Fairness and trust in structured populations

Tarnita, Corina E. Article Fairness and trust in structured populations Games Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Tarnita, Corina E. (2015) : Fairness and trust in structured populations, Games, ISSN 2073-4336, MDPI, Basel, Vol. 6, Iss. 3, pp. 214-230, https://doi.org/10.3390/g6030214 This Version is available at: https://hdl.handle.net/10419/167942 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Games 2015,6, 214-230; doi:10.3390/g6030214 OPEN ACCESS games ISSN 2073-4336 www.mdpi.com/journal/games Article Fairness and Trust in Structured Populations Corina E. Tarnita Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ 08544, USA; E-Mail: [email protected]; Tel.: +1-609-258-3896 Academic Editor: Martin A. Nowak and Christian Hilbe Received: 5 June 2015 / Accepted: 13 July 2015 / Published: 20 July 2015 Abstract: Classical economic theory assumes that people are rational and selfish, but behavioral experiments often point to inconsistent behavior, typically attributed to “other regarding preferences.” The Ultimatum Game, used to study fairness, and the Trust Game, used to study trust and trustworthiness, have been two of the most influential and well-studied examples of inconsistent behavior. Recently, evolutionary biologists have attempted to explain the evolution of such preferences using evolutionary game theoretic models. While deterministic evolutionary game theoretic models agree with the classical economics predictions, recent stochastic approaches that include uncertainty and the possibility of mistakes have been successful in accounting for both the evolution of fairness and the evolution of trust. Here I explore the role of population structure by generalizing and expanding these existing results to the case of non-random interactions. This is a natural extension since such interactions do not occur randomly in the daily lives of individuals. I find that, in the limit of weak selection, population structure increases the space of fair strategies that are selected for but it has little-to-no effect on the optimum strategy played in the Ultimatum Game. In the Trust Game, in the limit of weak selection, I find that some amount of trust and trustworthiness can evolve even in a well-mixed population; however, the optimal strategy, although trusting if the return on investment is sufficiently high, is never trustworthy. Population structure biases selection towards strategies that are both trusting and trustworthy trustworthy and reduces the critical return threshold, but, much like in the case of fairness, it does not affect the winning strategy. Further considering the effects of reputation and structure, I find that they act synergistically to promote the evolution of trustworthiness. Keywords: Ultimatum Game; Trust Game; reputation; evolutionary game theory; population structure Games 2015,6215 1. Introduction Game theorists have traditionally assumed that people act fully rationally to maximize their own financial gains. However, behavioral data has been inconsistent with these expectations. The Ultimatum Game (UG) and the Trust Game (TG) are two of the most influential examples of such inconsistent behavior, usually attributed to “other-regarding preferences” [1]. In the UG, two players have to divide a certain sum of money between them. One player (the proposer) makes an offer. The other player (the responder) can either accept the offer, in which case each receives the money as proposed, or reject the offer, in which case neither player receives anything. In a one-shot anonymous UG, a rational self-interested proposer will offer the minimum amount that she believes will be acceptable to the responder. A rational self-interested responder will accept any nonzero offer. Thus, the rational decision is for the proposer to make the minimum possible offer, and for the responder to accept it. To evaluate these predictions, many behavioral experiments have been conducted using the UG (see [1] for a review). Although there is considerable quantitative variation across studies, two clear qualitative deviations from rational self-interest are always observed: (i) many responders choose to reject low (but nonzero) offers; and (ii) many proposers offer more than the minimum amount required to avoid rejection. In the TG, the investor begins with an initial amount of one monetary unit and can either keep it or transfer it (or some part of it) to the trustee. To represent the value created by interactions based on trust, whatever gets transferred is multiplied by a factor b > 1. The trustee then chooses how much to return to the investor. In a one-shot anonymous trust game, there is no reason for a self-interested trustee to return anything [2]. Hence, there is no reason for a self-interested investor to make the transfer, and the potential gains of trust and exchange are lost. In all behavioral experiments with the trust game, however, investors do make transfers and trustees return significant amounts [3]. These inconsistencies have similarly been attributed to “other-regarding preferences”. While these are perfectly acceptable proximate explanations, the question that still remains is: what is the source of these other-regarding preferences? [4] Evolutionary biologists have attempted to explain these phenomena from an evolutionary game theoretic perspective in which agents are not assumed to be rational; instead, they reproduce proportional to how well they do in the specific games, their offspring inherit their strategies, and natural selection chooses the winning strategies [5]. Traditional, deterministic models of evolutionary game theory agree with the classical game theoretic predictions: in the one-shot anonymous UG and TG, natural selection favors rationality: low offers and demands in the UG and no investment or return in the TG [6,7]. However, more complex evolutionary models have showed promising results. Rand and collaborators [8] showed using stochastic evolutionary game theory, where agents make mistakes when judging the payoffs and strategies of others, that natural selection can favor fairness in the UG. Gale and collaborators [9] have shown that assuming that learning processes are subject to constant perturbations and noise can lead to fairness. Page and collaborators [10] found that fairness can evolve in the UG in a spatial setting, in which interactions are non-random. Manapat and collaborators [7] showed that when individuals’ strategies evolve in a context in which investors sometimes have knowledge about trustees’ reputations before transactions, natural selection can favor both trust and trustworthiness. Games 2015,6216 Here I examine the same stochastic evolutionary setup proposed by [8], but extended to apply to a structured population playing the UG and the TG respectively. An underlying population structure simply means that individuals are not equally likely to interact with each other, but that interactions are more likely occurring with one’s “neighbors” [11]. The neighbors could be geographical but they could also be individuals with similar strategies, preferences, physical features, cultural or genetic backgrounds etc. Population structure has been very powerful to explain the evolution of cooperation and social behavior both theoretically [12–14] and empirically [15,16]. However, its effects on fairness and trust have not been much studied with the exception of one promising experimental study in which a behavioral clustering mechanism—pairing trusting individuals with trustworthy ones—leads to an increase in the levels of cooperation compared to random pairings of investors and trustees [17]. I begin by developing a general theory and analytical results for the study of a continuous strategy space in structured populations, in the limit of weak selection. I then apply these general results to the study of fairness and the study of trust and trustworthiness with and without access to reputation. 2. General Model Description and Results I consider a structured population of Nplayers able to choose from a continuum of strategies situated on the n-dimensional hypercube. A strategy p∈[0,1]nis an n-dimensional vector of numbers between 0 and 1. The underlying population structure determines who interacts with whom to accumulate payoff and who competes with whom for reproduction. The expected payoff for an interaction between any two strategies is given by a function Edetermined by the specific game played. Individuals that accumulate higher payoff are more likely to reproduce: the rate of reproduction of each individual is proportional to 1 + δE, where δis a constant that measures the intensity of selection. The higher the intensity of selection, the more likely agents with higher payoffs are to be imitated (to reproduce). At the extreme of δ→ ∞, only those who obtain the highest payoff are imitated (strong selection). At the other extreme, δ→0selection is weak; in this case, all strategies have almost the same effective payoff and the dynamics is dominated by neutral drift. Weak selection is a natural situation that can arise in different ways: (i) payoff differences are small; (ii) strategies are similar; or (iii) individuals are confused about payoffs when updating their strategies. In such situations, the particular game makes only a small contribution to the overall reproductive success of an individual. Finally, reproduction is subject to symmetric mutation: with probability 1−uthe offspring inherits the strategy of the parent, but with probability ua random strategy is chosen. This process leads to a stationary distribution characterizing the mutation-selection equilibrium. I am interested in the dynamics of this process for large but finite population size and weak selection, which here will mean δ1/N. In this case, although the frequencies of the strategies can widely fluctuate in time, all strategies have approximately the same abundance on average in the stationary distribution of the mutation-selection process. I am interested in the small deviations from this uniform distribution. I say that strategy pis favored on average in the mutation-selection equilibrium, if its abundance exceeds the mean. To calculate this deviation I use a perturbation theory in the selection Games 2015,6217 strength, δand, for a strategy to be selected, I require that the first order term with respect to δis positive [8,18,19]. I show in Appendix A that this is equivalent to: Ep=˜ Lp+σ2˜ Hp>0(1) where ˜ Lp=Z[0,1]n [σ1E(p,p) + E(p,p0)−E(p0,p)−σ1E(p0,p0)] dp0 ˜ Hp=Z[0,1]nZ[0,1]n [E(p,p00)−E(p0,p00)] dp0dp00, (2) and E(p0,p00)is the expected payoff that strategy p0receives from strategy p00 in a given game. The parameters σ1and σ2are structural coefficients that need to be calculated for the specific evolutionary process under investigation [20]. These parameters depend on: the population structure that determines who interacts with whom; the update rule that determines whose strategy gets imitated and how; and the mutation rate. The first term, ˜ Lp, integrates over all pairwise competitions that involve strategy p, each pairwise comparison including the first structural coefficient, σ1. Thus, σ1encapsulates how much more likely a strategy is to play its own kind than the opposite kind in a pairwise encounter; henceforth I will call σ1the pairwise structural coefficient and I will consider only structures for which individuals are more likely to interact with their own kind, i.e.,σ1>1. The second term, σ2˜ Hp, evaluates the competition between strategy pand all other strategies simultaneously, weighted by the second structural coefficient σ2. Thus, σ2captures the interaction in the other extreme case, when all strategies are simultaneously present in the population; henceforth I will call σ2the mean structural coefficient and I will consider only structures for which σ2>0. In the limit of low mutation only one structural coefficient is needed and Condition (1) becomes: E0 p=Z[0,1]n [σ0E(p,p) + E(p,p0)−E(p0,p)−σ0E(p0,p0)] dp0>0(3) where σ0is the low mutation limit of σ= (2σ1+σ2)/(2 + σ2), the structure coefficient for games with two strategies [20,21]. Finally, the most favored (optimal) strategy is determined by maximizing Ep(or E0 pin the limit of low mutation). 3. Evolution of Fairness: Ultimatum Game I model the ultimatum game by imagining two players who have to split an amount summing to unity. In any given interaction, players are randomly assigned to the roles of proposer and responder. An agent’s strategy is given by the two dimensional vector S= (p, r)∈[0,1]2, where pis the amount offered when acting as proposer, and ris the minimum amount demanded when acting as responder, or the “rejection threshold.” An offer pis accepted by a responder with the rejection threshold rif and only if p≥r. Let U(S1, S2)be the expected payoff that strategy S1= (p1, r1)gets from strategy S2= (p2, r2) in the ultimatum game. Since I assume that in the interaction between a player using strategy S1and a Games 2015,6218 player using strategy S2, each player can be in the role of the proposer with equal probability, U(S1, S2) is given (up to a 1/2factor which I henceforth omit) by the function: U(S1, S2) =          1−p1+p2if p1≥r2and p2≥r1 1−p1if p1≥r2and p2< r1 p2if p1< r2and p2≥r1 0if p1< r2and p2< r1 (4) Depending on whether p≥ror p<r, the payoff function U(S, S)takes two different values (1 and 0 respectively) and hence I find the condition for strategy Sto be favored by selection to be ES=˜ LS+σ2˜ HS=σ1I(p≥r)−1 2+p−2p2+r−r2+σ2(p−p2−r2 2)>0 E0 S=σ0I(p≥r)−1 2+p−2p2+r−r2>0 (5) where I(condition)is one if condition is true and is zero if condition is false. In [8] it was shown that stochasticity and uncertainty can select for fairness in a well-mixed population. To recover their results I simply use the values of the structure coefficients for a well-mixed population, σ1= 1 and σ2=Nu. Adding population structure expands the region of strategies selected for to include increasingly more fair strategies (Figure 1). Already σ≥2is sufficient to select for the entire p≥rregion. However, not all strategies selected for have the same abundance in the stationary distribution, so next I focus on the optimum strategy (most abundant in the stationary distribution and hence, by the measure above, most favored by selection). Maximizing ESand E0 SI conclude that the optimal strategy is achieved when p≥rand is given by: (popt, ropt) = ((1 3,1 3)if u→0or 0≤σ2≤1 1+σ2 4+2σ2,1 2+σ2if u >> 0and σ2>1 Note that for low mutation the pairwise structural coefficient σ0has no effect on the optimum strategy. Thus, in the limit of weak selection and low mutation the optimum strategy is (1/3,1/3) – one that offers 33% and also rejects any offer lower than 33% – which is the same optimum obtained for the well-mixed population [8]. As mutation increases, the second structural coefficient, σ2, that captures the effect of mutation and structure in the center of the hypercube, starts to influence the optimal strategy, but only when σ2is larger than 1. However, the pairwise structural coefficient σ1continues to have no effect on the outcome. As σ2increases, the proposal increases and the rejection threshold decreases. For high σ2, the most frequent strategy is (1/2,0); thus, the proposal is 50% and the rejection threshold is 0. This outcome is achieved in the high mutation limit, where all strategies are present in the population simultaneously with approximately equal frequency. Hence, the optimum strategy is the one that maximizes its expected absolute payoff against a randomly chosen opposing strategy. As has been shown previously [6,8], it is intuitive that this strategy is (1/2,0) since the offer p= 1/2maximizes the proposer’s expected payoff of p(1 −p)when playing against a randomly chosen opponent; the demand r= 0 maximizes the expected payoff as responder because any nonzero demand results in lost profit. Games 2015,6219 Figure 1. Structure increases the region of strategies that are selected for: light blue region gets added to the yellow-green gradient region corresponding to a well-mixed population. Results are shown for weak selection with σ0= 1.5; here the optimum strategy remains the same (1/3,1/3). The important conclusion for the ultimatum game is that, in the limit of weak selection, the population structure has an impact on the space of strategies that get selected, increasing the density of fair strategies, but has little-to-no impact on the optimum strategy. The optimal strategies observed are very close to those selected for in the well-mixed population and depend only on the mean structural coefficient σ2. This is because the pairwise σ1affects the overall effect of selection on a strategy S= (p, r)via two terms: U(S, S)which is either 0 if p<ror 1 if p≥r(independent of the actual values of pand r) and R[0,1]2U(S0, S0)which is always equal to 1/2. Thus the effect of structure on pairwise interactions is the same for all strategies above and below the diagonal, respectively. This therefore can not affect the relative ordering of the strategies; however, it can change the region of strategies that are selected for. 4. Evolution of Trust: Trust Game I model the trust game by imagining two players that have an investor-trustee transaction. In any given interaction, players are randomly assigned to the roles of investor and trustee. An agent’s strategy is given by the two dimensional vector S= (p, r)∈[0,1]2, where pis the amount invested when the agent acts as investor and ris the fraction returned when the agent acts as trustee. The investor begins with an initial stake of one unit and can choose to either keep the stake or transfer some fraction 0≤p≤1of it to the trustee. To represent the value created by interactions based on trust, the transferred amount is multiplied by a factor b > 1, which I will refer to as the return on investment. The trustee then chooses what fraction rof the enhanced transfer amount pb to return to the investor. Let T(S1, S2)be the expected payoff that strategy S1= (p1, r1)gets from strategy S2= (p2, r2). Since I assume that in Games 2015,6220 the interaction between a player using strategy S1and a player using strategy S2, each player can be in the role of the investor with equal probability, T(S1, S2)is given (up to a 1/2factor which I henceforth omit) by the function: T(S1, S2) = 1 −p1+p1br2+p2b(1 −r1)(6) 4.1. Well-Mixed Population Here it is worth first investigating the results obtained in this stochastic game theoretic framework, in the limit of weak selection, in a well mixed population. This is the analog of the study conducted in [8] for the evolution of fairness. Although a very similar framework has been employed in [7], the analysis was not for weak selection. Using the payoff function in Equation (6) I find for any mutation: ES=2 + Nu 2−br +p(b−2) + 1(7) which is positive for the same set of strategies for which its low mutation version (obtained for Nu →0) is positive. Thus, mutation does not affect the selective outcome for a well-mixed population at weak selection. From this I conclude that any strategy with 0≤p≤1and 0< r < 1/b + (1 −2/b)pis selected for (Figure 2). Games 2015,xx 9 Pairwise outcompeted by (1,0) Pairwise outcompete (1,0) 1/b Trustworthiness, r Trust, p Selected for Selected against 1/b 1-1/b b > 2 Selected for Selected against 1/b 1-1/b 1 < b < 2 Pairwise outcompeted by (0,0) Pairwise outcompeted by (0,0) Pairwise outcompeted by (1,0) Pairwise outcompete (1,0) 1/b Trust, p Trust, p Trustworthiness, r 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 Figure 2. In a well-mixed population, in the limit of weak selection, both trust and trustworthiness are selected for and, for sufficient returns, the trusting but not trustworthy strategy, (1,0), is the optimal strategy. Upper panels: 1<b<2. Left: the most abundant strategy (green dot) is (0,0) – no trust and no trustworthiness. However, strategies with any amount of trust but low trustworthiness are also selected for, albeit less abundant. Middle: strategy (0,0) outcompetes all other strategies. Right: strategy (1,0) (blue dot) outcompetes a large number of strategies in pairwise interactions; for a subset of these strategies, (1,0) wins more in pairwise interactions than does (0,0) (grey shaded area). Lower panels: b > 2. Left: the most abundant strategy (green dot) is (1,0) – complete trust and no trustworthiness. However, strategies with any amount of trust and significant levels of trustworthiness are also selected for, albeit less abundant. Middle: strategy (0,0) outcompetes all other strategies. Right: strategy (1,0) (blue dot) outcompetes a large number of strategies in pairwise interactions; for a subset of these strategies, (1,0) wins more in pairwise interactions than does (0,0) (grey shaded area). Figure 2. In a well-mixed population, in the limit of weak selection, both trust and trustworthiness are selected for and, for sufficient returns, the trusting but not trustworthy strategy, (1,0), is the optimal strategy. Upper panels: 1< b < 2. Lower panels: b > 2. Left panels: the most abundant strategy (green dot). Middle panels: strategy (0,0) outcompetes all other strategies. Right panels: strategy (1,0) (blue dot) outcompetes a large number of strategies in pairwise interactions; for a subset of these strategies, (1,0) wins more in pairwise interactions than does (0,0) (grey shaded area). Games 2015,6221 Thus, as was the case for fairness [8], stochasticity and mistakes alone can promote the evolution of some amount of trust and trustworthiness even in a well-mixed population. In particular, strategy (1,0) that trusts but is not trustworthy is always selected for. This is because at weak selection all strategies have similar abundances and no strategies of the hypercube, even seemingly non-sensical ones – are a priori excluded from the game. Only natural selection eventually removes the strategies that do not perform well. In this case, strategies that display some trust and trustworthiness can win many pairwise encounters with strategies that are more trusting and trustworthy than they are (Figure 2). And although the (0,0) strategy that neither trusts nor is trustworthy outcompetes all other strategies pairwise (Figure 2 middle panels), the (1,0) strategy that trusts but is not trustworthy gains more from certain pairwise interactions than does (0,0) (Figure 2 right panels). Therefore, on average, as long as the return is sufficiently high (b > 2), strategy (1,0) can be the optimum strategy. To conclude, for weak selection, stochasticity and mistakes can lead to the evolution of trust and low-to-intermediate levels of trustworthiness. Finally, here I used a one-population formulation where individuals are equally likely to be found in the two roles. However, a two-population formulation in which one population is made of investors and the other of trustees can also be employed when both populations are well-mixed. Ohtsuki et al. [22] derived analytical conditions for a pair of strategies to be favored in the limit of weak selection in bimatrix well-mixed games and [8] found that the one-population and two-populations formulations yield identical results for the well-mixed ultimatum game. Applying the approach in [22] to the trust game between two well-mixed populations I similarly find that the two-populations formulation yields identical results to the one population formulation (Appendix B). However, no extensions have been made for the study of bimatrix games with population structure. 4.2. Structured Populations Next, I move on to the study of population structure. Using the same payoff Function (6), I find for any mutation: ES=1 2−br(2 + σ2) + p(2bσ1+bσ2−2(1 + σ1+σ2)) + 1 + b−(b−1)σ1+σ2(8) and in the limit of low mutation E0 S=1 2−2br +p(−2 + 2(b−1)σ0) + 1 + b−(b−1)σ0(9) Here σ0is the low mutation limit of the structural coefficient for a game with only two strategies σ= (2σ1+σ2)/(2 + σ2). From these I conclude that any strategy (p, r)with 0≤p≤1 0< r < 1 b−(b−1)(σ−1) 2b+pσ1−2−σ−1 σ b(10) is selected for. For low mutation, one simply needs to replace σby σ0in the above. Note that σ1and σ2 do not influence the selection outcome independently. They only influence it together via σ. Population structure affects the well-mixed results in two ways. First of all, it changes the region of strategy space Games 2015,6228 I find L(p, r) + 2(N−1)uH(p, r) = (1 + (N−1)u)(−br +p(b−2) + 1) (19) This is greater than zero for the same strategy space as derived in Section 4.1, Equation (7). Furthermore, optimizing, I obtain the same condition as for the one-population formulation: (popt, ropt) = ((0,0) if 1<b<2 (1,0) if b≥2 Thus, using the two population formulation for the trust game when both populations are well-mixed yields the same outcome as using the one population formulation. C. Calculation of ESfor the Trust Game with Reputation Because ˆ T(S, S)depends on whether r > 1/b, I find: for r≤1/b ES=1 2−b(1 −q)(2 + σ2)r+p(1 −q)(b(2σ1+σ2)−2(1 + σ1+σ2)) + C E0 S=1 2−2b(1 −q)r+ 2p(1 −q)(bσ0−2(1 + σ1)) + C0 and for r > 1/b ES=1 2−b(1 + q)(2 + σ2)r+p(1 −q)(b(2σ1+σ2)−2(1 + σ1+σ2)) + C+ +q(1 −σ1+b(1 + σ1+σ2)) E0 S=1 2−2b(1 −q)r+ 2p(1 −q)(bσ0−2(1 + σ1)) + C0+q(1 −σ1+b(1 + σ1)) (20) where C0and Care constants with respect to the strategy S= (p, r). The trustful and trustworthy strategy can be the winning strategy if its expected payoff is higher than that of the trusting strategy that is not trustworthy, i.e., if E(1,0) < E(1,1/b+). This is the case if and only if: q≥(1 + b)1 σ b(1 + 1 σ)−b1 σ−1(21) which is less than 1 if ≤(b−1)(σ+ 1) 2b(22) A sufficient condition for qto satisfy Equation (21) is that 2 3b−σ−1 3b(2σ+ 1) ≤q≤1(23) as long as  < σbσ−(1 + σ) b(1 + 2σ) + (1 + σ)(24) Games 2015,6229 References 1. Camerer, C.F. Behavioral Game Theory: Experiments in Strategic Interaction; Princeton University Press: Princeton, NJ, USA, 2003. 2. Berg, J.; Dickhaut, J.; McCabe, K. Trust, reciprocity, and social history. Games Econ. Behav. 1995,10, 122–142. 3. Johnson, N.D.; Mislin, A.A. Trust games: A meta-analysis. J. Econ. Psychol. 2011,32, 865–889. 4. Wilson, D.S.; Gowdy, J.M. Evolution as a general theoretical framework for economics and public policy. J. Econ. Behav. Organ. 2012,90, S3–S10. 5. Maynard, S.J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982. 6. Nowak, M.A.; Page, K.M.; Sigmund, K. Fairness versus reason in the ultimatum game. Science 2000,289, 1773–1775. 7. Manapat, M.L.; Nowak, M.A.; Rand, D.G. Information, irrationality and the evolution of trust. J. Econ. Behav. Organ. 2013,90, S57–S75. 8. Rand, D.G.; Tarnita, C.E.; Ohtsuki, H.; Nowak, M.A. Evolution of fairness in the one-shot anonymous Ultimatum Game. Proc. Natl. Acad. Sci. USA 2013,7, 2581–2586. 9. Gale, J.; Binmore, K.G.; Samuelson, L. Learning to be imperfect: The ultimatum game. Games Econ. Behav. 1995,8, 56–90. 10. Page, K.M.; Nowak, M.A.; Sigmund, K. The spatial ultimatum game. Proc. Roy. Soc. B 2000, 267, 2177–2182. 11. Lieberman, E.; Hauert, C.; Nowak, M.A. Evolutionary dynamics on graphs. Nature 2005,433, 312–316. 12. Ohtsuki, H.; Hauert, C.; Lieberman, E.; Nowak, M.A. A simple rule for the evolution of cooperation on graphs and social networks. Nature 2006,441, 502–505. 13. Nowak, M.A. Five rules for the evolution of cooperation. Science 2006,314, 1560–1563. 14. Nowak, M.A.; Tarnita, C.E.; Antal, T. Evolutionary dynamics in structured populations. Phil. Trans. R. Soc. Lond. B 2006,365, 19–30. 15. Rand, D.G.; Arbesman, S.; Christakis, N.A. Dynamic social networks promote cooperation in experiments with humans. Proc. Natl. Acad. Sci. USA 2011,108, 19193–19198. 16. Rand, D.G.; Nowak, M.A.; Fowler, J.H.; Christakis, N.A. Static network structure can stabilize human cooperation. Proc. Natl. Acad. Sci. USA 2014,111, 17093–17098. 17. McCabe, K.A.; Rigdon, M.L.; Smith, V.L. Sustaining Cooperation in Trust Games. Econ. J. 2007, 117, 991–1007. 18. Antal, T.; Traulsen, A.; Ohtsuki, H.; Tarnita, C.E.; Nowak, M.A. Mutation-selection equilibrium in games with multiple strategies. J. Theor. Biol. 2009,258, 614–622. 19. Tarnita, C.E.; Antal, T.; Nowak, M.A. Mutation-selection equilibrium in games with mixed strategies. J. Theor. Biol. 2009,261, 50–57. 20. Tarnita, C.E.; Wage, N.; Nowak, M.A. Multiple strategies in structured populations. Proc. Natl. Acad. Sci. USA 2011,108, 2334–2337. Games 2015,6230 21. Tarnita, C.E.; Ohtsuki, H.; Antal, T.; Fu, F.; Nowak, M.A. Strategy selection in structured populations. J. Theor. Biol. 2009,259, 570–581. 22. Ohtsuki, H. Stochastic evolutionary dynamics of bimatrix games. J. Theor. Biol. 2010,264, 136–142. 23. King-Casas, B.; Tomlin, D.; Anen, C.; Camerer, C.F.; Quartz, S.R.; Montague, P.R. Getting to know you: Reputation and trust in a two-person economic exchange. Science 2005,308, 78–83. 24. Cavaliere, M.; Sedwards, S.; Tarnita, C.E.; Nowak, M.A.; Csikasz-Nagy, A. Prosperity is associated with instability in dynamical networks. J. Theor. Biol. 2012,299, 126–138. 25. Wu, B.; Garcia, J.; Hauert, C.; Traulsen, A. Extrapolating weak selection in evolutionary games. PLoS Comp. Biol. 2013,9, e1003381. 26. Manapat, M.L.; Rand, D.G. Delayed and Inconsistent Information and the Evolution of Trust. Dyn. Games Appl. 2012,2, 401–410. c 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).