We analyzes the logit dynamics of softmax policy gradient methods. We derive the exact formula for the L2 norm of the logit update vector: $$ \|Δ\mathbf{z}\|_2 \propto \sqrt{1-2P_c + C(P)} $$ This equation demonstrates that update magnitudes are determined by the chosen action's probability ($P_c$) and the policy's collision probability ($C(P)$), a measure of concentration inversely related to entropy. Our analysis reveals an inherent self-regulation mechanism where learning vigor is automatically modulated by policy confidence, providing a foundational insight into the stability and convergence of these methods.

Evolutionary game theory provides the principal tools to model the dynamics of multi-agent learning algorithms. While there is a long-standing literature on evolutionary game theory in strategic-form games, in the case of extensive-form games few results are known and the exponential size of the representations currently adopted makes the evolutionary analysis of such games unaffordable. In this paper, we focus on dynamics for the sequence form of extensive-form games, providing three dynamics: one realization equivalent to the normal-form logit dynamic, one realization equivalent to the agent-form replicator dynamic, and one realization equivalent to the agent-form logit dynamic. All the considered dynamics require polynomial time and space, providing an exponential compression w.r.t. the dynamics currently known and providing thus tools that can be effectively employed in practice. Moreover, we use our tools to compare the agent-form and normal-form dynamics and to provide new "hybrid" dynamics.