Abstract
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 $${(\vec{\sigma},\mu)}$$ 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 $${\sigma_i'}$$ assigns positive, although possibly infinitesimal, probability to all actions at every information set) that differs only infinitesimally from $${\vec{\sigma}}$$ such that at each information set I for player i, σ i is an $${\epsilon}$$ -best response to $${\vec{\sigma}'_{-i}}$$ 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 $${\vec{\sigma}'_{-i}}$$ conditional on having reached I. Yet another variant is used to characterize proper equilibrium.
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