Why learning doesn’t add up: equilibrium selection with a composition of learning rules

Abstract

In this paper, we investigate the aggregate behavior of populations of learning agents. We compare the outcomes in homogenous populations learning in accordance with imitate the best dynamics and with replicator dynamics to outcomes in populations that mix these two learning rules. New outcomes can emerge. In certain games, a linear combination of the two rules almost always attains an equilibrium that homogenous learners almost never locate. Moreover, even when almost all weight is placed on one learning rule, the outcome can differ from homogenous use of that rule.Thus, allowing even an arbitrarily small chance of using an alternative learning style can shift a population to select a different equilibrium.

Dynamics of large cooperative pulsed-coupled networks

“…where m is the number of cells of the network. By hypothesis, Inequality (14) holds. Therefore m ≥ 3, which implies m − 1 ≥ √ m. We deduce that:…”

“…, m} (the cells), and whose edges ∆ i,j (the interactions) are directed and weighted. The hypothesis of cooperativity among the individuals, namely ∆ i,j ≥ 0 for all i = j makes the game work in an imitate the best strategy, which produce players that adopt a myopic behaviour ( [14]). In fact, by hypothesis, each cell or player i just knows its own actions to the other players j = i, the value of its own satisfaction variable, and the actions it receives from the other cells.…”

“…In spite that the network can be studied as a coalitional evolutive game, it has some important differences from the models that are usually investigated in Game Theory, even from those that focus on evolutive games (e.g. [14,1,19]): For instance, in our model, each cell or player i decides the instants when it acts in the network, by integrating the states of its own internal dynamics with the actions (or payoffs) that it had received from the other players. So, there does not exist a forcing external mechanism making the players interact sharing a resource.…”

Evolutionary imitative dynamics with population-varying aspiration levels

“…Letg * (t) := max z∈Z g[z](t) and Z * (t) := argmax z∈Z g[z](t).Then g * : R + → R is Lipschitz continuous, and for almost all t ∈ R + , we have that ġ * (t) = g[z](t) for each z ∈ Z * (t).Proof of Lemma 4.2. (i) This is immediate from Lemma 4.1 iii) 9. For ii), observe that, if x i = 0, then xp i = 0 and thus y p i = m px i − x p i = 0 for all p ∈ P.…”

Quantal response equilibria with heterogeneous agents

“…The representative agent is what we can estimate in an experiment. Thus, a representativeagent model enables us to capture aggregate behavior while recognizing underlying 1 Further motivation to consider heterogeneity in a population of quantal responders comes from recent findings that models of heterogeneous learners often cannot be adequately approximated by representative-agent models with common parameter values for all [24,8,6].…”