Crowd-anticrowd theory of multi-agent market games

Abstract

We present a dynamical theory of a multi-agent market game, the so-called Minority Game (MG), based on crowds and anticrowds. The time-averaged version of the dynamical equations provides a quantitatively accurate, yet intuitively simple, explanation for the variation of the standard deviation ('volatility') in MG-like games. We demonstrate this for the basic MG, and the MG with stochastic strategies. The time-dependent equations themselves reproduce the essential dynamics of the MG.Agent-based games have great potential application in the study of fluctuations in financial markets. Challet and Zhang's Minority Game (MG) [1,2] offers possibly the simplest example and has been the subject of much research [2]. The MG comprises an odd number of agents N choosing repeatedly between option 0 (e.g. buy) and option 1 (e.g. sell). The winners are those in the minority group, e.g. sellers win if there is an excess of buyers. The outcome at each timestep represents the winning decision, 0 or 1. A common bit-string of the m most recent outcomes is made available to the agents at each timestep [3]. The agents randomly pick s strategies at the beginning of the game, with repetitions allowedeach strategy is a bit-string of length 2 m which predicts the next outcome for each of the 2 m possible histories. Agents reward successful strategies with a (virtual) point. At each turn of the basic MG, the agent uses her most successful strategy, i.e. the one with the most virtual points. Here we develop a dynamical theory for MG-like games based on the formation of crowds and anticrowds.The number of agents holding a particular combination of strategies can be written as a D × D × . . . (s terms) dimensional matrix Ω, where D is the total number of available strategies. For s = 2, this is simply a D × D matrix where the entry (i, j) represents the number of agents who picked strategy i and then j. The strategy labels are given by the decimal representation of the strategy plus unity, for example the strategy 0101 for m = 2 has strategy label 5+1=6. Ω is fixed at the beginning of the game ('quenched disorder') and can represent either the full strategy space or the reduced strategy space [1], depending on the choice of D. Σ is another time-independent matrix, containing all the strategies in the required space in their binary form: Σ r,h+1 describes the prediction of strategy r given the history h (where h is the decimal corresponding to the m-bit binary history string).We introduce a vector n(t): this contains the number of agents using each strategy at time t, in order of increasing strategy label. The vector S(t) contains the virtual score for each strategy at time t in order of increasing strategy label. The vector R(t) lists the strategy label in order of best-to-worst virtual points score at time t; if any strategies are tied in points then the strategy with the lower-value label is listed first. The vector ρ(t) shows the rank of the strategy listed in order of increasing strategy label at time t. Hence R(t) and ρ(t) can...

Statistical mechanics of socio-economic systems with heterogeneous agents

“…5.3. Other possibilities, like the 'crowd-anticrowd' theory [50] will not be discussed here (an account can be found in [6]). It is helpful for a start to introduce the auxiliary variables [47] …”

Wealth inequality in the minority game

“…Our numerical simulation shows that both the maximal cooperation point and the point of maximum wealth inequality occur around 2 M+1 ≈ N S. This confirms our suspicion that the apparent cooperation of players shown in the σ 2 (A) does not tell us the complete story. In fact, we are able to explain the trend of a modified Gini index qualitatively using the crowd-anticrowd theory [12,13,14]. In particular, we find that the cooperation comes along with wealth inequality partially because poorly-performing players cannot change their strategies in the MG.…”

“…On the other hand, those players picking the crowd of high ranking strategies have a higher winning probability and keep on using those strategies. Note that the ranking of the strategies is almost unchanged when α ≈ α c [12,13,14]. As a result, the wealth distribution of players would become relatively diverse and the Gini index G Ξ of the population attains its maximum value when α → α c .…”

How to attain maximum profit in minority game?

“…Since the dynamics of MG minimizes a global function related to market predictability, we may regard MG as a disordered spin glass system [7,8]. Recently, Hart et al introduced the socalled crowd-anticrowd theory to explain the dynamics of MG [9,10]. Their theory stated that fluctuations arised in the MG is controlled by the interplay between crowds of like-minded agents and their perfectly anti-correlated partners.…”

“…In each of these strategy ensemble, the action of a strategy is counter-balanced by its anticorrelated strategies. Therefore, the step size of the random walk of a mutually anti-correlated strategy ensemble is equal to the difference between the number of players using a single strategy from the mean number of players using the strategies in this ensemble [9,10]. This random walk idea can be readily extended to the case of multiple alternatives.…”