Abstract
We consider whether the evolution of inequality in a large population game generates a Kuznets inverted-U curve. There are different types of agents, based on the cost of using a technology. Agents migrate and change their type, either by moving to a favorable location or by adopting a better version of the technology. This generates the replicator dynamic. We apply indirect evolution by assuming agents always play the Nash equilibrium. All agents migrate to the lowest-cost type, causing inequality to approach zero eventually. However, if initial access to the technology is limited, inequality increases before decreasing. In that case, we obtain the Kuznets curve. Otherwise, inequality may decline monotonically. As a robustness check, we also consider a multilevel selection model in which agents revise strategies based on the best-response dynamic and migrate based on the replicator dynamic. The conclusions are similar.
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Notes
See Kanbur [22] for a survey of the literature on the Kuznets curve and, more generally, on the post-war literature on inequality and development.
Thus, in these models, the shape of the Kuznets curve arises from the proportion of agents in the rural or urban sector. They do not explain what determines those proportions or how much effort agents may choose to exert in whichever sector they are in.
Myopia implies agents make strategic decisions based on current social conditions. This is a standard assumption in evolutionary game theory.
In advanced industrial economies, such phenomena as rural–urban migration or diffusion of electricity to rural areas have largely been completed. However, they remain relevant in many other parts of the world.
The fact that migration is based on current payoffs is an example of myopic behavior.
The indirect evolutionary approach considers the survival of preferences incorporating non-individualistic traits like altruism, spite or Kantian morality. In preference evolution, agents play the Nash equilibrium of the game generated by subjective payoffs. The resulting differences in material or objective payoffs drive the evolutionary success of different types of preferences. See Alger and Weibull [8] for a review. We clarify that although our methodology is one of indirect evolution, our model is not one of preference evolution. All agents have the standard individualistic preference.
Thus, initially, most agents may be in a disadvantageous location like a rural area.
See also Lahkar [27] for a similar application of the multilevel selection approach to coordination games.
Equivalently, \(\frac{\mu _p(B)}{m_p}\in [0,1]\) is the proportion of agents in population p playing strategies in B. Thus, \(\frac{\mu _p(\mathcal {S})}{m_p}=1\) or \(\mu _p(\mathcal {S})=m_p\).
In this example, suppose p and q denote electric and traditional technology types. Assuming \(k_p0\) implies there is an initial mass of agents in population 1. Otherwise, by (17), \(\dot{m}_1=0\), which would imply \(m_1(\tau )=0\) at all \(\tau >0\).
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Appendix
Appendix
1.1 A.1 Proofs
Proof of Proposition 2.2
As \(\bar{x}\) is arbitrarily large, we focus on solutions in \([0,\bar{x})\). Hence, taking \(b_p(\alpha )=\left( \frac{\theta }{\gamma k_p}\right) ^{\frac{1}{\gamma -1}}\alpha ^{\frac{\rho }{\gamma -1}}\) from (5) and applying it to (6) gives us two solutions, 0 and \(\alpha ^{*}\) defined by (8).
To obtain the Nash equilibrium \(\mu ^0\), we note that \(b_p(0)=0\). Hence, when the aggregate strategy \(\alpha =0\), the best response for all agents is 0. The Nash equilibrium \(\mu ^{*}\) derives from (8). Applying \(\alpha ^{*}\) to (5) gives us \(b_p(\alpha ^{*})=\alpha _p^{*}\) in (7).\(\square \)
Proof of Proposition 2.3
We obtain the individual equilibrium payoff (10) by applying \(x=\alpha _p^{*}\) from (7) and \(A(\mu ^{*})=\alpha ^{*}\) from (8) to (2). As \(k_p1\), it follows that \(F_{\alpha _p^{*},p}(\mu ^{*})>F_{\alpha _q^{*},q}(\mu ^{*})\) if \(p0\), then \(m_p(\tau )\rightarrow 0\) as \(\tau \rightarrow \infty \) for all \(p\in \mathcal {P}\setminus \{1\}\).
Equivalently, \(m_1(\tau )\rightarrow 1\).Footnote 22 Moreover, \(m_1(\tau )\) must increase monotonically due to the strict dominance of \(p=1\). Hence, \(\phi _1(m)>\bar{\phi (m)}\) and so, \(\dot{m}_1>0\) at all m by (17).\(\square \)
Proof of Corollary 3.4
We know from (14) that \(\phi _p(m)=F_{\alpha _p^{*},p}(\mu ^{*}(m))\). Hence, (18) follows from (10) by applying \(m_1=1\) and \(m_p=0\) for all \(p\ne 1\). Since all agents obtain the payoff (18), it must also be the aggregate payoff \(\bar{\phi }(m^I)\). It is the highest possible aggregate payoff (11) at a Nash equilibrium \(\mu ^{*}(m)\) because \(k_1k_p\) as \(p0\). Then, from every such initial point, \(\alpha (\tau )\ge \min \{\alpha (0), \underline{\alpha }\}>0\) for all \(\tau >0\). Hence, \(\alpha (\tau )\) remains bounded away from 0 along every such solution trajectory.
Proof
The argument is independent of the \(\lambda \)-replicator dynamic (22). Hence, we focus on the ABR dynamic (20). We know from Fig. 1 that if the population state m is fixed, then the trajectory of the ABR dynamic (9) converges to \(\alpha ^{*}(m)\) from every initial point \(\alpha (0)>0\).
Denote the population mass distribution m with \(m_n=1\) as \(\underline{m}\). Now, note from (5) that for every \(\alpha >0\), \(b_1(\alpha )\ge b_2(\alpha )\ge \cdots {\ge }b_n(\alpha )\). Therefore, \(\sum _{p\in \mathcal {P}}m_pb_p(\alpha )\ge b_n(\alpha )\). Hence, at every \(\alpha >0\), the ABR at an arbitrary distribution m must be at least as high as the ABR at \(\underline{m}\). Since \(\underline{\alpha }\) is the strictly positive solution to \(b_n(\alpha )=\alpha \), Fig. 1 implies \(\alpha ^{*}(m)\ge \underline{\alpha }\). We divide the rest of the argument into two parts.
-
1.
Suppose \(\alpha (\hat{\tau })\ge \underline{\alpha }\) for some fixed \(\hat{\tau }\in [0,\infty )\). Consider the ABR dynamic (9) for the fixed population mass distribution \(\underline{m}\). This is
$$\begin{aligned} \dot{\alpha }=b_n(\alpha )-\alpha . \end{aligned}$$(26)Since \(\underline{\alpha }\) is the unique positive solution to \(b_n(\alpha )=\alpha \), it must be that \(b_n(\alpha (\hat{\tau }))-\alpha (\hat{\tau })\le 0\), with equality holding if \(\alpha (\hat{\tau })=\underline{\alpha }\). Hence, \(\alpha (\tau )\) falls under the dynamic (26). But \(\alpha (\tau )\) cannot fall below \(\underline{\alpha }\) as in that case, we would have \(\dot{\alpha }>0\). Therefore, \(\alpha (\tau )\ge \underline{\alpha }\) for all \(\tau \ge \hat{\tau }\). Now consider the ABR dynamic (20) for any arbitrary population distribution \(m(\tau )\) for any \(\tau \ge \hat{\tau }\). The argument in the second paragraph of this proof implies \(\sum _{p\in \mathcal {P}}m_p(\tau )b_p(\alpha (\tau ))\ge b_n(\alpha )\). But this means that the rate of change in \(\alpha (\tau )\) under (20) must be greater than the rate of change under (26). Hence, if \(\alpha (\tau )\ge \underline{\alpha }\) under (26), it must be that \(\alpha (\tau )\ge \underline{\alpha }\) under (20) for all \(\tau \ge \hat{\tau }\).
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2.
Suppose \(\alpha (\hat{\tau })\in (0,\underline{\alpha })\) for some fixed \(\hat{\tau }\in [0,\infty )\). Then, \(\dot{\alpha }>0\) under (26). Therefore, \(\alpha (\tau )\rightarrow \underline{\alpha }\) under (26). In particular, \(\alpha (\tau )\ge \alpha (\hat{\tau })\) for all \(\tau \ge \hat{\tau }\). But by a similar argument as above, the rate of change in \(\alpha (\tau )\) under (20) must be greater than under (26). Hence, it follows that \(\alpha (\tau )\ge \alpha (\hat{\tau })\) under (20) from \(\alpha (\hat{\tau })\).
Together, parts 1 and 2 suffice to establish that \(\alpha (\tau )\ge \min \{\alpha (0), \underline{\alpha }\}>0\) for all \(\tau >0\) under the dynamical system (20) and (22). \(\square \)
Proof of Proposition 5.2
By Lemma 5.1, \(\psi _{1}(\alpha )>\psi _{p}(\alpha )\) for all \(p\ne 1\) and \(\alpha >0\). We now use Lemma A.1 to establish a stronger result. Along any trajectory \(\{\alpha (\tau ),m(\tau )\}_{\tau \ge 0}\) of the dynamical system (20) and (22) from an initial point such that \(\alpha (0)>0\), \(\psi _{1}(\alpha (\tau ))-\psi _{p}(\alpha (\tau ))\) remains bounded away from zero, for all \(p\ne 1\).
Using (24) and (25), we obtain that for all \(p\ne 1\),
Together with Lemma A.1, (5) and (27) implies
Thus, (28) implies \(\psi _{1}(\alpha (\tau ))-\psi _{p}(\alpha (\tau ))\) remains bounded away from zero. Thus, along the entire trajectory \(\alpha (\tau )\), the option of migrating to population 1 is strictly dominant. We now apply the well-known proof of elimination of strictly dominated strategies under the replicator dynamic to establish that \(m_1(\tau )\rightarrow 1\) as \(\tau \rightarrow \infty \).
For \(p\ne 1\) and \(m_1(\tau )>0\), consider the function
Applying the \(\lambda \)-replicator dynamic (22) and taking the derivative of (29), we obtain
where the inequality in (30) follows from (28). Hence, applying (30), we obtain
Therefore, by (31), \(L_p(\tau )\rightarrow -\infty \) as \(\tau \rightarrow \infty \). From (29), we then obtain \(\frac{m_p(\tau )}{m_1(\tau )}\rightarrow 0\) as \(\tau \rightarrow \infty \). Since the denominator is bounded above by 1, this is possible only if \(m_p(\tau )\rightarrow 0\). Since this is true for every \(p\ne 1\) and \(\sum _{p=1}^nm_p(\tau )=1\), we obtain \(m_1(\tau )\rightarrow 1\) if \(m_1(0)>0\).
Equivalently, \(m(\tau )\rightarrow m^I\). Hence, as \(\tau \rightarrow \infty \), the ABR dynamic (20) becomes
But if \(\alpha (\tau )>0\) throughout the solution trajectory, then it must be that \(\alpha (\tau )\rightarrow \alpha ^{**}\) under (32). Thus, if \(\alpha (0)>0\) and \(m_1(0)>0\), \(\{\alpha (\tau ),m_1(\tau )\}\rightarrow \{\alpha ^{**},1\}\) under (20) and (22).\(\square \)
1.2 A.2 Further Simulations in Sect. 5
We define the Gini coefficient in the multilevel selection model as follows. Similar to (12), we define
with \(\Sigma _{n+1}=0\). Thus, \(\Sigma _p\) is the aggregate payoff of the populations \(\{p,p+1,\ldots ,n\}\) at the aggregate effort level \(\alpha \) under the type distribution \((m_1,m_2,\ldots ,m_n)\). The associated Gini coefficient is then
Hence, we apply (34) in simulating the trajectory of the Gini coefficient in Fig. 7. This requires the trajectories \(\alpha (\tau )\) and \(\psi _p(\alpha (\tau ))\). We obtain the former from the differential equations (20) and (22), while the latter is the type-specific payoff (19) under multilevel selection.
Figure 8 presents trajectories of aggregate effort and aggregate payoff under multilevel selection for Example 4.1. The initial conditions are the same as in Fig. 7 and, therefore, as in Fig. 2. The trajectory of aggregate effort is \(\alpha (\tau )\) from (20) and (22), while the aggregate payoff is \(\sum _{p\in \mathcal {P}}m_p\psi _p(\alpha (\tau ))\). We compare the left panel of Fig. 8 with the right panel of Fig. 3, and the right panel of Fig. 8 with the right panel of Fig. 4.Footnote 23 In each case, we observe that the trajectory under multilevel selection is similar to that under indirect evolution.
Figure 9 is analogous to Fig. 6. It presents trajectories of the Gini coefficient from the initial values in Fig. 6. Thus, in each case, \(\alpha (0)=\alpha ^{*}\) as defined in (8). Again, the trajectories are broadly similar, strengthening the insight from the indirect evolutionary analysis of Sect. 4.
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Lahkar, R., Mahanta, A. An Evolutionary Game Theoretic Analysis of the Kuznets Inequality Curve. Dyn Games Appl (2026). https://doi.org/10.1007/s13235-026-00719-w
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DOI: https://doi.org/10.1007/s13235-026-00719-w