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Replicator dynamics and evolutionary game of social tolerance: The role of neutral agents
Economics Letters 159 (2017) 10–14
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Replicator dynamics and evolutionary game of social tolerance: The role of neutral agents✩ Yingying Shi a,b , Min Pan a, *, Daiyan Peng c a b c
School of Economics and Management, Wuhan University, Wuhan, Hubei 430072, PR China School of Economics and Business Administration, Central China Normal University, Wuhan, 430079, PR China School of Economics, Huazhong University of Science and Technology, Wuhan, 430074, PR China
highlights • • • •
We study the evolutionary game of tolerance including neutral agents. We show that neutral agents plays an important role in the dynamics of tolerance. Tolerance can be a natural consequence of economic integration. Evolutionary game with neutral agents is consistent with economic integration.
article
info
Article history: Received 17 April 2017 Received in revised form 27 June 2017 Accepted 6 July 2017 Available online 13 July 2017 MSC: C7 D7 Keywords: Replicator dynamics Tolerance Economic interaction model
a b s t r a c t The role of neutral agents on evolutionary tolerance between two differentiated groups is discussed based on the replicator game model proposed recently. We show that, very different from the pure opposing case studied previously, dynamics of social tolerance with neutral agents is more positive and exhibiting rich interesting effects. The full intolerance steady state (0, 0) is unstable when neutral agents are taken into consideration and there are two type of evolution trajectory according to the population of the neutral agents. Especially, phase trajectories reach to the stable full tolerance steady state (1, 1) at any starting point if the population of the neutral agents is large enough, and the tolerance between different social groups can be a natural consequence of economic integration in the present of neutral agents. We show that neutral agents may remove the contradiction between the traditional idea of economic integration and the evolutionary game point of view. © 2017 Published by Elsevier B.V.
1. Introduction Tolerance, which is defined as a generic attitude to accept diversity (Akerlof and Kranton, 2000; Darity Jr. et al., 2006), is seen as a vital element of a liberal political order from the philosophical viewpoint of the liberal political thought: shifting into a Hobbesian state of anticipation is unlikely in a tolerant society due to the tolerant behavior of accepting conflicting political values (Muldoon et al., 2012). Cross-country and cross-region differences with respect to tolerance have been studied by sociologists resorting to the theory of cultural modernization (Berggren and Elinder, 2012; Berggren and Nilsson, 2013). The conjecture that social tolerance ✩ The work was supported in part by the Key Projects of Philosophy and Social Sciences Research, Ministry of Education of China (Grant number 15JZD013), the National Social Science Fund Project of China (Grant No. 10AZD019), and the China Postdoctoral Science Foundation funded project (Grant No. 2015M582250). Corresponding author. E-mail address: [email protected] (M. Pan).
*
http://dx.doi.org/10.1016/j.econlet.2017.07.005 0165-1765/© 2017 Published by Elsevier B.V.
is beneficial for population growth and technological performance has been demonstrated empirically (Bjørnskov, 2004, 2008), and social tolerance is shown positively related to social and economic development in many cases (Becchetti et al., 2010). Recently, the discussion on tolerance at the individual level reveals that economic reasoning can offer original and unique insights into the determinants of tolerance (Corneo and Jeanne, 2009; Garofalo et al., 2010; Shi and Pan, 2017). It has been believed that the fairer distribution of wealth among people, which also means economic integration, is of great importance to tolerance between different social groups (Becchetti et al., 2010). However, from the evolutionary game point of view, tolerance requires an unequal distribution of aggregate wealth among people, and fairness without a corresponding balance in the perception of diversity leads to a society with full intolerance (Cerqueti et al., 2013). Here we study the evolutionary game of tolerance, where the neutral agents is taken into consideration. We show that the tolerance between different social groups can be a natural consequence
Y. Shi et al. / Economics Letters 159 (2017) 10–14
of economic integration in the present of neutral agents, and in this case there is no contradiction between the traditional idea of economic integration and the evolutionary game point of view. 2. Coupled equations of tolerance dynamics An evolutionary game model of social tolerance with two differentiated groups has been proposed recently (Cerqueti et al., 2013). The population of each group are N1 and N2 , respectively, and assumed to be large enough and changeless with time. Each individual can be tolerant or intolerant towards the agents with the opposite group. If using the notation xi and xˆ i be the share of tolerant and intolerant agents in group i respectively, then xi + xˆ i = 1 and xi , xˆ i ∈ [0, 1], for each i = 1, 2. The replicator dynamics and the random pairwise matching assumption are used to discuss the tolerance dynamics, where two agents interact after being randomly matched (such economic interaction can be, for example, a business deal), producing aggregate wealth Rij = Rji which depend on the contribution of capital of the two agents with i, j = 1, 2. The evolutionary game model implicitly assumes that tolerant and intolerant behavior spreads based on the payoff of the strategy. The relative contribution of capital of agents in group i when she interact with agents in group j is δij ≡ ki /(ki + kj ) which determines the shares of the aggregate wealth with ki the contribution of capital of agents in group i: the agent in group i shares δij Rij when she interact with the agent in group j. Based on the evolutionary dynamics of social tolerance, it has been shown that tolerance requires an unequal distribution of aggregate wealth among people, and fairness without a corresponding balance in the perception of diversity leads to a society with full intolerance (Cerqueti et al., 2013). From the philosophical viewpoint of the liberal social thought, differences (such as ethnicity, religion, country of origin and social class) are only applied to specific groups and neutral agents which are always tolerant can exist in general. In the present work, we keep the two-group assumption while the neutral agents with population Ne is introduced in the evolutionary game model of social tolerance. We use an evolutionary game model of social tolerance similar to the model proposed by Cerqueti et al. (2013), where the replicator dynamics is used, to discuss the tolerance dynamics with neutral agents. We also use the random pairwise matching assumption that two agents interact after being randomly matched, producing aggregate wealth Rij = Rji or Rie = Rei with i, j = 1, 2, which depend on the contribution of capital of the two agents. We write the contribution of capital of agents in group i as ki and the contribution of capital of neutral agents as ke . The relative contribution of capital of agents in group i when she interact with agents in group j is δij ≡ ki /(ki + kj ), which determines the shares of the aggregate wealth. Similarly, the relative contribution of capital of agents in group i when she interact with neutral agents is δie ≡ ki /(ki + ke ). The agent in group i shares δij Rij when she interact with the agent in group j, while shares δie Rie when she interact with a neutral agent. The social tolerance influences the net gain obtained by each agents in the following cases: 1. For the case that the two unneutral agents are of the same group, whether tolerant or not, δii = 1/2, and each of them obtains Rii /2. 2. For the case that the two unneutral agents are of different group and both tolerant, them suffer a psychological cost α = Rii /2 and a social cost c = β (1 − xi xj ) with the exception of δij Rij , with δij + δji = 1, for each i, j = 1, 2. The parameter β is greater than zero, and a higher β leads a higher social costs, so the parameter β describes the social reaction of intolerant agents adverse to the agents of the opposite group. 3. For the case that the two unneutral agents are of different group and if one or both of them is intolerant, which rules out any
11
interaction between them, there is no wealth produced, and each of them obtains 0. 4. For the case that one of the agents is neutral while the another is of group i and tolerant, the neutral agent obtains δei Rei while the another agent of group i obtains δie Rie . 5. For the case that one of the agents is neutral while the another is of group i and intolerant, which rules out any interaction between them, there is no wealth produced, and each of them obtains 0. 6. For the case that the two agents are both neutral, each of them obtains Ree /2. The evolutionary dynamics of social tolerance can be modeled by the theory of replicators in which the payoff functions described above serve as the fitness function, and the evolution of tolerant population in group i can be described by x˙ i = xi xˆ i (E [xi ] − E [ˆxi ]),
(1)
where E [xi ] and E [ˆxi ] are the expected net gain of tolerant and intolerant individuals in group i respectively, and can be calculated by using the following expression: E [x1 ] = Px1 x1 R11 /2 + Px1 xˆ 1 R11 /2 + δ1e R1e Px1 Ne ,
+ [δ12 R12 − R11 /2 − β (1 − x1 x2 )]Px1 x2 , E [ˆx1 ] = Pxˆ 1 x1 R11 /2 + Pxˆ 1 xˆ 1 R11 /2, E [x2 ] = Px2 x2 R22 /2 + Px2 xˆ 2 R22 /2 + δ2e R2e Px2 Ne
+ [δ21 R21 − R22 /2 − β (1 − x1 x2 )]Px2 x1 , E [ˆx2 ] = Pxˆ 2 x2 R22 /2 + Pxˆ 2 xˆ 2 R22 /2,
(2)
where Pxi xj the probability for a tolerant agent of group i matches a tolerant agent of group j, and can be calculated as follows: P x1 x1 = Px1 xˆ 2 = P x2 x2 = Pxˆ 1 x1 = Pxˆ 2 x1 = Pxˆ 2 xˆ 2 =
x1 N1 − 1 N −1 xˆ 2 N2
,
N −1 x2 N2 − 1 N −1 x1 N1 N −1 x1 N1
,
Px1 xˆ 1 =
P x2 x1 =
,
Pxˆ 1 xˆ 2 =
,
Pxˆ 2 xˆ 1 =
N −1
,
N −1 x1 N1
,
N −1 xˆ 2 N2
Px2 xˆ 2 =
,
N −1 xˆ 2 N2 − 1
xˆ 1 N1
N −1 xˆ 2 N2
N −1 xˆ 1 N1
P x1 x2 =
Px2 xˆ 1 =
,
Pxˆ 1 xˆ 1 =
,
Pxˆ 2 x2 =
N −1
,
x2 N2
N −1 xˆ 1 N1
,
,
N −1 x2 N2
Pxˆ 1 x2 =
,
N −1 Ne
P xi N e =
,
N −1 xˆ 1 N1 − 1 N −1 x2 N2
,
,
N −1 Ne
Pxˆ i Ne =
,
N −1
,
(3)
with N = N1 + N2 + Ne . It is indeed possible that people develop tolerant attitudes for some reasons other than the tendency of mixed interaction in economic incentive. From natural continuation of economic studies on fundamentalism, here we assume that the behavior (tolerant and intolerant) spreads in the society often the selection process with higher payoff. Given the above probabilities produces a system of two differential equations which describes the evolution of tolerant populations in each groups:
) [δ12 R12 − R11 /2 − β (1 − x1 x2 )]x2 N2 + δ1e R1e Ne , N −1 ( ) x2 xˆ 2 x˙ 2 = [δ21 R21 − R22 /2 − β (1 − x1 x2 )]x1 N1 + δ2e R2e Ne . (4) N −1
x˙ 1 =
x1 xˆ 1
(
These differential equations give a complete description of the evolutionary dynamics of social tolerance. With payoff functions serve as the fitness function, the frequency of tolerance in a large, well-mixed society changes at a per capita rate equal to the difference between its expected payoff and the average payoff of the population. In the replicator dynamics, the rate of change in xi , which describes the proportion of tolerant members in group i, is
12
Y. Shi et al. / Economics Letters 159 (2017) 10–14
proportional to the product xi (1 − xi ), so that, ceteris paribus, this rate is greatest when exactly half the members of the group are tolerant. In what follows, we will discuss evolutionary dynamics of the tolerance, and show that neutral agents can greatly influence the steady states and dynamics of social tolerance. 3. Steady states and dynamics of tolerance In general, the steady states of the dynamical system (4) can be obtained as follows: P1 = (0, 0),
P2 = (0, 1), P3 = (1, 0), P4 = (1, 1), ) ( ) η12 − γ12 η21 − γ21 , P6 = ,1 , P5 = 1, β β ( ) η12 x2 − γ12 η21 x1 − γ21 , P7 = , β x22 β x21
(
occurs while each of them obtains 0 if no interaction. However, the agents of two opposing groups suffer a psychological cost and a social cost which may be high enough to prevent the interaction. When neutral agents are added to the model, there is greater scope for the evolution of tolerance because there is no psychological and social costs when the unneutral agents interact with the neutral agents. Compared with obtains 0, all unneutral agents tend to choose the strategy that take interaction with the neutral agents. Under the random pairwise matching assumption, the unneutral agents may match an agent of the same group, an agent of the opposing group, and a neutral agent, with the probability depending on the population of each group. If the population of neutral agents is large enough, there is a relatively high probability to match a neutral agent, and payoff of the tolerant strategy become dominant.
(5)
where ηij = β + Rii /2 − δij Rij , γij = Ne δie Rie /Nj . These steady states have precise economic and social meanings. In these steady states, The point P1 depicts that all agents in these groups are all intolerant. Steady states P2 –P3 depict situations that one group is wholly populated by intolerant agents while the other groups are wholly populated by tolerant agents. The steady state P4 depicts that all agents in the society are all tolerant. Economic and social meanings for these steady states are relatively simple, and independent with the properties of groups. P1 –P4 are always exist, while P5 –P7 are conditionally exist. Steady states P5 –P7 describes the society steady states determined by the properties of groups. Among them, P5 can exist when γ21 ≤ η21 ≤ β + γ21 while P6 can exist when γ12 ≤ η12 ≤ β + γ12 . So steady states of P5 –P7 are determined by the distribution of aggregate wealth and group populations. Steady states of two-group tolerance dynamics with neutral agents [Fig. 1(b)] is shown to be more positive and interesting compared with the steady states in the two-group model without neutral agents [shown in Fig. 1(a)]. In the two-group case without neutral agents, there is no economic interaction between agents of different groups for the steady states P1 , P2 , and P4 , which are the most common steady states in the system. A typical phase diagram of social tolerance in an evolutionary two-group game model without neutral agents is shown in Fig. 2(a). The phase trajectories fall into the steady states P1 , P2 , and P4 after a period of evolution unless both components (x01 , x02 ) of the starting point are large enough. It is shown that only phase trajectories with starting point (x01 , x02 ) located in the upper right corner evolving to the full tolerance steady state (1, 1). Things are quite different in the case that neutral agents is taken into consideration. We choose (x01 , x02 ) = (0.6, 0.6) as an example [shown in Fig. 2(d)]. Without neutral agents, the share of tolerant agents in both groups are decrease with time and the phase trajectory falls into the steady states P1 . If neutral agents is taken into consideration, tolerance spreads in both groups and falls into the full tolerance steady state (1, 1). In this case, there are two type of evolution according to the population of neutral agents. If the population of neutral agents is not large enough, most of phase trajectories fall into the full tolerance steady state (1, 1) while there are still phase trajectories [shown in Fig. 2(b)] evolving to the steady state (1, 0) and (0, 1). When Ne is large enough, x1 and x2 will reach to the stable full tolerance steady state (1, 1) at any starting point [shown in Fig. 2(c)]. A sufficient condition in order that tolerance spreads in both groups will be presented later. Here we give an intuitive explanation of how the introduction of neutral agents changes the evolution dynamic and why it tends to favor the full-tolerance steady state. In the original evolutionary game model proposed by Cerqueti et al., the agents of two opposing groups producing aggregate wealth R12 when the interaction
4. Full tolerance steady state The full tolerance steady state is of particular interest. Proposition 1. The necessary condition in order that tolerance spreads in both groups is R11 + R22 − 2(R12 + γ12 + γ21 ) < 0.
(6)
Proof. To verify the stability of steady state (1,1), we introduce infinitesimals δ x1 and δ x2 with 0 ≤ δ xi ≪ 1 and write x1 = 1 − δ x1 and x2 = 1 − δ x2 , thus the evolution equations can be linearized to
δ˙x1 = − δ˙x2 = −
δ x1 N −1 δ x2 N −1
[(δ12 R12 − R11 /2)N2 + δ1e R1e Ne ], [(δ21 R21 − R22 /2)N1 + δ2e R2e Ne ],
(7)
where higher-order small quantity of δ xi , such as δ x2i , are ignored, and the solution can be obtained as follows:
δ x1 ∝ exp(−λ1 t), δ x2 ∝ exp(−λ2 t),
(8)
where λ1 = [(δ12 R12 − R11 /2)N2 + δ1e R1e Ne ]/(N − 1) and λ2 = [(δ21 R21 − R22 /2)N1 +δ2e R2e Ne ]/(N − 1). So the necessary condition in order that tolerance spreads in group i is λi > 0 which ensures that δ xi → 0 for t → ∞. Simplification of these inequalities leads to the following inequalities:
δ12 R12 − R11 /2 + γ12 > 0,
(9)
δ21 R21 − R22 /2 + γ21 > 0.
(10)
Addition of inequalities (9) and (10) leads to the inequality (6). □ If Ne = 0, then inequality (6) reduce to R11 + R22 − 2R12 < 0,
(11)
which is exactly the result obtained by Cerqueti et al. (2013). Tolerance is impossible if R11 + R22 − 2R12 > 0 where R12 is not sufficiently high to produce a tendency of mixed interaction in economic incentive. However, the things become very different if neutral agents are introduced. In this case, although Ree + R11 − 2R1e > 0 where R1e is not sufficiently high to produce a tendency of mixed interaction in economic incentive, tolerance is still possible due to the additional terms γ12 + γ21 . A sufficient condition in order that tolerance spreads in both groups is summarized as follows:
Y. Shi et al. / Economics Letters 159 (2017) 10–14
13
Fig. 1. (a) Steady states of two-group tolerance dynamics in an evolutionary game model without neutral agents; (b) Steady states of two-group tolerance dynamics with neutral agents. The parameters used in the calculation are η12 = η21 = 0.6β and γ12 = γ21 = 0.028β .
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.2
0.4
0.6
0.8
1
0
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.2
0.4
0.6
0.8
1
0
0
0
0.2
0.5
0.4
0.6
1
0.8
1.5
1
2
Fig. 2. (a) Phase diagram of social tolerance in an evolutionary two-group game model without neutral agents; (b) Phase diagram of two-group social tolerance with neutral agents (Ne = 100); (c) Phase diagram of two-group social tolerance in the present of neutral agents with population Ne = 200; (d) Evolutionary dynamics of social tolerance with and without neutral agents. The parameters used in the calculation are N1 = N2 = 500, β = 20, δ12 = δ21 = δ1e = δ2e = 0.5, R11 = 35, R22 = 40, and R12 = R1e = R2e = 60.
Proposition 2. A sufficient condition in order that tolerance spreads in both groups at any starting point (x01 , x02 ) is
5. Conclusions
δ12 R12 − R11 /2 − β + γ12 > 0,
(12)
δ21 R21 − R22 /2 − β + γ21 > 0,
(13)
We discuss the dynamics of tolerance in a society with neutral agents, which is very different from the pure opposing case discussed by Cerqueti et al. (2013). In the pure opposing case, tolerance requires an unequal distribution of aggregate wealth among people, and fairness without a corresponding balance in the perception of diversity leads to a society with full intolerance. In the economic interaction model with neutral agents, the full intolerance steady state (0, 0) is unstable and phase trajectories reach to the stable full tolerance steady state (1, 1) at any starting point if the population of the neutral agents is large enough. These results show how neutral agents affects tolerance growth and may
Proof. [δ12 R12 − R11 /2 − β (1 − x1 x2 )]x2 N2 + δ1e R1e Ne > 0 ensures x˙ 1 > 0 for any x01 while [δ21 R21 − R22 /2 − β (1 − x1 x2 )]x1 N1 + δ2e R2e Ne > 0 ensures x˙ 2 > 0 for any x02 , so [δ12 R12 − R11 /2 − β]N2 + δ1e R1e Ne > 0 and [δ21 R21 − R22 /2 − β]N1 + δ2e R2e Ne > 0 give a sufficient condition of achieving full tolerance at any starting point. After some simplification we can obtain inequality (12) and (13). □
14
Y. Shi et al. / Economics Letters 159 (2017) 10–14
remove the contradiction between the traditional idea of economic integration and the evolutionary game point of view. References Akerlof, G.A., Kranton, R.E., 2000. Economics and identity. Quart. J. Econ 115 (3), 715–753. Becchetti, L., Rossetti, F., Castriota, S., 2010. Real household income and attitude toward immigrants: an empirical analysis. J. Soc. Econ. 39 (1), 81–88. Berggren, N., Elinder, M., 2012. Is tolerance good or bad for growth? Publ. Choice 150 (1–2), 283–308. Berggren, N., Nilsson, T., 2013. Does economic freedom foster tolerance? Kyklos 66 (2), 177–207. Bjørnskov, C., 2004. Inequality, Tolerance, and Growth. Aarhus School of Business. Department of Economics, WP N 8.
Bjørnskov, C., 2008. The growth-equality association: government ideology matters. J. Dev. Econ. 87 (2), 300–308. Cerqueti, R., Correani, L., Garofalo, G., 2013. Economic interactions and social tolerance: A dynamic perspective. Econom. Lett. 120 (3), 458–463. Corneo, G., Jeanne, O., 2009. A theory of tolerance. J. Publ. Econ. 93 (5), 691–702. Darity Jr., W.A., Mason, P.L., Stewart, J.B., 2006. The economics of identity: the origin and persistence of racial identity norms. J. Econ. Behav. Organ. 60 (3), 283–305. Garofalo, G., Di Dio, F., Correani, L., 2010. The evolutionary dynamics of tolerance. Theoret. Pract. Res. Econ. Fields 2 (2), 218–230. Muldoon, R., Borgida, M., Cuffaro, M., 2012. The conditions of tolerance. Polit. Philos. Econ. 11 (3), 322–344. Shi, Y., Pan, M., 2017. Evolutionary dynamics of social tolerance in the economic interaction model with local social cost functions. Appl. Econ. Lett. 24 (2), 75–79.
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Replicator dynamics and evolutionary game of social tolerance: The role of neutral agents✩ Yingying Shi a,b , Min Pan a, *, Daiyan Peng c a b c
School of Economics and Management, Wuhan University, Wuhan, Hubei 430072, PR China School of Economics and Business Administration, Central China Normal University, Wuhan, 430079, PR China School of Economics, Huazhong University of Science and Technology, Wuhan, 430074, PR China
highlights • • • •
We study the evolutionary game of tolerance including neutral agents. We show that neutral agents plays an important role in the dynamics of tolerance. Tolerance can be a natural consequence of economic integration. Evolutionary game with neutral agents is consistent with economic integration.
article
info
Article history: Received 17 April 2017 Received in revised form 27 June 2017 Accepted 6 July 2017 Available online 13 July 2017 MSC: C7 D7 Keywords: Replicator dynamics Tolerance Economic interaction model
a b s t r a c t The role of neutral agents on evolutionary tolerance between two differentiated groups is discussed based on the replicator game model proposed recently. We show that, very different from the pure opposing case studied previously, dynamics of social tolerance with neutral agents is more positive and exhibiting rich interesting effects. The full intolerance steady state (0, 0) is unstable when neutral agents are taken into consideration and there are two type of evolution trajectory according to the population of the neutral agents. Especially, phase trajectories reach to the stable full tolerance steady state (1, 1) at any starting point if the population of the neutral agents is large enough, and the tolerance between different social groups can be a natural consequence of economic integration in the present of neutral agents. We show that neutral agents may remove the contradiction between the traditional idea of economic integration and the evolutionary game point of view. © 2017 Published by Elsevier B.V.
1. Introduction Tolerance, which is defined as a generic attitude to accept diversity (Akerlof and Kranton, 2000; Darity Jr. et al., 2006), is seen as a vital element of a liberal political order from the philosophical viewpoint of the liberal political thought: shifting into a Hobbesian state of anticipation is unlikely in a tolerant society due to the tolerant behavior of accepting conflicting political values (Muldoon et al., 2012). Cross-country and cross-region differences with respect to tolerance have been studied by sociologists resorting to the theory of cultural modernization (Berggren and Elinder, 2012; Berggren and Nilsson, 2013). The conjecture that social tolerance ✩ The work was supported in part by the Key Projects of Philosophy and Social Sciences Research, Ministry of Education of China (Grant number 15JZD013), the National Social Science Fund Project of China (Grant No. 10AZD019), and the China Postdoctoral Science Foundation funded project (Grant No. 2015M582250). Corresponding author. E-mail address: [email protected] (M. Pan).
*
http://dx.doi.org/10.1016/j.econlet.2017.07.005 0165-1765/© 2017 Published by Elsevier B.V.
is beneficial for population growth and technological performance has been demonstrated empirically (Bjørnskov, 2004, 2008), and social tolerance is shown positively related to social and economic development in many cases (Becchetti et al., 2010). Recently, the discussion on tolerance at the individual level reveals that economic reasoning can offer original and unique insights into the determinants of tolerance (Corneo and Jeanne, 2009; Garofalo et al., 2010; Shi and Pan, 2017). It has been believed that the fairer distribution of wealth among people, which also means economic integration, is of great importance to tolerance between different social groups (Becchetti et al., 2010). However, from the evolutionary game point of view, tolerance requires an unequal distribution of aggregate wealth among people, and fairness without a corresponding balance in the perception of diversity leads to a society with full intolerance (Cerqueti et al., 2013). Here we study the evolutionary game of tolerance, where the neutral agents is taken into consideration. We show that the tolerance between different social groups can be a natural consequence
Y. Shi et al. / Economics Letters 159 (2017) 10–14
of economic integration in the present of neutral agents, and in this case there is no contradiction between the traditional idea of economic integration and the evolutionary game point of view. 2. Coupled equations of tolerance dynamics An evolutionary game model of social tolerance with two differentiated groups has been proposed recently (Cerqueti et al., 2013). The population of each group are N1 and N2 , respectively, and assumed to be large enough and changeless with time. Each individual can be tolerant or intolerant towards the agents with the opposite group. If using the notation xi and xˆ i be the share of tolerant and intolerant agents in group i respectively, then xi + xˆ i = 1 and xi , xˆ i ∈ [0, 1], for each i = 1, 2. The replicator dynamics and the random pairwise matching assumption are used to discuss the tolerance dynamics, where two agents interact after being randomly matched (such economic interaction can be, for example, a business deal), producing aggregate wealth Rij = Rji which depend on the contribution of capital of the two agents with i, j = 1, 2. The evolutionary game model implicitly assumes that tolerant and intolerant behavior spreads based on the payoff of the strategy. The relative contribution of capital of agents in group i when she interact with agents in group j is δij ≡ ki /(ki + kj ) which determines the shares of the aggregate wealth with ki the contribution of capital of agents in group i: the agent in group i shares δij Rij when she interact with the agent in group j. Based on the evolutionary dynamics of social tolerance, it has been shown that tolerance requires an unequal distribution of aggregate wealth among people, and fairness without a corresponding balance in the perception of diversity leads to a society with full intolerance (Cerqueti et al., 2013). From the philosophical viewpoint of the liberal social thought, differences (such as ethnicity, religion, country of origin and social class) are only applied to specific groups and neutral agents which are always tolerant can exist in general. In the present work, we keep the two-group assumption while the neutral agents with population Ne is introduced in the evolutionary game model of social tolerance. We use an evolutionary game model of social tolerance similar to the model proposed by Cerqueti et al. (2013), where the replicator dynamics is used, to discuss the tolerance dynamics with neutral agents. We also use the random pairwise matching assumption that two agents interact after being randomly matched, producing aggregate wealth Rij = Rji or Rie = Rei with i, j = 1, 2, which depend on the contribution of capital of the two agents. We write the contribution of capital of agents in group i as ki and the contribution of capital of neutral agents as ke . The relative contribution of capital of agents in group i when she interact with agents in group j is δij ≡ ki /(ki + kj ), which determines the shares of the aggregate wealth. Similarly, the relative contribution of capital of agents in group i when she interact with neutral agents is δie ≡ ki /(ki + ke ). The agent in group i shares δij Rij when she interact with the agent in group j, while shares δie Rie when she interact with a neutral agent. The social tolerance influences the net gain obtained by each agents in the following cases: 1. For the case that the two unneutral agents are of the same group, whether tolerant or not, δii = 1/2, and each of them obtains Rii /2. 2. For the case that the two unneutral agents are of different group and both tolerant, them suffer a psychological cost α = Rii /2 and a social cost c = β (1 − xi xj ) with the exception of δij Rij , with δij + δji = 1, for each i, j = 1, 2. The parameter β is greater than zero, and a higher β leads a higher social costs, so the parameter β describes the social reaction of intolerant agents adverse to the agents of the opposite group. 3. For the case that the two unneutral agents are of different group and if one or both of them is intolerant, which rules out any
11
interaction between them, there is no wealth produced, and each of them obtains 0. 4. For the case that one of the agents is neutral while the another is of group i and tolerant, the neutral agent obtains δei Rei while the another agent of group i obtains δie Rie . 5. For the case that one of the agents is neutral while the another is of group i and intolerant, which rules out any interaction between them, there is no wealth produced, and each of them obtains 0. 6. For the case that the two agents are both neutral, each of them obtains Ree /2. The evolutionary dynamics of social tolerance can be modeled by the theory of replicators in which the payoff functions described above serve as the fitness function, and the evolution of tolerant population in group i can be described by x˙ i = xi xˆ i (E [xi ] − E [ˆxi ]),
(1)
where E [xi ] and E [ˆxi ] are the expected net gain of tolerant and intolerant individuals in group i respectively, and can be calculated by using the following expression: E [x1 ] = Px1 x1 R11 /2 + Px1 xˆ 1 R11 /2 + δ1e R1e Px1 Ne ,
+ [δ12 R12 − R11 /2 − β (1 − x1 x2 )]Px1 x2 , E [ˆx1 ] = Pxˆ 1 x1 R11 /2 + Pxˆ 1 xˆ 1 R11 /2, E [x2 ] = Px2 x2 R22 /2 + Px2 xˆ 2 R22 /2 + δ2e R2e Px2 Ne
+ [δ21 R21 − R22 /2 − β (1 − x1 x2 )]Px2 x1 , E [ˆx2 ] = Pxˆ 2 x2 R22 /2 + Pxˆ 2 xˆ 2 R22 /2,
(2)
where Pxi xj the probability for a tolerant agent of group i matches a tolerant agent of group j, and can be calculated as follows: P x1 x1 = Px1 xˆ 2 = P x2 x2 = Pxˆ 1 x1 = Pxˆ 2 x1 = Pxˆ 2 xˆ 2 =
x1 N1 − 1 N −1 xˆ 2 N2
,
N −1 x2 N2 − 1 N −1 x1 N1 N −1 x1 N1
,
Px1 xˆ 1 =
P x2 x1 =
,
Pxˆ 1 xˆ 2 =
,
Pxˆ 2 xˆ 1 =
N −1
,
N −1 x1 N1
,
N −1 xˆ 2 N2
Px2 xˆ 2 =
,
N −1 xˆ 2 N2 − 1
xˆ 1 N1
N −1 xˆ 2 N2
N −1 xˆ 1 N1
P x1 x2 =
Px2 xˆ 1 =
,
Pxˆ 1 xˆ 1 =
,
Pxˆ 2 x2 =
N −1
,
x2 N2
N −1 xˆ 1 N1
,
,
N −1 x2 N2
Pxˆ 1 x2 =
,
N −1 Ne
P xi N e =
,
N −1 xˆ 1 N1 − 1 N −1 x2 N2
,
,
N −1 Ne
Pxˆ i Ne =
,
N −1
,
(3)
with N = N1 + N2 + Ne . It is indeed possible that people develop tolerant attitudes for some reasons other than the tendency of mixed interaction in economic incentive. From natural continuation of economic studies on fundamentalism, here we assume that the behavior (tolerant and intolerant) spreads in the society often the selection process with higher payoff. Given the above probabilities produces a system of two differential equations which describes the evolution of tolerant populations in each groups:
) [δ12 R12 − R11 /2 − β (1 − x1 x2 )]x2 N2 + δ1e R1e Ne , N −1 ( ) x2 xˆ 2 x˙ 2 = [δ21 R21 − R22 /2 − β (1 − x1 x2 )]x1 N1 + δ2e R2e Ne . (4) N −1
x˙ 1 =
x1 xˆ 1
(
These differential equations give a complete description of the evolutionary dynamics of social tolerance. With payoff functions serve as the fitness function, the frequency of tolerance in a large, well-mixed society changes at a per capita rate equal to the difference between its expected payoff and the average payoff of the population. In the replicator dynamics, the rate of change in xi , which describes the proportion of tolerant members in group i, is
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Y. Shi et al. / Economics Letters 159 (2017) 10–14
proportional to the product xi (1 − xi ), so that, ceteris paribus, this rate is greatest when exactly half the members of the group are tolerant. In what follows, we will discuss evolutionary dynamics of the tolerance, and show that neutral agents can greatly influence the steady states and dynamics of social tolerance. 3. Steady states and dynamics of tolerance In general, the steady states of the dynamical system (4) can be obtained as follows: P1 = (0, 0),
P2 = (0, 1), P3 = (1, 0), P4 = (1, 1), ) ( ) η12 − γ12 η21 − γ21 , P6 = ,1 , P5 = 1, β β ( ) η12 x2 − γ12 η21 x1 − γ21 , P7 = , β x22 β x21
(
occurs while each of them obtains 0 if no interaction. However, the agents of two opposing groups suffer a psychological cost and a social cost which may be high enough to prevent the interaction. When neutral agents are added to the model, there is greater scope for the evolution of tolerance because there is no psychological and social costs when the unneutral agents interact with the neutral agents. Compared with obtains 0, all unneutral agents tend to choose the strategy that take interaction with the neutral agents. Under the random pairwise matching assumption, the unneutral agents may match an agent of the same group, an agent of the opposing group, and a neutral agent, with the probability depending on the population of each group. If the population of neutral agents is large enough, there is a relatively high probability to match a neutral agent, and payoff of the tolerant strategy become dominant.
(5)
where ηij = β + Rii /2 − δij Rij , γij = Ne δie Rie /Nj . These steady states have precise economic and social meanings. In these steady states, The point P1 depicts that all agents in these groups are all intolerant. Steady states P2 –P3 depict situations that one group is wholly populated by intolerant agents while the other groups are wholly populated by tolerant agents. The steady state P4 depicts that all agents in the society are all tolerant. Economic and social meanings for these steady states are relatively simple, and independent with the properties of groups. P1 –P4 are always exist, while P5 –P7 are conditionally exist. Steady states P5 –P7 describes the society steady states determined by the properties of groups. Among them, P5 can exist when γ21 ≤ η21 ≤ β + γ21 while P6 can exist when γ12 ≤ η12 ≤ β + γ12 . So steady states of P5 –P7 are determined by the distribution of aggregate wealth and group populations. Steady states of two-group tolerance dynamics with neutral agents [Fig. 1(b)] is shown to be more positive and interesting compared with the steady states in the two-group model without neutral agents [shown in Fig. 1(a)]. In the two-group case without neutral agents, there is no economic interaction between agents of different groups for the steady states P1 , P2 , and P4 , which are the most common steady states in the system. A typical phase diagram of social tolerance in an evolutionary two-group game model without neutral agents is shown in Fig. 2(a). The phase trajectories fall into the steady states P1 , P2 , and P4 after a period of evolution unless both components (x01 , x02 ) of the starting point are large enough. It is shown that only phase trajectories with starting point (x01 , x02 ) located in the upper right corner evolving to the full tolerance steady state (1, 1). Things are quite different in the case that neutral agents is taken into consideration. We choose (x01 , x02 ) = (0.6, 0.6) as an example [shown in Fig. 2(d)]. Without neutral agents, the share of tolerant agents in both groups are decrease with time and the phase trajectory falls into the steady states P1 . If neutral agents is taken into consideration, tolerance spreads in both groups and falls into the full tolerance steady state (1, 1). In this case, there are two type of evolution according to the population of neutral agents. If the population of neutral agents is not large enough, most of phase trajectories fall into the full tolerance steady state (1, 1) while there are still phase trajectories [shown in Fig. 2(b)] evolving to the steady state (1, 0) and (0, 1). When Ne is large enough, x1 and x2 will reach to the stable full tolerance steady state (1, 1) at any starting point [shown in Fig. 2(c)]. A sufficient condition in order that tolerance spreads in both groups will be presented later. Here we give an intuitive explanation of how the introduction of neutral agents changes the evolution dynamic and why it tends to favor the full-tolerance steady state. In the original evolutionary game model proposed by Cerqueti et al., the agents of two opposing groups producing aggregate wealth R12 when the interaction
4. Full tolerance steady state The full tolerance steady state is of particular interest. Proposition 1. The necessary condition in order that tolerance spreads in both groups is R11 + R22 − 2(R12 + γ12 + γ21 ) < 0.
(6)
Proof. To verify the stability of steady state (1,1), we introduce infinitesimals δ x1 and δ x2 with 0 ≤ δ xi ≪ 1 and write x1 = 1 − δ x1 and x2 = 1 − δ x2 , thus the evolution equations can be linearized to
δ˙x1 = − δ˙x2 = −
δ x1 N −1 δ x2 N −1
[(δ12 R12 − R11 /2)N2 + δ1e R1e Ne ], [(δ21 R21 − R22 /2)N1 + δ2e R2e Ne ],
(7)
where higher-order small quantity of δ xi , such as δ x2i , are ignored, and the solution can be obtained as follows:
δ x1 ∝ exp(−λ1 t), δ x2 ∝ exp(−λ2 t),
(8)
where λ1 = [(δ12 R12 − R11 /2)N2 + δ1e R1e Ne ]/(N − 1) and λ2 = [(δ21 R21 − R22 /2)N1 +δ2e R2e Ne ]/(N − 1). So the necessary condition in order that tolerance spreads in group i is λi > 0 which ensures that δ xi → 0 for t → ∞. Simplification of these inequalities leads to the following inequalities:
δ12 R12 − R11 /2 + γ12 > 0,
(9)
δ21 R21 − R22 /2 + γ21 > 0.
(10)
Addition of inequalities (9) and (10) leads to the inequality (6). □ If Ne = 0, then inequality (6) reduce to R11 + R22 − 2R12 < 0,
(11)
which is exactly the result obtained by Cerqueti et al. (2013). Tolerance is impossible if R11 + R22 − 2R12 > 0 where R12 is not sufficiently high to produce a tendency of mixed interaction in economic incentive. However, the things become very different if neutral agents are introduced. In this case, although Ree + R11 − 2R1e > 0 where R1e is not sufficiently high to produce a tendency of mixed interaction in economic incentive, tolerance is still possible due to the additional terms γ12 + γ21 . A sufficient condition in order that tolerance spreads in both groups is summarized as follows:
Y. Shi et al. / Economics Letters 159 (2017) 10–14
13
Fig. 1. (a) Steady states of two-group tolerance dynamics in an evolutionary game model without neutral agents; (b) Steady states of two-group tolerance dynamics with neutral agents. The parameters used in the calculation are η12 = η21 = 0.6β and γ12 = γ21 = 0.028β .
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.2
0.4
0.6
0.8
1
0
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.2
0.4
0.6
0.8
1
0
0
0
0.2
0.5
0.4
0.6
1
0.8
1.5
1
2
Fig. 2. (a) Phase diagram of social tolerance in an evolutionary two-group game model without neutral agents; (b) Phase diagram of two-group social tolerance with neutral agents (Ne = 100); (c) Phase diagram of two-group social tolerance in the present of neutral agents with population Ne = 200; (d) Evolutionary dynamics of social tolerance with and without neutral agents. The parameters used in the calculation are N1 = N2 = 500, β = 20, δ12 = δ21 = δ1e = δ2e = 0.5, R11 = 35, R22 = 40, and R12 = R1e = R2e = 60.
Proposition 2. A sufficient condition in order that tolerance spreads in both groups at any starting point (x01 , x02 ) is
5. Conclusions
δ12 R12 − R11 /2 − β + γ12 > 0,
(12)
δ21 R21 − R22 /2 − β + γ21 > 0,
(13)
We discuss the dynamics of tolerance in a society with neutral agents, which is very different from the pure opposing case discussed by Cerqueti et al. (2013). In the pure opposing case, tolerance requires an unequal distribution of aggregate wealth among people, and fairness without a corresponding balance in the perception of diversity leads to a society with full intolerance. In the economic interaction model with neutral agents, the full intolerance steady state (0, 0) is unstable and phase trajectories reach to the stable full tolerance steady state (1, 1) at any starting point if the population of the neutral agents is large enough. These results show how neutral agents affects tolerance growth and may
Proof. [δ12 R12 − R11 /2 − β (1 − x1 x2 )]x2 N2 + δ1e R1e Ne > 0 ensures x˙ 1 > 0 for any x01 while [δ21 R21 − R22 /2 − β (1 − x1 x2 )]x1 N1 + δ2e R2e Ne > 0 ensures x˙ 2 > 0 for any x02 , so [δ12 R12 − R11 /2 − β]N2 + δ1e R1e Ne > 0 and [δ21 R21 − R22 /2 − β]N1 + δ2e R2e Ne > 0 give a sufficient condition of achieving full tolerance at any starting point. After some simplification we can obtain inequality (12) and (13). □
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