Perfect Regular Equilibrium

Suppose the player chooses its strategy δ such that 1) δ¹(α) = δ¹(^) = ɪ and 2) for any
t ≥ 2, δ^t(α¹. ....α^{t 1}: a) = 1 if a^l' ¹ = a and δ^t(a¹. ....a^l' ¹: β) = 1 if a^l' ¹ = β. Then, the
expected utility value with respect to δ is obviously ɪ. However, according to our definition
of the expected utility functional E, E(δ) cannot be ɪ since E(δ^,) = 0 or 1 for any arbitrary
strategy δ^,. Accordingly, we cannot define an expected utility functional for this game. In
fact, this assumption regarding the utility function is a weak requirement in that it is always
satisfied in finite-period games and also satisfied in repeated games that consist of infinitely
repeated finite-period games. Nevertheless, this assumption is so potent that we can define
an expected utility functional by using only finitely iterated integrals.

Based on this expected utility functional, the Nash equilibrium by Nash (1951) and the
subgame perfect Nash equilibrium by Selten (1975) are extended in the general multi-period
games with observed actions. In this paper, we suggest two conditions for rational solution
concepts in the general games. One is the subgame perfect Nash equilibrium condition. The
other is weak consistency introduced by Myerson (1991, 4.3). This weak consistency is a
criterion of a consistent relation between players’ beliefs and players’ actual strategies. A
formal definition of the weak consistency is presented in Section 5.

Definition 1 A strategy profile δ = (δ₁..... δ₁ ) is a Nash equilibrium if δ satisfies E_i(δ) =
maxy._eπ. E_i(δ^l_i. δ~f) for each i < I. A Nash equilibrium is subgame perfect if it induces a
Nash equilibrium in every subgame⁶ .

⁶ For a formal definition of the subgame, please refer to Selten (1975, Section 5).

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