## 1. Introduction

#### Social Classes in Terms of Reward Clusters

## 2. Shopping Trouble Model

**Basic utility.**The basic utility function ${v}_{i}$, which is identical for all agents, encodes the notion that options come with different payoffs. Mapping options to qualities ${q}_{i}\phantom{\rule{-0.166667em}{0ex}}\in \phantom{\rule{-0.166667em}{0ex}}[0,2]$, we will use a simple inverted parabola, ${v}_{i}=1-{(1-{q}_{i})}^{2}$, for the basic utility.**Competition.**There is a flat penalty $\kappa $ for agents competing heads-on. Payoff reduction is proportional to the probability ${p}_{i}^{\beta}$ that other agents select the option in question. The respective combinatorial factors are approximated linearly in (3), as given by the sum ${\sum}_{\beta \ne \alpha}{p}_{i}^{\beta}$.**Envy.**One’s own success with respect to the mean reward, ${R}^{\alpha}/\overline{R}$, induces a psychological reward component.

#### 2.1. Correspondence to Confined Interacting Classical Particles

agents | ↔ | classical particles |

${q}_{i}$ | ↔ | states |

$-v\left({q}_{i}\right)$ | ↔ | confining potential |

$\kappa $ | ↔ | Coulomb repulsion |

$\u03f5$ | ↔ | energy-dependent interaction |

#### 2.2. Strategy Evolution

#### 2.3. Pure vs. Mixed Strategies

## 3. Results

#### 3.1. Monetary Incomes—Everybody Loses

#### 3.2. Analytic Properties of the Class-Stratified State

#### 3.3. Monarchy vs. Communism

## 4. Terminology

**Options, qualities, and strategies.**Options correspond to possible actions, such as making a purchase in a shop. The numerical value associated with option i is the quality ${q}_{i}$. Furthermore, we differentiate between option and strategy, which is defined here as the probability distribution function ${p}_{i}=p\left({q}_{i}\right)$ to pursue a given option.

**Pure vs. mixed strategies.**A strategy is pure when the agent plays the identical option at all times, and mixed otherwise, viz when behavior is variable.

**Evolutionary stable strategies.**Taking the average payoff received as an indicator for fitness, a given strategy is evolutionary stable if every alternative leads to lower fitness. Evolutionarily stable strategies are Nash-stable.

**Support.**Strategies are positive definite for all options, ${p}^{\alpha}\left({q}_{i}\right)\ge 0$. In reality, ${p}^{\alpha}\left({q}_{j}\right)$ is finite only for a subset of options, the support of the strategy. Strategies are pure/mixed when the size of the support is one/larger than one.

**Payoff/reward.**The payoff function is a real-valued function of the qualities (options). The mean payoff, as averaged over the current strategy, is the reward.

**Competitive/cooperative game.**Parties may coordinate their strategies in cooperative games, but not in competitive games. For the shopping trouble game, voluntary cooperation is not possible.

**Collective effects/phase transition.**The state of a complex system, like a society of agents, may change qualitatively upon changing a parameter, f.i. the strength of envy. Such a transition corresponds in physics terms to a phase transition. Phase transitions are in general due to collective effects, which means that they are the result of the interaction between the components, here the agents.

**Forced cooperation/class stratification.**Forced cooperation is present when agents seemingly cooperate by avoiding each other, as far as possible. It is forced when, in reality, agents optimize just their individual fitness. Forced cooperation and the class-stratified state are separated by a collective phase transition.

**Envy.**Envy is postulated to have opposite effects on agents with high/low rewards. When their reward is above the average, agents take this as an indication that they are doing well and that the best course of action is to enhance the current strategy. In contrast, agents with below-average rewards are motivated to search for alternatives, viz to change the current strategy.

**Monarchy and communism.**Monarchy and communism are used throughout this study exclusively for the labeling of states defined by specific constellations of strategies. Secondary characterizations in terms of political theory are not implied. Monarchy is present in a class-stratified society when all but one or two agents belong to the lower class. All members of the society are part of a unique class in communism, with everybody receiving identical rewards and following the same mixed strategy.

## 5. Discussion

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Reeve, C. Plato: Republic. In Hackett, Indianapolis; Hackett Publishing: Indianapolis, IN, USA, 2004. [Google Scholar]
- Swidler, A. The ideal society. Am. Behav. Sci.
**1991**, 34, 563–580. [Google Scholar] [CrossRef] - Myerson, R.B. Game Theory; Harvard University Press: Cambridge, MA, USA, 2013. [Google Scholar]
- Nguyen, T.T.; Rothe, J. Minimizing envy and maximizing average Nash social welfare in the allocation of indivisible goods. Discret. Appl. Math.
**2014**, 179, 54–68. [Google Scholar] [CrossRef] - Hopkins, E.; Kornienko, T. Running to keep in the same place: Consumer choice as a game of status. Am. Econ. Rev.
**2004**, 94, 1085–1107. [Google Scholar] [CrossRef] - McBride, M. Relative-income effects on subjective well-being in the cross-section. J. Econ. Behav. Organ.
**2001**, 45, 251–278. [Google Scholar] [CrossRef] - Clark, A.E.; Senik, C. Who compares to whom? The anatomy of income comparisons in Europe. Econ. J.
**2010**, 120, 573–594. [Google Scholar] [CrossRef] - Sen, A. Poor, relatively speaking. Oxf. Econ. Pap.
**1983**, 35, 153–169. [Google Scholar] [CrossRef] - Wagle, U. Rethinking poverty: Definition and measurement. Int. Soc. Sci. J.
**2002**, 54, 155–165. [Google Scholar] [CrossRef] - Conte, R.; Gilbert, N.; Bonelli, G.; Cioffi-Revilla, C.; Deffuant, G.; Kertesz, J.; Loreto, V.; Moat, S.; Nadal, J.P.; Sanchez, A.; et al. Manifesto of computational social science. Eur. Phys. J. Spec. Top.
**2012**, 214, 325–346. [Google Scholar] [CrossRef] - Lane, J.E.; Shults, F.L. Cognition, culture, and social simulation. J. Cogn. Cult.
**2018**, 18, 451–461. [Google Scholar] [CrossRef] - Janssen, M.A.; Holahan, R.; Lee, A.; Ostrom, E. Lab experiments for the study of social-ecological systems. Science
**2010**, 328, 613–617. [Google Scholar] [CrossRef] - Gächter, S.; Herrmann, B. Reciprocity, culture and human cooperation: Previous insights and a new cross-cultural experiment. Philos. Trans. R. Soc. B
**2009**, 364, 791–806. [Google Scholar] [CrossRef] [PubMed] - Fehr, E.; Gächter, S. Fairness and retaliation: The economics of reciprocity. J. Econ. Perspect.
**2000**, 14, 159–181. [Google Scholar] [CrossRef] - Bowles, S. Group competition, reproductive leveling, and the evolution of human altruism. Science
**2006**, 314, 1569–1572. [Google Scholar] [CrossRef] [PubMed] - Bicchieri, C. The Grammar of Society: The Nature and Dynamics of Social Norms; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Smith, J.M. The theory of games and the evolution of animal conflicts. J. Theor. Biol.
**1974**, 47, 209–221. [Google Scholar] [CrossRef] - Newton, J. Evolutionary game theory: A renaissance. Games
**2018**, 9, 31. [Google Scholar] [CrossRef] - Gros, C. Entrenched time delays versus accelerating opinion dynamics: Are advanc ed democracies inherently unstable? Eur. Phys. J. B
**2017**, 90, 223. [Google Scholar] [CrossRef] - Ridley, D.; de Silva, A. Game Theoretic Choices Between Corrupt Dictatorship Exit Emoluments and Nation-Building CDR Benefits: Is There a Nash Equilibrium? Am. Econ.
**2020**, 65, 51–77. [Google Scholar] [CrossRef] - Scheepers, D.; Ellemers, N. Social identity theory. In Social Psychology in Action; Springer: Berlin, Germany, 2019; pp. 129–143. [Google Scholar]
- Hecht, S. The visual discrimination of intensity and the Weber-Fechner law. J. Gen. Physiol.
**1924**, 7, 235–267. [Google Scholar] [CrossRef] - Dehaene, S. The neural basis of the Weber–Fechner law: A logarithmic mental number line. Trends Cogn. Sci.
**2003**, 7, 145–147. [Google Scholar] [CrossRef] - Howard, M.W. Memory as Perception of the Past: Compressed Time in Mind and Brain. Trends Cogn. Sci.
**2018**, 22, 124–136. [Google Scholar] [CrossRef] - Gros, C.; Kaczor, G.; Marković, D. Neuropsychological constraints to human data production on a global sca le. Eur. Phys. J. B
**2012**, 85, 28. [Google Scholar] [CrossRef] - Gros, C. Self induced class stratification in competitive societies of agents: Nash stability in the presence of envy. R. Soc. Open Sci.
**2020**, 7, 200411. [Google Scholar] [CrossRef] [PubMed] - Congleton, R.D. Efficient status seeking: Externalities, and the evolution of status games. J. Econ. Behav. Organ.
**1989**, 11, 175–190. [Google Scholar] [CrossRef] - Haagsma, R.; van Mouche, P. Equilibrium social hierarchies: A non-cooperative ordinal status game. BE J. Theor. Econ.
**2010**, 10, 1–49. [Google Scholar] [CrossRef] - Shi, B.; Van Gorder, R. Nonlinear dynamics from discrete time two-player status-seeking games. J. Dyn. Games
**2017**, 4, 335–359. [Google Scholar] [CrossRef] - Courty, P.; Engineer, M. A pure hedonic theory of utility and status: Unhappy but efficient invidious comparisons. J. Public Econ. Theory
**2019**, 21, 601–621. [Google Scholar] [CrossRef] - Hauert, C.; Szabó, G. Game theory and physics. Am. J. Phys.
**2005**, 73, 405–414. [Google Scholar] [CrossRef] - Hofbauer, J.; Sigmund, K. Evolutionary game dynamics. Bull. Am. Math. Soc.
**2003**, 40, 479–519. [Google Scholar] [CrossRef] - Gros, C. Complex and Adaptive Dynamical Systems, a Primer; Springer: Berlin, Germany, 2015. [Google Scholar]
- Van de Ven, N. Envy and its consequences: Why it is useful to distinguish between benign and malicious envy. Soc. Personal. Psychol. Compass
**2016**, 10, 337–349. [Google Scholar] [CrossRef]

**Figure 1.**Social classes as reward clusters. When ordering the rewards ${R}^{\alpha}$ obtained by agents $\alpha $ as a function of size, the resulting reward spectrum may be characterized either by a continuous distribution (black), or by one or more gaps (red). Clustering rewards with regard to proximity consequently allows for a bare-bone definition of social classes.

**Figure 2.**Correspondence to interacting classical particles. (

**Left**) Agents selecting a strategy i receive a bar utility $v\left({q}_{i}\right)$ (inverted parabola), which is reduced by a flat amount $\kappa $, the competition term, if another agent selects the same option. Utilities are to be maximized. (

**Right**) Classical particles in a confining potential $-v\left({q}_{i}\right)$ (parabola) repel each other by an amount $-\kappa $. Energy is minimized.

**Figure 3.**Envy-induced transition from pure to mixed strategies. Illustration of the case of two agents that can select between two options, $a/b$, with basic utilities ${v}_{a}$ and ${v}_{b}$. Here, ${v}_{a}>{v}_{b}$. In the absence of envy, $\u03f5=0$, both agents play pure strategies, here with the first/second agent selecting $a/b$. It would be unfavorable for the second agent to invade option a, as ${v}_{a}-\kappa <{v}_{b}$, and vice versa, where $\kappa $ is the strength of the competition. In this state, rewards are ${R}_{a}={v}_{a}$ and ${R}_{b}={v}_{b}$ and ${R}_{a,b}/\overline{R}=2{v}_{a,b}/({v}_{a}+{v}_{b})$. For the second agent, the envy term $\u03f5{p}_{i}^{\alpha}log({R}^{\alpha}/\overline{R})$ is negative for the b-option, vanishing for the a-option. The second agent starts to play a mixed strategy (green shaded area) when the payoff ${E}_{b}^{2}={v}_{b}+\u03f5log(2{v}_{b}/({v}_{a}+{v}_{b}))$ (red solid line) becomes smaller than the ${E}_{a}^{2}={v}_{a}-\kappa $ (red dashed line).

**Figure 4.**Envy-induced class stratification. Simulation results for $M=N=100$ and $\kappa =0.3$. (

**Top**) For $\u03f5=0.4$ (black) the reward spectrum is continuous, with agents receiving varying rewards. For $\u03f5=0.8$ (red) two strictly separated reward clusters emerge. Members of the same class receive identical rewards, which implies intra-class communism. (

**Bottom**) The respective spectrum of monetary incomes ${I}^{\alpha}$, as defined by Equation (6). The gap between the lower and upper classes is substantial. Note that everybody’s monetary income drops when envy is increased from 0.4 to 0.8. Percentage-wise, the loss is comparatively small for top-income agents.

**Figure 5.**Evolution of mixed strategies. For $N\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}100$ options and $M\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}100$ agents, the fraction of agents playing pure and mixed options respectively. For small envy, the number of mixed strategies rises, in agreement with the mechanism illustrated in Figure 3 for the case of two players. Mixed strategies played by distinct agents merge into a single mixed strategy for the entirety of lower-class agents once a critical density of mixed strategies is reached. The shaded region denotes bistability. When starting from random initial strategies and values of $\u03f5$ in the shaded region, the evolutionary dynamics (5) lead to either of two possible Nash equilibria, forced cooperation, and class stratification. The fraction of pure strategies drops for all $\u03f5$, until only one or two upper-class members remain, the monarchy state. Adapted from [26].

**Figure 6.**Payoffs in the class-stratified state. Numerically obtained payoff functions ${E}_{i}^{\alpha}={E}^{\alpha}\left({q}_{i}\right)$, for a system with 10 options/agents. The strength of competition/envy is $\kappa \phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0.3$ and $\u03f5\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0.8$. Shown is the payoff function for the two pure upper-class strategies (red), and for the single mixed lower-class strategy (green), played by eight agents. For the functional form of the bare utility, ${v}_{i}=v\left({q}_{i}\right)$, an inverse parabola has been selected (black squares). Also shown are the analytic expressions (8) and (7) for the upper-/lower-class rewards, ${R}_{\mathrm{U}}$ and ${R}_{\mathrm{L}}$ (dashed horizontal lines). Indicated by ${q}_{\mathrm{U}}$ and ${q}_{\neg \mathrm{U}}$ are qualities played/not played by the upper class.

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