Fragmentation in trader preferences among multiple markets: Market coexistence versus single market dominance
FFragmentation in trader preferences among multiple markets:Market coexistence versus single market dominance
Robin Nicole , Aleksandra Alori´c ∗ , and Peter Sollich Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, UnitedKingdom Scientific Computing Laboratory, Center for the Study of Complex Systems, Institute ofPhysics Belgrade, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia Institut f¨ur Theoretische Physik, Georg-August-Universit¨at G¨ottingen,Friedrich-Hund-Platz 1, D-37077 G¨ottingen, Germany
Technological advancement has lead to an increase in number and type of trading venues anddiversification of goods traded. These changes have re-emphasized the importance of understandingthe effects of market competition: does proliferation of trading venues and increased competitionlead to dominance of a single market or coexistence of multiple markets? In this paper, we addressthese questions in a stylized model of Zero Intelligence traders who make repeated decisions at whichof three available markets to trade. We analyse the model numerically and analytically and findthat parameters that govern traders’ decisions – memory length and intensity of choice, e.g. howstrongly decisions are based on past success – make the key distinctions between consolidated andfragmented steady states of the population of traders. All three markets coexist with equal sharesof traders only when either learning is too weak and traders choose randomly, or when marketsare identical. In the latter case, the population of traders is fragmented across the markets. Forthe more general case of markets with different biases, we note that market dominance is the moretypical scenario. These results are interesting because previously either strong differentiation ofmarkets or heterogeneity in the needs of traders was found to be a necessary condition for marketcoexistence. We show that, in contrast, these states can emerge simply as a consequence of co-adaptation of an initially homogeneous population of traders.
The possible risks and benefits of market competition have been the subject of a long standing de-bate, which is often expressed as “market consolidation versus market fragmentation”[1, 2]. Whenthe New York Stock Exchange had by far the strongest influence on price formation, the finan-cial trading system was much closer to a consolidated state (Hasbrouck’s [3]), but more recently ∗ [email protected] a r X i v : . [ q -f i n . T R ] D ec echnological progress has created a variety of trading venues and led to ever increasing marketfragmentation. Particularly interesting in this regard are so-called dark pools . These trading venueshave gained a certain notoriety from their lack of transparency and the possibility to trade largevolumes without large price impacts, and they frequently offer a greater variety of market mecha-nisms compared to the conventional exchanges. Shorter and Miller [4] noted that in only five years(from 2008 to 2013) the US market share traded in dark pools increased from 4% to 15%, signalinga distinct increase in market fragmentation. Gomber et al. [1] suggest that the main driver ofmarket fragmentation is the heterogeneity of traders’ needs, which will be more easily satisfied byvariety of different markets rather than a single trading venue. In this paper we show that evenwhen identical markets compete, economic agents can develop loyalties to specific markets, thuseffectively fragmenting trading. Conversely we find in the case of competition of markets that arebiased towards different classes within the population of traders, single market dominance is thetypical outcome.To tackle this question of market coexistence versus single market dominance we build onprevious work [5, 6, 7, 8] where we introduced and analysed a system consisting of double auctionmarkets and a large number of traders choosing between them. What we showed in this settingis that for a range of parameters describing the markets and agents, the agents split into groupswith a strong loyalty towards one of the markets, often giving an overall market coexistence withan equal share of traders at both markets. When the agents have a long memory to previoustrading outcomes, other steady states with single market dominance also exist and are in factstable, whereas the system state with markets splitting trades roughly equally between them is onlymetastable [6, 8]. While these initial studies focused on settings with two markets for simplicity,traders do in general have a choice between multiple markets (see e.g. [1]) and this feature wasalso present in the CAT game [9] that originally motivated our research into market-trader co-fragmentation. We therefore extend the double auction market model from two to three markets inthis paper, and use the results to formulate conjectures for the expected behaviour in cases wheremore than three markets compete.There is a large body of work that uses the JCAT library [10] to explore competition between continuous double auction markets [11, 12, 13]. In a spirit similar to our work they use simplelearning algorithms such as Zero-intelligence [14] or Zero-Intelligence-Plus [15] for both markets andtraders, and analyse the allocation efficiency of double auction markets when they are competingagainst each other. Multi-agent based simulations have mostly been used in this context and allowadditional layers of complexity such as adaptive markets and heterogeneous agents to be added.We pursue instead a modelling approach that strips out as much detail as possible [6, 7, 8] to allowfor detailed theoretical analysis, which can often reveal features that would be missed when relyingexclusively on numerical simulations. In this spirit, while the market mechanisms implemented inthe JCAT library are continuous double auctions, we use in our model a mechanism more similar toa clearing house where the clearing process takes place at discrete time steps. This makes a largelyanalytical approach possible, which reveals the learning process of the agents as the main driverof fragmentation. This conclusion was shown in [6] to carry over to models with more complexmarket mechanisms and more sophisticated agent strategies, based e.g. on [16].Authors such as Ellison et al. [17] and Shi et al. [18] have focused on studying the competitionbetween markets and the conditions under which this lead to multiple market coexistence or theemergence of a market monopoly. The authors name two significant effects in the competition ofdouble auctions, one of them is the positive size effect, i.e. agents prefer trading in a market where2here are already many traders of the opposite type (e.g. sellers like trading at markets wherethere are many buyers), as the choice among offers is better. The authors additionally suggestthe existence of the negative size effect in a double auction market, as agents will prefer being inthe minority group to trade more often (e.g. buyers see the benefit of trading at a market wherethere are not many buyers, see e.g. [19]). Ellison et al. [17] point out that due to this negative sizeeffect, coexistence of many markets is possible. On the other hand Shi et al. [18] investigate whichof the two effects is stronger and finds that due to more substantial positive effects, a monopolywill in many situations be the preferred outcome. When there is strong market differentiation, theauthors of [18] argue that market coexistence is possible, especially for markets that have differentpricing policies, e.g. where one market charges a fixed participation fee while another charges aprofit fee. Although in what follows we will consider markets without fee charging policies, wewill find nonetheless there are system parameter ranges that enable coexistence, where marketsare populated by roughly the same numbers of traders; conversely we also identify the parameterregimes for which one market is dominant. It is important to note that the studies cited abovehave focused on finding either the Nash Equilibria or states favored by the replicator dynamics. Incontrast we consider dynamics based on agents learning to improve their market choosing strategy,which we believe is more appropriate in the context of agents engaging in economic interactions.In this study we show that fragmentation can arise even in an initially homogeneous population oftraders, only because the traders adapt to their past record of successful trades.
Here we summarize the basic assumptions and properties of the model introduced in [5, 6, 8] andextend it to include multiple markets.
Traders.
We study a population of agents without sophisticated trading strategies, essentiallyzero-intelligence traders [14, 20, 21]. The orders to buy at a certain price (bids) and orders tosell at a certain price (asks) are assumed to be unrelated to previous trading success or any otherinformation. We assume that bids, b , and asks, a , are normally distributed ( a ∼ N ( µ a , σ a ) and b ∼ N ( µ b , σ b )), where µ b > µ a , in line with [6]. After each round of trading each agent receivesa score, reflecting their payoff in the trade. The scores of agents who do trade are assigned aselsewhere in the literature [14, 22]: buyers value paying less than they offered ( b ), and so theirscore is S = b − π , where π is the trading price. Sellers value trading for more than their ask ( a ),and so S = π − a is a reasonable model for their payoff. Markets.
The role of a market is to facilitate trades so we define markets in terms of their price-setting and order matching mechanisms. We consider a single-unit discrete time double auctionmarket where all orders arrive simultaneously and market clearing happens once every period afterthe orders are collected. We also assume that a uniform price is set by the market – once all ordershave arrived, these are used to determine average bid (cid:104) b (cid:105) and average ask (cid:104) a (cid:105) and then set a globaltrading price in between the two: π = (cid:104) a (cid:105) + θ ( (cid:104) b (cid:105) − (cid:104) a (cid:105) ) (1)where θ fixes the price closer to the average bid ( θ > .
5) or the average ask ( θ < . θ thus represents the bias of the market towards sellers (they earn more when θ > .
5) or buyers(earn more when θ < . . Once the trading price has been set, all bids below this price, and all Note that traders are not informed about these market biases, nor the market mechanism in general; they only π . Note that we assume here that each order is for a single unit of the good traded.The most efficient resource allocation happens when demand equals supply, i.e. at the equilib-rium trading price. In a setup like ours where the bids and asks are Gaussian random variables withequal variances ( σ a = σ b ) and when the number of buyers is equal to the number of sellers at a givenmarket, the equilibrium trading price corresponds to θ = 0 .
5, i.e. the price is π eq = ( (cid:104) b (cid:105) + (cid:104) a (cid:105) ) / fair as θ = 0 . θ (cid:54) = 0 . Learning rules.
Agents trade repeatedly in our model, and they adapt their preferences forthe various choices at disposal from one trading period to the next. We assume that each agentdecides where to trade (which of many markets) at the beginning of each trading period, onlybased on his or her past experience. To formalize this we introduce a set of attractions A m foreach player, one for each market m = 1 , ,
3. The attractions will generally differ from playerto player, but we suppress this in the notation for now. The attractions are updated after everytrading period, n , using the following reinforcement learning rule (similar to Q-learning [23] andthe experience-weighted attraction rule [24, 25]): A m ( n + 1) = (cid:26) (1 − r ) A m ( n ) + rS m ( n ) if the agent chose market m in round n (1 − r ) A m ( n ) otherwise (2)The quantity S m ( n ) is the score gained trading at market m in the n -th trading period. The lengthof the agents’ memory is set by r : effectively an agent takes into account a sliding window of lengthof order 1 /r for the weighted averaging of past returns.Once each preference is updated, traders use the multinomial logit function to choose at whichmarket to trade in the next round: P ( M = m ) = exp( βA m ) (cid:80) m (cid:48) exp( βA m (cid:48) ) (3)This is inspired by the experience-weighted attraction literature [24, 25], where β is the intensityof choice and regulates how strongly the agents bias their preferences towards actions with highattractions. For β → ∞ the agents choose the option with the highest attraction, while for β → p B , which we call their buying preference. We will study a population of tradersconsisting of two classes of agents with fixed buying preferences p B = p (1) B and p B = p (2) B , respectively.The attractions of agents from different classes will be denoted by A ( c ) m with c ∈ { , } .We will frequently study a setup with symmetric markets (i.e. θ = 1 − θ < .
5) and a populationconsisting of two symmetrically biased classes (i.e. p (1) B = 1 − p (2) B > . θ , θ , p (1) B , p (1) B ) = (0 . , . , . , . β , the uniquefixed point of the learning dynamics is such that agents develop a higher attraction to the market obtain information through the scores they receive. β . When β is increased, this fixed point becomes unstable as buyers and sellers would congregate in differentmarkets and so lose many trading opportunities. Instead the population fragments: agents of bothclasses self-organize to divide into two groups within each class. One of these groups is returnoriented (e.g. buyers at market 1) and the corresponding agents earn more per single trade; theother group can be characterized as volume oriented (e.g. sellers at market 1), earning less pertrade but having the opportunity to trade more often. Numerical simulations
To motivate the use of this stylised model of agents choosing between multiple markets, we startwith multi-agent simulations of the system. We look at a default population of traders consistingof two classes – some tend to act more as buyers ( p B = 0 . p B = 0 . θ . We show an example ofthree qualitatively different distributions of the attractions of the agents in Fig. 1. To facilitate theinterpretation of these distributions we mark by coloured regions in each panel which market anagent prefers at the given attraction (differences), i.e. which market s/he chooses with the highestprobability. − . − . . . . A − A − . − . . . . A − A M2 M1M3 ( a ) θ = θ = θ = 0 . , /β = 0 . − . − . − . − . . . A − A − . − . − . . . ( b ) θ = 0 . , θ = 0 . , θ = 0 . , /β = 0 . − . − . − . . . . . A − A − . − . . . . . ( c ) θ = 0 . , θ = 0 . , θ = 0 . , /β = 0 . Figure 1: Distribution of attraction differences of population of traders for market and learningparameters as indicated in each graph title. In (a), the population is strongly fragmented into threegroups of equal size. In (b), the population is weakly fragmented, the distribution has two peaks:one large peak and one peak that (as we will later see) becomes exponentially small as the memorylength increases. In (c), the population is strongly fragmented, but only across two markets. Toobtain those graphs, we ran simulations with r = 0 .
01 and N/ ,
000 traders in each class untila steady state was reached. Traders from class 1 have preference to buy p (1) B = 0 . p (2) B = 0 .
2. The ( A − A , A − A ) plane is shown subdivided intothree zones that indicate which market an agent with the corresponding attractions chooses mostoften. The zones are coloured blue, red and green for markets 1, 2 and 3, respectively, as indicatedin (a).We now give a brief description of the attractions distributions in each of the panels and explainthe difference between (i) strong fragmentation, which persists in the large memory limit, and (ii)weak fragmentation, which disappears in the same limit; similar results for two market systemsare discussed in [6, 26, 8]. In panel (a) of Fig. 1, one sees that the distribution of attractions has5hree peaks, all of which have a size of order O (1) and correspond to subpopulations of traderswho choose to trade mainly at a single market. In other words, the trader population (in the classshown in the figure) splits into three subpopulations that are more attracted to one market overthe others, e.g. traders developed individual loyalties to one of the markets. Such distributions ofattractions with more than one peak with a size of order one are called strongly fragmented [8]. Asdiscussed in previous works, this does not mean the traders’ preferences are frozen: they do changetheir preferred market but only after a long persistence time [6]. We also note that in the stateshown, i.e. for the given parameters, three identical markets coexist and receive and equal share oftraders, on average.The second distribution, shown in panel (b), corresponds to a population divided into two loyaltygroups but with different sizes: one large (order N ) subpopulation is attracted to the second market(the fair market, θ = 0 . r → r , in the large memory limit,market two has a monopoly. When attraction distributions are multimodal but only one peak hasa weight of order 1 (i.e. fragmentation is only present at finite r ) we call them weakly fragmented .The distribution plotted in panel (c) corresponds to a strongly fragmented population, but con-trary to the case depicted in panel (a) the third market has now lost the competition. Additionally,the share of attracted traders is not the same between the markets (as in panel (a)), but both peakspersist in the long memory limit.The above simulation results offer a glimpse into a rich variety of qualitatively different struc-tures of the attraction distributions (number and size of peaks) and consequently different outcomesof a three market competition. To study these in more detail, we focus on the analytical and nu-merical methods described previously [7, 8] for large populations of traders and in the large memorylimit ( r → To proceed with the analysis, in line with our earlier studies [6, 7, 8], we start from the fact that thesystem is Markovian and accordingly the master equation introduced in [6] is an exact and completedescription of the evolution of agents in the limit of an infinite population N and large memory 1 /r .We focus here on the steady states of this dynamical evolution. For a population with fixed buy/sellpreferences, this is specific by a steady state distribution P ( A | p B ) where A is an M -dimensionalvector of attractions and conditioning on the buying preference distinguishes the different classesof traders. When we study more than two markets the distribution is multivariate, though we canintroduce attraction differences and look for a solution in the resulting M − n is not astandard linear Chapman-Kolmogorov equation as the transition kernel K depends on the tradingprobabilities, which in turn depend on P n ( A | p B ). This self-consistent nature of the descriptionarises from the reduction from a description in terms of the attractions of all N agents to onefor a single agent; this reduction becomes exact for N → ∞ . In principle, a steady state couldthen be found by tracking the evolution in time from the initial condition P ( A | p B ) = δ ( A ), whichcorresponds to all agents having zero attraction to all markets. We take a different route and firsttransform the time evolution equation to a Fokker-Planck description using the Kramers-Moyalexpansion. This is appropriate for small r , i.e. for agents with long memory.6ven after the simplification to a Fokker-Planck equation, the dimensionality of the problemmakes finding the steady state a non-trivial task. But we can make progress by considering thelimit r →
0; this will allow us to evaluate the onset of fragmentation. We do this by analysingthe drift µ ( c ) m in the Fokker-Planck equation, defined in the Appendix A. To find the single agentsteady state, we will search for zeros of the drift assuming fixed market order parameters, i.e.trading probabilities. We start by assuming that the two classes have homogeneous preferencesfor the markets (i.e. P ( A ( c ) | p ( c ) B ) is a delta-distribution). This is the expected solution in the low β -limit, when the steady state is unfragmented. With this assumption, the expressions for themarket order parameters simplify, and we can solve the simultaneous equations for the two classes.At any fixed point solution ( A (1) ∗ , A (2) ∗ ) we evaluate the market order parameters and check if thesingle agent dynamics is consistent with the homogeneous population assumption: when we solve µ ( c ) m ( A ) = 0 we expect only one zero that coincides with A ∗ ). The onset of fragmentation (weak orstrong) is then given by the intensity of choice where the single agent dynamics first has multiplezeros when evaluated at the homogeneous population market order parameters, which indicatesthat for r > /r , and large peaks, whose weight remains finite and oforder unity when the r → M = 3 markets and wedescribe each of the two classes in terms of the two attraction differences ∆ A = A − A and∆ A = A − A . We perform a Kramers-Moyal expansion of the trader’s learning dynamics andobtain two Fokker-Planck equations (one for each class c ∈ { , } of traders) for the distribution ofattraction differences P ( ∆A ( c ) , t ): ∂ t P ( ∆A ( c ) , t ) = − (cid:88) m =2 ∂ ∆ A ( c ) m (cid:104) µ ( c ) m ( ∆A ( c ) , f , f , f ) P ( ∆A ( c ) , t ) (cid:105) + r (cid:88) m,m (cid:48) =2 ∂ ∆ A ( c ) m ∂ ∆ A ( c ) m (cid:48) (cid:104) Σ ( c ) mm (cid:48) ( ∆A ( c ) , f , f , f ) P ( ∆A ( c ) , t ) (cid:105) (4)Here the time variable t = nr is a rescaled number of trading rounds, ∆A ( c ) = (∆ A ( c )2 , ∆ A ( c )3 )and f m is the market order parameter, i.e. the ratio of buyers to sellers at market m (effectivelythe demand-to-supply ratio). The expressions for the drift vectors µ ( c ) m ( ∆A ( c ) , f , f , f ) and thecovariance matrices Σ ( c ) mm (cid:48) ( ∆A ( c ) , f , f , f ) for each class are given in Appendix A. Three fair markets
We start by looking at what happens when the three markets available are all fair, i.e. θ = θ = θ = 0 .
5. This means they set their trading price to be exactly the mean of the average bid andthe average ask. As mentioned previously, the fair market corresponds to a market mechanismdelivering the equilibrium trading price, provided the number of buyers equals number of sellers.Based on intuition from similar physical systems one might expect spontaneous symmetry break-ing, where random fluctuations lead the whole population to select only one of the possible sym-metric markets. However, in stochastic multi-agent simulations we observe instead steady states7ith fragmented populations within each class; we therefore focus on steady states of the traders’learning dynamics without symmetry breaking.Since the three markets have the same bias θ , in a symmetric solution, they should attract thesame number of agents, irrespective of their class. On the other hand, as we study classes of agentswith symmetric preferences to buy p (1) B = 1 − p (2) B , the difference between the number of buyers andthe number of sellers at a single market is of order √ N , N B = N S + O ( √ N ). As a consequence,in the large size limit, the ratio of the number of buyers to the number of sellers in each market isequal to 1. This simplification is the reason why we choose to start the analysis with the simplecase of three fair markets, which allows one to explore the phenomenon of fragmentation acrossthree double auction markets without the need for a self-consistent determination of market orderparameters [7, 26, 8].We start by looking at the fixed point structure of the single agent dynamics when the intensityof choice β is small. As expected, the only fixed point of the learning dynamics is A − A = A − A = 0 and corresponds to a trader who chooses to randomise between the three markets (seeFig. 2(a)). When the intensity of choice β reaches a critical value β c = 1 / . θ = 0 .
5. In the more general case where thethree markets are different, we expect the appearance of each pair of new fixed points to take placeat a different value of β .When looking at the deterministic dynamics for low intensity of choice (see Fig. 2(a)) it isobvious that the system is not fragmented and there is only one stable fixed point. At largerintensities of choice as in Fig. 2(b, c, d), knowing the deterministic dynamics is not sufficient todistinguish between “stable” fixed points (the ones where, in our terminology, large peaks will becentred) and “metastable” ones (which for us indicate the positions of the small peaks). To assessthe stability of fixed points in Fig. 2 and weight sizes of potential peaks, we use the Freidlin-Wentzellapproach detailed in Appendix B.As an example of an attraction distribution that has both small and large peaks we considerthe range 1 / . ≥ β ≥ / .
254 for the intensity of choice, where the system is weakly fragmented(as in Fig. 2(b)). The central fixed point is stable and a large peak in the attraction distributionis located at this fixed point, while the three outer fixed points are metastable and correspondto small peaks. As β is increased to a second critical value of β (cid:48) c = 1 / . β above this second fragmentation threshold, the system will be strongly fragmented asthe distribution of preferences of the traders will have three peaks of equal weight, each of whichcorresponds to a stable fixed point of the single agent dynamics (red points in Fig. 2(c,d)). For1 / . ≤ β ≤ / . ,
0) but the weight of this peak will become exponentially small as the memory lengthincreases (see Fig. 2(c)). This metastable fixed point and the associated small peak in the attractiondistribution then disappear for β ≥ β (cid:48)(cid:48) c = 1 / .
237 (see Fig. 2(d)).We summarize briefly the intuitive meaning of the above results for the attraction distributionsin a system of agents with long memory choosing between three fair markets. When the intensityof choice is small the agents cannot develop strong attractions to any particular market as low β implies that they choose a market largely randomly. With increasing β , three small subpopulationsof the agents in each class develop a loyalty to one of the markets, signaled by increased attractions,8 d) (a) (b) (c) Figure 2: Flow diagram and fixed points of the learning dynamics of a single trader with p (2) B = 0 . β = 1 / . β reachesthe weak fragmentation threshold β c = 1 / . β (cid:48) c = 1 / . β increases, the meta-stable fixed points eventually becomes unstable.Above each graph we indicate in triangular notation the category to which each of the fixed pointstructures belongs (see main text for details).but the random choice strategy remains dominant. These loyal subpopulations grow until (beyond β (cid:48) c ) they encompass most of each agent class.To help with understanding variety of different steady states, we introduced attraction distribu-tion notation in shape of triangles, as depicted in panels of Fig. 2. We focus on the number and sizeof the peaks, rather than their exact position, and use the triangle to visualize attraction to anyof the three markets (circle close to the corner) or market indifference (star shape). To distinguishbetween large and small peaks we use filled or empty objects (both stars and circles).In the simple case of three competing markets considered so far, we find that they alwayscoexist, but in different scenarios ranging from all traders choosing a market randomly to traderssplitting into subpopulations with persistent market loyalties. An obvious question is then whetherthis fragmentation is critically dependent on the fact that all the markets are identical. To answerthis, we next extend our analysis to markets with different biases. Each market bias θ , θ , θ is between zero and one, i.e. the market parameter space is a unitcube. Of course the phenomenon of fragmentation is independent under permutation of the marketbiases as this effectively just changes the labelling of the markets. We can therefore restrict ouranalysis to 1 / θ ≤ θ ≤ θ and can reconstruct the behaviour in the rest of theparameter space by symmetry. We will mostly follow this scheme but sometimes allow a different9arameter ordering to get simpler 2-D phase diagrams, with a typical bias along the x -axis andthe inverse intensity of choice along the y -axis. We study three different types of scenarios, guidedby explorations in our previous work: (i) one fair market θ = 0 . θ = 1 − θ , with θ as a free parameter varying between 0 and 1 /
2, shown in Fig. 3, (ii)two symmetrically biased markets θ = 0 . θ = 0 . θ varied as a free parameter, shownin Fig. 4, (iii) θ = 0 . θ = 0 . θ again ranging from 0 to 1, shown in Fig. 6. As willbecome clear in the rest of this section, these parameter settings allow for the analysis of the effectof a number of properties on the occurrence of fragmentation, such as the market symmetry, the“distance” between market biases and the effect of market fairness. Following the reasoning we utilized in the case of three fair markets, we continue to focus on solutionsthat do not break the market symmetries. This assumption is supported by stochastic multi-agent simulations in which we do not observe market symmetry breaking. We use the symmetriesto restrict the possible values of the “market aggregates”, i.e. the demand-to-supply ratios. Inparticular, we can show that these ratios are inverses of each other for the symmetrically biasedmarkets, and that the ratio is unity at the fair market as before. To see this, note first that when θ = 1 − θ and θ = 0 .
5, for traders with symmetric preferences to buy, the role played by market1 for traders from class 1 is the same as the role played by market 3 for traders from class 2 andvice-versa. As a consequence, the probability of trading at the first market for a trader from class1 (resp. 2) is equal to the probability of trading at the third market for a trader of class 2 (resp.1). We can write the buyer/seller ratios in market 1 and 3 as f = P (1) ( M = 1) p (1) B + P (2) ( M = 1) p (2) B P (1) ( M = 1)(1 − p (1) B ) + P (2) ( M = 1)(1 − p (2) B ) f = P (1) ( M = 3) p (1) B + P (2) ( M = 3) p (2) B P (1) ( M = 3)(1 − p (1) B ) + P (2) ( M = 3)(1 − p (2) B ) (5)When substituting into these expressions the equalities P (1) ( M = 1) = P (2) ( M = 3), P (2) ( M =1) = P (1) ( M = 3) and remembering that p (1) B = 1 − p (2) B , one sees that f = 1 /f . The fact thatthe ratio of buyers to sellers at the fair market (market 2) is unity follows by analogous reasoning.Let us first calculate the value of the intensity of choice at which traders start to fragmentweakly. To do so, for a given value of the free parameter θ , we start from low values of β andgradually increase the intensity of choice until it reaches a critical value where the single agentdynamics has two stable fixed points. Those values of β are shown by the upper solid line in Fig. 3.The natural continuation of this analysis is to look – if it exists – for the strong fragmentationthreshold. While thanks to our previous analysis of symmetric markets we know that for θ = 0 . β = 1 / . θ < .
48, strong fragmentation does not take place across the entire rangeof values of β that we consider numerically for our phase diagram. For θ between 0 .
48 and 0 .
5, ournumerics suggest possible strong fragmentation but a definite conclusion cannot be reached giventhe numerical precision limits of the required action minimizations.To distinguish between different types of steady states in the following analysis – the number ofemergent loyalty groups, their market preferences and sizes, we now introduce a triangle notation10
Preferences fordifferent markets:Large peakpositionsSmall peakpositions
Figure 3: Peak structure of the steady state distribution of traders’ preferences when they learn tochoose between three markets, two of which have symmetric market biases θ = 1 − θ and one ofwhich is fair. The three insets on the top show the fixed point structure for an agent from class2 ( p (2) B = 0 . θ = 0 . β as indicated in the phase diagram by grey points. Fullcircles in the phase diagram correspond to a stable (large peak) and empty circles to a metastablefixed point (small peak), colors (black = class 1, red = class 2) differentiate between the agentclasses. The grey band at θ = 0 . θ = 0 . θ , /β ) phase diagram there. Each of the trianglecorners represent preferences for one of the three markets, while full and empty circles representlarge/small peaks; different colours denote the different trader classes. This notation allows us toquickly realise whether some markets lost the competition, which markets are dominant, and whichmight attract only a single class of traders. Additionally, we use a star to denote an attractiondistribution peak without preferences for a specific market. This is present only for the scenariowith three fair markets, as depicted in the right band of the phase diagram in Fig. 4. The triangularrepresentations shown on the right correspond to the flow diagrams with fixed points depicted in11ig. 2.In Fig. 3 we see that for any value of β and θ < .
5, the majority of the traders will preferto trade at the fair market (market number two), so that this market will have a monopoly in the r → β is greater than theweak fragmentation threshold, but market two still attracts the majority of trades. Interestingly, inthe region of the phase diagram with intermediate β (see inset (b)), all three markets coexist, butmarkets one and three are visited by only a single class, despite the fact that trading opportunitiesare lower that way.In summary, the results depicted in Fig. 3 tell us that, apart from the particular case when thethree markets are all fair, strong fragmentation does not take place when a fair market competesagainst two symmetrically biased markets. We therefore move next to an even less symmetricsituation. We continue exploration of the space of market biases by considering two symmetric markets withfixed market biases θ = 0 . θ = 0 .
7; this is the market set up we mostly studied in previousworks. Without the third market, when the two classes of traders adaptively choose between twosymmetric markets one finds both weak and strong fragmentation above β c = 1 / .
28 [8]. Here, weadd the third market and vary its bias, which as Fig. 4 shows leads to a range of different steadystate attraction distributions.We first note that strong fragmentation appears, and does so across a reasonably broad rangeof market biases (grey zone in Fig. 4). This range excludes the case studied above where market2 is fair: strong fragmentation occurs only for θ (cid:54)∈ [0 . , . θ < .
45 (resp. θ > .
55) the traders from the first (resp. second) class strongly fragment across the two markets that maximise average profit per trade for each class. Forexample, in the case of θ = 0 .
4, buyers (traders in class 1, who have p (1) B = 0 .
8) will prefer tradingat markets 1 and 2 while the sellers remain unfragmented.We do not explore the phase diagram below the first strong fragmentation threshold as thiswould require the numerical solution of self-consistency conditions for multiple aggregates in thepresence of two (or more) strong fragmentation peaks in the traders’ attraction distributions. Thisis numerically very challenging and so we leave it for future work. However, it is possible to get anintuition about the shape of the phase diagram below this threshold by extrapolating the zones ofweak fragmentation in the range of θ where the second market is close to fair.We show in Fig. 4 graphically the types of steady state attraction distribution within thedifferent regions of the phase diagram. These predictions are obtained using single agent flowdiagrams as shown in Figs. 2 and 3. We show an exemplary comparison to stochastic multi-agent simulations in Fig. 5 and find excellent qualitative agreement. The agent class that mostlybuys (class 1, left panel) fragments into two subpopulations mainly trading at markets 1 and 2,respectively, where they maximize their profit because θ , θ < .
5. Agents in the class thatmostly sells prefer market 2 as the less biased of the two markets that are populated by the buyers.We conjecture that it is the asymmetry imposed by two markets favoring buyers that lead to aconsolidation around markets favouring buyers, while sellers do not develop attractions toward themarket that favours them.Having described the range of values of θ for which strong fragmentation takes place, weinspect more closely the range of parameters for which only weak fragmentation occurs (see Fig. 4).12 trongly fragmented strongly fragmented Figure 4: Types of attraction distributions in the populattion choosing between markets θ =1 − θ = 0 . θ . The grey zone indicates the region in parameter space where thedistribution of attractions has two large peaks for at least one class of agents, i.e. where strongfragmentation occurs. Note that between every unfragmented and strongly fragmented region(appearance of large loyalty groups at market 1 and 3) there is always a weakly fragmented region(where the same loyalty group, i.e. peak in the distribution, is small), but these regions are mostlytoo narrow to be visible. The grey line in the centre corresponds to the dashed line in Fig. 3.To do so, we look at how the attraction distributions of both classes of traders evolve at fixed θ = 0 .
47 when β increases. For values of β small enough in relation to the agents’ attractions,they will essentially randomise their market choice, with a weak preference towards the market thatis closest to fair, market 2. This preference increases with β so that traders from the two classeseffectively coordinate at market 2, providing a good trade-off between profit and trading volume. As β grows further, additional small peaks arise in the attraction distributions while most of the tradersremain in the fairer market. In particular, at β = 1 / .
246 a peak corresponding to the strategy“trading at the profit maximizing market” (market 1, which has θ = 0 .
3) appears for class 1. Thenat β = 1 / . θ = 0 .
7) appears in the attraction distribution of the agents from the secondclass. After those two successive appearances of weak fragmentation between the fairer marketand the profit maximizing market for both class 1 and class 2, further peaks in the attractiondistribution – which correspond to the strategy “trading at the volume maximizing market” –13 . − . − . . . . − . − . . . . − . − . − . . . . . − . − . . . . . Figure 5: Distribution of attraction differences of traders who choose between three markets withmarket biases ( θ , θ , θ ) = (0 . , . , . N/ traders with symmetric buy-sell preferences p (1) B = 1 − p (2) B = 0 .
8, inverse memory length r = 0 . β = 1 / .
21. We see that the attraction distribution of the first class isstrongly fragmented (panel (a)), while the second one is unfragmented (panel (b)), as predicted bythe phase diagram in Fig. 4.appear successively for class 2 at β = 1 / .
207 and then for class 1 at β = 1 / . θ close to 0.5, for larger β (lower 1 /β ) than investigated in the phase diagram of Fig. 4.Interestingly, addition of the third market leads to trade shifting away from one of the symmetricmarkets, throughout the entire strong fragmentation region in Fig. 4. Only when the added marketis close to fair can the two symmetric markets continue to coexist, though with both receiving onlya small fraction of trades. Market 2 in fact has the largest market share throughout Fig. 4.We can summarize the intuition behind the above results as follows. As the intensity of choiceincreases, each class of agents will first fragment weakly between a market that is close to fair(market 2) and the market that maximizes profit for them, and then fragment weakly across allthree markets. On the other hand, if the second market is not fair, the class for which this marketis more profitable will fragment strongly between their two profit maximizing markets, while theother class will only trade at the market that is closest to fair. The results of this subsectionsuggest that as soon as traders have at their disposal a reasonably fair market, they are not goingto fragment and will prefer to trade with the fair market; when they have no fair market they willalways prefer the profit maximizing market, and will visit the volume maximizing market (whichbrings lower profits but typically more trades) only as a last resort. The two examples presented in subsections 5.1 and 5.2 lead to the conjecture that the presence ofa fair or nearly fair market – which provides a good trade-off between profit in individual tradesand trading volume – can suppress fragmentation. To confirm this conjecture, we consider three14arkets where the first one is biased toward buyers ( θ = 0 .
3) and the second one is fair ( θ = 0 . β and the bias of thethird market θ ∈ [0 , r → /β than shown in Fig. 6 suggeststhat this situation does not change at even larger intensity of choice.) Only one peak has weightof order one and, depending on the values of β and θ , the steady state is either unfragmented orweakly fragmented, having one or two small peaks that disappear in the r → θ = 0 . θ = 0 . p (1) B =1 − p (2) B = 0 .
8. The solid/dashed lines show weak fragmentation transitions where subpopulationsemerge that favour markets 1 or 3 (line colours denote class of agents in which transition occurs).One notes that once the intensity of choice increase above a certain threshold value shown bythe full black line in Fig. 6, a weak peak corresponding to the strategy “trading at market one”appears in the distribution of attractions of the first class of agents, whose attractions are markedby black circles; recall here that market 1 provides buyers, who are more frequent among agents ofthe first class, with higher returns. When β crosses the second fragmentation threshold (red line inFig. 6), the same type of weak peak emerges in the distribution of attractions of the second classof agents (denoted by a red empty circle as before).15he fact that the two sold lines just described are close to horizontal reflects the fact thatsince almost all of the population trades at the fair market, the bias of the third market will notsignificantly influence the preference of traders. This is the reason why the intensity of choice atwhich traders of class 1 (resp. class 2) will weakly fragment between markets 1 and 2 is almostindependent of the bias of the third market. The same is not true of the thresholds for theappearance of a peak corresponding to the strategy “trade at market 3”, which are indicated bythe sloping dashed lines in Fig. 6.Consistently with previously discussed results, the existence of fair market suppressed strongfragmentation and within the space of parameters depicted in Fig. 6 we note only weak fragmen-tation. This means that across the investigated parameter range fair market attracts most of thetraders. We also note that the third market loses the competition when it is very biased and theintensity of choice is not large enough (note regions where market three either attracts none or onlyone class). However, it is interesting to see that for sufficiently large intensity of choice β all threemarkets coexist independently of third market bias. M So far we have discussed various cases of fragmentation in the three market setup. We found thatabove some critical value of the intensity of choice β , the solution in which the population remainsindecisive towards the markets is never stable and at least one market loyalty group is formed.The obvious question is now whether we can say something about the number of distinct agentsubpopulations in the general case of M markets.The theoretical description of the population’s adaptation in the most general case, withoutmarket or agent symmetry requires the self consistent procedure of calculating order parameters(one per market) and the steady state distribution of the agent attractions. This is a non-trivialtask in higher dimensions but the general existence of solutions can be rationalized within a simplecounting argument.In the following we make the assumption that for all M there is a fragmentation threshold β s above which the drift in the Fokker-Planck representation of the dynamics has multiple zeros.However, even when this is the case it is not clear whether all agent classes will develop loyaltygroups towards each of the markets (and the corresponding attraction distribution peaks), whetherthe peaks will be small or large; in the latter case fragmentation persists by definition in the r → M markets so that in the limit r → M delta peaks with weightsof order unity. We can find the peak positions by locating the zeros of the drift, but without theFokker-Planck solution, we cannot obtain the peak weights and the Freidlin-Wentzell approachbecomes difficult. We therefore ask how many non-zero peak weights can exist in general, for C agent classes and M markets. As explained, we assume the general shape of the steady statedistribution: P ( c ) ( A ) = M (cid:88) m =1 ω ( c ) m δ ( A − A ( c ) m ) . Each of the agent classes is described by peak weights ω ( c )1 , . . . , ω ( c ) M that satisfy the normalizationcondition (cid:80) Mm =1 ω ( c ) m = 1, thus in the absence of any symmetry we have M − f m , thus the system ofequations we need to solve to find a strongly fragmented solution is: F m ( ω (1)1 , ω (1)2 , . . . , ω (1) M , . . . , ω ( C ) M ) = f m , Here F m denotes the relation between the peak weights and market order parameters; an exampleof this for C = 2 and M = 3 is written explicitly in Eq. 5. Without symmetries, when all theequations and variables are independent, this system of M equations and C ( M −
1) variables hasa unique solution only when the number of equations is equal to the number of variables, i.e. M = C ( M − M and classes C is equal to two, ( C, M ) = (2 , C = 2 agent classes requires 2( M −
1) weights for strong fragmentation across M markets, andequating the number of variables 2( M −
1) and the number of equations M gives M = 2 markets,which is the case studied in [6].Since we have seen that full fragmentation, with all agent classes developing separate loyaltygroups for all markets, can only happen (without symmetries) in systems with two markets and twoagent classes, we next relax the assumption on the number of loyalty groups. Let us suppose thereare M markets and two agent classes, each of them fragmenting into η ( c ) subgroups (i.e. havingonly η ( c ) non zero peak weights), the system of equations for these weights has a unique solutionwhen η (1) + η (2) − M . This shows that if one class divides into M loyalty groups, the secondclass will fragment only across two markets; other combinations satisfying η (1) + η (2) = M + 2 arealso possible. For a general number of agent classes, the analogous constraint reads η (1) + η (2) + · · · + η ( C ) = M + C (6)As an example, if one class develops loyalty groups to all M markets, the other C − C such subpopulations in total, equating to one bimodal and C − η (1) + η (2) + · · · + η ( C ) ≤ M . E.g. in the case C = 2, there will be at leasttwo markets for which both classes have loyalty groups; the overlap will be even greater if somemarkets lose out and have no associated loyalty group.Summarizing, the conclusion of our counting argument is that in the r → C + M loyalty groups can coexist. In the three market scenario with two classes, this is at most five loyaltygroups. We saw an exception in the case of three fair markets, where six loyalty groups can exist;this is because of the symmetry between the markets, which our general argument excludes. It isremarkable how the simple counting argument gives a variety of new conjectures for the systemswith multiple markets. It provides a maximal number of loyalty groups; it tells us that all marketscan in principle coexist, and that the overlaps of different agent classes must overlap at C marketsat least. An interesting consequence is the emergence of a state where some markets are persistentlyonly visited by a subset of the overall population of traders. In this paper we investigated whether market coexistence is possible in the systems with morethan two markets when agents with fixed buy/sell preferences adapt dynamically to optimize their17hoice of market. To this end we studied the possible steady states of the agent dynamics, inparticular with regards to the occurrence of fragmentation, where a homogeneous class of agentsspontaneously forms subpopulations with long-lived market preferences.Motivated by the wide variety of structures of the attraction distributions that one observes inmulti-agent simulations, we explored different combinations of market biases and their influence onthe phenomenon of fragmentation. First we studied fragmentation across three fair markets, i.e.with θ = θ = θ = 0 .
5. This was the only scenario where we found that all three markets coexistacross the full range of the intensity of choice β of the agents. As β increases we nonetheless see achange, from an indecisive population (where agents visit all three markets randomly) to a stronglyfragmented population where each agent class splits into three equal-sized loyalty groups with adistinct preference for one market.We continued by exploring different market configurations to get an intuition for the factorsthat drive fragmentation. This enabled us to identify two principal causes of fragmentation: (i) the similarity between the markets’ biases , (ii) the average volume of trade and average profit earned ata market . The similarity between two markets is going to enhance fragmentation because tradersare more likely to split across two markets if they effectively cannot tell them apart. This effect isvisible in Sec. 5.3 where the strong and weak fragmentation thresholds are the highest (in termsof 1 /β ) when the second market and the fair market have the same bias. The ordering of theappearance of the peaks in the traders’ attraction distributions suggests – as we pointed out inSec. 5.2 – that traders will have an initial preference for markets that provide a good balancebetween trading volume and profit, then as the intensity of choice increases they will first spreadto the market that maximises their profit and then subsequently to the one that maximises theirtrading volume.The concepts of positive and negative size effects introduced previously [17, 18] are useful whenthinking about traders who develop loyalty for markets that do not reward them highly. At thesemarkets, traders benefit from many trading options available (positive size effects), and the factthat they are in the minority group (negative size effects). However, contrary to the findings of theauthors of Refs. [17, 18], we note that market coexistence is more prevalent when the markets aresimilar – the fragmentation region shrinks with increased market distance.Apart from the case of three identical markets, we find that once β is large enough for agents tostop choosing markets at random, the three markets never coexist fully in the large memory limit,i.e. at least one of them will have a market share that vanishes for r →
0. At most, we observethat the population fragments strongly across two markets (see strong fragmentation in Fig. 4).These markets then each have a finite share of the trading volume for r →
0, though with one beingsubdominant because it is visited only by (some of the) agents from a single class.From a general counting argument we found further that full market coexistence, where allagent classes develop the (joint) maximal number of loyalty groups, leads to apparently specialisedmarkets: some agent classes develop loyalties only to a subset of all markets (as in Fig. 4) andconversely some markets are not visited by agents from all classes. This is not a consequence of amarket explicitly targeting some subset of the agent population, but rather of the limited numberof market loyalties the different agent classes can support.We mostly considered moderate values of β driven by our interest in finding domains of differentsteady states, and for those purposes our straightforward implementation of the action minimizationalgorithm served us well. However, for large values of β it occasionally fails to find minimal actionpath, thus robustness and accuracy improvements are needed if one is interested particularly in18his regime. One possibility might be to use the geometric minimum action method [27].Although the analytical and numerical methodology we have proposed to study agents whochoose between multiple markets is valid for any number of markets M , it is challenging for tworeasons: (i) the parameter space dimension grows with M thus making numerical exploration ofall possible behaviours difficult, (ii) analytical approaches also become harder to implement as theanalysis is done in the space of attraction differences of dimension M − β in the large memory limit ( r → r > r (effectively for short memory) only a strongly fragmented steady stateexists [6] instead of two weakly fragmented and metastable strongly fragmented states; it would beinteresting to investigate if similar results also hold for multiple markets.In this and previous studies, we have investigated how agents adapt based on their explorationof markets; the adaptation mechanism implicitly assumes that markets do not change. Realistically,one would expect that a market tries to adapt as well once the number of traders using it decreases.If markets only try to maximise this number of traders, one could speculate that by adapting their θ biases they would converge to all-fair markets (similarly to the Hotelling paradox [28]). If on theother hand markets were to adapt to optimize the number of successful trades, by e.g. chargingfixed or profit dependent fees, then it would be intriguing to know what types of steady stateswould be realized in the overall system of agents and markets.Finally, a broad implication of our study is that fragmentation (weak or strong) can emergespontaneously within a class of homogeneous traders, in contrast to statements elsewhere [1] arguingthat heterogeneity among traders is the reason for market fragmentation. This we think is a veryinteresting result as it demonstrates that structure in the preferences of economic agents mightemerge out of adaptation rather than being present from the start. Nonetheless, it would bean interesting next step to investigate how heterogeneities in terms of trading strategies, budgetconstraints, or adaptation parameters such as β would affect fragmentation. Acknowledgement
The authors are grateful to Peter McBurney for useful discussions. PS acknowledges the stimulatingresearch environment provided by the EPSRC Centre for Doctoral Training in Cross-DisciplinaryApproaches to Non-Equilibrium Systems (CANES, EP/L015854/1). AA acknowledges the fundingprovided by the Institute of Physics Belgrade, through the grant by the Ministry of Education,Science, and Technological Development of the Republic of Serbia.
Authors’ contributions
R.N., A.A. and P.S. conceptualized the model and formalized the mathematical framework. R.N.implemented the numerical simulations and prepared the figures. R.N, A.A. and P.S. analysed theresults and discussed their implications. All authors wrote the manuscript and gave final approvalfor publication. 19
Kramers-Moyal expansion
In this appendix we give the expression of the drift and covariance matrix that appear in theKramers-Moyal expansion in Eq. (4). We only give the results here; the steps of the derivation canbe found in the thesis of Alori´c [29]. First, the drifts of the attraction differences are: µ ( c )2 ( ∆A ( c ) , f , f , f ) = (cid:16) P ( c )1 ( f ) P ( M = 1) − P ( c )2 ( f ) P ( M = 2) (cid:17) − ∆ A ( c )2 (7) µ ( c )3 ( ∆A ( c ) , f , f , f ) = (cid:16) P ( c )1 ( f ) P ( M = 1) − P ( c )3 ( f ) P ( M = 3) (cid:17) − ∆ A ( c )3 (8)Here P ( c ) m ( f m ) is the average payoff of a trader from class c at market m and P ( M = m ) is theprobability to trade at market m , which depends on the vector ∆ A ( c ) of attraction differences. Wedo not write this dependence explicitly to lighten the notations. The f m are the market aggregates,i.e. buyer-to-seller ratios, at the three markets. In order to check the validity of our calculationswe compared the dynamics of the aggregate f during a multi-agent simulation with the evolutionof the aggregates under the homogeneous population dynamics as detailed in [7], finding goodagreement as shown in Fig. 7.We next look at the covariance matrix of the effective noise acting on the attraction differences, (cid:32) Σ ( c )22 Σ ( c )23 Σ ( c )23 Σ ( c )33 (cid:33) (9)which is given byΣ ( c )22 ( ∆A ( c ) , f , f , f ) = (cid:16) Q ( c )1 ( f ) − A ( c )2 P ( c )1 ( f ) (cid:17) P ( M = 1)+ (cid:16) Q ( c )2 ( f ) − A ( c )2 P ( c )2 ( f ) (cid:17) P ( M = 2)+ ∆ A ( c )2 2 (10)Σ ( c )33 ( ∆A ( c ) , f , f , f ) = (cid:16) Q ( c )1 ( f ) − A ( c )3 P ( c )1 ( f ) (cid:17) P ( M = 1)+ (cid:16) Q ( c )3 ( f ) − A ( c )3 P ( c )3 ( f ) (cid:17) P ( M = 3)+ ∆ A ( c )3 2 (11)Σ ( c )23 ( ∆A ( c ) , f , f , f ) =∆ A ( c )2 (cid:16) P ( M = 3) P ( c )3 ( f ) − P ( M = 1) P ( c )1 ( f ) (cid:17) + ∆ A ( c )3 (cid:16) P ( M = 2) P ( c )2 ( f ) − P ( M = 1) P ( c )1 ( f ) (cid:17) + P ( M = 1) Q ( c )1 ( f ) + ∆ A ( c )2 ∆ A ( c )3 (12)where Q ( c ) m ( f m ) is the average squared payoff, see [7]. B Freidlin-Wentzell theory
We describe in this section the large deviation methods we use to study multimodal attractiondistributions in the steady state of our agents’ learning dynamics. As explained in more details20
50 100 150 200 t . . . . . . . φ multiagents simulationsdeterministic dynamics Figure 7: Comparison between the time series of the aggregate (ratio of buyers to sellers) at thefirst market during a multi-agent simulations (with r = 0 .
01 and 10 agents in each class) and itsevolution under the homogeneous population dynamics. The parameters for the plots in this figureare ( θ , θ , θ ) = (0 . , . , . β = 1 / . p (1) B = 1 − p (2) B = 0 . r will be peaked around the stable fixed pointsof the single agent dynamics. The shape of these peaks becomes Gaussian for r →
0, with acovariance matrix proportional to r that is straightforward to determine. Much more difficult tofind are the weights of the peaks as these involve rare fluctuations of an agent making the transitionfrom one peak to another. In one dimension the problem is tractable as an explicit formula for thesteady state distribution of attractions can be given [6]. In higher dimensions detailed balance [30]would have a similar simplifying effect, but our single agent dynamics in the two-dimensionalattraction space (for each class of agents) does not have this property.In our approach we therefore consider the peak weights in an attraction distribution as a resultof the balance between transitions between the various peaks. We therefore need to find the ratesfor these transitions. To do this, note from the Kramers-Moyal expansion that the single agentlearning is described by a Langevin equation with noise variance O ( r ). For r → reidlin-Wentzell theory We use Freidlin-Wentzell theory in the form developed in [32, 33], which generalises the Eyring-Kramers [34] formula for the rates of noise-activated transitions to non-conservative dynamics. Wegive a brief summary of those aspects of Freidlin-Wentzell theory that we use in our numericalapplication and refer to [31] for a mathematically rigorous description and to [32] for a morestatistical physics-oriented summary.Freidlin-Wentzell theory is concerned with the transition rates between two stable states (here A (cid:63) and A (cid:63) ; below we drop the ∆ from the notation for the attraction differences for brevity) of anon-conservative stochastic dynamics in the low noise limit. A general Langevin equation can bewritten in the form ˙ A ( t ) = µ ( A ( t )) + √ r [ Σ ( A ( t ))] / ξξξ ( t ) (13)where ξξξ ( t ) is white noise with unit covariance matrix. The drift µ and the covariance matrix Σ ofthe noise in the Langevin equation are given in [7] for our learning dynamics. In the generic versionabove we have omitted the superscript ( c ) indicating the class of agents we are considering, as wellas the dependence of drift and noise covariance on the market aggregates.Associated with the Langevin dynamics is an Onsager-Machlup action S [ A ] for any path A ( t ): S [ A ] = (cid:90) t t (cid:16) ˙ A ( t ) − µ ( A ( t )) (cid:17) T Σ − ( A ( t )) (cid:16) ˙ A ( t ) − µ ( A ( t )) (cid:17) d t (14)The action determines the probability of observing any path [ A ( t )] according toΓ → ∼ exp( −S [ A ] /r ) (15)where ∼ means that the equality is true up to a prefactor (which depends on the time discretisationused). The main Freidlin-Wentzell result we need is that the rate Γ → for a transition from A (cid:63) to A (cid:63) ( forward path ) is [31, 35] Γ → ∼ exp( −S (cid:63) → /r ) (16)where S (cid:63) → is the minimal action achievable by any path from A (cid:63) to A (cid:63) in the infinite timeinterval ( t , t ) = ( −∞ , ∞ ). The rate Γ → for the reverse transition from A (cid:63) to A (cid:63) is similarlyΓ → ∼ exp( −S (cid:63) → /r ).The attraction distributions we are after will consist of narrow (for small r ) peaks at A (cid:63) and A (cid:63) . The weights ω and ω of these two peaks, which represent the probability for an agent to bewithin each peak, must then be such that forward and backward transitions balance: ω Γ → = ω Γ → (17) ω ω ∝ exp (cid:18) S (cid:63) → − S (cid:63) → r (cid:19) (18)This expression shows that when the forward and backward minimal actions are not equal, thenone of the two peaks will have an exponentially small weight as r →
0. In practice this is true whenthe action difference inside the exponential in (17) is large compared to r . If it is only of order r or smaller, then we cannot say anything about the weights as we do not determine the prefactor in(17), though we would expect them to be of order unity.22 inding the minimal action path numerically Following the method of Bunin et al. [35], we find the minimal action by discretising the path[ A ( t )], evaluating the action as a function of this discretised path and then minimising with respectto the (discretised) path. The path is discretised into 10 equally spaced timesteps between t = 0and t = 10; we found this choice of parameters to be a reasonable trade-off between the precisionof our result and the complexity of minimising the discretised action.There are other methods for finding the minimal value of the action defined in Eq. (14), suchas solving a Hamilton-Jacobi equation [32], but we chose to use the path discretisation methodbecause we found this to be more robust with respect to changes of model parameters. Thediscretisation approach could also be improved further, using for example the geometric minimumaction method [27], but we found that this was not necessary to achieve the desired precision. Wetested this e.g. by benchmarking against closed-form results that can be obtained for M = 2 [6].The numerical path optimisation can be simplified by restricting attention to the activation part of the path. Generally, for a system with two stable fixed points A (cid:63) and A (cid:63) and one saddlepoint ¯ A between them, the optimal path starting from A (cid:63) will pass through the saddle point¯ A and then relax to A (cid:63) following the relaxation dynamics ˙ A ( t ) = µµµ ( A ( t )), Eq. (14) shows thatthe relaxation dynamics does not contribute to the total action as the integrand (the Lagrangian)vanishes identically along this section of the path. As a consequence, the problem of findinga minimal action path between A (cid:63) and A (cid:63) can be reduced to finding the minimal action pathbetween A (cid:63) and ¯ A , i.e. from the initial fixed point to the saddle. This restriction significantlyimproves the precision of the numerical path optimisation.With the above method, we can work out the action difference between any two fixed points ofthe single agent dynamics, as a function of the market aggregates. The values of these aggregateswhere the action difference between two single agent fixed points vanishes identify the points wherethe steady state attraction distribution of our learning can have more than one peak. Either sideof these values, a single peak is dominant in the attraction distribution; which peak this is changesdiscontinuously at a zero action difference value of the market aggregates. References [1] Peter Gomber, Satchit Sagade, Erik Theissen, Moritz Christian Weber, and Christian West-heide. Competition between equity markets: A review of the consolidation versus fragmenta-tion debate.
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