Strategic Interaction in Trend-Driven Dynamics

Abstract

We propose a discrete-time stochastic dynamics for a system of many interacting agents. At each time step agents aim at maximizing their individual payoff, depending on their action, on the global trend of the system and on a random noise; frictions are also taken into account. The equilibrium of the resulting sequence of games gives rise to a stochastic evolution. In the limit of infinitely many agents, a law of large numbers is obtained; the limit dynamics consist in an implicit dynamical system, possibly multiple valued. For a special\ud model, we determine the phase diagram for the long time behavior of these limit dynamics and we show the existence of a phase, where a locally stable fixed point coexists with a locally stable periodic orbit

Measuring Brand Awareness in a Random Utility Model

“…In other words, we impose rational expectations: each agent has a correct belief about others' preferences; moreover, we assume that each agent shares the same expectation about other player's actions f . For a fixed population of N agents, where N → ∞, at least one Nash equilibrium in pure strategies exists (see [6]). The result of this game theoretical setting -when we let N → ∞-is therefore the emergence of a Nash equilibrium characterized by levels x * and s * .…”

“…g All these results are summarised in the next proposition: Proposition 1. Assume a population of N agents as described by equations (3)- (5) where {β, p A , p B , q A , q B } are fixed and where η A and η B have the form (6). Then, at least one Nash equilibrium…”

“…Proof. The existence of Nash equilibria and the convergence of (x * N , s * N ) to the points solving (7) can be deduced from arguments developed in [6]. We now concentrate on the shape of the functions f (x, s) and g(x, s), hence on P A and P B .…”

Strategic Interaction in Interacting Particle Systems

“…In this paper, elaborating on ideas contained in [3], we propose a formulation of what we will refer to as strategic interaction, i.e. a mechanism of interaction between individuals which includes forecasting of other individuals' behavior in the near future.…”

“…Then the resulting stochastic process (m N (σ (n))) n≥0 is tight, and each weak limit point satisfies a.s. the implicit equationm(n) = (1 + m(n − 1))η(k m(n) − k m(n − 1) + µ) +(1 − m(n − 1))η(k m(n) − k m(n − 1) − µ) − 1 =: G 3 (m(n), m(n − 1)). (11)The analysis of the steady state solutions of (11) has been done in[3]. There is a constant A(β µ) ≤ 1+e 2β µ 4 such that:1. the equation y = G 3 (y, x) uniquely determines a function y = ψ(x) around (x, y) = (0, 0).…”