Persistence of discrimination: revisiting Axtell, Epstein and Young
aa r X i v : . [ phy s i c s . s o c - ph ] J un Persistence of discrimination : revisitingAxtell, Epstein and Young.
Gérard WeisbuchEcole normale superieure, 24, rue Lhomond, Paris, FranceLaboratoire de physique statistique, Département de physique de l’ENS,École normale supérieure, PSL Research University, Université Paris Diderot,Sorbonne Paris Cité, Sorbonne Universités, UPMC Univ. Paris06, CNRS, 75005 Paris, France24 juillet 2018 email : [email protected] : Socio-Physics ; Social Cognition ; Discrimination ; Dynamics ;Attraction basins. AbstractWe reformulate an earlier model of the "Emergence of classes..." proposedby Axtell et al. (2001) using more elaborate cognitive processes allowing astatistical physics approach. The thorough analysis of the phase space andof the basins of attraction leads to a reconsideration of the previous socialinterpretations : our model predicts the reinforcement of discriminationbiases and their long term stability rather than the emergence of classes.
During the 90’s social scientists introduced several thought provocativemodels of social phenomena, most often using numerical simulations (multi-agent simulations). These models have later been extended by methods andconcepts derived from statistical physics such as Master Equations and MeanField Approximation. A few examples include voters models and imitationprocesses Nowak et al. (1990) and the review of Castellano et al. (2009) , ElFarol and the minority game (Arthur, 1994) and Challet et al. (2013), dif-fusion of cultures (Axelrod, 1997) and (Castellano et al. , 2000). Revisiting1hese models provided deeper insight, more precise results and even some-times corrections.The questions of the emergence and persistence of classes and discrimi-nation received a lot of attention from social scientists, ethnographers andeconomists, see e.g. (Bowles & Naidu, 2006) and references within. A veryinspiring model entitled "Emergence of Classes in a Multi-Agent BargainingModel" was proposed by Axtell, Epstein and Young (Axtell et al. (2001)).We here propose to revisit their approach using a more elaborate model ofagent cognition and to compare a mean field approach to our agent basedsimulation results.
Let us briefly recall the original hypotheses and the main results of Axtell,Epstein and Young (Axtell et al. (2001)).— Framework : pairs of agents play a bargaining game introduced byNash Jr (1950) and Young (1993). During sessions of the game, eachagent can, independently of his opponent, request one among threedemands : L(ow) demand 30 perc. of a pie, M(edium) 50 perc. andH(igh) 70 perc. As a result, the two agents get at the end of the sessionwhat they demanded when the sum of both demands is less than the100 perc. total ; otherwise they don’t get anything. The correspondingpayoff matrix is written table (1). At each step a random pair of agentsis selected to play the bargaining game. The iterated game is playedfor a large number of sessions, much larger that the total number ofagents which could then learn from their experience how to improvetheir performance.— Learning and memory : Agents keep records of the previous demandsof their opponents, e.g. for 10 previous moves.— Choosing the next move : at each time step, pairs of agent are ran-domly selected to play the bargaining game. They most often choosethe move that optimises their expected payoff using the memory ofprevious encounters as a surrogate for the actual probability distribu-tion of their opponent’s next moves. With a small probability ǫ , e.g.0.1, they choose randomly among L, M, H.The main results obtained by Axtell et al. (2001) from numerical simula-tions are :— They observe different transient configurations which they interpret2s "norms", e.g. the equity norm is observed when all agents play M.Because of the constant probability of random noise, the system neverstabilises on an attractor, even in the sense of Statistical Physics. Theduration of the transients increases exponentially with the memorysize and /ǫ .— Their most fascinating result is obtained when agents are dividedinto two populations with arbitrary tags say e.g. one red and oneblue. When agents take into account tags for playing and memorisinggames (in other words when agents play separately two games, oneintra-game against agents with the same tag and another inter-gameagainst agents with a different tag) one observes configurations in theinter-game such that one population always play H while the other po-pulation plays L ; they interpret such inequity norm as the emergenceof classes, the H playing population being the upper class.Equivalent results are obtained when agents are connected via a socialnetwork as observed by (Poza et al. , 2011) on a square lattice as opposed tothe full connection structure used by (Axtell et al. , 2001). For some instances,domains with different norms occupy different parts of the lattice. Otherwise,one single domain of agents playing the same norm covers the entire lattice,depending upon the initial conditions.From now on, we follow a plan starting with the exposition of our ownmodel (section 2.2). The use of a mean field approximation allows to simplydescribe the attractors of the dynamics and the different dynamical regimes(section 3). These results are then compared with those obtained by directagent based simulations (section 4), including a thorough survey of the at-traction basins. We further proceed with the analysis of the two tagged po-pulations version (section 5). The discussion compares our results to those ofprevious models and to magnetic systems. A short conclusion stresses the dif-ference in interpretation of the models in terms of social phenomena (section6). We start from the same bargaining game as (Axtell et al. , 2001) with apayoff matrix written in table (1), but using different coding of past expe-rience (moving average of past profits) and choice function (Boltzman func-tion). 3 M HL 0.3 0.3 0.3M 0.5 0.5 0H 0.7 0 0
Table H . The first row representsthe move of her opponent. The figures in the matrix represent the payoffobtained by the first player.The present model is derived from standard models of reinforcement lear-ning in cognitive science, see for instance Weisbuch et al. (2000).Rather than memorising a full sequence of previous games, agents update3 "preference coefficients" J j for each possible move j , based on a movingaverage of the profits they made in the past when playing j . J is the prefe-rence coefficient for playing H , J for M and J for L . The updating processfollowing time interval τ after a transaction is : J j ( t + τ ) = (1 − γ ) · J j ( t ) + π j ( t ) , ∀ j, (1)The decrease term in − γ corresponds to discounting the importance ofpast transactions, which makes sense in an environment varying with thechoices of the other players. π j ( t ) is the actual profit made during the chosentransaction j ; the 2 other J j ′ corresponding to the 2 other choices j ′ aresimply decreased.These preference coefficients are then used to choose the next move in thebargaining game. Agents face an exploitation/exploration dilemma : they candecide to exploit the information they earlier gathered by choosing the movewith the highest preference coefficient or check possible evolutions of profitsby trying randomly other moves. Rather than using a constant rate of randomexploration ǫ as in Axtell et al. (2001), the probability of choosing demand j is based on the logit function : P j = exp( βJ j ) P j exp( βJ j ) , ∀ j, (2)where β , the discrimination rate, measures the non-linearity of the relation-ship between the probability P j and the preference coefficient J j . Large β values results in always playing the choice j with the largest J j , small β va-lues to indifference among the three choices. Economists use the name logitfor the Boltzmann distribution. We have earlier shown Nadal et al. (1998)4hat the Boltzmann distribution can be derived by maximising a linear com-bination of expected profits and information gained through exploration, see(Bouchaud, 2013) for a thorough discussion.Comparing our model with the one proposed by (Axtell et al. , 2001) :— The moving average corresponds to a gradual rather than abrupt de-crease of previous memories, it is based on agent’s own experience interms of profit rather than the observation of her opponents’ movesand it uses less memory.— Boltzman choice has a random character as the constant probabilitynoise introduced in (Axtell et al. , 2001), but furthermore the choicedepends upon the differences in experienced profits ; we might expectagents to be less hesitant when their previous experience resulted invery different preference coefficients. The difference equation (1) can be changed to a differential equation inthe limit of a slow dynamics : dJ j dt = − γJ j ( t ) + π j ( t ) (3)where the time unit is the average time between the agent’s bargaining pro-cesses. π j ( t ) is the profit made by the agent if he chose demand j .The Mean Field approximation consists in replacing π j ( t ) by its expectedvalue < π j > , thereby transforming the stochastic differential equation intoa deterministic differential equation.The time evolution of J ij is thus approximated by the following set ofequations : dJ j dt = − γJ j + < π j > (4)where < π j > is given by : < π j > = P i π ij exp( βJ i ) P i exp( βJ i ) ; (5) j is the agent’s move, i are the 3 possible moves of her opponent, and the π ij (0, 0.3, 0.5, 0.7) are the coefficients of the pay-off matrix. The mean fieldapproximation neglects fluctuations among agents representations, their J j .5ence agent j evaluates the probability of her opponent’s moves accordingto her own estimations, using Boltzman functions of her own J .Using statistical physics notation Z : Z = X i exp( βJ i ) (6)the internal representation of the agent is thus vector ( J , J , J ) which com-ponents obey dynamics : dJ dt = − γJ + ( exp ( βJ ) ∗ . ∗ exp ( βJ )) /Z/Z (7) dJ dt = − γJ + ( exp ( βJ ) ∗ . exp ( βJ ) + exp ( βJ )) /Z/Z (8) dJ dt = − γJ + exp ( βJ ) ∗ . /Z (9)Taking the exponentials as new variables simplifies expressions (7-9) andallows to deduce scaling properties. Let : x = exp ( βJ ) , J = log ( x ) /β (10) y = exp ( βJ ) , J = log ( y ) /β (11) z = exp ( βJ ) , J = log ( z ) /β (12)The new equations are : dxβdt = x ( − αlog ( x ) + 0 . xzs ) (13) dyβdt = y ( − αlog ( y ) + 0 . y ( y + z ) s ) (14) dzβdt = z ( − αlog ( z ) + 0 . zs ) (15)with s = x + y + z .Expressions (13-15) show that a single parameter α = γβ determines equi-librium conditions, an improvement on (Axtell et al. , 2001) who needed twoparameters ǫ and memory size. Phase transition diagrams will then be drawnvarying β while keeping γ = 0 . constant. β plays the role of a kinetic coeffi-cient, increasing the characteristic time towards equilibrium. The magnitudeof the J coefficients at equilibrium scales as /γ .6 .2 Mean field analysis : Attractors and transitions The state of the system is described by the set of the preference coeffi-cients J , J and J of the agents, i.e. their estimated profit divided by γ for the three possible moves resp. H, M and L. This is an improvement withrespect to (Axtell et al. , 2001) which space phase dimension was three timesthe memory size. Our analysis can then proceed using the more powerful me-thods of dynamical systems and statistical mechanics rather the Markovianformulation proposed in (Axtell et al. , 2001).Trajectories in the J phase space are obtained by solving the mean fieldequations (4-5) using a Rosenbrock integrator (GRIND et al. , 2017). Grids oftrajectories help to figure out attractors and attraction basins. Since we can-not draw sets of 3 trajectories, we display their projections in plans ( J , J ), ( J , J ) and ( J , J ) for a given choice of β = 2 , γ = 0 . in figure (1). Thetrajectories start at regular interval in the projection plan with the samethird J coordinate.Three attractors can be observed : one with large J when move M isthe preferred choice by all agents, to be called the M attractor ; one withlarge J when move L is the preferred choice by all agents, to be called theL attractor ; and one with lower values of J , J and J , to be called the HLattractor.The Mean Field analysis readily tells us that two of the attractors aresuch that all agents always play the same strategy either M or L.The dependence of the J’s upon the reduced parameter β / γ is displayedon the continuation plot of figure 2. We clearly identify the 3 same attractorsin the ordered regime above β = 0 . , and only one attractor left with moveM as the preferred choice for lower β values . < β < . . The bifurcationis observed around β ≈ . . A steep, but not abrupt, transition furtheroccurs when β ≈ . to a disordered regime such that agents do not displaystrong preferences for any choice. 7
10 5 10J2 0510 J10 5 10J3 0510J30 5 10J2 0510 J10 1.5 3J3 036
Figure J , J ) plan starting from J = 5 (above left), on the ( J , J ) plan starting from J = 0 . (above right)and on the ( J , J ) plan starting from J = 6 (below left). The black trianglesfigure the initial conditions of the individual trajectories. β = 2 , γ = 0 . .The fourth set of projections on the ( J , J ) plan starting from J = 0 . (below right) was obtained just above the bifurcation for β = 0 . . and γ = 0 . . The L and HL attractors get closer and the basin of attractionof the HL attractor is strongly reduced by the widening of the M attractor.GRIND et al. (2017) software. 8 J3J1J3J1J1J2 J3 J20 1 2J’s 0510
Figure γ = 0 . . (The continuation algorithm failed to converge to theexact position of the bifurcation around β ≈ . ). Attractor M is colou-red blue, attractor L is coloured red and attractor HL is coloured green.GRIND et al. (2017) software. 9 Agent-based simulations
Let us now compare the above results with those directly obtained byagent-based simulations. At each time step a pair of agents is randomly cho-sen. They play the bargaining game choosing their move with a probabilitygiven by equation (2) using their own specific J j (not an average J j as in themean field approximation), which they update after the session. And so on.We here report 4 types of results :— On figure 3 the phase transition diagram (to be compared with figure2).— On figure 4 individual trajectories in the J simplex.— On figure 5 the distribution of individual J ’s on 4 attractors.— On figure 6 a sketch of the attraction basins.We first monitor the different J j averaged over the whole population atequilibrium when β is scanned downward and upward between 0 and 2 (fi-gure 3). For the β decreasing branches J id , we start at β = 2 from initialdistributions of J j close to one of the attractors for each branch and carry onintegration until the attractor is reached. The branches are continued when β is lowered, taking as initial conditions the previous values of J j on theattractor. The equivalent method is applied when β is increased from 0 forthe J im branches. For the sake of clarity, attractors HL and attractor L arerepresented resp. on the upper and lower plots.Only attractor M can be reached when β is increased from the disorderedattractor. When β > . , the path is reversible, but a hysteresis cycle isobserved when β crosses the bifurcation.In the ordered region, the transition from attractor HL to attractor M isdirect. By contrast, one first observes a continuous transition from L to HLaround β = 0 . above the sharp transition at the bifurcation.Some attractor levels can be readily obtained from equilibrium conditionsof equations (10-12). When only one move j is chosen by the agents, thefraction involving exponentials equals 1 and the value of J j is given by : J j = π i γ (16)in accordance with simulation results : on attractor M J reaches . / .
05 =10 and on attractor L J reaches . / .
05 = 6 . In the case of the disorderedattractor for low β values, the exponentials are close to one and the J j aredirectly computed from equations (10-12).Phase diagrams of the mean field approximation and of the agent-basedsimulations look pretty similar with the same attractors ; the main difference10 e s ti m a t e d g a i n β J1dJ2dJ3dJ1mJ2mJ3m 0 2 4 6 8 10 12 0 0.5 1 1.5 2 2.5 3 e s ti m a t e d g a i n β J1dJ2dJ3dJ1mJ2mJ3m
Figure β decrease between 0 and 2 (J1d e.g.) and then increase (J1m e.g.) from 0 to 2.The decrease is started from the mixed attractor HL on the upper plot, andfrom the uniform L attractor on the lower plot. Reversibility is only observedin the low β or in the high β region but not across the β = 0 . transition.11s the dependence of J upon β for the HL attractor observed in the MeanField Approximation. The previous results concerned global features. Let us now examine in-dividual agent choices. We use a simplex representation as in Axtell et al. (2001). At any time step, preference coefficients J of an agent are displayedon the simplex by a point which position corresponds to the center of gravityof masses proportional to J1, J2, J3 placed at vertices H, M, L. For instancean agent positioned close to the center of the simplex is indifferent to choiceH, M or L, while any agent close to one of the vertices has strong preferencesand mostly plays according to that vertex.Figure (4) represents a typical set of 30 agents trajectories in the simplexfor β = 2 and γ = 0 . during 10000 integration steps (each agent has beensampled 666 times on average). We started from a uniform distribution ofinitial J ’s of width 2 centered around (2.0, 2.0, 2.0). Their positions areindicated by a small square. Trajectories are distinctively coloured. After afew initial wanderings, they diverge in the direction of the closest vertex. Theyremain fixed in the case of L and M vertices. Some might fluctuate aroundvertex H because of possible encounters with high demanding opponentswhich result in the decrease of J .Each of the 16 simplices on figure (5) is a snapshot of agents’ J s after agiven integration time for a given value of β and for γ = 0 . .Each red point represents the set of preference coefficients of a singleagent. Each line of vertices displays the evolution of agents preferences atincreasing iteration times towards one of the 4 asymptotic configurations fora given value of β .The initial conditions where chosen to favour the attractor to be dis-played. We used uniform distributions of width 1.0 around (0.9, 0.9, 0.9)for the disordered attractor, D, around (1.0, 0.6, 1.0) for the M attractor,around (1.0, 0.6, 3.0) for the HL attractor and around (0.6, 0.6, 6.5) for theL attractor.In agreement with previous observations, we see on the first line of figure(5) that for low β values the agents positions remain dispersed inside thesimplex even for long iteration times, which corresponds to a disorderedphase.By contrast when β is increased, agents in the ordered phase gather to-wards one or two vertices, even after much smaller iteration times. We have12 ML Figure β = 2 , γ = 0 . , 10000 integra-tion steps. 13hosen intermediate values of β to avoid the accumulation of representativepoints on simplex vertices which would be observed at larger β values, e.g. β = 2 .A physical interpretation of the above results would be a comparison witha condensed phase with thermal excitations above the ground state. When β further increases, an equivalent of temperature decrease in physical systems,agents preferences condense exactly on the vertices (see further figure (6)), aproperty which helps us to check the basins of attraction. The next question concerns the extension of the basins of attraction of thedifferent attractors. In fact, a systematic search for β = 2 , γ = 0 . displaysmany more attractors than expected from our preliminary scans.Figuring basins of attraction in a 3D phase space is not obvious andwe once again use a vertex representation. The data are obtained by a triplescan of initial conditions. Each initial condition is a uniform distribution of J’svalues of width 1.0 around a given center. For instance, an initial distributioncentered on the center of gravity of the simplex is randomly drawn in thecube : . < J < . , . < J < . , . < J < . . The upper circle offigure 6 corresponds to the initial distribution : . < J < . , . < J < . , . < J < . etc.Figure (6) describes data gathered by 3 nested loops across initial J , J , J ,where distribution centers are varied from 0.4 to 6.4 by a factor 2. The posi-tions of the centers of the circles in the simplex codes the initial distributioncenters. We used the condensation of agents preferences on the 3 vertices todisplay final distributions by pie charts. Red sectors represent the percentageof agents with choice H, blue sectors represent the percentage of agents withchoice M, green sectors represent the percentage of agents with choice L.Rare and narrow white sectors represent the percentage of agents inside thesimplex. We checked that they are located in the immediate neighbourhoodof vertices.Several important conclusions about ordered phases for large β can bedrawn :— Mixed strategies inside the vertex are unstable.— The attractors are distributions of agents on the vertices or very close.Agents have strong opinions about the value of their choice and don’tchange them frequently.— J space is paved with basins of attraction surrounding the attractors.The dynamics collapse choices to the nearest vertex, with the excep-tion of vertex H. 14 =0.2 D 200000 400000 800000 1600000 β=0.4
M 2000 4000 8000 16000 β=0.8
HL 2000 4000 8000 16000 β=0.8
L 2000 4000 8000 16000
Figure β . γ =0 . . Each 4 simplex line represents the position of agents J’s after differentiteration times written at the bottom of the simplex. β values are 0.2, 0.4and twice 0.8 for the different lines. Initial conditions are given in the text.15 Figure β = 2 . , γ = 0 . , 100 agents, 100000 iteration steps.16 No distribution consists of only H preferences - which would give nogain to the agents. When the initial conditions are close to vertex H,the attractors can only be mixed distributions, with very few agentsplaying M.— (Axtell et al. , 2001) and (Poza et al. , 2011) report the existence of"fractious attractor" such that agents oscillate between choices H andL for long transient. We never observed such "fractious attractor",even after a specific search, and we suspect that they were due totheir choice of a constant noise term. The most striking result in Axtell et al. (2001) is the existence of an in-equity norm sustainable among two a priori equivalent tagged groups. Theirinequity norm corresponds in our settings to an attractor such that all mem-bers of one population with tag T1 play H against any member of the othertagged population (T2) who always plays L against them.To investigate the basins of attractions for the two tagged populations weproceed with the same scan of initial conditions of population with tag T2as above, but maintain the same initial conditions for population with tagT1 around the center of the simplex : . < J < . , . < J < . , . < J < . .Figure 7 displays the attractors of the inter-population dynamics, thesimplex T1 above simplex T2 with the same conventions as in figure 5, ex-cept that the positions on both simplices correspond to the scan of initialconditions of population T2. The colour codes are the same as for figure 6and reflects the attractors of each population.The pie charts close to the left vertices of the simplices e.g. are colouredgreen for T1 and red for T2 ; the attractors of the dynamics are then L forT1 and H for T2. The pie charts close to the top vertices correspond to astable mixture of the 3 possible moves H,M,L for T1 and to pure L for T2The pie charts close to the right vertices correspond to a stable mixture of 2possible moves M,L for T1 and to pure M for T2.Not surprisingly, the attractors reflect the initial conditions of populationT2 since the initial conditions of T1 were kind of neutral.The main difference with the no-tag simulation is the appearance of apure strategy attractor such that all agents with tag T2 play H against agentswith tag T1 who play L. This asymmetrical attractor parallel the findings of(Axtell et al. , 2001) who refer to a "discriminatory norm". But our analysischaracterises an attractor reached from initial conditions that were already17iased towards inequity with larger values of J . And this condition is anattractor of the dynamics, not a transient. The reformulation of the iterated bargaining game of (Axtell et al. , 2001)using more elaborate cognitive processes such as taking moving averages ofpast gains and choosing next moves according to Boltzman probabilities al-lows a more precise description of the dynamics in terms of attractors, regimetransitions and basins of attraction. The number of parameters was reducedfrom two to one. Several transitions are observed between a disordered stateand several stable ordered configurations when β increases. Because we useBoltzman choice function, agents end-up using mostly pure strategies forlarger values of β . We never observed any "fractious state" such that agentsremain in the interior of the simplex changing randomly their choice betweenH and L as reported in (Axtell et al. , 2001) and (Poza et al. , 2011). Our guessis that such behaviour is due to their hypothesis of a random choice with aconstant non-zero probability.As discussed earlier in sections 4.3 and 5, J space is paved with basinsof attraction surrounding the attractors and dynamics collapse preferencecoefficients to the nearest vertex. Unbiased random initial conditions nevergenerate H/L attractors.Hence our interpretation in terms of social phenomena : game interactionsand cognitive processes can increase and stabilise discrimination and inequa-lity among tagged populations, even when tag were a priori neutral. Onthe other hand, inequality attractors never "emerge" spontaneously fromrandom unbiased initial conditions.The changes we introduced in (Axtell et al. , 2001) also allow to figure outwhich properties can be considered as generic, that is to say independent fromthe details of each model, and which are specific to the exact formulation ofthe model.The two versions of the iterated bargaining game, (Axtell et al. , 2001)and the present paper, agree that unfair social institutions such as classesand discrimination can result as the downside of a rational cognitive practice,namely memorising or coding previous events to take present decisions. Andfurthermore, that taking into account a priori irrelevant tags can lead to a
1. (Axtell et al. , 2001) specify in a footnote that they "use the term "emergent" ... tomean simply "arising from the local interactions of agents." ". But the word Emergence inthe title of their paper evokes the idea of emergence of Classes in a previously egalitariansociety. percentages of choices: H red, M blue, L greenpercentages Figure for both groups while initial conditions of tag T1 playersremain identical around the center of the simplex. β = 2 . , γ = 0 . , 100 000iteration steps per data. 19issociation of the two tagged populations into an upper and a lower class.But we differ by our interpretation in terms of social phenomena : gameinteractions and cognitive processes can increase and stabilise discrimination,but inequality attractors never "emerge" spontaneously from random unbia-sed initial conditions. History has taught us that wars and invasions oftenresult into discriminations that are maintained long after these events. Acknowledgments
We thank Sophie Bienenstock, Bernard Derrida, Alan Kirman, Jean-Pierre Nadal and Jean Roux for helpful discussions and David Poza forproviding his Netlogo program of the lattice version of (Axtell et al. , 2001)model.
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