On the discrete-time origins of the replicator dynamics: From convergence to instability and chaos

We consider three distinct discrete-time models of learning and evolution in games: a biological model based on intra-species selective pressure, the dynamics induced by pairwise proportional imitation, and the exponential / multiplicative weights (EW) algorithm for online learning. Even though these models share the same continuous-time limit - the replicator dynamics - we show that second-order effects play a crucial role and may lead to drastically different behaviors in each model, even in very simple, symmetric $2\times2$ games. Specifically, we study the resulting discrete-time dynamics in a class of parametrized congestion games, and we show that (i) in the biological model of intra-species competition, the dynamics remain convergent for any parameter value; (ii) the dynamics of pairwise proportional imitation exhibit an entire range of behaviors for larger time steps and different equilibrium configurations (stability, instability, and even Li-Yorke chaos); while (iii) in the EW algorithm, increasing the time step (almost) inevitably leads to chaos (again, in the formal, Li-Yorke sense). This divergence of behaviors comes in stark contrast to the globally convergent behavior of the replicator dynamics, and serves to delineate the extent to which the replicator dynamics provide a useful predictor for the long-run behavior of their discrete-time origins.Comment: 22 pages, 8 figure

Two results on entropy, chaos, and independence in symbolic dynamics

We survey the connections between entropy, chaos, and independence in topological dynamics. We present extensions of two classical results placing the following notions in the context of symbolic dynamics: 1. Equivalence of positive entropy and the existence of a large (in terms of asymptotic and Shnirelman densities) set of combinatorial independence for shift spaces. 2. Existence of a mixing shift space with a dense set of periodic points with topological entropy zero and without ergodic measure with full support, nor any distributionally chaotic pair. Our proofs are new and yield conclusions stronger than what was known before.Comment: Comments are welcome! This preprint contains results from arXiv:1401.5969v

Measuring income inequality in social networks

We present a new index for measuring income inequality in networks. The index is based on income comparisons made by the members of a network who are linked with each other by direct social connections. To model the comparisons, we compose a measure of relative deprivation for networks. We base our new index on this measure. The index takes the form of a ratio: the network’s aggregate level of relative deprivation divided by the aggregate level of the relative deprivation of a hypothetical network in which one member of the network receives all the income, and it is with this member that the other members of the network compare their incomes. We discuss the merits of this representation. We inquire how changes in the composition of a network affect the index. In addition, we show how the index accommodates specific network characteristics

Consensus income distribution

In determining the optimal redistribution of a given population’s income, we ask which factor is more important: the social planner’s aversion to inequality, embedded in an isoelastic social welfare function indexed by a parameter alpha, or the individuals’ concern at having a low relative income, indexed by a parameter beta in a utility function that is a convex combination of (absolute) income and low relative income. Assuming that the redistribution comes at a cost (because only a fraction of a taxed income can be transferred), we find that there exists a critical level of beta below which different isoelastic social planners choose different optimal allocations of incomes. However, if beta is above that critical level, all isoelastic social planners choose the same allocation of incomes because they then find that an equal distribution of incomes maximizes social welfare regardless of the magnitude of alpha

A class of proximity-sensitive measures of relative deprivation

We introduce a new class of generalized measures of relative deprivation. The class takes the form of a power mean of order p. A characteristic of the class is that depending on the value of the proximity-sensitive parameter p, the class is capable of accommodating both a decreasing weight (the case of p > 1), and an increasing weight (the case of p ∈ (0,1)) accorded to given changes in the incomes of the individuals who are wealthier than the reference individual, depending on their proximity in the income distribution to the reference individual

Family of chaotic maps from game theory

From a two-agent, two-strategy congestion game where both agents apply the multiplicative weights update algorithm, we obtain a two-parameter family of maps of the unit square to itself. Interesting dynamics arise on the invariant diagonal, on which a two-parameter family of bimodal interval maps exhibits periodic orbits and chaos. While the fixed point b corresponding to a Nash equilibrium of such map f is usually repelling, it is globally Cesàro attracting on the diagonal, that is, limn→∞1n∑n−1k=0fk(x)=b for every x∈(0,1). This solves a known open question whether there exists a ‘natural’ nontrivial smooth map other than x↦axe−x with centres of mass of all periodic orbits coinciding. We also study the dependence of the dynamics on the two parameters

Follow-the-Regularized-Leader Routes to Chaos in Routing Games

We study the emergence of chaotic behavior of Follow-the-Regularized Leader (FoReL) dynamics in games. We focus on the effects of increasing the population size or the scale of costs in congestion games, and generalize recent results on unstable, chaotic behaviors in the Multiplicative Weights Update dynamics to a much larger class of FoReL dynamics. We establish that, even in simple linear non-atomic congestion games with two parallel links and \emph{any} fixed learning rate, unless the game is fully symmetric, increasing the population size or the scale of costs causes learning dynamics to becomes unstable and eventually chaotic, in the sense of Li-Yorke and positive topological entropy. Furthermore, we prove the existence of novel non-standard phenomena such as the coexistence of stable Nash equilibria and chaos in the same game. We also observe the simultaneous creation of a chaotic attractor as another chaotic attractor gets destroyed. Lastly, although FoReL dynamics can be strange and non-equilibrating, we prove that the time average still converges to an \emph{exact} equilibrium for any choice of learning rate and any scale of costs