) is the following: LL → LNLN → LNNL → NLNN → NN (8)Unfortunately, this protocol is non-symmetric, and hence non-Pavlovian. . Bournez, J. Chalopin, J. Cohen, X. Koegler Remark 4.
Actually, the problem is with the first rule, since one wants two leaders to becomeonly one. If the two leaders are identical, this is clearly problematic with symmetric rules.
However, the leader election problem can actually be solved by a Pavlovian protocol, at theprice of a less trivial protocol.
Proposition 3.
The following Pavlovian protocol solves the leader election problem, as soon asthe population is of size ≥ . L L → L NL N → NL L N → NL NN → NNL L → NL NL → L NNL → L NL L → L L L L → L L (9) Proof.
Indeed, starting from a configuration containing not only N s, eventually after some timeconfigurations will have exactly one leader, that is one agent in state L or L .Indeed, the first rule and the fifth rule decrease strictly the number of leaders whenever thereare more than two leaders. Now the other rules, preserve the number of leaders, and are madesuch that an L can always be transformed into an L and vice-versa, and hence are made suchthat a configuration where first or fifth rule applies can always be reached whenever there aremore than two leaders. The fact that it solves the leader election problem then follows from thehypothesis of fairness in the definition of computations.This is a Pavlovian protocol, since it corresponds to the following payoff matrix, with thresh-old D = Opponent L L N Player L L N Proposition 4.
The majority problem (given some population of s and s, determine whetherthere are more s than s) can be solved by a Pavlovian population protocol. If one prefers, the predicate n ≥ n on counts of input agents can be computed by a Pavlovianpopulation protocol.2 Playing With Population Protocols
Proof.
We claim that the following protocol outputs 1 if there are more s than s in the initialconfiguration and 0 otherwise, NY → YYY N → YYN → Y N → YY → N Y → N → NY → Y N (10)taking • S = { , } , Q = { , , Y , N } , • w ( ) = w ( Y ) = , • w ( ) = w ( N ) = .In this protocol, the states Y and N are “neutral” elements for our predicate but they shouldbe understood as Yes and No . They are the “answers” to the question: are there more s than s. This protocol is made such that the number of s and s is preserved except when a meetsa . In that latter case, the two agents are deleted and transformed into a Y and a N .If there are initially strictly more s than s, from the fairness condition, each will bepaired with a and at some point no will left. By fairness and since there is still at least a , a configuration containing only and Y s will be reached. Since in such a configuration, norule can modify the state of any agent, and since the output is defined and equals to in sucha configuration, the protocol is correct in this caseBy symmetry, one can show that the protocol outputs if there are initially strictly more sthan s.Suppose now that initially, there are exactly the same number of s and s. By fairness,there exists a step when no more agents in the state or left. Note that at the moment wherethe last is matched with the last , a Y is created. Since this Y can be “broadcast” over the N s, in the final configuration all agents are in the state Y and thus the output is correct.This protocol is Pavlovian, since it corresponds to the following payoff matrix with thresh-old . Opponent N Y 0 1N
Player Y . Bournez, J. Chalopin, J. Cohen, X. Koegler We proved that predicates on counts of input agents n ≥ , n = , n ≥ m , where n , m are somecounts of input agents, can be computed by some Pavlovian population protocols.It is clear that the subset of the predicates computable by Pavlovian population protocolsis closed by negation: just switch the value of the individual output function of a protocolcomputing a predicate to get a protocol computing its negation.However, some work remains to be done to fully characterize which predicates can be com-puted by a Pavlovian population protocol. The first steps would be to understand the followingquestions. Question 1. Is mod , or equivalently the predicate n ≡ ( mod ) , computable by a Pavlovianpopulation protocol? Question 2. Is ≥ k , or equivalently the predicate n ≥ k , for fixed k , computable by a Pavlovianpopulation protocol? Notice that, unlike what happens for general population protocols, composing Pavlovianpopulation protocols into a Pavlovian population protocol is not easy. It is not clear whetherPavlovian computable predicates are closed by conjunctions: classical constructions for generalpopulation protocols can not be used directly.As we said, Pavlovian Population protocols are symmetric. We however know that assumingpopulation protocols symmetric is not a restriction.
Proposition 5.
Any population protocol can be simulated by a symmetric population protocol,as soon as the population is of size ≥ . Before proving this proposition, we state the (immediate) main consequence.
Corollary 1.
A predicate is computable by a symmetric population protocol if and only if it issemilinear.Proof (of proposition):
To a population protocol ( Q , S , i , w , d ), with Q = { q , · · · , q n } associatepopulation protocol ( Q ∪ Q ′ , S , i , w , d ′ ) with Q ′ = { q ′ , · · · , q ′ n } , w ( q ′ ) = w ( q ) for all q ∈ Q , and forall rules qq → ab in d , the following rules in d ′ : qq ′ → ab q ′ q → ba qq → q ′ q ′ q ′ q ′ → qqq g → q ′ g q ′ g → q gg q → g q ′ g q ′ → g q for all g ∈ Q ∪ Q ′ , g = q , g = q ′ , and for all pairs of rules (cid:26) qr → ab rq → de Playing With Population Protocols with q , r ∈ Q , the following rules in d ′ : qr ′ → ab r ′ q → ba rq ′ → de q ′ r → ed . The obtained population protocol is clearly symmetric. Now the first set of rules guaranteesthat a state in Q can always be converted to its primed version in Q ′ and vice-versa. By fairness,whenever a rule qq → ab (respectively qr → ab ) can be applied, then the corresponding twofirst rules of the first set of rules (resp. of the second set of rules) can eventually be fired afterpossibly some conversions of states into their primed version or vice-versa. References [1] Dana Angluin, James Aspnes, Zo¨e Diamadi, Michael J. Fischer, and Ren´e Peralta. Computation innetworks of passively mobile finite-state sensors. In
Twenty-Third ACM Symposium on Principlesof Distributed Computing , pages 290–299. ACM Press, July 2004.[2] Dana Angluin, James Aspnes, and David Eisenstat. Stably computable predicates are semilinear.In
PODC ’06: Proceedings of the twenty-fifth annual ACM symposium on Principles of distributedcomputing , pages 292–299, New York, NY, USA, 2006. ACM Press.[3] James Aspnes and Eric Ruppert. An introduction to population protocols. In
Bulletin of the EATCS ,volume 93, pages 106–125, 2007.[4] Robert M. Axelrod.
The Evolution of Cooperation . Basic Books, 1984.[5] Bruno Beaufils.
Mod`eles et simulations informatiques des probl`emes de coop´eration entre agents .PhD thesis, Universit´e de Lille I, 2000.[6] Pierre Br´emaud.
Markov Chains, Gibbs Fields, Monte Carlo Simulation, and Queues . Springer-Verlag, New York, 2001.[7] DJ Daley and DG Kendall. Stochastic Rumours.
IMA Journal of Applied Mathematics , 1(1):42–55,1965.[8] Z. Diamadi and M.J. Fischer. A simple game for the study of trust in distributed systems.
WuhanUniversity Journal of Natural Sciences , 6(1-2):72–82, 2001.[9] Martin E. Dyer, Leslie Ann Goldberg, Catherine S. Greenhill, Gabriel Istrate, and Mark Jerrum.Convergence of the iterated prisoner’s dilemma game.
Combinatorics, Probability & Computing ,11(2), 2002.[10] Laurent Fribourg, St´ephane Messika, and Claudine Picaronny. Coupling and self-stabilization. InRachid Guerraoui, editor,
Distributed Computing, 18th International Conference, DISC 2004, Ams-terdam, The Netherlands, October 4-7, 2004, Proceedings , volume 3274 of
Lecture Notes in ComputerScience , pages 201–215. Springer, 2004.[11] D.T. Gillespie. A rigorous derivation of the chemical master equation.
Physica A , 188(1-3):404–425,1992.[12] Herbert W. Hethcote. The mathematics of infectious diseases.
SIAM Review , 42(4):599–653, De-cember 2000.[13] J. Hofbauer and K. Sigmund. Evolutionary game dynamics.
Bulletin of the American MathematicalSociety , 4:479–519, 2003.[14] D. Kraines and V. Kraines. Pavlov and the prisoner’s dilemma.
Theory and Decision , 26:47–79,1988.[15] Ouassila Labbani. Comparaison des th´eories des jeux pour l’´etude du comportement d’agents. Mas-ter’s thesis, Universit´e de Lille I, 2003. . Bournez, J. Chalopin, J. Cohen, X. Koegler [16] James Dickson Murray. Mathematical Biology. I: An Introduction . Springer, third edition, 2002.[17] M. Nowak and K. Sigmund. A strategy of win-stay, lose-shift that outperforms tit-for-tat in thePrisoner’s Dilemma game.
Nature , 364(6432):56–58, 1993.[18] M. Presburger. Uber die Vollstandig-keit eines gewissen systems der Arithmetik ganzer Zahlen, inwelchemdie Addition als einzige Operation hervortritt.
Comptes-rendus du I Congres des Mathe-maticians des Pays Slaves , pages 92–101, 1929.[19] J¨orgen W. Weibull.