Distributional Stability and Deterministic Equilibrium Selection under Heterogeneous Evolutionary Dynamics

Transcription

1 Learning, Evolution and Games 2018 (Lund, Sweden) June 2018 Distributional Stability and Deterministic Equilibrium Selection under Heterogeneous Evolutionary Dynamics Dai Zusai Philadelphia, U.S.A. Introduction Highlights Basic idea Heterogeneous population game Nonaggregability Distributional stability Put to work What dynamic? Selected equilibrium Implications Interpretation of the model & results Literature review

2 Introduction Zusai, Distributional Stability 2 Highlights Deterministic evolutionary dynamics in a population of heterogeneous agents Two layers to describe the current state of the population: Strategy composition x: action distribution conditional on type (or, joint dist of types & actions) Aggregate strategy x Ex: action distribution among all the agents, regardless of types. Pay attention to a subtle difference in topologies/local stability concepts Composition x is locally stable xconverges to x as long as x is close enough to x Then, aggregate strategy x Exalso converges to close to x Ex Even if aggregate strategy x is close to x, x may not converge to a locally stable x. It may be typically the case when x is not sorted enough. x is distributionally stable x converges to x as long as x is close enough to x regardless of the underlying composition x. Here I propose distributional stability as a criterion for equilibrium selection In a binary coordination game with random matching, risk dominant eqm is selected. More selected by making the dynamic (esp. switching rates) more sensitive to payoff differences.

3 Basic idea Zusai, Distributional Stability 3 Population game with heterogeneous payoffs A binary-choice large population game: A society consists of a unit mass of continuously many agents ω Ω: P Ω 1. Each agent is assigned to a type Θand chooses either action In or Out: a I, O. Strategy composition x: Θ Δ : x is an action distribution among type- agents. Aggregate strategy x Ex Δ is an action distribution among all agents in Ω. Notation: Δ a simplex on R ; P probability measure on Θ; E aggregator/expected value: x Ex x dp Aggregate game: F x R is the payoff vector for type When assuming additive separability of payoff heterogeneity, F x F x and F x. x is an equilibrium composition x 1 0 if F Ex x is an aggregate equilibrium x PF x (under continuity). x: Θ Δ Strategy composition Shares Θ type Aggregate In x x Out x 1x x 1x mass dp 1 x Δ Aggregate strategy

4 Basic idea Zusai, Distributional Stability 4 Zusai Heterogeneity and aggregation in evolutionary dyn (a companion paper) Payoff vector Revision protocol r, R π π Str composition x x Action distribution Game (Payoff function) π Fx Aggregation x Ex Aggregability: The transition of aggregate strategy x is completely identified from the current state of the aggregate strategy x, independently of composition x. If individual agents adopt standard/perturbed best response for their revision protocols, switching rate: constant Aggregable under heterogeneity x reduces to a homogenized pbrd: heterogeneity is only transitory (i.i.d.), just perturbation to BR Other revision protocols: Individual agent s switching rate r Aggregate strategy Deterministic mean dynamic x, x, r, x, r, (Ely & Sandholm, 2005 GEB) switching rate payoff gain from a switch Nonaggregable Incentive matters for when to switch, not only what to switch However, in a potential game, Yet, convergence may depend on x, not only x. x Ex locally stable in the homogenized pbrd x local max of potential function x locally stable in any (admissible) hetero dynamics In H&A paper, the assumption of binary actions or additive separability is not needed; agents can have different revision protocols. x

5 Basic idea Zusai, Distributional Stability 5 Equilibrium selection by distributional stability Definition: distributional stability x Ex is distributionally stable under a heterogeneous dynamic, if Ex converges to x whenever Ex is close enough to x (regardless of how far the composition x is from x ). Ex is distributionally stable x locally stable Under aggregable dynamics, they are equivalent. Use distributional stability for a selection criterion 1. First, find locally stable aggregate equilibria from a homogenized pbrd, assuming an aggregable dynamic as the underlying heterogeneous dynamic. 2. Then, consider a nonaggregable dynamic. Choose an equilibrium by distributional stability. That is, check convergence from an unsorted composition. What nonaggregable dynamic to use? Even under a nonaggregable dynamic, all the locally stable equilibria may remain distributionally stable. To strengthen the selection power, the switching rate should become more sensitive to payoffs.

6 Put to work Zusai, Distributional Stability 6 What dynamic to use? Tempered BRD (Zusai, 2018, IJGT) the rate to switch from action a to the optimal action is an increasing function Qπ of the payoff gain π For a general tbrd, we obtain a sufficient condition for distributional stability. (See the paper.) Parameterize payoff sensitivity: Qπ strictly increases until π reaches q q 0 Q π constant for all π the tbrd reduces to the standard BRD. By raising q, we can select one distributionally stable eqm. Consider a binary coordination game and introduce payoff heterogeneity. Assume additive separability of payoff heterogeneity, and linearity of payoff function in x (random matching). When one equilibrium is finally selected by increasing q, it is the one perturbed from the risk dominant eqm. In other words, risk dom eqm is dynamically stable even if there is heterogeneity in agents payoffs, their initial choices are not sorted according to the payoffs, and agents adjust their choices in a way sensitive to payoff differences. max all actions π π 1 Q Standard BRD Payoff PayoffInsensitive Insensitive sensitive sensitive to payoff to payoff q q π

7 Implications Zusai, Distributional Stability 7 Interpretation of the model & results No revisit unlike stochastic evolution, steady escape from dist lly unstable eqm Possibly found and checked from simulations Selection by checking transitions from unsorted strategy compositions Interpretation of the initial unsorted composition: a large preference shock (but no immediate change in the choices) from the state in which they were settled to eqm/or, simply introducing an economic incentive scheme to start a game Starting from the extreme: similar to the idea of contagion in global games? But the process of contagion is visible through off-equilibrium revisions of strategies, unlike the introspective reasoning process behind global game theory. Sorting pressure by individual rational choices Payoff-sensitive dynamic: They gradually adjust their choices based on the new payoffs, in a way that more discontented agents move earlier than those less discontented. The tbrd requires rationality/payoff-based decision not only to choices of new actions (revision protocols) but also to timing of revisions.

8 Implications Zusai, Distributional Stability 8 Comparison with preceding models Preceding models of deterministic eqm selection: homogeneous population, BRD with a constant switching rate Kreindler & Young, Fast convergence in evolutionary equilibrium selection (GEB, 2013). Robustness to greater noise level in perturbed BRD (adding dynamics to McKelvey & Palfrey s eqm selection by QRE) Note that, in the perturbed BRD, the revision rate is constant and, once an agent gets observation of perturbed payoffs, the agent must switch to the perturbed BR, just like the standard BRD. Oyama, Sandholm & Tercieux, Sampling BRD (TE, 2015) Robustness to errors by limited finite sampling for an agent s estimate of action distribution (to calculate payoffs) The selection criterion here may be comparable with Kreindler & Young s. KY: increasing the degree of noise level, while keeping the constant revision rate Zusai: making the revision rate more sensitive to payoff gains from revisions. while the payoff heterogeneity (the distribution of idiosyncratic payoffs) fixed, Possibly, after selecting an eqm, the degree of payoff heterogeneity can be reduced. KY: rigorously implying stochastic stability in logit evolution in finite population, thanks to the deterministic approximation theorem (Benaim & Weibull) for dynamics on a finite strategy space. Zusai: possible but needs such a theorem for a continuous strategy space.