we define two more items, namely, a system of beliefs and expected utility functionals. We
use all seven items to define the solution concept, namely, the perfect regular equilibrium.
Finally, based on this setting of the general game, we extend the definitions of the Nash
equilibrium and the subgame perfect Nash equilibrium. Consequently, this section is devoted
to defining the setting of the general multi-period game with observed actions and the basic
solution concepts.
In a general multi-period game with observed actions, there are a finite number of players
denoted by i = 1,2,...,/. Each player i has its type θi ∈ Θi and this type is its private
information as in Harsanyi (1967-68). In addition, there exists a state θo ∈ Θo and the
players do not have information about the actual state. Thus, each player has information
about its type θi, but no information about the other players’ types and the state θ i ∈ Θ i =
Θo × (×i'≠iΘi'). We assume that Θ = ×{=oΘi is a non-empty metric space. Realizations θ
∈ Θ are governed by a probability measure η on the class of the Borel subsets2 ×i=oβ(Θi)
of Θ. Given players’ types θi, a conditional probability measure of η exists and is denoted
by η i : Θi × (×j=iβ(Θj)) > [0,1] so that for each θi ∈ Θi and B ∈ ×j=iβ(Θj), η-i(θi; B)
represents a probability of B given θi.
The players play the game in periods t = 1,2, ...,T where T ∈ N U {∞}. In each period
t, all players simultaneously choose actions, and then their actions are revealed at the end of
the period. We assume, for simplicity, that each player’s available actions are independent
of its type so that each player i’s action space in period t is Ati regardless of its type. In
2 Given a metric space X, the class of the Borel sets β(X) is the smallest class of subsets of X such that
i) β(X) contains all open subsets of X and ii) β(X) is closed under countable unions and complements.