A nonstandard characterization of sequential equilibrium, perfect equilibrium, and proper equilibrium

New characterizations of sequential equilibrium, perfect equilibrium, and proper equilibrium are provided that use nonstandard probability. It is shown that there exists a belief system μ such that $(\stackrel{\rightharpoonup }{\sigma },\mu )$ is a sequential equilibrium in an extensive game with perfect recall iff there exist an infinitesimal $\epsilon$ and a completely mixed behavioral strategy profile σ′ (so that ${\sigma }_i\prime$ assigns positive, although possibly infinitesimal, probability to all actions at every information set) that differs only infinitesimally from $\stackrel{\rightharpoonup }{\sigma }$ such that at each information set I for player i, σ i is an $\epsilon$ ‐best response to $\stackrel{\rightharpoonup }{\sigma }{\prime }_{‐i}$ conditional on having reached I. Note that the characterization of sequential equilibrium does not involve belief systems. There is a similar characterization of perfect equilibrium; the only difference is that σ i must be a best response to $\stackrel{\rightharpoonup }{\sigma }{\prime }_{‐i}$ conditional on having reached I. Yet another variant is used to characterize proper equilibrium.