addition, we assume that At = ×=1 At is a non-empty metric space3 for each t. Finally, we
consider only the perfect recall games introduced by Kuhn (1950).
A strategy is defined as follows. For each i = 1,..., I and t = 1,..., T ,let δt be a measure
from θi × A1 × ∙ ∙ ∙ ×At 1 × β(At) to [0,1]. Then, a behavioral strategy δi is an ordered list
of measures δi = (δ1,...,δτ ) such that 1) for each (θ⅛, a1,..., at l) ∈ θ⅛ × A1 × ∙ ∙ ∙ × At l,
δti(θi,a1, ...,at~β ∙) is a probability measure on β(Ati) and 2) for every B ∈ β(Ati), δt(∙; B)
is β(θi) × U-) ∙' ~1β(Ati,)) measurable. The condition 1) requires that each δti(θi,a1,
..., al' 1 : ∙) specify what to play at each information set θ~j ×{(θi, a1, ..., at~1)}. The condition
2) requires that δt allow a well-defined expected utility functional, which is defined later.
Hereafter, we simply call a behavioral strategy a strategy. Let ∏i be the set of strategies
for player i and let ∏ be the set of strategy profiles, that is, ∏ = ×f=1∏⅛. Note that these
definitions originated from Milgrom and Weber (1985) and Balder (1988) and are adapted
to the general multi-period games with observed actions.
A system of beliefs is a set of probabilistic assessments about other players’ types condi-
tional on reaching each of the information sets. It therefore consists of conditional probability
measures over each of the information sets and each measure denotes players’ beliefs about
the others’ types given actions taken before and private information on their own types. Its
formal definition is similar to that for the strategy. For each i and t, let μti be a measure on
θi × A1 ×∙∙∙ ×At 1 × (×j=iβ(θj )) into [0,1]. In addition, for each t, let μt denote (μt1,..., μt1 ).
Then, a system of beliefs μ is an ordered list of measures μ = (μ1, ...,μτ) such that 1) for
3 Therefore, the space θ × A1 × ∙ ∙ ∙ × Aτ is a non-empty metric space. On this space, expected utility
functionals are well-defined according to Ash (1972, 2.6).