The Kullback-Liebler Divergence as a Lyapunov Function for Incentive Based Game Dynamics

It has been shown that the Kullback-Leibler divergence is a Lyapunov function for the replicator equations at evolutionary stable states, or ESS. In this paper we extend the result to a more general class of game dynamics. As a result, sufficient conditions can be given for the asymptotic stability of rest points for the entire class of incentive dynamics. The previous known results will be can be shown as corollaries to the main theorem. 1 Information Theory and The Replicator Dynamics Information theory was originally developed by Claude Shannon and Warren Weaver [Sha01, SW49] as a mathematical framework to describe problems in communication including, but not limited to, data compression and storage. They introduced measures of information called entropy. Shannon’s entropy, denoted H(P ), is a measure of the average uncertainty in a random variable, P . It can be interpreted as the average number of bits needed to encode a message drawn i.i.d. from P. Maximizing the entropy can be used to give a lower bound on this average number of bits needed for encryption. For our purposes, the concepts of cross entropy and relative entropy will be of great use. The Kullback-Leibler divergence (KL divergence or DKL) [KL51], or relative entropy is a measure of information gain (loss) from one state to another. More precisely, it is an average measure of the additional bits needed 1In fact, the Shannon entropy is simply the Boltzmann entropy [Jay65] without the constants 1 ar X iv :1 20 7. 00 36 v1 [ m at h. D S] 3 0 Ju n 20 12 to store y given a code optimized to store x. It is defined as