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The adaptive learning mechanism is implemented as follows The strategy π, is associated with a score, U, via a monotonic function that tends to 0as the score tends to negative (positive) infinity. The specific function we use is π, = eΓUi+/1+eΓUi+, where Γ is the learning ratesimilarly for the strategy π, The scores are updated at each period by adding (, R, p, − 1) to, U, if, k, = +1 and to U, if kiω= −1
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More precisely, the conditions for stationarity, in terms of the dynamic variables π, are equivalent to the first-order conditions for the minimization of a function, H, In the limit Γ → 0 it is also possible to derive a continuum time dynamics that admits a Lyapunov function, H, i.e., a function of π, which is bounded from below and monotonically decreasing along the trajectories of the dynamics. Hence, the asymptotic dynamics is described by the minima of, H, which, for, N, → ∞, is simply the squared, distance H, p, R,) of prices from returns
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ω) of prices from returns.
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To study analytically the minima of, H, and thus the long-term dynamics of the system, we use the replica method, originally developed to describe disordered magnetic alloys (so-called spin-glasses). We consider, H, as the energy function in a statistical mechanics problem, where the variables π, play the role of phase-space variables, like spins in the model of a magnet. The minima of, H, are found by taking, the “temperature” in this problem to zero. The same analysis applies when the wealth of agents evolves. Now the phase space variables are, z, w, z, w, Because unlike π, these variables are not bounded, the phase space is largerconsequently the value of α, is larger with respect to the model with constant wealth. Some details on the analytical calculations may be found in [24]
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c is larger with respect to the model with constant wealth. Some details on the analytical calculations may be found in [24].
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