The Opinion Pool

Abstract

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{maxd'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 ∑k 1 λi = 1); construct the pooled density function psλ(θ) = ∑k 1 λip si(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 \begin{equation*}\tag{1.1}u\lbrack d_{s\lambda} \mid p_a(\theta)\rbrack \geqq \min_{i = 1, \cdots, k} u\lbrack d_{si} \mid p_a(\theta)\rbrack.\end{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.