Lyapunov Functions for Time-Scale Dynamics on Riemannian Geometries of the Simplex

Abstract

We combine incentive, adaptive, and time-scale dynamics to study multipopulation dynamics on the simplex equipped with a large class of Riemannian metrics, simultaneously generalizing and extending many dynamics commonly studied in dynamic game theory and evolutionary dynamics. Each population has its own geometry, method of adaptation (incentive), and time-scale (discrete, continuous, and others). Using information-theoretic measures of distance we give a widely-applicable Lyapunov result for the dynamics.

Periodic orbit can be evolutionarily stable: Case Study of discrete replicator dynamics

“…It is easy to observe that mixed ESS is the unique HSO(1). In passing, we remark that the concept of incentive stable state equilibrium [22] to describe incentive dynamics is simply HSO(1). We also observe that one could in principle replace states by strategies and use appropriate payoff matrix (U, say) in Inequality (17) to analogously define heterogeneity stable strategy (HSS).…”

“…This is the case for other monotone dynamics in evolutionary game theory as well. In literature there are many different evolutionary dynamical models: some of them can be obtained by varying revision protocol [16,17], e.g., replicator dynamics [14], best response dynamics [18], Brownvon Neumann-Nash dynamics [19], Smith dynamics [20] and logit dynamics [21]; whereas some can be seen as variants of incentive dynamics [22], e.g., replicator dynamics [14], best response dynamics [18], logit dynamics [21] and projection dynamics [23]. All the aforementioned dynamics have the common property of converging towards NE [24][25][26][27].…”

Evolution representation with limited hosting capacities: A comparison between MSD and IEED

“…As a result, the product is not monotonically increasing but is still positive. Authors of [17] proved the stability of the escort evolutionary dynamics with positive monotonically increasing escort functions. Even if it is out of the scope of this paper, it is possible to prove that the following divergence-like function,…”

Stationary Stability for Evolutionary Dynamics in Finite Populations

“…The Fermi process of Traulsen et al is the q-Fermi for q = 1 [25], and q = 0 is called the logit incentive, which is used in, e.g., [42]. The q-replicator incentive has previously been studied in the context of evolutionary game theory [30,33] and derives from statistical-thermodynamic and information-theoretic quantities [43]. Recently human population growth in Spain has been shown to follow patterns of exponential growth with scale-factors q = 1 [44].…”

“…While it is true that on the interior of the probability simply many incentives can be re-written as nonlinear fitness functions, there are significant advantages to the change in perspective from fitness-proportionate selection to incentive proportionate-reproduction. In particular, the authors showed in [33] that incentives ultimately lead to a deeper understanding of evolutionary stability and a substantial improvement in the ability to find Lyapunov functions (via formal and geometric considerations) for a wide-range of evolutionary dynamics. Moreover, when dynamics interact with the boundary, such as innovative and non-forward invariant dynamics, incentives yield a better description of evolutionary stability.…”