GGames in rigged economies
Lu´ıs F Seoane
1, 2 Departamento de Biolog´ıa de Sistemas, Centro Nacional de Biotecnolog´ıa (CSIC), C/ Darwin 3, 28049 Madrid, Spain. Instituto de F´ısica Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB), Palma de Mallorca, Spain.
Modern economies evolved from simpler human exchanges into very convoluted systems. Today,a multitude of aspects can be regulated, tampered with, or left to chance; these are economic degrees of freedom which together shape the flow of wealth. Economic actors can exploit them,at a cost, and bend that flow in their favor. If intervention becomes widespread, microeconomicstrategies of different actors can collide or resonate, building into macroeconomic effects. Howviable is a ‘rigged’ economy, and how is this viability affected by growing economic complexityand wealth? Here we capture essential elements of ‘rigged’ economies with a toy model. Nashequilibria of payoff matrices in simple cases show how increased intervention turns economic de-grees of freedom from minority into majority games through a dynamical phase. These stages arereproduced by agent-based simulations of our model, which allow us to explore scenarios out ofreach for payoff matrices. Increasing economic complexity is then revealed as a mechanism thatspontaneously defuses cartels or consensus situations. But excessive complexity enters abruptlyinto a regime of large fluctuations that threaten the system’s viability. This regime results fromnon-competitive efforts to intervene the economy coupled across degrees of freedom, becomingunpredictable. Thus non-competitive actions can result in negative spillover due to sheer eco-nomic complexity. Simulations suggest that wealth must grow faster than linearly with economiccomplexity to avoid this regime and keep economies viable in the long run. Our work providestestable conclusions and phenomenological charts to guide policing of ‘rigged’ economic systems.
Keywords: Agent based modeling, economic equilibrium, rigged economies, economic complexity
I. INTRODUCTION
The existence of ‘rigged’ economic scenarios is am-ply acknowledged. Most notable examples are non-competitive markets [1, 2], legal or illegal, such as cartels,or natural monopolies [3]. In these, all actors usuallycooperate to secure similar profits. This entails ‘hand-crafting’ some aspects of the economic games in whichthey engage. In competitive markets we also find illegalschemes (e.g. inside trading) or innovative, often border-line legal, enterprises to explore unprecedented economicpossibilities – e.g. anticipating a broker’s moves withfaster internet cables [4]. Such out-of-the-box thinkingis part of the economy’s open-ended nature [5, 6]. Itredesigns the rules of the game and easily results in asentiment that “the market is rigged” [4]. Even if all ac-tors stick to the norms and do not innovate, competitivemarkets are strongly regulated. Some conditions (e.g. de-manding a minimum equity to participate) are designedby governments or international institutions. They mightchange due to democratic consensus or lobbying. If pow-erful firms bend the rules systematically, regulatory cap-ture happens [7–9] threating democracy at large [10, 11].As transnational markets grow ever more complex andfaster, slow public bureaucracies might lag behind andabdicate into nimbler private regulators [12, 13].Through and through, economies are ‘rigged’. Avail-able games are somehow manufactured. Once estab-lished, they remain open to manipulations that might i)impact costs and rewards of economic games, ii) cap theinformation available, or iii) limit the number of playersallowed to partake. This can be achieved through public- ity, bribes, threats, imposing tariffs, etc. More abstractly,we can think of degrees of freedom that can be harnessedin economic systems. Each degree of freedom is a pocketof opportunity that can be exploited, contested or uncon-tested, at some cost. Envelop theorems assess changes oflikely payoffs when a game is altered externally [14–16].Wolpert and Grana [17] recently wondered how much anagent should pay if she (and no other actor involved) wasgiven this control before playing a game. The decisionboils down to a positive payoff balance with versus with-out intervention. Thus a single agent is offered control,at a cost, over a single economic degree of freedom.Here we study what happens when multiple actors areallowed, also at a cost, to manipulate several economicdegrees of freedom. Different efforts might align or not,yielding uncertain returns. A single agent’s decision torig one game (as per [17]) might be of limited conse-quence in isolation. But effects may be amplified, mit-igated, or produce emergent phenomena when coupledacross games and players. We are interested in how mi-croscopic fates scale up to macroeconomic trends, so weadopt a systemic perspective. More available degrees offreedom result in more complex economies – interven-tion possibilities grow combinatorially and more exter-nal variables become relevant if extra degrees of freedomare left unchecked. How do system-wide dynamics of arigged economy depend on its complexity? How muchcan such economies grow? Open-endedly, perhaps? Dothey collapse, unable to sustain their participants? Howis this affected by the amount of wealth generated anddistributed? What is a natural level of intervention de-pending on these aspects? a r X i v : . [ phy s i c s . s o c - ph ] J u l Agent 1 Agent 2 Agent 3 a1a2a3a4
Time dc b FIG. 1
A rigged economy and its dynamics. a
Threeagents choose (red shade) whether to rig each game (blackboxes) or not (empty boxes), and (gray shade) what to playin each game ( a ik = 1 or 0). b-d Dynamics of model mea-surements for C C = 10, n = 20, and B = 100 (thus b = 5). b Population (black, left scale) and wealth (red, right scale).
We could tackle these questions rigorously throughutility functions that discount intervention costs, as in[17], extended to multiple agents and games; but thisquickly becomes untreatable. Instead, inspired by agent-based models and complex adaptive systems [18–23], wecapture essential elements that affect our research ques-tions into a toy economy. We assume a population ofagents who engage in n economic games. Each game hasa rule that randomly determines its winning strategy.Agents can pay to intervene each game’s rule, affectingthe winning strategy for all (figure 1 a ). These games constitute our degrees of freedom , thus n is a proxy forthe economy’s complexity. An amount of wealth, B , isdistributed among winners. B reminds us of a GDP andis a proxy for our economy’s size. The model is describedin detail in section II. We miss some important ways ofmanipulating real economies – such shortages and modelextensions are discussed in section IV.We write complete payoff matrices for some simple sce-narios. Their analysis (section III.A) shows that increas-ing intervention switches isolated degrees of freedom fromminority to coordination games. In between, Nash equi-libria are mixed strategies, anticipating dynamic strug-gles. We explore increasing economy size and complexitywith simulations based on agents of bounded rational-ity and Darwinian dynamics to select successful strate-gies. We argue (section IV) that our results should notdepend critically on the agent’s rationality and the Dar-winism. We simulate model dynamics for a small, fixednumber of degrees of freedom as economy size grows (sec-tion III.B). This reveals the same progression: from mi-nority, through dynamic, to coordinating regimes. Thelater remind us of cartels. Adding degrees of freedomabruptly halts within-game coordination, suggesting anempirical test: increased economic complexity should dis-solve cartels spontaneously. We study our toy economy’sviability as its complexity grows large and its size scalesappropriately (section III.C). Economies whose size doesnot grow fast enough with their complexity fall in alarge-fluctuations regime that threatens their viability –thus non-competitive actions can have negative spilloversas agents and degrees of freedom become coupled enmasse. Our toy model allows us to find limit regimes(e.g. within-game coordination, large-fluctuations, etc.)that emerge from essential elements potentially commonto any ‘rigged’ economy. We lay out comprehensive mapsof such regimes (section III.D). Their occurrence is tiedto a few, yet abstract parameters. In section IV we spec-ulate how we might link them to real-world economies. II. METHODSA. Model description
Our toy economy (Figure 1 a ) consists of a fixed num-ber of games, n ; and a population of N ( t ) ∈ [0 , N max ]agents that changes over time. At each iteration, everyagent has to play all games, which admit strategies 0 or1. The strategies played by agent A i are collected in anarray: a i ≡ [ a ik , k = 1 , . . . , n ]. Besides, a second array r i ≡ [ r ik , k = 1 , . . . , n ] codifies whether A i attempts torig game k ( r ik = 1) or not ( r ik = 0). The combination( a ik , r ik ) constitutes the proper strategy of agent A i to-wards game k . However, to aid the model’s discussion,we use the word ‘strategy’ only to name a ik .At each iteration a rule exists, common to all agents,that determines the winning strategy for each game: R ( t ) ≡ [ R k ( t ) ∈ { , } , k = 1 , . . . , n ]. If any agents at-tempt to rig game k , R k ( t ) takes the most common actionamong those rigging agents (Figure 1 a4 ): R k ( t ) = argmax ¯ a ∈{ , } (cid:0) || (cid:8) A i , a ik = ¯ a, r ik = 1 (cid:9) || (cid:1) . (1)In case of draw (including no intervention), R k ( t ) is setrandomly (Figure 1 a2-3 ). Each agent pays an amount C R for each intervention attempt – successful or not. If A i has a wealth w i ( t ) at the beginning of a round, aftersetting R ( t ) this becomes: w i ( t + ∆ t rig ) = w i ( t ) − C R (cid:88) k r ik . (2)Each round, an amount b is ruffled at each game –a total wealth B = nb is potentially distributed. Theamount allocated to game k is split between all agentswho played the winning strategy, R k ( t ). After this: w i ( t + ∆ t play ) = w i ( t + ∆ t rig )+ b (cid:88) k δ ( a ik , R k ( t )) N wk ( t ) , (3)where δ ( · , · ) is Kronecker’s delta and N wk ( t ) is the numberof winners of game k . If w i ( t +∆ t play ) <
0, the i -th agentis removed, decreasing the population by 1.If w i ( t + ∆ t play ) > C C , A i has a child and an amount C C is subtracted from w i . A new agent is generatedwhich inherits a i and r i . Each of the bits in these arraysflips once with a probability p µ . After this, both arraysremain fixed throughout the new agent’s lifetime. Wegenerate an integer number j ∈ [1 , N max ] to allocate thenew individual. If j ≤ N ( t ), the new agent becomes A j .The former agent in that position is removed, its wealthis lost, and the population size remains unchanged. If j > N ( t ), the new individual is appended at the end ofthe pool and the population grows by 1. B. Measurements on model dynamics
For each simulation we set model parameters ( C R = 1, C C = 10, p µ = 0 .
1, and N max = 1000; but variationsare explored in Appendix C to show the generality of ourresults). We explore ranges of n and B to address themain questions – i.e. “how do rigged economies behaveas their complexity and size change?”Model simulations start with a single agent A withrandom strategies and no interventions ( r k = 0 , ∀ k ). Inaverage, A accrues half of the distributed wealth until w ( t ) > C C . As new descents fill the population, rein-forcing or competing strategies unfold. After a rapid ini-tial growth, population size and wealth reach an attractor(Figure 1 b ). We asses these attractors numerically. Sim-ulations run for 5000 iterations. We take averages (noted (cid:104)·(cid:105) ) of diverse quantities over the last 500 iterations. Forexample, population size (cid:104) N (cid:105) , for which we also reportnormalized fluctuations σ ( N ) / (cid:104) N (cid:105) , where σ ( · ) indicates standard deviation. Unless B is very small, 5000 itera-tions suffice to observe convergence (Supporting Figures13, 14, and 15).To measure the heterogeneity of strategies in the pop-ulation, we take the fraction f k ( t ) of agents with a k = 1: f k ( t ) = N ( t ) (cid:88) i =1 δ ( a ik , N ( t ) , (4)from which we compute the entropy: h ak ( t ) = − (cid:104) f k ( t ) log ( f k ( t ))+(1 − f k ( t )) log (1 − f k ( t )) (cid:105) (5)and mean entropy across games (Figure 1 c ): h a ( t ) = 1 n n (cid:88) k =1 h ak ( t ) . (6)If h ak ( t ) = 0, all agents play the same strategy in game k .This quantity is maximal ( h ak ( t ) = 1) if the populationsplits in half around that game. If h a ( t ) = 0, agents playthe same strategy in each game, but not necessarily thesame one across games. If h a ( t ) = 1, agents are splitin half at each game, but this split is not necessarilyconsistent across games.Finally, we introduce the rigging pressure on a game: r k ( t ) = 1 N ( t ) N ( t ) (cid:88) i =1 r ik , r k ( t ) ∈ [0 , d ): r ( t ) = n (cid:88) k =1 r k ( t ) , r ( t ) ∈ [0 , n ]; (8)and average rigging pressure per game r ( t ) /n ∈ [0 , III. RESULTSA. Intervention turns minority into majority games
Before looking at model dynamics we can gain someinsight from payoff matrices in simple cases. Populationsize affects these matrices: earnings are split among win-ners; and more agents imply more distinct, possible cor-relations between strategies and rigging choices. Hence,utility functions rapidly become very complex. In ap-pendix A we discuss payoff matrices for a single gameand one player (Supporting Table I) and for one gameand three players (Supporting Tables II, III, and IV). Allmatrices show average earnings over time if strategies,rigging choices, and population size are fixed.Table I presents the payoff matrix for one game withtwo players. If C R > b/
2, rigging the game is prohibitive.
Agent 1Agent 2
TABLE I
Payoff matrix of one game with two players.
Table entries are labeled by each agent’s strategy a = 0 , r = 0 ,
1. Each cell displays average payoff with no death or reproduction for fixed options. Gray cells are Nashequilibria if C R > b/
2. Gray circuits indicate possible dynamic situations that emerge for C R < b/ Then, the system has the Nash equilibria marked in gray– both agents try to take opposite actions a (cid:54) = a . Withno intervention, we deal with a minority game. For largerpopulations it pays even more to be in the minority (Sup-porting Table II). These equilibria disappear if interven-tion is cheap enough ( C R < b/ n >
3, if all agents are intervening(red frame, Supporting Table IV), the sub-game’s Nashequilibrium is a full coordination. This is not a globalequilibrium, but large coordinations emerge in our sim-ulations for rising intervention levels (see next section).Note how all agents rigging a game in an agreed-uponway to share profits resembles a cartel.Payoff matrices are equal for all games. If n games wereplayed in isolation (i.e. wealth earned by manipulatinga game could not be invested into another), we wouldobserve the same transition to within-game coordinationfor each degree of freedom as intervention takes hold.What happens when we lift such compartmentalization? B. Fixed complexity and growing wealth
We now study model dynamics and stability for a fixednumber of games and varying economy size. Discussionof the rich behavior uncovered follows in Appendix B.Figure 2 a shows (cid:104) N (cid:105) for n = 2 games. Circles overthe plots indicate values of B for which a stretch of thedynamics is plotted in Supporting Figure 14. Generally, (cid:104) N (cid:105) increases with the economy size – i.e. as more moneybecomes available to sustain more agents or to invest intorigging more games. Indeed, the rigging pressure pergame (Figure 2 b ) grows more or less monotonously. (cid:104) N (cid:105) is not so parsimonious. For roughly B <
750 it growssteadily. At B ∼
750 it jumps swiftly, then remainssimilar but slightly declining up until B ∼ c shows that the strategy entropy (cid:104) h a (cid:105) drops sharply before the first boost (shaded area).Before that drop, resources are scarce and rigging theeconomy is difficult. Either strategy is equally likely towin, so agents playing either option are equally abundant(Supporting Figure 14 a ). As B grows, more resources be-come available to rig the games. Either 1 or 0 becomesthe winning strategy over longer time stretches, resultingin temporary selective preferences for one strategy overthe other, and oscillatory dynamics ensue (SupportingFigure 14 b-c ). As (cid:104) h a (cid:105) falls definitely, agents coordi-nate their strategies (Supporting Figure 14 d-e ). Thesedynamics shifts happen simultaneously in all games –as if, so far, payoff matrices were essentially indepen-dent for each degree of freedom. By B ∼ B ∼ (cid:104) N (cid:105) , (cid:104) r (cid:105) /n , and (cid:104) h a (cid:105) as economy size grows for different, fixed n . Withmore games, more discrete jumps in (cid:104) N (cid:105) appear. Thesearise, potentially, from the combinatorially growing co-ordination possibilities across games. They happen afterthe oscillatory phases (Supporting Figures 3 and 15 c-e for n = 3). This again suggests that within-game co-ordination happens first, simultaneously for all games;then degrees of freedom start coupling with each other.Some regimes have similar (cid:104) N (cid:105) for different n (horizontaldashed lines, Supporting Figure 4 a ), suggesting that theyare effectively similar. Population boosts succeed eachother more rapidly for larger n , approaching a continu-ous buildup instead of discrete jumps (Supporting Fig-ure 5 a ). This is not reflected by (cid:104) h a (cid:105) , which only dropsonce due to within-game coordination. The (cid:104) h a (cid:105) plateauis higher for larger n (Supporting Figure 5 c ), indicating a b c FIG. 2
Model behavior for n = 2 games and growing wealth, B . a (cid:104) N (cid:105) , b (cid:104) r (cid:105) /n , and c (cid:104) h a (cid:105) . Open circles indicate B values for which we plot sample dynamics in Supporting Figure 14. Red vertical lines loosely indicate discrete jumps in (cid:104) N (cid:105) . that across-game correlations weaken or interrupt within-game coordination. Above, we compared such coordina-tion to cartels: games are consensually rigged to favormost actors. Our results suggest that increasing eco-nomic complexity prevents the formation of such consen-sus, defusing cartels, even with rising intervention levels.This is a testable conclusion of our model. C. Growing wealth and economic complexity
We now change the number of games as wealth scalesas B = B ( B , n ). The constant B is a normalizing factorto facilitate comparisons. We explore four cases: I : A fixed wealth B I = B is split evenly between allgames: b I = B /n . Returns per game drop as theeconomy becomes more complex. II : Each game distributes a fixed amount b II = B , to-tal wealth grows linearly B II = B · n . Returns pergame remain constant against growing complexity. III : Each degree of freedom revalues previously exist-ing games logarithmically: b III = B (log( n ) + 1).Total wealth grows as B III ∼ B · n (log( n ) + 1). IV : Each degree of freedom revalues previously existinggames linearly: b IV = B · n . Total wealth growsquadratically: B IV = B · n .Figure 3 a-b shows (cid:104) N (cid:105) for each scenario. Extremecases I (black curves) and IV (green) are relatively un-interesting: Stable population size declines quickly for I . As the economic complexity grows and returns pergame drop, more intervention is needed to secure thesame earnings. Such rigged economies collapse if theybecome too complex. For IV , wealth grows so quicklywith n that, promptly, population saturates.Intermediate cases II (blue curves) and III (red) aremore interesting. With B = 50 (Figure 3 a ) and B =100 (Figure 3 b ), (cid:104) N (cid:105) in declines slowly for II . Thus, ingeneral, a rigged economy’s wealth must grow faster thanlinearly with its complexity to remain viable. In case III , (cid:104) N (cid:105) saturates for B = 100; but not for B = 50,for which population seems stagnant.For case II , and III with B = 50, fluctuationsin population size reveal the existence of thresholds, n ∗ II/III ( B ), at which system dynamics change abruptly(Figure 3 c-d ). This affects (cid:104) N (cid:105) marginally (arrows inFigure 3 a-b ), but the increase in σ ( N ) / (cid:104) N (cid:105) is alwayssalient. For n < n ∗ , fluctuations are small ( < n > n ∗ ( B ) large fluctuations ( ∼
25% for case II and ∼
15% for case
III ) set in. There is an absorbing stateat N ( t ) = 0, thus fluctuations of 15 −
25% system sizecan compromise its viability.We explore the transition to large fluctuations by simu-lating case II below ( n = 40 ,
60) and above ( n = 80 , n ∗ . We ran the model for 5000 iterationsand discarded the first 1000. Figure 3 e shows the prob-ability of finding the system with population N . Be-low n ∗ we see a neat Gaußian; above, the distributionpresents two balanced modes. Transitions between themcontribute to the large fluctuations. We also plot to-tal wealth (Figure 3 f ) and wealth per agent (Figure 3 g ).Their averages fall, first, and grow, eventually, as n in-creases. Below n ∗ clear Gaußians appear again. Above,we observe broad tails, indicating inequality. Despiterisking collapse, the average agent can be wealthier inthe large-fluctuations regime. D. Charting rigged economies
We run simulations of case I ( B I ≡ B , b I ≡ B /n )for ranges of economic complexity, n , and distributedwealth, B . This renders maps (Figure 4) where the fourscalings above can be read as curved sections. Trivially,case I traces a horizontal line (solid, red; bottom of eachmap). Case II traces a line with slope B (dashed blacklines). Cases III (dotted black) and IV (dash-dottedblack) trace curves growing faster than linearly. Read-ing (cid:104) N (cid:105) (Figure 4 a ) or σ ( N ) / (cid:104) N (cid:105) (Figure 4 b-c ) alongsuch curves renders the plots from figures 3 a-b and 3 c-d .Results for fixed n and growing B from Figure 2 resultfrom vertical cuts of the map. Other possible progres-sions B = B ( n ) can be charted similarly. d c a b efg FIG. 3
Population size and fluctuations for growing n and B ≡ B ( B , n ) . a-b show (cid:104) N (cid:105) and c-d show σ ( N ) / (cid:104) N (cid:105) . B = 50 in a , c ; B = 100 in b , d . Arrows mark the onset of large-fluctuations in (cid:104) N (cid:105) . Dotted, vertical lines and circles in a , c indicate n values examined in e-g . e Probability density functions of population size, N ; f total wealth, w ; and g wealth percapita w/n . a b c FIG. 4
Comprehensive maps of (cid:104) N (cid:105) and σ ( N ) / (cid:104) N (cid:105) . Maps result from simulating case I (i.e. B I = B , b I = B /n ) forranges of economic complexity, n , and distributed wealth, B . Black curves represent trajectories B = B ( B = 50 , n ) for cases II , III , and IV (dashed, dotted, and dot-dashed respectively). Horizontal red lines at the bottom of maps a-b represent case I with B = 50. a shows (cid:104) N (cid:105) and b-c show σ ( N ) / (cid:104) N (cid:105) over two ranges of n and B . The large-fluctuations regime is a salient anomaly (Fig-ure 4 b-c ) expanding up- and right-wards (perhaps un-boundedly) over a broad range of ( n, B ) values. Its con-tour constraints dependencies, B = B ( n ), that couldavoid this regime. Its upper bound seems to grow fasterthan n · log( n ) suggesting that case III with B = 50 willnot scape large-fluctuations despite sustained growth.Supporting Figure 6 shows maps for (cid:104) h a (cid:105) and (cid:104) r (cid:105) /n .The dent of low (cid:104) h a (cid:105) due to within-game coordinationin simple yet wealthy setups is notable (Supporting Fig-ure 6 a ). We argued that such cartel-like cases could bedefused by increasing complexity. But this map showsthat, if n grows too much without raising B , (cid:104) h a (cid:105) dropsgradually – consensus strategies build up again. It is in-tuitive that (cid:104) r (cid:105) /n grows alongside B (Supporting Figure6 b ), since more available resources can be dedicated torigging the games. Less intuitively, our map shows (cid:104) r (cid:105) /n growing with n as well, even if returns per game diminish. We speculate that, for low n , different agents meddlingare likely to collide, resulting in uncertain returns. Withlarger n , different agents can intervene different degreesof freedom, lowering the chance of mutual frustration. IV. DISCUSSION
It is difficult to pinpoint what an ‘unrigged’ economyis. We model economies as containing degrees of freedomthat can be controlled at a cost by its actors. Uncheckeddegrees favor economic agents at random. An economywith more ‘riggable’ facets is more complex. We studieddynamics, stability, and viability of a rigged economy toymodel as its complexity and total wealth change.Simple scenarios allow a study of equilibria in payoffmatrices. We find that individual degrees of freedom turnfrom minority into majority games, through a dynamicalphase, as intervention raises. Agent-based simulationsconfirm these regimes. They also show new behaviors assynergies develop between degrees of freedom. These newbehaviors (difficult to capture with payoff matrices) haltwithin-game coordination. Within-game coordination insimple yet wealthy markets resembles cartels: most eco-nomic actors with decision power bend the rules homoge-neously in their favor. Our results suggest that this con-sensus is spontaneously defused if the system becomescomplex enough, which can be empirically tested.We study our toy economies as their complexity in-creases and the wealth they distribute remains constant,grows linearly, or faster than linearly with the number ofeconomic degrees of freedom. In general, wealth shouldgrow faster than linearly. Against raising complexity,stagnant or slowly growing wealth only sustains a de-creasing ensemble of actors sharing ever more meager re-sources. An unlucky fluctuation can kill them off. Thisbecomes more pressing as our model predicts that largefluctuations build up abruptly above a complexity thresh-old. These large fluctuations remind us of chaotic regimesin the El Farol and similar problems [18–21]. In them,agents with sufficient rationality anticipate a market, buttheir own success turns the market unpredictable. In ourmodel, above a complexity threshold, non-competitiveintervention choices become intertwined across games.Birth and death of agents ripple system-wide, makingsuccessful strategies hard to track. Even though agentsare exploring non-competitive strategies, large fluctua-tions ( ∼ −
20% population size) ensue, compromisingthe system’s viability – thus non-competitive actions canresult in negative spillover by sheer market complexity.This is another testable conclusion.Behavioral economics offers a prominent chance to testour findings. We see stable states with raising riggingpressure as expected returns grow. This is consistentwith empirical data on cheating : while different profilesexist (including people hardly corrupted), cheating even-tually ensues for large enough rewards [24], especiallyafter removing the concern of being caught [25]. Furtherexperiments reveal that cheating is more likely as a part-nership [26]. This resounds with our model’s “cartels” insimple yet wealthy economies. Such simple experimentsare perfect to test our predictions for growing complex-ity: Does coordination fall apart swiftly? Does riggingpressure grow with complexity in the long run? Moreambitiously, we could emulate recent implementations ofPrisoner dilemmas and other simple games [27–36].This work did not aim at specific realism, but atcapturing elements that we find essential about ‘rigged’economies, and thus derive qualitative regimes andwealth-complexity scalings that keep our toy economiesviable. Exploring lesser model parameters (Appendix C),the same phenomenology features consistently. This en-courages us to think that we are unveiling general resultsof ‘rigged’ economies. But we made important simplifi-cations to keep our model tractable. All agents partici-pate of all games, while real economic actors might walk out or be banned from a specific market. We model alldegrees of freedom with a similar game. Real manipula-tions might treat agents with a same strategy differently.Some pay off only the first intervention, others rewardnon-linearly a varying investment. Exploring these andother alternatives is easy and might uncover new systemicregimes. Our results constitute solid limit behaviors thatshould be recovered under appropriate circumstances.In our model, wealth is generated externally – the eco-nomic games merely distribute it. An important varia-tion should create wealth organically, depending on pop-ulation size, strategies explored, and degrees of freedomavailable. These, like technological niches, are devel-oped and sustained at a cost. Rigged economies mightthen correct themselves by losing complexity if neces-sary. Similar feedbacks can poise complex systems nearcritical regimes [37–44], which proved relevant to ratio-nalize some phenomenology in economics [21, 23, 45] –at criticality we observe fat tails in wealth distributionsor dynamic turnover of complex markets.An important design choice are the Darwinian dynam-ics that propagate successful strategies. We could havemodeled boundedly rational agents that learn, similarlyspreading successful behaviors. A key parameter thenwould be a learning rate, instead of our replication cost, C C . Similar models show that certain regimes dependtangentially on the cognitive mechanism [18–21]. Dif-ferent implementations might move around the onset ofunpredictable regimes (as C C does, Supporting Figure7). When unpredictability is intrinsic to the phenom-ena studied, rational agents cannot perform better ei-ther. Our results suggest that rigged economies mightbe intrinsically uncomputable in certain limits.Our work is designed in economic terms, but it has anobvious political reading – e.g. construction and con-trol of power structures. More pragmatically, in ourmodel wealth redistribution is achieved through low rig-ging pressures. Empirical measurements of redistributionmight help us map real economies into our framework, ashas been done for similarly abstract models [22, 23]. Atlarge, the evolutionary stability of fair governance [13] isunder scrutiny. In ecosystems that bring together wealth,people, and economic games, all subjected to Darwinism:What lasting structures emerge? Do fair rules survive?Under which circumstances does unfairness prevail? Acknowledgments
The author wishes to thank Roberto Enr´ıquez (BobPop) for his deep insights in socio-economic systems,which prompted this work. The author also wishesto thank Juan Fern´andez Gracia, V´ıctor Egu´ıluz, Car-los Meli´an, and other IFISC (Institute for Interdisci-plinary Physics and Complex Systems) members, as wellas V´ıctor Notivol, Paolo Barucca, David Wolpert, JustinGrana, and, very especially, Federico Curci for indispens-able feedback about economic systems. This work hasbeen funded by IFISC as part of the Mar´ıa de MaeztuProgram for Units of Excellence in R&D (MDM-2017-0711), and by the Spanish National Research Council(CSIC) and the Spanish Department for Science and In-novation (MICINN) through a Juan de la Cierva Fellow-ship (IJC2018-036694-I).
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Appendix A: Payoff matrices for simple cases
Let us note that the model is actually grounded ongame theory by building payoff matrices for simple sce-narios.Take one game ( n = 1) and a fixed population ofone player N ( t ) = 1 (i.e. even if the agent accumu-lates wealth, she does not have descendants, so she neverpays C C ; she is not removed either if she accumulatesnegative wealth). Supporting Table I shows the averagepayoff that a player earns if she plays the same game re-peatedly with fixed behavior (i.e. fixed strategy a andrigging choice r ). If there is no intervention ( r = 0),the winning rule R ( t ) is set randomly at each iterationand the expected payoff per round is b/
2. If the agent at-tempts to rig the game ( r = 1), she always succeeds (be-cause there is no opposition) and sets R ( t ) equals to itsown strategy ( R ( t ) = a ). Thus she ensures earning anaverage b per round, from which C R must be subtracted.The optimal strategy is to intervene if C R < b/ N ( t ) = 2) while, again, playing only one game.This case was discussed in the main text. Let us take acloser look. There are three scenarios worth consideringseparately, and each corresponds to a 2 × • No player attempts any rigging (upper-leftblock matrix in Table I). In this case the winningrule is set randomly, so that both players win halfof the time. If they play the same strategy, when-ever they win (i.e. half of the rounds), they mustsplit the earnings. If they play different strategies,each agent still wins half of the time but they al-ways get to keep all the earnings. In other words,in this case the model reduces to a minority game.If the winning rule behaves randomly because thereis no intervention, the preferable strategy is to stayin the minority. This is true also when there aremore players (see below), since the only varyingfactor that reduces a player’s earnings is the num-ber of others with a same strategy, among whomthe benefit is split. • Only one of the players attempts to rig thegame (either off-diagonal block matrices in TableI). The intervening agent pays C R to ensure thatthe winning strategy R ( t ) always matches her own.Assuming that only one agent (e.g. agent 1, thuslook at the top-right block matrix in Table I) isgiven the option to rig the game, doing it becomesalways favorable if C R < b/
4, disregarding of whataction agent 2 takes. If b/ < C R < b/
2, thenrigging the game is favorable only if agent 2 playsa different strategy. If C R > b/
2, it never becomesfavorable to rig the game. • Both agents attempt to rig the game (bottom-right block matrix in Table I). Both agents pay C R Player’s behavior a = 0, r = 0 a = 1, r = 0 a = 0, r = 1 a = 1, r = 1Payoff b/ b/ b − C R b − C R TABLE I
Payoff matrix for one game with one player.
The agent’s behavior is coded by two bits. A first one ( a )indicates the agent’s strategy (0 or 1). A second bit ( r ) indicates whether the agent attempts to rig the game or not. in this case. But they only intervene the game suc-cessfully if both play the same strategy. Note thatthis has the effect of turning the minority gameinto a neutral one regarding the agent’s strategies a i : If both players are attempting to intervene thegame, they will always receive the same payoff dis-regarding of whether a = a or not. The receivedpayoff is always less than the best scenario with nointervention. But, if C R < b/
4, it is better thanthe scenarios with no intervention and matchingstrategies.It becomes cumbersome to write payoff matrices whenmore players are involved, but it is still feasible for N ( t ) = 3. We do so in Supporting Tables II, III, andIV. In them, we group up the behaviors of players 2 and3, assuming that, whenever one of the three agents playsa different strategy (i.e. not all a i are the same), it isalways player 1 (either a = 0 and a , = 1 or a = 1and a , = 0). We call player 1 the minority player andplayers 2 and 3 the majority players.Supporting Table II shows the average payoff matrixwhen only the minority player is allowed to rig the game.If she is not meddling with the rules (left half of Support-ing Table II) we deal again with a minority game. TheNash equilibria of this subgame ( r i = 0 ∀ i and a (cid:54) = a , )are Nash equilibria of the whole game if C R > b/
4. Forcheaper cost of rigging, the global Nash equilibria disap-pear as it becomes favorable to one of the majority agentsto intervene (for which we have to look at SupportingTable III). This sets on a dynamic situation similar tothe one discussed in the main text. Finally, SupportingTable IV has both majority players attempting (and suc-ceeding, since they are in the majority) to rig the game.Interestingly, if all three agents are trying to manipulatethe winning rule (red frame), the model turns into a ma-jority game in which all three players earn b/ − C R . If anagent decided to change its strategy a i , this would puther in the minority, in which (according to SupportingTable IV), it would earn 0 per round – thus full coor-dination is a Nash equilibrium of the subgame in whicheverybody intervenes. This, however, is not a Nash equi-librium of the complete game: in full coordination, itwould pay off to a single agent to stop rigging the game.This suggests that the way that our model reaches largelevels of coordination (as discussed in the main text) is atragedy-of-the-commons scenario.In a static situation (i.e. population is fixed and agentsalways choose the same actions and whether to rig eachgame or not), games are independent of each other. Wecould take these payoff matrices and compute averages over many games. The situation becomes more difficultwhen dynamics are included. Because new agents can beborn and older ones may die, averages over time shouldkeep into account that agent’s actions feed back on eachother. For example, a possible good strategy for agent 1may be to rig games that favor a third agent (say, agent3) who, in turn, is rigging games that favor agent 1. Insuch a way, games can become coupled to each other andresult in much more complicated payoff functions. Appendix B: Supporting plots and discussion for increasingeconomy size and fixed complexity
Despite its simplicity, the model turned out to havevery rich dynamics. Its behavior changes, sometimesdrastically, with the economy complexity (as measuredby the number of games, n ) or with its size (as mea-sured by wealth distributed at each round, B ). In thisappendix we take a closer look to what happens whenthe number of games is fixed, but the wealth distributedin each game grows. We saw an example of this (with n = 2) in the main text, and we saw that increasing thenumber of resources drives agents to coordinate with eachother in different manners around the available strategiesand whether to rig them or not.Let us start with the simplest case now, with n = 1.We plot average population size in the steady state (Sup-porting Figure 1 a ), rigging pressure over the only game(Supporting figure 1 b ), and the strategy entropy (Sup-porting Figure 1 c ) as more resources become available.Circles over these plots show values of B for which weshow 1000 iterations of the dynamics in Supporting Fig-ure 13.As we saw for n = 2 in the main text, if there are veryfew resources, spending them in intervening the economyis not a favored behavior in the steady state. This impliesthat the winning strategy is randomly 0 or 1, likely chang-ing from one iteration to the next. As a consequence, thesteady population does not settle for either strategy. Thesecond row of Supporting Figure 13 a shows f k the frac-tion of agents playing 1 over time in game k = 1. Wesee that this number moves around 0 .
5, indicating thatroughly half the population is choosing 1 and the otherhalf is choosing 0. This results in a high strategy entropy (cid:104) h ak (cid:105) in game k = 1, as shown in the third row of Sup-porting Figure 13 a . The fourth row shows that, indeed,the rigging pressure is negligible for low values of B .Above some amount of available resources, an effectivelevel of rigging pressure starts to build up periodically(lower row of Supporting Figure 13 b-c ). Conceive a situ- Agent 1Agent 2, 3
TABLE II
Payoff matrix of one game with three players – only the minority player can rig.
We assume thatplayer 1 is in the minority when there is no consensus. In this table, only player 1 is allowed to rig the game, so she alwayssucceeds. Entries marked in gray are global Nash equilibria when rigging is very expensive C R >> b . Agent 1Agent 2 Agent 1Agent 3
TABLE III
Payoff matrix of one game with three players – only one of the majority players (player ) rigs. This is the only situation in which the symmetry between the majority players is broken. ation in which no rigging exists, but a mutation producesa single agent that decides to rig the game. She securesthe next rounds played, and all agents playing her samestrategy are consequently benefited. She will replicate,producing more agents that rig the game in the samedirection. But those parasitic agents playing her samestrategy without paying C R will earn more and producea slightly larger descent. Darwinian dynamics expandslight differences exponentially over time, thus eventu-ally the rigging agents are driven off to extinction. Someagents playing the other option might have survived –perhaps because they had some savings. After all riggingagents were removed from the population, those playingthe minority option will earn more money (as our analy-sis of payoff matrices indicates), and start making a comeback. Eventually, a mutation in their descendants mightproduce an agent rigging the game to favor the minor-ity. This would start a cycle all over again. SupportingFigure 13 b-c shows that this oscillating behavior takesplace for a range of economy sizes B .Vertical, dashed blue lines in this figure show the pointat which population reaches a maximum, which happensas the rigging pressure peaks as well. Interestingly, this isalso the point at which the proportion of agents playingeither strategy is well balanced. The population min-imum (indicated by vertical, dashed red lines) happenswhen the rigging pressure is minimal as well and the pop-ulation presents a more homogeneous strategy. Increas-ing the economy size results in longer alternating cyclesof this nature. Perhaps, hoarding more resources mightmake it more difficult to remove agents rigging the games in one direction over the other.Eventually, these cycles become infinitely long sothat most of the population ends up adopting a samestrategy (Supporting Figure 13 d ). This homogeneousstate supports a larger population than the cycling orfully random regimes. However, convergence to themajority is not full. A reservoir of agents playing theminority strategy survives, suggesting that these canoccasionally succeed in rigging the game and upset themajority.Regarding n = 2, in Supporting Figure 2 we reproducethe first three panels of Figure 2. Here we have addederror bars (which indicate the standard deviation of eachquantity over the last 500 iterations of each simulation)to give an idea of the variation that we can find. Er-ror bars are of similar relative magnitude in all examplesshown (Supporting Figure 1 for n = 1 and SupportingFigure 3 for n = 3, as well as Supporting Figures 4 and5 which compare several n ). Hence, we omitted errorbars anywhere else for clarity. Figure 2 and Support-ing Figure 2 both show a transition (similar to the oneobserved for n = 1) from unintervened games (sampledynamics are shown in Supporting Figure 14 a ), throughcycles of growing and declining coordination (Supportingfigure 14 b-c ), to more homogeneous states (SupportingFigure 14 d-g ).In the main text we indicated that this cyclic regime(which we called a shift towards within-game coordina-tion) is transited simultaneously for both games. Thereare some nuances, though. Supporting Figures 14 b-c Agent 1Agent 2
TABLE IV
Payoff matrix of one game with three players – both majority players rig the game.
Since they are inthe majority, they always succeed in their attempt to set the winning rule. If all three players rig the game simultaneously (redframe), the model turns into a majority game – i.e. the best strategy is to play what everybody else is playing. Gray squaresindicate Nash equilibria of this sub-game. a b c
SUP. FIG. 1
Fixed economy complexity, n = 1 , and growing distributed wealth, B . a Average population size in thesteady state. b Rigging pressure over the only game in the steady state. c Strategy entropy in the only game in the steadystate. Circles over the plots curves indicate values of B for which we show samples of the dynamics in Supporting Figure 13. show that for both cases the amount of rigging in bothgames is very similar (bottom row). But (cid:104) h ak (cid:105) reveals im-portant asymmetries which, furthermore, change as weincrease B . For the lowest B value shown with cyclicbehavior ( B = 500, Supporting Figures 14 b ), in aver-age, the population does not converge on persistent ho-mogeneous strategies for neither of the games. But itdoes not stay divided randomly either (as it happens for B = 300, Supporting Figure 14 a ). In the second examplewith oscillating behavior ( B = 700, Supporting Figure14 c ), in average, the population has converged regardingthe strategy of one of the games. The dynamics move to-wards converge for the other game as well, but they failperiodically or, if they succeed, then the strategy for theother game (formerly homogeneous) breaks apart. Sum-ming up: while the level of intervention is similar in bothgames, this symmetry is broken regarding how homoge-neous the population is about each strategy.After the cyclic behavior, Figure 2 and SupportingFigure 2 still show two more regimes separated each bya large boost in stable population size. As noted in themain text, this last regime shift is not accompanied bylarge changes in action entropy, (cid:104) h ak (cid:105) , of neither game.Hence, we conclude that within-game coordination hasbeen exhausted and that new kinds of correlations, nowacross games, are taking place. Supporting Figure 14 d-g are samples of the dynamics in those two regimes afterthe cyclic behavior. Panels 14 d-e sample the regimebetween B ∼
800 and 1600, and panels 14 f-g sample the regime for
B > (cid:104) h ak (cid:105) as well, which remains lowthroughout. We appreciate the population boost alreadydiscussed in the main text (average population in thetop panels of Supporting Figure 14 d-e is lower than inSupporting Figure 14 f-g ). We also appreciate that thefluctuations in population is larger when the populationis smaller. The other significant difference between thesetwo last regimes is that the rigging pressure becomes no-tably higher in the regime with larger B . This suggeststhat the difference between both regimes lies in a moreefficient coordination between the rigging efforts acrossgames, allowing the population to extract more wealthin average. For example, we see that both games sustaina minority of agents playing the minority option even ifthe population has broadly converged about each game’sstrategy. But these minority-playing agents might notbe the same in both games if the conditions allow it. Atransition to a higher across-game coordination mighthappen if the agents playing the minority in both gamesbecome the same.Finally, for n = 3 too, we show average populationsize in the steady state (Supporting Figure 3 a ), riggingpressure over each game (Supporting Figure 3 b ), and ac- a b c SUP. FIG. 2
Fixed economy complexity, n = 2 , and growing distributed wealth, B . Average population size in thesteady state. b Average rigging pressure per game. c Average strategy entropy over the two games. Circles over the plotscurves indicate values of B for which we show samples of the dynamics in Supporting Figure 14. Error bars indicate thestandard deviation over the last 500 iterations of the corresponding simulation. a b c SUP. FIG. 3
Fixed economy complexity, n = 3 , and growing distributed wealth, B . Average population size in thesteady state. b Average rigging pressure per game. c Average strategy entropy over the three games. Circles over the plotscurves indicate values of B for which we show samples of the dynamics in Supporting Figure 15. tion entropy (Supporting Figure 3 c ) as more resourcesbecome available (i.e. as B grows). We observe moreregime shifts (as identified by boosts in population size)than for n = 2, which is compatible with more availablegames and more possibilities for across-game coordina-tion.Supporting Figure 15 shows samples of the dynamicsfor various of these regimes, including the oscillatoryregime. We see again that the rigging pressure overall three games is simultaneous, even if the symmetryregarding strategy coordination is broken. In the exam-ple shown we see that, in average, the population hasconverged regarding the strategies of two out of threegames. As for n = 2, when the population convergesalso about the third game, one of the former agreementsbreaks apart. There are also values of B for whichthere is convergence of, at most, one game in average(not shown). Again, the extra regime shifts (whichwe identify by abrupt boosts of (cid:104) N (cid:105) ) happen afterwithin-game coordination has been exhausted. This isconsistent with the idea that more games bring in morepossible across-game coordination, which are exploredonly for large values of B .As briefly discussed in the main text, adding more games has two different effects: One the one hand moreregimes seem to become available (as duly noted); andon the other hand, more consecutive regimes seem to bevisited within a smaller range of B . This means that, aswe increase B , regimes progress more rapidly into each-other (Supporting Figure 4). This effect is exaggerated ifeven more games are available (Supporting Figure 5). Somuch so that, instead of regime shifts, we approximatea continuous progression. The increase in rigging pres-sure per game becomes parsimonious as well (while for asmall number of games it presented some discrete boostsassociated to regime shifts).Supporting Figure 4 c shows that exhausting thewithin-game coordination results in a drop of strategyentropy, (cid:104) h a (cid:105) . We see that this drop becomes less ac-centuated for larger n (Supporting Figure 5 c ). This sug-gests that the onset of interactions across games some-how thwarts within game coordination. In other words, amore complex economy seems to enable populations withmore diverse strategies within single games. a b c SUP. FIG. 4
Fixed economy complexity, n , and growing distributed wealth, B – comparison of cases with smallcomplexity. a Average population size in the steady state. Horizontal dashed lines help us identify cases which, despite havingdifferent n and B , reach similar stable population size. This suggests that some scenarios might be essentially equivalent. b Average rigging pressure per game. c Average strategy entropy over the n games in each case. a b c SUP. FIG. 5
Fixed economy complexity, n , and growing distributed wealth, B – comparison of cases with largecomplexity. a Average population size in the steady state. Boost in stable population size are smoothed into a continuousbuildup as complexity increases. b Average rigging pressure per game. c Average strategy entropy over the n games in eachcase. Appendix C: Robustness of results against variations ofmodel parameters
The maps that we develop allow us to chart our modeleasily. Similarly to the maps built for (cid:104) N (cid:105) and σ ( N ) / (cid:104) N (cid:105) in Figure 4 of the main text, it is possible to build mapsfor other quantities such as the average strategy entropy, (cid:104) h a (cid:105) (Supporting Figure 6 a ); or the rigging pressure pergame, (cid:104) r (cid:105) /n (Supporting Figure 6 b ). Such maps canhelp us reveal regimes and phenomenology in arbitrarymeasurements but: How general are these phenomena?Do they depend critically on model parameters?The model has 6 parameters: One sets the wealth dis-tributed by the economy in each round ( b , B , and B are univocally linked in each case); another one sets theavailable number of degrees of freedom and thus the econ-omy’s complexity (i.e. the number of games, n ); two pa-rameters set up costs (of attempting to rig a game, C R ;and of producing descent, C C ); a mutation rate p µ ; andthe parameter N max that sets an external upper limitto the population (this acts similarly to a load capac-ity in ecological models). We designed our model withthe hope of pinning down essential features of riggedeconomies. We hope that the elements involved in themodel introduce as few additional effects as possible. Inthat sense, an abundance of parameters is not desired.Furthermore, we hoped that the most interesting phe- nomenology would depend on B and n . These parame-ters capture respectively the economy’s size (as measuredby distributed wealth) and complexity, which are at thecenter of our research questions. Luckily, as we show inthis appendix, the observed phenomenology is not muchaltered when toying with the remaining parameters. Thissuggests that the regimes and phenomenology discoveredfor varying economy size and complexity should be foundover again for a range of model options – which speaksstrongly in favor of the minimalism of our approach.First we note that the cost of rigging a game C R setsa scale with respect to the wealth allotted to each game b = B/n . In our simulations we set C R = 1. If wewould try a different value of C R , we could normalize˜ b ≡ b/C R , ˜ C C ≡ C C /C R , and ˜ C R ≡ C R /C R = 1 andmap the parameter choice back to a case that we havealready studied. Thus actually our model has only 5 freeparameters.Supporting Figure 7 shows what happens to (cid:104) N (cid:105) and σ ( N ) / (cid:104) N (cid:105) as C C changes. Interestingly, the effect in (cid:104) N (cid:105) seems negligible for the values explored (Figure Sup-porting Figure 7 a-c ). More notably, this parameter hasthe effect of displacing the onset of the large-fluctuationsregime (Supporting Figure 7 d-f ). If C C is smaller, thisregime ensues for a lower economy complexity, n . The pa-rameter C C tells us how cheap it is to have descendants. b a SUP. FIG. 6
Comprehensive maps of strategy entropyand rigging pressure. a
Average strategy entropy, (cid:104) h a ( t ) (cid:105) ,shows a dent for small n and large B – a situation thatwe compared to cartels in the main text. This coordinationregime falls apart swiftly as complexity grows a little bit. If wemove to very large n without increasing B , coordination startsto build up again – yet very smoothly. b Rigging pressure pergame, (cid:104) r ( t ) /n (cid:105) . Unsurprisingly, it grows with the amount ofwealth distributed. More interestingly, it also grows with thenumber of degrees of freedom in the system. When it is cheaper, it is easier to trigger large fluctu-ations; probably because a large descent explores morebehaviors (both in rigging decisions and game strategies)simultaneously, as well as it displaces a bigger propor-tion of former agents. Both these actions result in majordisruptions of the winning rules. Thus, cheaper descentmore easily brings up a scenario in which agents are con-tinuously deceiving each other into bankruptcy. If this iscorrect, other parameters that promote behavior diver-sity or population renewal among agents should have asimilar effect.Supporting Figure 8 shows, indeed, that the mutationrate p µ prompts this expected outcome too. As for C C , (cid:104) N (cid:105) is mostly unaffected by variations of p µ (not shown),but increasing p µ has the predicted result of advancingthe onset of the large-fluctuations regime. In SupportingFigure 8 we show a range of p µ with C C = 20. This choiceof C C is different from the value taken in simulationsin the main text ( C C = 10) to show that, with a largeenough p µ , we can advance the onset of large fluctuationsto the point where it was with C C = 10 and p µ = 0 . C C on riggingpressure and rigging pressure per game. The effectis mostly uninteresting, as it just smooths or slightlydisplaces a map similar to the settings commented in the main text (Supporting Figure 6). The key outcomeis that rigging pressure grows both if more resources, B , and more games, n , are available – as discussed inthe main text. A similar structure is found for differentvalues of p µ (not shown). The strategy entropy (notshown) is largely unchanged by C C and p F – while largeenough p F (i.e. very large noise) has the expected effectof weakening the convergence to a majority strategy forlow n and large B (arrows in Supporting Figure 6 a ).The model is a bit more sensitive to the parameter N max that sets an external maximum size to the pop-ulation. This parameter makes sense as population sizemight be constrained by actual physical limits – e.g., theamount of people that can occupy a territory. It couldalso be seen as a manufactured (‘rigged’) limit to the sizeof the market. This is an important kind of economic ma-nipulation not studied in the model. Even if we were tolook at N max from this perspective, model agents cannotmodify it, so we would not be studying such manipula-tion organically, as we do with other degrees of freedom.A more technical reason to set a parameter N max isthat it solves parsimoniously the problem of a maximumaverage lifespan. In the limit N max → ∞ , the first agent(who does not rig any game, hence does not pay C R )never dies because it never ends up with negative wealth.The same would happen to any descent that does not rigany games. This is unrealistic and undesired. In thecurrent model, agents that last too long are naturallyand randomly replaced by newborns thanks to the finite N max . If we would set an infinite N max , we would needto introduce other mechanisms to remove unrealisticallylong-lasting agents – e.g. an average life-time or remov-ing agents with a wealth below a threshold w − >
0. Itwould be interesting to try these and other variations inthe future – noting that they introduce new parametersnevertheless.Ideally, we would like to find intrinsic limits to popu-lation size that emerge out of the model dynamics alone.To achieve this, we would need to simulate the modelin the large N max limit. The value chosen to reportour results ( N max = 1000) is a compromise between afairly large maximum population size and an ability torun simulations within a reasonable time. We ran oneadditional simulation for N max = 5000 and another onefor N max = 10000 (reported next). Each of these tookmore than ten days in a fairly powerful computer cluster.Supporting Figure 10 shows (cid:104) N (cid:105) for N max =100 , , , n (around n ∼ B (around B ∈ [1000 , N max . Itmight be interesting to look at this, but it is not so rele-vant for our discussion. We see that the region associatedto the large-fluctuations regime for large n , one of themost salient features of the model, is not very much af-fected by N max . Average population sizes in the steadystate in this regime (Figure 4 a ) as well as its fluctuations(Figure 4 b-c ) fall well below the chosen N max = 1000. a b c d e f SUP. FIG. 7
Comprehensive maps of (cid:104) N (cid:105) and σ ( N ) / (cid:104) N (cid:105) for varying replication cost, C C . Maps result fromsimulating case I (i.e. B I = B , b I = B /n ) for ranges of economic complexity, n , and distributed wealth, B . Black curvesrepresent trajectories B = B ( B = 50 , n ) for cases II , III , and IV (dashed, dotted, and dot-dashed respectively). Horizontalred lines at the bottom of each map represent case I with B = 50. a-c shows (cid:104) N (cid:105) and d-f show σ ( N ) / (cid:104) N (cid:105) . a , d , C C = 5; b , e , C C = 10; c , f , C C = 20. a b cd e f SUP. FIG. 8
Comprehensive maps of σ ( N ) / (cid:104) N (cid:105) for varying mutation, p µ . Maps result from simulating case I (i.e. B I = B , b I = B /n ) for ranges of economic complexity, n , and distributed wealth, B . Black curves represent trajectories B = B ( B = 50 , n ) for cases II , III , and IV (dashed, dotted, and dot-dashed respectively). Horizontal red lines at the bottomof each map represent case I with B = 50. a , p µ = 0 . b , p µ = 0 . c , p µ = 0 . d , p µ = 0 . e , p µ = 0 . f , p µ = 0 . All this suggests that the reported (cid:104) N (cid:105) are intrinsic limitsemerging from the model.The onset of the large-fluctuations regime, however, ismore affected (Supporting Figure 11). Increasing N max results in a displaced onset of this regime. It happens fornotably larger values of n and B for increasing N max .Within the plotted range, however, we still appreciatefluctuations as large as 25% of population size for some( n, B ) combinations. Note that for N max = 10000 the maximum population size is around two orders of magni-tude bigger than the stable population size ( ∼
100 forthe affected area). This strongly suggests that large-fluctuations are an intrinsic regime of the model, evenif its precise location in the map is affected by N max .Nevertheless, it is affected by N max , which indicatesthat this phenomenon is enhanced by having a finite,maximum population size. Let us compare the shift ofthis onset with N max to the shifts observed when we var- a b c SUP. FIG. 9
Comprehensive maps of rigging pressure per game for varying replication cost, C C . Maps resultfrom simulating case I (i.e. B I = B , b I = B /n ) for ranges of economic complexity, n , and distributed wealth, B . Blackcurves represent trajectories B = B ( B = 50 , n ) for cases II , III , and IV (dashed, dotted, and dot-dashed respectively).Horizontal red lines at the bottom of each map represent case I with B = 50. a , C C = 5; b , C C = 10; c , C C = 20. a b c d SUP. FIG. 10
Comprehensive maps of (cid:104) N (cid:105) for varying maximum population size, N max . Maps result from simulatingcase I (i.e. B I = B , b I = B /n ) for ranges of economic complexity, n , and distributed wealth, B . Black curves representtrajectories B = B ( B = 50 , n ) for cases II , III , and IV (dashed, dotted, and dot-dashed respectively). Horizontal red lines atthe bottom of each map represent case I with B = 50. a , N max = 100; b , N max = 1000; c , N max = 5000; d , N max = 10000. ied C C (Supporting Figure 7) and p µ (Supporting Figure8). About these, we argued that the onset of the regimewas advanced by mechanisms that result in more diversestrategies competing closer together or a higher popula-tion turnover. Thus, lower C C (cheaper reproduction)and higher p µ (increased mutation) both advanced theonset of the regime because an array of diverse strate-gies is promptly forced to compete, potentially alteringthe winning rules in an unpredicted fashion. A larger N max has the effect of diluting this competence becausethere is less replacement of older agents. Accordingly,higher N max displaces the large-fluctuations regime tolarger values of n and B . Oppositely, note that the pop-ulation renewal introduced by smaller N max results in a higher uncertainty about winning strategies. This isconsistent, as discussed in the main text, with an onsetof large fluctuations associated to a cognitive transitionof the population as a whole: at some point, it becomescognitively impossible to keep track of winning strategiesas agents attempt to deceive each other.Another sensible area of the map is that with large B and small n , which showed the transition to within-gamecoordination that results in most of the populationconverging to a same strategy for each game. Sup-porting Figure 10 shows that for such combination ofparameters (large B and small n ), there are enoughresources to sustain a very large average population size,often saturating even for large N max values. Supporting0 a b c d SUP. FIG. 11
Comprehensive maps of σ ( N ) / (cid:104) N (cid:105) for varying maximum population size, N max . Maps result fromsimulating case I (i.e. B I = B , b I = B /n ) for ranges of economic complexity, n , and distributed wealth, B . Black curvesrepresent trajectories B = B ( B = 50 , n ) for cases II , III , and IV (dashed, dotted, and dot-dashed respectively). Horizontalred lines at the bottom of each map represent case I with B = 50. a , N max = 100; b , N max = 1000; c , N max = 5000; d , N max = 10000. a bc d SUP. FIG. 12
Comprehensive maps of strategy entropy, (cid:104) h a (cid:105) for varying maximum population size, N max . Mapsresult from simulating case I (i.e. B I = B , b I = B /n ) for ranges of economic complexity, n , and distributed wealth, B .Black curves represent trajectories B = B ( B = 50 , n ) for cases II , III , and IV (dashed, dotted, and dot-dashed respectively).Horizontal red lines at the bottom of each map represent case I with B = 50. a , N max = 100; b , N max = 1000; c , N max = 5000; d , N max = 10000. Figure 12 shows that this homogeneous regime remainspresent as we increase N max . Some isolated cases in N max = 10000 (Supporting Figure 12 d ) and, morenotably, N max = 5000 (Supporting Figure 12 c ) show up. These cases appear in the plots as outstanding pixelsof huge (cid:104) h a (cid:105) in their otherwise smoother neighborhoodwith lower entropy (further supporting that these areoddballs).1All in all, the results summarized in this appendixstrongly suggest that the relevant phenomenology of themodel is the one reported in the main text, and that thisphenomenology is robust against reasonable variations ofall parameters. Furthermore, changes on the onset of thisphenomenology as we vary these extra parameters areparsimonious and follow logical explanations. All this, once again, is a strong reassurance of the minimalism ofthe model. This phenomenology likely underlies morecomplicated models that could study additional effectssuch as those discussed in the final section of the pa-per. We would expect to find at least similar regimesand regime shifts to the ones described here, even if theexact numerical values at which they happen are altered.2 . . . . . . .
201 0 . . . T i m e . . .
75 1 0 . . . . . . .
20 0 200012001400160018001000 T i m e . . . . . . . . . .
20 200012001400160018001000 T i m e . . . . . . .
20 00 . . .
15 200012001400160018001000 T i m e Population size Preferred action Action entropy Presure per game a b c d S U P . F I G . S a m p l e s o f t h e m o d e l d y n a m i c s f o r fi x e d e c o n o m y c o m p l e x i t y ( n = ) a ndd i ff e r e n t e c o n o m y s i z e s . T h e m o d e l d y n a m i c s o v e r i t e r a t i o n s a r e s h o w n f o r o n e ga m e a nd e c o n o m y s i z e s B = ( a ) , B = ( b ) , B = ( c ) , a nd B = ( d ) . P a n e l s i n t h e t o p r o w s h o w t h e p o pu l a t i o n s i z e o v e r t i m e , s e c o nd r o w s h o w s t h e f r a c t i o n f ( t ) o f ag e n t s p l a y i n g s t r a t e g y , t h i r d r o w s h o w s t h e a c t i o n e n t r o p y ( h a = − f · l o g ( f ) − ( − f ) · l og ( − f )) w h i c h i s m a x i m a l w h e n t h e ag e n t s d i s ag r ee m a x i m a ll y o v e r t h e i r s t r a t e g i e s r e ga r d i n g t h e o n e ga m e , b o tt o m r o w s h o w s t h e r i gg i n g p r e ss u r e o v e r t i m e . .
20 00 . . .
15 20001200140016001800 T i m e .
20 00 . . .
15 20001200140016001800 T i m e .
20 00 . . .
15 20001200140016001800 T i m e .
20 00 . . .
15 20001200140016001800 T i m e .
20 00 . . .
15 20001200140016001800 T i m e .
20 0 . . .
15 20001200140016001800 T i m e a b c d e f g . . .
41 00 . . . . . .
41 00 . . . . . .
41 00 . . . . . .
41 00 . . . . . .
41 00 . . . . . .
41 00 . . . .
20 00 . . .
15 100020001200140016001800 T i m e . . .
41 00 . . . Population size Preferred action Action entropy Presure per game S U P . F I G . S a m p l e s o f t h e m o d e l d y n a m i c s f o r fi x e d e c o n o m y c o m p l e x i t y ( n = ) a ndd i ff e r e n t e c o n o m y s i z e s . T h e m o d e l d y n a m i c s o v e r i t e r a t i o n s a r e s h o w n f o r o n e ga m e a nd e c o n o m y s i z e s B = ( a ) , B = ( b ) , B = ( c ) , B = ( d ) , B = ( e ) , B = ( f ) , a nd B = ( g ) . P a n e l s i n t h e t o p r o w s h o w t h e p o pu l a t i o n s i z e o v e r t i m e , s e c o nd r o w s h o w s t h e f r a c t i o n f k ( t ) o f ag e n t s p l a y i n g s t r a t e g y i n t h e k - t h ga m e , t h i r d r o w s h o w s t h e c o rr e s p o nd i n g a c t i o n e n t r o p y ( h a = − f k · l o g ( f k ) − ( − f k ) · l og ( − f k )) w h i c h i s m a x i m a l w h e n t h e ag e n t s d i s ag r ee m a x i m a ll y o v e r t h e i r s t r a t e g i e s r e ga r d i n g t h e o n e ga m e , b o tt o m r o w s h o w s t h e r i gg i n g p r e ss u r e o v e r t i m e . . . .
75 1 0 . . . . . .
75 1 0 . . . . . .
75 1 0 . . . .
20 00 . . . . . .
75 1 0 . . . .
20 00 . . . .
20 00 . . .
15 20001200140016001800 T i m e .
20 00 . . .
15 2000120014001600180010002000120014001600180010002000120014001600180010001000 800850900950 1 00 . . .
75 1 0 . . . .
20 00 . . .
15 20001200140016001800 T i m e a b c d e Population size Preferred action Action entropy Presure per game S U P . F I G . S a m p l e s o f t h e m o d e l d y n a m i c s f o r fi x e d e c o n o m y c o m p l e x i t y ( n = ) a ndd i ff e r e n t e c o n o m y s i z e s . T h e m o d e l d y n a m i c s o v e r i t e r a t i o n s a r e s h o w n f o r o n e ga m e a nd e c o n o m y s i z e s B = ( a ) , B = ( b ) , B = ( c ) , B = ( d ) , a nd B = ( e ) . P a n e l s i n t h e t o p r o w s h o w t h e p o pu l a t i o n s i z e o v e r t i m e , s e c o nd r o w s h o w s t h e f r a c t i o n f k ( t ) o f ag e n t s p l a y i n g s t r a t e g y i n t h e k - t h ga m e , t h i r d r o w s h o w s t h e c o rr e s p o nd i n ga c t i o n e n t r o p y ( h a = − f k · l o g ( f k ) − ( − f k ) · l og ( − f k )) w h i c h i s m a x i m a l w h e n t h e ag e n t s d i s ag r ee m a x i m a ll y o v e r t h e i r s t r a t e g i e s r e ga r d i n g t h e o n e ga m e , b o tt o m r o w s h o w s t h e r i gg i n g p r e ss u r ee
15 20001200140016001800 T i m e a b c d e Population size Preferred action Action entropy Presure per game S U P . F I G . S a m p l e s o f t h e m o d e l d y n a m i c s f o r fi x e d e c o n o m y c o m p l e x i t y ( n = ) a ndd i ff e r e n t e c o n o m y s i z e s . T h e m o d e l d y n a m i c s o v e r i t e r a t i o n s a r e s h o w n f o r o n e ga m e a nd e c o n o m y s i z e s B = ( a ) , B = ( b ) , B = ( c ) , B = ( d ) , a nd B = ( e ) . P a n e l s i n t h e t o p r o w s h o w t h e p o pu l a t i o n s i z e o v e r t i m e , s e c o nd r o w s h o w s t h e f r a c t i o n f k ( t ) o f ag e n t s p l a y i n g s t r a t e g y i n t h e k - t h ga m e , t h i r d r o w s h o w s t h e c o rr e s p o nd i n ga c t i o n e n t r o p y ( h a = − f k · l o g ( f k ) − ( − f k ) · l og ( − f k )) w h i c h i s m a x i m a l w h e n t h e ag e n t s d i s ag r ee m a x i m a ll y o v e r t h e i r s t r a t e g i e s r e ga r d i n g t h e o n e ga m e , b o tt o m r o w s h o w s t h e r i gg i n g p r e ss u r ee o v ee
15 20001200140016001800 T i m e a b c d e Population size Preferred action Action entropy Presure per game S U P . F I G . S a m p l e s o f t h e m o d e l d y n a m i c s f o r fi x e d e c o n o m y c o m p l e x i t y ( n = ) a ndd i ff e r e n t e c o n o m y s i z e s . T h e m o d e l d y n a m i c s o v e r i t e r a t i o n s a r e s h o w n f o r o n e ga m e a nd e c o n o m y s i z e s B = ( a ) , B = ( b ) , B = ( c ) , B = ( d ) , a nd B = ( e ) . P a n e l s i n t h e t o p r o w s h o w t h e p o pu l a t i o n s i z e o v e r t i m e , s e c o nd r o w s h o w s t h e f r a c t i o n f k ( t ) o f ag e n t s p l a y i n g s t r a t e g y i n t h e k - t h ga m e , t h i r d r o w s h o w s t h e c o rr e s p o nd i n ga c t i o n e n t r o p y ( h a = − f k · l o g ( f k ) − ( − f k ) · l og ( − f k )) w h i c h i s m a x i m a l w h e n t h e ag e n t s d i s ag r ee m a x i m a ll y o v e r t h e i r s t r a t e g i e s r e ga r d i n g t h e o n e ga m e , b o tt o m r o w s h o w s t h e r i gg i n g p r e ss u r ee o v ee r t i m ee