Collective strategy condensation: When envy splits societies
EEntropy , xx , 1-x; doi:10.3390/—— OPEN ACCESS entropy
ISSN 1099-4300
Article
Collective strategy condensation: When envy splits societies
Claudius Gros
Institute for Theoretical Physics, Goethe University Frankfurt, Germany * Author to whom correspondence should be addressed; gros07[@]itp.uni-frankfurt.de
Received: xx / Accepted: xx / Published: xx
Abstract:
Human societies are characterized, besides others, by three constituent features.(A) Options, as for jobs and societal positions, differ with respect to their associatedmonetary and non-monetary payoffs. (B) Competition leads to reduced payoffs whenindividuals compete for the same option with others. (C) People care how they are doingrelatively to others. The latter trait, the propensity to compare one’s own success with thatof others, expresses itself as envy.It is shown that the combination of (A)-(C) leads to spontaneous class stratification. Societiesof agents split endogenously into two social classes, an upper and a lower class, when envybecomes relevant. A comprehensive analysis of the Nash equilibria characterizing a basicreference game is presented. Class separation is due to the condensation of the strategiesof lower-class agents, which play an identical mixed strategy. Upper class agents do notcondense, following individualist pure strategies.Model and results are size-consistent, holding for arbitrary large numbers of agents andoptions. Analytic results are confirmed by extensive numerical simulations. An analogy tointeracting confined classical particles is discussed.
Keywords:
Self-Organization; Sociophysics; Game Theory; Strategy Condensation; NashEquilibrium; Phase Transition; Envy; Social Classes; Complex Systems a r X i v : . [ phy s i c s . s o c - ph ] J a n ntropy , xx
21. Introduction
The notion of an ‘ideal society’ has always been controversial [1,2], especially regarding theconditions for social classes to arise endogenously when by-birth privileges and handicaps are absent,a feature commonly presumed to be desirable. In this regard one may consider a society to be ‘ideal’when the playing ground is fair, which means that members have equal access to societal options andpositions. Here we examine this situation using a generalized game theoretical setting.Two building blocks constitute the core of most abstract games [3]: competition and that differentoptions yield distinct rewards. In this study we examine what happens if a third element is added,postulating that agents desire to compare their individual success reciprocally, a trait usually termed‘envy’ [4]. We show that envy splits ideal societies. Two distinct social classes, an upper and a lowerclass, form endogenously when the desire to compare success becomes substantial.The notion of envy is based on the observation that the satisfaction individuals receive from havingand spending money depends not only on the absolute level of consumption, but also on how one’s ownconsumption level compares with that of others [5]. This view, which is at the heart of relative incometheory [6,7], is taken for granted, to give an example, when poverty is defined not in absolute, but inrelative terms [8,9].Key to our research is the notion that class structures may emerge from class-neutral interactionsbetween individual agents. On an equivalent basis, a large body of social computation research [10,11]has investigated to which extent cooperation [12,13], reciprocity [14], altruism [15] and social norms[16] are emergent phenomena. The model investigated in this study is formulated directly in terms ofstrategies, as usual for animal conflict models [17], like the war of attrition. A corresponding agent-basedsimulation setup would also be possible, with the differences vanishing in the limit of large numbers ofagents and behavioral options. The adaptive game-theoretical formulation used here comes with theadvantage that the properties of the class stratified state can be studied analytically. Our study can beseen as a generalization of evolutionary game theory, which is dedicated in good part to the origins ofbehavioral traits [18], to the emergence of class structures. Other alternatives include dynamical systemsinvestigations of the stability of societies [19], and game theoretical approaches centered on selected keysocietal players [20].
We consider a society of N agents, with every agent able to select from M options. The payofffunction E αi , for option i and agent α , is agent specific, but only to the extent that it depends explicitlyon the strategies p αi ≥ . Strategies are normalized, (cid:80) i p αi = 1 , with p αi ≥ denoting the probabilitythat agent α selects option i . Rewards R α are defined as the expected payoffs, R α = (cid:88) i E αi p αi = (cid:104) E αi (cid:105) { ρ αi } . (1)The number of options M can be both larger or smaller than the number of agents N , with size-consistentlarge- N limits being recovered for constant ratios ν = M/N .In this study we define social classes in terms of reward clusters. Agents within the same classreceive rewards similar in magnitude, which are separated by a gap from the rewards of other classes, ntropy , xx agents - α r e w a r d s - R α only one reward cluster / social classtwo reward clusters / social classes lower class upper class Figure 1. Social classes as reward clusters.
When ordering the rewards R α obtained byagents α as a function of size, the resulting reward spectrum may be characterized either by acontinuous distribution (black), or by one or more gaps (red). Clustering rewards with regardto proximity allows consequently for a bare-bone definition of social classes.as illustrated in Fig. 1. In human societies, social groups are shaped in particular also by the notion ofsocial identity [21], which is absent in the bare-bone definition of social classes used here. A politicaltheory for social classes is beyond the scope of the present study.
2. Shopping trouble model
We define with ¯ R = 1 M (cid:88) α R α , (2)the mean reward ¯ R of all agents. The payoff function E αi = v i − κ (cid:88) β (cid:54) = α p βi + ε p αi log (cid:18) R α ¯ R (cid:19) (3)of our reference model contains three terms:• Basic utility.
The basic utility function v i , which is identical for all agents, encodes the notion thatoptions come with different payoffs. Mapping options to qualities q i ∈ [0 , , we will use a simpleinverted parabola, v i = 1 − (1 − q i ) , for the basic utility.• Competition.
There is a flat penalty κ for agents competing heads on. Payoff reduction isproportional to the probability p βi that other agents select the option in question. The respectivecombinatorial factors are approximated linearly in (3), as given by the sum (cid:80) β (cid:54) = α p βi .• Envy.
One’s own success with respect to the mean reward, R α / ¯ R , induces a psychological rewardcomponent.The log-dependency log( R α / ¯ R ) of the envy term in (3) reflects the well established observation that thebrain discounts sensory stimuli [22], numbers [23], time [24], and data sizes [25] logarithmically. Inaddition, the envy term is proportional to the current probability p αi to select option i , which encodesa straightforward rational. When everything is fine, when log( R α / ¯ R ) > , the current strategy is ntropy , xx options p a yo ff s utilityfunction v(q i ) - κ v(q i ) qualities q i confiningpotential -v(q i ) + κ -v(q i ) Figure 2. Correspondence to interacting classical particles.
Left:
Agents selecting astrategy i receive a bar utility v ( q i ) (inverted parabola), which is reduced by a flat amount κ ,the competition term, if another agent selects the same option. Utilities are to be maximized. Right:
Classical particles in a confining potential − v ( q i ) (parabola) repel each other by anamount − κ . Energy is minimized.reinforced, and suppressed when log( R α / ¯ R ) < . The effect is that agents with high/low rewards tend topursuit pure/mixed strategies. Eq. (3) is called the ‘shopping trouble model’, as it can be applied, besidesthe general social context, to the case that agents need to optimize their shopping list [26]. Note that thatagents have only a single goal within the shopping trouble game, reward maximisation, in contrast tomost status seeking games [27,28], for which both status and utility are separately important [29,30]. The shopping trouble model (3) can be interpreted in terms of interacting and confined classicalparticles, with the correspondenceagents ↔ classical particles q i ↔ states − v ( q i ) ↔ confining potential κ ↔ Coulomb repulsion (cid:15) ↔ energy-dependent interactionas is illustrated in Fig. 2. Without the energy-dependent interaction term, (cid:15) (envy), particles settle into therespective lowest energy states, which are given by − v ( q i )+( n i − κ , where n i is the occupation numberof state q i (the number of agents selecting the quality q i ). Strategies p αi correspond in physics terms tothe occupation distribution. At finite temperature particles can always swap places, which implies thatstrategies are identical. This is however not the case at zero temperature. Finite temperatures correspondin game theory that agents select alternative, lower-reward strategies, with a probability given by therespective Boltzmann factors. Here we work with strictly rational, zero-temperature agents, which goalways for the best choice. Agents interact in the shopping trouble model via two averaging fields [31]. The first coupling termis a scalar quantity, the average reward ¯ R . It quantifies the envy term in (3). The second coupling term, ntropy , xx envy ε b - κ v a - κ v b v a E a1 - first agentE b1 - first agentv a + ε log(2v a /(v a +v b ))v b + ε log(2v b /(v a +v b )) E a2 - second agentE b2 - second agent κ - c o m p e t i t i o n options a / b mixed strategysecond agent Figure 3. Envy-induced transition from pure to mixed strategies.
Illustration of thecase 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, (cid:15) = 0 , both agents play pure strategies, here withthe first/second agent selecting a/b . It would be unfavorable for the second agent to invadeoption a , as v a − κ < v b , and vice versa, where κ is the strength of the competition. Inthis state rewards are R a = v a and R b = v b and R a,b / ¯ R = 2 v a,b / ( v a + v b ) . For the secondagent the envy term (cid:15)p αi log( R α / ¯ 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 = v b + (cid:15) log(2 v b / ( v a + v b )) (red solid line) becomes smaller than the E a = v a − κ (red dashedline).the mean strategy ¯ p i = (cid:80) β p βi /M , is in contrast a function of the available options. It enters the penaltyterm via (cid:88) β (cid:54) = α p βi = M ¯ p i − p αi . (4)Numerically, the shopping trouble model is solved using standard evolutionary dynamics [32], p αi ( t + 1) = p αi ( t ) E αi ( t ) (cid:80) j p αj ( t ) E αj ( t ) . (5)In practice, a constant offset E is added on the right-hand side, which acts as a smoothing factor. The support of a strategy p αi is given by the set of options selected with finite probabilities p αi > .The smallest possible support is one, the case of a pure strategy, p αi = δ i,k . Supports larger than onecorrespond to mixed strategies. Without envy, viz when (cid:15) = 0 , the Nash stable strategies of the shoppingtroubling game are all pure. Agents just compare the payoff options v i − κ ( n i − of distinct options,where n i is the occupation factor, viz the number of times option i has been selected by all agents. If ntropy , xx agents - α r e w a r d s - R α envy: ε = 0.4envy: ε = 0.8 lower class upper class M = N = 100 κ = 0.3 agents - α m on e t a r y i n c o m e s - I α envy: ε = 0.4envy: ε = 0.8 lower class upper class M = N = 100 κ = 0.3 Figure 4. Envy induced class stratification.
Simulation results for M = N = 100 and κ = 0 . . Top:
For (cid:15) = 0 . (black) the reward spectrum is continuous, with agents receivingvarying rewards. For (cid:15) = 0 . (red) two strictly separated reward clusters emerge. Membersof the same class receive identical rewards, which implies intra-class communism. Bottom:
The respective spectrum of monetary incomes I α , as defined by Eq. (6). The gap betweenlower and upper class is substantial. Note that everybody’s monetary income drops whenenvy is increased from 0.4 to 0.8. Percentage-wise the loss is comparatively small for topincome agents.not favorable, agents will avoid occupied options and settle for lower basic utilities. The situation isillustrated for two players in Fig. 3. By avoiding each other, agents seemingly cooperate, a state called‘forced cooperation’ [26].Relative payoff magnitudes change when envy is introduced. The own option becomes progressivelyless attractive when the envy term is negative, which is the case for agents below ¯ R , see Fig. 3. Eventuallythe payoff for the own option levels with that of an occupied option with a higher basic utility and mixedstrategies appear. For larger numbers of agents, and options, we find that the evolution of mixed strategieswith increasing levels of envy leads to a class stratified state, as discussed further below.
3. Results
In Fig. 4 representative reward distributions for the shopping trouble model are given. The resultsare obtained iterating (5) recursively for · times. The initial strategies are random, which impliesthat chance determines the fate of individual agents, in particular the final reward. Equivalent resultsare obtained for smaller and larger numbers of agents and options. Changing the density of agents per ntropy , xx envy ε s t r a t e g i e s / a g e n t s M=N=100 κ = 0,3forcedcooperation classstratified mixedpure Figure 5. Evolution of mixed strategies.
For N = 100 options and M = 100 agents thefraction of agents playing respectively pure and mixed options. For small envy the numberof mixed strategies raises, in agreement with the mechanism illustrated in Fig. 3 for thecase of two players. Mixed strategies played by distinct agents merge into a single mixedstrategy for the entirety of lower-class agents once a critical density of mixed strategies isreached. The shaded region denotes bistability. When starting from random initial strategiesand values of (cid:15) in the shaded region, the evolutionary dynamics (5) leads to either of twopossible Nash equilibria, forced cooperation and the class stratification. The fraction of purestrategies drops for all (cid:15) , until only one or two upper class members remain, the monarchystate. Adapted from [26].option, ν = M/N , leads to quantitative, but not qualitative changes. Larger values of ν increase theinfluence of competition, κ , and hence also of envy. The same holds when increasing κ directly.The transition from forced cooperation at κ = 0 . to class separation, for κ = 0 . , observed in Fig. 4induces a striking self-organized reorganization of the reward spectrum. The distribution of rewards iscontinuous, but otherwise inconspicuous below the transition. A finite competition of κ = 0 . inducescooperation in the sense that it is in general favorable for agents to select different options. Two flatbands arise in contrast in the class stratified state, one for the upper and one for the lower class.The observation that all lower-class agents receive identical rewards has a relatively simpleexplanation. The number of mixed strategies first raises with (cid:15) , in order to drop to one in the classstratified state. Compare Fig. 5. Inspecting the individual strategies one by one reveals that an identicalmixed strategy is played by the entirety of lower-class agents. Colloquially speaking one becomes amember of the masses when joining the lower class. This result explains that a single mixed strategyremains in the class stratified state and that all members of the lower class receive the same reward.In contrast to the lower class, upper-class agents play pure strategies. Members of the upper classavoid each others, their strategies are hence individualistic, as illustrated in Fig. 6 for a small system.Why is it then, as evident from Fig. 4, that upper-class agents have identical rewards? This effect isdue to the interaction with the lower-class mixed strategy, which adapts itself autonomously, until thecontribution from competition, the term ∼ κ in (3), exactly cancels the reward differential arising fromdifferences in the respective basic utilities v i . One can trace analytically, as discussed further below, whythis remarkable self-organized process takes place. ntropy , xx qualities/options q i p a yo ff s E α ( q i ) utility v(q i ) κ = 0.3 ε = 0.8 pure, upper class N = M = 10 mixed, lower class q U q U R U R L Figure 6. Payoffs in the class stratified state.
Numerically obtained payoff functions E αi = E α ( q i ) , for a system with ten options/agents. The strength of competition/envy is κ = 0 . and (cid:15) = 0 . . Shown is the payoff function for the two pure upper-class strategies (red), andfor the single mixed lower-class strategy (green), played by eight agents. For the functionalform of the bare utility, v i = v ( q i ) , an inverse parabola has been selected (black squares).Also shown are the analytic expressions (8) and (7) for the upper-/lower-class rewards, R U and R L (dashed horizontal lines). Indicated by q U and q ¬ U are qualities played/not played bythe upper class. It is evident from the top panel of Fig. 4 that envy induces the formation of two well definedreward clusters. The question arises, if the gap between the lower- and the upper-class cluster is purelypsychological, viz exclusively due to the envy term in (3). For this purpose we define with I α = (cid:88) i (cid:16) v i − κ (cid:88) β (cid:54) = α p βi (cid:17) p αi (6)the monetary income I α , which represents the reward R α minus the envy contribution. Fig. 4 shows, thata gap opens both for the reward and for the monetary income. Everybody loses when envy increases, inthe sense that monetary incomes drop for all agents, also for those at the top, when increasing levels ofenvy force the society to class separate. The agent-to-agent interaction is mediated in the shopping trouble model by two averaging fields, ¯ R and ¯ p i , as discussed further above. It can be shown [26], that this property allows to derive analyticexpressions for the rewards of the lower and of the upper class, respectively R L and R U , R L = ε − f L e κ/ε − (cid:18) e κ/ε − f L − f L (cid:19) (7)and R U = ε − f L e − κ/ε − e − κ/ε log (cid:18) e κ/ε − f L − f L (cid:19) . (8)Remarkably, above expressions are not explicitly dependent on the basic utility v i . The only freeparameter in (7) and (8) is the fraction f L of agents in the lower class, which can be determined ntropy , xx numerically. For the class stratified state shown in Fig. 4 one has, as an example, f L = 80 /
100 = 0 . ,as the number of lower- and upper-class agents is respectively 80 and 20. Using f L = 0 . in (7) and (8),the resulting values for R L and R U coincide exactly with the values obtained numerically. The continuous downsizing of the upper class observable in Fig. 5 raises an interesting question. Isthere a critical envy (cid:15) , beyond which the upper class vanishes altogether? In this case agents wouldplay exclusively the mixed strategy of the former lower class, a telltale characteristics of a communiststate. Rewards would be equally the same for everybody. This hypothesis can be tested numerically byusing the extracted lower-class mixed strategy as the starting strategy for all M agents. Even for large (cid:15) ,we performed simulations up to (cid:15) = 20 , the communist state is found to be numerically unstable. Thesystem converges without exception to a class-separated state containing one or two upper-class agents.Within the shopping trouble model communism is unstable against monarchy.
4. Terminology
The notation used throughout this study is summarized below. The aim of the compendium below isto provide an overview, not complete and detailed definitions.
Options, qualities & strategies.
Options correspond to possible actions, such as making a purchase ina shop. The numerical value associated with option i is the quality q i . Furthermore we differentiatebetween option and strategy, which is defined here as the probability distribution function p i = p ( q i ) topursue a given option. Pure vs. mixed strategies.
A strategy is pure when the agent plays the identical option at all times, andmixed otherwise, viz when behavior is variable.
Evolutionary stable strategies.
Taking the average payoff received as an indicator for fitness, a givenstrategy is evolutionary stable if every alternative leads to a lower fitness. Evolutionary stable strategiesare Nash stable.
Support.
Strategies are positive definite for all options, p α ( q i ) ≥ . In reality, p α ( q j ) is finite only for asubset of options, the support of the strategy. Strategies are pure/mixed when the size of the support isone/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 incompetitive 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, maychange qualitatively upon changing a parameter, f.i. the strength of envy. Such a transition correspondsin physics terms to a phase transition. Phase transitions are in general due to collective effects, whichmeans that they are the result of the interaction between the components, here the agents.
Forced cooperation / class stratification.
Forced cooperation is present when agents seeminglycooperate by avoiding each other, as far as possible. It is forced when agents optimize in reality just ntropy , xx their individual fitness. Forced cooperation and the class stratified state are separated by a collectivephase transition. Envy.
Envy is postulated to have opposite effects on agents with high/low rewards. When their rewardis above the average, agents take this as an indication that they are doing well and that the best courseof action is to enhance the current strategy. Agents with below-the-average rewards are motivated incontrast to search for alternatives, viz to change the current strategy.
Monarchy & communism.
Monarchy and communism are used throughout this study exclusively forthe labeling of states defined by specific constellations of strategies. Secondary characterizations interms of political theory are not implied. Monarchy is present in a class stratified society when all butone or two agents belong to the lower class. All members of the society are part of a unique class incommunism, with everybody receiving identical rewards and following the same mixed strategy.
5. Discussion
The payoff function of the shopping trouble model (3) is not static, as in classical games, but highlyadaptive. The payoff received when selecting a certain action i depends dynamically on the strategiesof the other agents. At its core this is typical for social simulations studies [10], with the twist thatthe shopping trouble model is formulated directly in terms of strategies. The resulting evolutionarystable strategies are hence to be determined self-consistently, which implies that certain aspects of themulti-agent Nash solutions may have emergent character [33]. This is indeed observed.The shopping trouble model studied here incorporates specific functionalities. We believe, however,that alternative models based on the same three principles, payoff diversity, competition, and inter-agentreward comparison, would lead to qualitatively similar results. Further we note that the widely useddistinction between benign and malicious envy [34] enters the shopping trouble model, albeit indirectly.Benign envy, the quest to reach a better outcome by improving oneself, can be said to be operative whenagents select the best pure strategy compatible with everybody else’s choice. Malicious envy, which aimsto pull somebody down from their superior position [34], is functionally operative when agents start toinvade somebody else’s zone by switching to mixed strategies, as shown in Fig. 3. In this interpretation,societies are pushed towards class stratification by malicious, and not by benign envy. On a societallevel, malicious envy is counterproductive.
6. Conclusions
We conclude by recapitulating the driving forces for the class stratification transition. Agents with lowrewards are constantly searching for better options (compare Sect. 4). However, more than one optioncan be sampled only when using mixed strategies, which implies that raising levels of envy induce acorresponding larger number of mixed strategies, as observed in Fig. 5. In the end, a large number oflow-reward agents are trying to explore extended ranges of options. At a certain level of envy theirrespective mixed strategies collide, collapsing at this point into a single encompassing strategy for theentire lower class. Class stratification is hence a result of a spontaneous condensation of strategies.In effect, class stratification results from the constant state of discontent of low-reward agents, takingplace right at the point when their continuing search for alternatives runs out of options. High-reward ntropy , xx agents have in contrast little incentive to do anything else. In order to keep their privileged position theyjust need to concentrate efforts on what they are doing best, their current strategies. Acknowledgment
The author thanks Carolin Roskothen, Daniel Lambach and Roser Valenti for comments.
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