The Opinion Pool

When a group of k individuals is required to make a joint decision, it occasionally happens that there is agreement on a utility function for the problem but that opinions differ on the probabilities of the relevant states of nature. When the latter are indexed by a parameter θ, to which probability density functions on some measure μ(θ) may be attributed, suppose the k opinions are given by probability density functions pₛ₁(θ), ⋯, pₛₖ(θ). Suppose that D is the set of available decisions d and that the utility of d, when the state of nature is θ, is u(d, θ). For a probability density function p(θ), write u d p(θ) = ∫ u(d, θ)p(θ) dμ(θ). The Group Minimax Rule of Savage [1] would have the group select that d minimising maxi = 1, ⋯, k'ε D u d' pₛᵢ(θ) - u d pₛᵢ(θ)\. As Savage remarks ([1], p. 175), this rule is undemocratic in that it depends only on the different distributions for θ represented in those put forward by the group and not on the number of members of the group supporting each different representative. An alternative rule for choosing d may be stated as follows: "Choose weights λ₁, ⋯, λₖ (λᵢ 0, i = 1, ⋯, k and ∑ᵏ₁ λᵢ = 1); construct the pooled density function psλ(θ) = ∑ᵏ₁ λᵢpₛᵢ(θ); choose the d, say dsλ, maximising u d psλ(θ)." This rule, which may be called the Opinion Pool, can be made democratic by setting λ₁ = ⋯ = λₖ = 1/k. Where it is reasonable to suppose that there is an actual, operative probability distribution, represented by an `unknown' density function pₐ(θ), it is clear that the group is then acting as if pₐ(θ) were known to be psλ(θ). If pₐ(θ) were known, it would be possible to calculate u dsλ pₐ(θ) and u dₛᵢ pₐ(θ), where dₛᵢ is the d maximising u d pₛᵢ(θ), i = 1, ⋯, k and then to use these quantities to assess the effect of adopting the Opinion Pool for any given choice of λ₁, ⋯, λₖ. It is of general theoretical interest to examine the conditions under which {equation*}{1.1}u dsλ p_a(θ) mini = 1, ⋯, k u dₛᵢ p_a(θ).{equation*} Theorems 2.1 and 3.1 provide different sets of sufficient conditions for (1.1) to hold. Theorem 2.1 requires $k = 2$ and places a restriction on pₐ(θ) (or, equivalently, on pₛ₁(θ) and pₛ₂(θ)); Theorem 3.1 puts conditions on D and u(d, θ) instead.