Generalized Reciprocity without Genetic. Linkage

Transcription

1 Generalized Reciprocity without Genetic Linkage Bernhard Voelkl April 1, 2013 Short Communication Running title: Generalized Reciprocity Department of Zoology, University of Oxford, South Parks Road OX1 3PS, Oxford UK. phone: , fax: University of Oxford, Department of Zoology University of Bern, Institute of Ecology and Evolution, Behavioural Ecology 1

2 1 Abstract Generalized reciprocity has been proposed as a possible mechanism for enabling cooperation between unrelated individuals. If reciprocators are themselves more often receivers of help than non-reciprocators, then they can gain higher net-payoffs than the non-reciprocators. While this is not possible in well mixed populations, van Doorn and Taborsky (2012) have shown that this can happen in structured populations. Yet, in their model the authors assumed complete linkage between reciprocation and spontaneous initiation of altruistic acts. Here, I extend the discussion by asking what happens if one does not assume such a linkage. In that case the dimensionality of the strategy space increases, allowing for more complex evolutionary dynamics. As a concequence, evolution can lead from a population dominated by defectors via a population dominated by conditional reciprocators who do not themselves initiate altruistic acts to a population of altruistic reciprocators. This evolutionary path is neutral in contrast to the previously suggested direct route from defectors to altruists, which required an initial walk against a selection gradient. 2

3 18 Introduction Generalized reciprocity (Pfeiffer et al., 2005) has been proposed as a mechanism for enabling continued cooperation between unrelated individuals. It can be described by the simple rule help somebody if you received help from someone and has also been studied under the headings of up-stream tit-fortat (Boyd and Richerson, 1989), up-stream indirect reciprocity (Nowak and Roch, 2007), or pay-it-forward reciprocity (Fowler and Christakis, 2010). Its appeal lies in its simplicity and in the fact that it does not require extensive memory capacities nor individual recognition. As individuals are considered being indiscriminative by helping others irrespective of their identity, such a system might be vulnerable to defectors who readily accept help but who fail to reciprocate. Though, if reciprocators are on average more often receivers of help than non-reciprocating defectors, they can gain higher net-payoffs and defectors cannot prosper in the population. Van Doorn and Taborsky (2012) have shown that this can happen in structured populations. Studying generalized reciprocity on artificial and real-world social networks they found that both sparsity and modularity enhanced the evolutionary stability of cooperation. In order to study under which circumstances generalized reciprocity can prosper, van Doorn and Taborsky devised a simple model of a population 3

4 with two strategy phenotypes: conditional reciprocators who, after receiving help from someone, help a randomly chosen individual by performing a behaviour that brings along a cost (c) for themselves and a benefit (b) for the chosen recipient, and defectors who never help others. As in their model a conditional reciprocator upon receiving help will offer help exactly one time, one gets a chain of reciprocation that terminates as soon as a reciprocator helps a defector. However, a critical question is, how these chains of reciprocation are started. Van Doorn and Taborsky make the assumption, that from time to time a reciprocator spontaneously initiates an altruistic act without having received help before, while defectors never do so. That is, their model implicitly assumes complete genetic linkage between conditional reciprocation and spontaneous initiation of altruistic acts. Here, I will ask what happens if one does away with this assumption and considers spontaneous initiation and conditional reciprocation as independent behavioural phenotypes. In order to keep the model analytically tractable I will restrict the treatment to a simple and symmetric graph the cycle. Extensions and relations to other graph structures are discussed. 4

5 55 The Model Considering no linkage between the genotypes for spontaneous initiation and conditional reciprocation strategies, there are four possible phenotypes: actors who initiate and reciprocate (A, as of altruist ), actors who do not initiate but who do reciprocate (L, as of lazy altruist ), actors who initiate but who do not reciprocate (S, as of starter ) and actors who do neither initiate nor reciprocate (D, as of defector ). When one individual spontaneously gives some help, this might trigger a chain of reciprocation that wanders through the population as a random walk, until the random walk hits a non-reciprocator (S or D). Given the four strategy phenotypes there are four possible scenarios: A R D, A R S, S R S, and S R D, where R is a chain of conditional reciprocators (A or L types) of length 0 or longer. In a population of size N where the strategy phenotypes A, L, S, D oc- 69 cur at frequencies a N, l N, s N, d N respectively, the expected payoff for a single 70 individual Π X of a strategy X {A, L, S, D} is given by Π X = ( s N Π X S + a N Π X A) 1 x, (1) 71 where Π X S is the expected payoff for all X individuals given that a 72 random walk was initiated by an S type and Π X A is the expected payoff for 5

6 all X given that a random walk was initiated by an A type. In the following I will consider the conditional expected payoffs for all four strategies. If a random walk is initiated by an S individual, this induces a cost of c for the population of S. In the long run the initiating individual will give its help directly to another S type with a frequency of s 1. In these cases the N 1 random walk terminates immediately and the gain for the population of S individuals is b. If the random walk is terminated by a D type, the benefits for S are zero. With a frequency of a+l N 1 the initiator will direct its help towards 81 a conditional reciprocator (A or L). In these cases the expected gain for S will be b if the random walk returns to the initiating S-type individual and s 1 b if the random walk reaches the other boundary of the segment of s+d 1 reciprocators. In general, the probability that a symmetric random walk in one dimension starting from y will reach m before 0 is given by y/m. If we denote the length of the chain of reciprocators by j and consider that y = 1, as we investigate the case where the first transmission was from the lower boundary 0, representing the S-type initiator, to the first member of a chain 1 of reciprocators, we get as the likelihood that the random walk reaches j+1 ( ) the upper boundary and 1 1 that it returns to the initiator. To get j+1 the expected long term payoff for S we have to consider the frequency φ j with which we encounter uninterrupted chains of conditional reciprocators of length j. The expected payoff for S given that the walk was initiated by an 6

7 94 S-type individual is therefore given by Π S S = a + l N 1 a+l j=1 ( 1 1 j s 1 j + 1 s + d 1 ) φ j b + s 1 b c. (2) N If a random walk is initiated by an A type, then the likelihood that it will be terminated by an S type is given by the relative frequency of S types in the sub-population of non-reciprocators. Hence, the expected payoff for S is given by Π S A = s b. (3) s + d D-type individuals never initiate nor reciprocate, thus they do not bear any costs but just reap off the benefits of being receivers of help. As soon as a D-type is the recipient of help, the random walk stops. The long term frequency at which that happens is the complement to the frequency for a termination by S. The conditional payoff for D given that the random walk was started by S is therefore given by Π D S = a + l N 1 a+l j=1 1 d j + 1 s + d 1 φ jb + d b. (4) N Likewise, the payoff for D given that the random walk was started by A 106 is 7

8 Π D A = d b. (5) s + d 107 For conditional reciprocators the payoff depends on the expected length 108 of the random walk. Any time an A or L individual receives help it will reciprocate by helping one neighbour. That is, for any such act the payoff for a conditional reciprocator is (b c). For a symmetric random walk in one dimension with boundaries 0 and m starting at y the expected length of the walk is given by y(m y). Substituting y by 1 and m by j+1 as previously, we get for chains of length j an expected total payoff for conditional reciprocators of j(b c). From this total payoff A-types will acquire a, while a+l l a+l will fall to L-type individuals. Above we have denoted φ j as the expected frequency of chains of reciprocators of length j for the case that a random walk was started by an S-type individual. The expected payoff for A given that the random walk was started by S is therefore Π A S = a + l N 1 a+l a jφ j (b c). (6) a + l j= For the case that the random walk was started by an A-type individual we have to consider several facts. First, we have to add c to the payoff for A for the cost of initiating the walk. Second, we have to reconsider the frequencies at which chains of reciprocators of length j occur. Previously we 8

9 have not distinguished between A and L but just asked for the frequencies of reciprocator chains of a certain length. Now, for the case that the random walk was initiated by an A individual we should consider only those chains 126 of reciprocators, that contain at least one A individual. For the case of j = 1 only a chains will contain an A type individual, while for chains of a+l j > l all chains must contain at least one A type. Thus, clearly, the relative frequencies of chains of reciprocators that contain at least one A of length j differs from φ j. Furthermore, the likelihood with which an initiator is part of a chain of length j differs as it is dependent on the relative frequency of As in chains of that length. I, therefore, denote with φ j the likelihood that an initiating A-type individual is a member of a chain of reciprocators of length j. A final fact to be considered is, that the expected number of visits differs for the different positions within a chain of reciprocators. In order to get the expected number of visits of the initiator and the remaining reciprocators given that the random walk was initiated by an A-type individual, we can consider the random walk as a Markov process with two absorbing states and j transient states. If we rearrange the transition matrix so that the transient states come first we will get 9

10 Q j,j R j,2 P =. (7) O 2,j I 2,2 142 The fundamental matrix N for P is then given by N = (I j,j Q j,j ) 1, (8) 143 and tr(n ) j 1 gives the expected number of re-visits of the initiator (Grin stead and Snell, 1997). For a simple, symmetric random walk with p = q = 1 2 and two absorbing boundaries we get a closed form for the trace of N as tr(n ) = j(j + 2). (9) This gives an expected number of re-visits of the initiator of E[visit initiator] = j(j+2) 3 j 1 = j 1 3. (10) As the expected total number of visits until absorption at one of the boundaries is (j+1)(j+2) 6 1, the expected number of visits of the remaining reciprocators is E[visit other reciprocators] = (j + 1)(j + 2) 6 1 j 1 3 = j2 + j 2. (11) 6 10

11 In the long term average all of the re-visits of the initiator and a 1 of a 1+l the remaining visits will fall on A individuals. The expected payoff for A given that the random walk was initiated by A is therefore a+l ( j 1 Π A A = 3 j=1 + j2 + j 2 6 ) a 1 a 1 + l φ j (b c) c. (12) The expected payoffs for L can be evaluated in the same way as for A with the only differences that one has to take the proportion of L in the chains of reciprocators, and that L-types never pay costs for initiating. The expected payoff for L given that the random walk was started by S is Π L S = a + l N 1 a+l l jφ j (b c), (13) a + l j=1 157 and the expected payoff for L given that the walk was started by A is Π L A = a+l j=1 j 2 + j 2 6 l a 1 + l φ j(b c). (14) What remains to be done is finding expressions for φ and φ. If an S-type individual initiates a random walk by helping a conditional reciprocator, the likelihood φ j that the chain of reciprocators will have length j equals the long term relative frequency of finding chains of length j by randomly placing a+l reciprocators on a cycle of size N. 11

12 φ j = σ j a+l i=1 σ, (15) i where σ j is the total number of chains of reciprocators of length j for all permutations of a + l reciprocators on a cycle of size N, which is given by a+l j N σ j = (N j k 1). (16) (a + l j)! k=1 In case that a random walk is started by an A-type individual, the like- lihood that the chain of reciprocators has length j is given by φ j = σ jγ j ϱ j a+l i=1 σ iγ i ϱ i, (17) where γ j is the ratio of all chains of length j, which can be built out of a A-type and l L-type individuals, that contain at least one A, 169 γ j = ( a+l j ) ( l j) ( a+l ), (18) j and ϱ j is the expected number of As in those chains, given by a 1 ϱ j = 1 + (j 1) a 1 + l. (19) Having found explicit expressions for φ and φ, we can directly evaluate expected payoffs for any population with given frequencies for A, L, S, and D. Figure 1a shows the evolutionary trajectories on the faces of the Simplex 12

13 S 4 for the case that the benefit b for receiving help is 2.5 times as large as the cost c for helping (that is, c = 1 and b = 2.5). 175 Results and Discussion The evolutionary dynamics on the faces of the simplex S 4 are visualized in figure 1a. Evolutionary trajectories are evaluated assuming that individual payoffs translate linearly into fitness. Selection gradients were evaluated for every possible integer combination of the four strategies for a population size of N=100 and linear interpolation was used to generate the surface plot. If one looks at the edge A-D one retrieves the dynamics reported by van Doorn and Taborsky (2012). Selection is frequency dependent: if the population consists predominantly of A-types, then A can gain higher average payoffs than D and, consequently, the selection gradient goes towards a pure A population. On the other side, if D dominates, then the gradient goes towards D. The location of the saddle point depends on the benefit-cost ratio: the larger the benefits in relation to the costs, the larger the basin of attraction for A. The same dynamics can be observed along the A-S edge. In populations consisting entirely of S and D individuals, the gradient goes always towards D. If a population consists only of A and L individuals, any chain of reciprocation would in principle continue ad infinitum. Phenotype A has 13

14 a small disadvantage due to the cost for the initial help, though this advantage vanishes in the limit with the consequence, that there is no selection gradient along the A-L edge. Likewise there are no selection forces along the L-D edge, because the payoff for both strategies is always zero, and along the L-S edge, because both strategies receive on average the same payoff of (b c)/n. The fact that there is no fitness gradient along these edges means that the system does not have a global attractor towards which the population converges. In the A-S-D plane there is one area in which all trajectories point towards A and one area where all trajectories point towards D. These two areas are separated by a ridge where the selection gradient is about zero and all changes in strategy frequencies are due to drift alone. Similarly, the other three planes are each divided into two areas with gradients towards the A-L, L-S, and S-D edges, respectively. If one looks at the dynamics in the interior of the simplex, one can identify a bulgy cone with its tip at L, within which all trajectories point either towards A or an edge incident to A. This cone has a cool outer shale, where the selection gradient is relatively low and drift dominates, and a hot inner core along the A-L edge, where the selection gradient is rather steep and evolution should accelerate. The larger the benefit-cost ratio, the larger this cone becomes. One interesting aspect of this four-phenotype model is that it offers new 14

15 213 routes towards enhanced cooperativeness. For the two-phenotype model represented by the A-D edge it could be shown that for even moderate benefit-cost ratios the basin of attraction for A can become larger than the one for D. A resident population of A-types is stable against D-type mutants, though the question how the population became cooperative in the 218 first place remains to be answered. How can A-type individuals invade a D-type resident population? The tentative answer given by van Doorn and Taborsky (2012) is, that mutation, drift or migration cause the frequency of A to increase beyond the saddle point on the A-D edge. Yet, this requires assuming either drift against a selection gradient or strong perturbations, wich is in general not an entirely convincing explanation. In the four-phenotype model there is also no direct trajectory from D to A and one still has to invoke random drift, though in contrast to the former model drift has not to be uphill against a selection gradient. For example, it is possible to draw a route on the A-L-D face starting from D, drifting along the L-D edge (where there are no fitness differences) up to a point where L dominates, and from there along the A-L edge again with nearly-neutral drift towards an A-dominated population. Numeric simulations have shown that this is a common scenario (Figure 1b). For example, in simulations with a population size of 500, a mutation rate of 0.01, a cost-benefit ratio of 0.5 and weak selection the route A L D (occurring in 81 percent of the simulations) 15

16 was approximately four times as likely as the the direct route A D. In the presented model strategies are discrete that is, during its lifetime an individual plays always the same strategy. Taking a quantitative genetics approach to social behaviour it seems more plausible that the propensities to initiate and reciprocate are continuous traits, where real-valued parameters can describe the likelihood with which an individual will initiate or reciprocate an altruistic act. Such a model was introduced by Nowak and Roch (2007) and might be a fruitful ground for further investigation. For example, in the present model I have assumed that initiation and reciprocation are separately coded behavioural acts. An alternative view on initiation would be to consider it as a perception failure of a reciprocator. Arguing this way, it would be interesting to know whether there is a selection pressure for making fewer perception errors. This question could be addressed by such a quantitative approach, where the continuous parameters describe the likelihood of making false-positive or false-negative perception errors. It must be noted, that the evolutionary dynamics for graphs other than the cycle can look quite differently. The cycle is the symmetric, non-trivial graph with the lowest density and the highest characteristic path length, and consequently the largest effective population size (Broom and Voelkl, 2012). Strategies that profit from localized interactions will often perform best on the cycle (e.g. Grafen, 2007). This is supposedly also the case with general- 16

17 ized reciprocity. Density and characteristic path length of real-world social networks fall usually between those for the complete graph representing the well mixed population and the cycle. Adding even a small number of random edges to a cycle will lead to a graph with substantially reduced characteristic path length. This phenomenon is known as the small-world effect (Watts and Strogatz, 1998) and it suggests that the evolutionary dynamics of most real-world networks should be far less favourable for the evolution of cooperativeness than those of the cycle. 263 Acknowledgements I would like to thank Peter Hammerstein, Benjamin Bossan, Matthias Flor, Sander van Doorn and Michael Taborsky for fruitful discussions and comments on the manuscript. The author of this manuscript declares no conflict of interests. 268 References Boyd, R. and P. J. Richerson, The evolution of indirect reciprocity. Social Networks 11: Broom, M. and B. Voelkl, Two measures of effective population size 17

18 272 for graphs. Evolution 66: Fowler, J. H. and N. A. Christakis, Cooperative behaviour cascades in human social networks. Proc. Natl. Acad. Sci. USA 107: Grafen, A., An inclusive fitness analysis of altruism on a cyclical 276 network. J. Evol. Biol. 20: Grinstead, C. M. and J. L. Snell, Introduction to Probability. American Mathematical Society, Providence, RI Nowak, M. A. and S. Roch, Upstream reciprocity and the evolution of gratitude. Proc. R. Soc. Lond. B 274: Pfeiffer, T. C., C. Rutte, T. Killingback, M. Taborsky, and S. Bonhoeffer, Evolution of cooperation by generalized reciprocity. Proc. R. Soc. Lond. B 272: van Doorn, G. S. and M. Taborsky, The evolution of generalized reci- procity on social interaction networks. Evolution 66: Watts, D. J. and S. H. Strogatz, Collective dynamics of small-world networks. Nature 393:

19 a A D A S b 1.0 L S Figure 1: (a) Evolutionary trajectories on the faces of the simplex S 4 for c = 1 and b = 2.5 for a cycle of N = 100. Colour reflects the selection gradient with hot colours (red) indicating steep gradients and cold colors (blue) showing areas with selection gradients close to zero, where random drift will dominate. (b) Relative frequencies of the four phenotypic strategies A (blue), L (purple), S (brown), D (green), in a numeric simulation assuming a haploid birthdeath process, weak selection (where the fitness of an individual is given by 1 ω + ω payoff, with ω=0.05), population size of N=500, mutation rate of 0.01, and a cost-benefit ratio of 0.5. The simulation was run over years, each year comprising 365 rounds with one random walk started by one spontaneous initiation of a randomly selected individual and reproduction of a single individual at the end of the round. 19