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, to which probability density functions on some measure () may be attributed, suppose the k opinions are given by probability density functions pₒ₁ (), , pₒ₊ (). Suppose that D is the set of available decisions d and that the utility of d, when the state of nature is, is u (d, ). For a probability density function p (), write u d p () = u (d, ) p () d (). The Group Minimax Rule of Savage 1 would have the group select that d minimising ₈ = ₁, , ₊\₃' ₃ u d' pₒ₈ () - u d pₒ₈ () \. As Savage remarks (1, p. 175), this rule is undemocratic in that it depends only on the different distributions for 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 ₁, , ₖ (ᵢ 0, i = 1, , k and ᵏ₁ ᵢ = 1) ; construct the pooled density function pₒ () = ᵏ₁ ᵢpₒ₈ () ; choose the d, say dₒ, maximising u d pₒ (). " This rule, which may be called the Opinion Pool, can be made democratic by setting ₁ = = ₖ = 1/k. Where it is reasonable to suppose that there is an actual, operative probability distribution, represented by an `unknown' density function pₐ (), it is clear that the group is then acting as if pₐ () were known to be pₒ (). If pₐ () were known, it would be possible to calculate u dₒ pₐ () and u dₒ₈ pₐ (), where dₒ₈ is the d maximising u d pₒ₈ (), i = 1, , k and then to use these quantities to assess the effect of adopting the Opinion Pool for any given choice of ₁, , ₖ. It is of general theoretical interest to examine the conditions under which equation*1. 1u dₒ pₐ () ₈ = ₁, , ₊ u dₒ₈ pₐ (). 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 () (or, equivalently, on pₒ₁ () and pₒ₂ () ) ; Theorem 3. 1 puts conditions on D and u (d, ) instead.