Statistical physics of adaptive correlation of agents in a market

Recent results and interpretations are presented for the thermal minority game, concentrating on deriving and justifying the fundamental stochastic differential equation for the microdynamics. Market economics poses several problems of potential interest and challenge to statistical physics, involving the co-operative behaviour of many agents whose actions involve mutual frustration and disorder, both quenched and stochastic. In a nutshell, speculators in an idealized stock market are made up of buyers and sellers, each having personal gain as their objectives, trying to buy low and sell high, making their decisions based on commonly available information using individual strategies, with their collective actions determining the (time-varying) ‘right choices’ and learning from experience. From the point of view of the market regulator, however, preference is for low volatility and market efficiency. The minority game (MG) is a simple encapsulation of some of the ingredients and issues of a market. It consists of N agents each of whom at each step of a parallel dynamical process makes either of two choices, with the objective of being in the minority overall. The agents have no direct knowledge of one another and make their decisions based on purely global information ~ I(t), available equally to all. Their decisions are determined through the application to ~ I(t) of individual strategy functions, each agent having a small number of such strategies, drawn randomly and independently from a large distribution at the outset and fixed throughout the game. At each time-step each agent employs (just) one of his or her strategies. Adaptation occurs through the development of functions which determine their choices of strategy. In the original formulation [1] the information ~ I(t) was the minority choice over the last m time-steps and the adaptation was achieved through the cumulative award of points at each time-step to the strategies which would have yielded the actual minority choice at that step. The strategy played by any agent at any timestep was that of his/her strategies which currently had the largest point-score. A remarkable observation in simulations [2] was that the variance in the minority choice became smaller than that of random choice for large enough m, indicating correlation of the agents’ actions. A critical memory length mc was observed for minimum variance, with agents appearing to be frozen in their choices for m > mc, non-frozen for m < mc. Moreover, it was shown that the dependence on m was through the scaling variable D/N , where D = 2 was the dimension of the space of strategies [2]. Further simulations showed (i) these results are unaffected by replacing the true history by a random ~ I(t) [3], indicating that as far as macroscopic observables are concerned the ‘information’ merely effectuates the correlation; (ii) replacing the deterministic strategy-choices by stochastic ones can significantly reduce the volatility for information vectors of less than the critical length [4]. Here we consider the determination of a fundamental analytic theory and report the derivation of the underlying stochastic differential equation for the microdynamics [5]. We concentrate on a continuous formulation in which ~ I(t) is a stochastically randomly chosen unit-length vector on a D-dimensional hypersphere, the strategies are quenched random vectors of length √ D in the same space, ~ R i , i = 1, . . . , N labeling the agents and the α = 1, . . . , s their strategies. The analogues of the binary choices above are bids bi (t) = ~ R i · ~ I(t). The strategies which are actually used are indicated by ~ R i (t). The total bid at time t is A(t) = ∑ i ~ R i (t) · ~ I(t). The point update rule is P α i (t+ 1) = P α i (t)− bi (t)A(t)/N. (1) For simplicity we specialize to s = 2 and define ~ ξi ≡ (~ R i − ~ R i )/2, ~ωi ≡ (~ R i + ~ R i )/2; pi(t) = P 1 i (t)− P 2 i (t) (2) In a generalized thermal minority game (TMG) the probability of strategy use is π i (t) ≡ [1 + exp(∓βf(pi(t)))] (3) and it is useful to define a ‘spin’ si(t) ≡ π i (t)− π i (t) = tanh(βf(pi(t))). (4) In [4] the choice f(p) = p was employed, but here we consider f(p) = sgn(p) ≡ z [5]. We are interested in coarse-grained average behaviour on a time-scale greater than the step-length in order to pass to a continuum-time theory. Equivalently, we take a time-scale ∆t with ~ I(t) a differential random noise ~ I(t) = ∆ ~ W (t) with zero mean and variance ∆t. In the limit ∆t → ∞ a Kramers-Moyal expansion yields [5] dpi(t) = −(ND) ~ R i (t) · ~ ξidt+O(dt) (5) so that to O(dt) the information noise has been eliminated in favour of an effective interaction between the agents and the averaged variance becomes σ ≡ N〈A(t)2〉 = (ND) ∑ ij 〈R i (t) · R j (t)〉, (6) where the 〈·〉 refer to a temporal average and the bar to an average over the quenched disorder of the strategies. At T = 0 Eqs. (2) are deterministic and to leading order in dt reduce to dp/dt = −∇sH|s=z ; p ≡ (p1, ...pN) (7)