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 $\theta$, to which probability density functions on some measure $\mu(\theta)$ may be attributed, suppose the $k$ opinions are given by probability density functions $p_{s1}(\theta), \cdots, p_{sk}(\theta)$. Suppose that $D$ is the set of available decisions $d$ and that the utility of $d$, when the state of nature is $\theta$, is $u(d, \theta)$. For a probability density function $p(\theta)$, write $u\lbrack d\mid p(\theta)\rbrack = \int u(d, \theta)p(\theta) d\mu(\theta)$. The Group Minimax Rule of Savage [1] would have the group select that $d$ minimising $\max_{i = 1, \cdots, k}\{\max_{d'\epsilon D} u\lbrack d' \mid p_{si}(\theta)\rbrack - u\lbrack d \mid p_{si}(\theta)\rbrack\}$. As Savage remarks ([1], p. 175), this rule is undemocratic in that it depends only on the different distributions for $\theta$ 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 $\lambda_1, \cdots, \lambda_k (\lambda_i \geqq 0, i = 1, \cdots, k$ and $\sum^k_1 \lambda_i = 1)$; construct the pooled density function $p_{s\lambda}(\theta) = \sum^k_1 \lambda_ip_{si}(\theta);$ choose the $d$, say $d_{s\lambda}$, maximising $u\lbrack d \mid p_{s\lambda}(\theta)\rbrack$." This rule, which may be called the Opinion Pool, can be made democratic by setting $\lambda_1 = \cdots = \lambda_k = 1/k$. Where it is reasonable to suppose that there is an actual, operative probability distribution, represented by an `unknown' density function $p_a(\theta)$, it is clear that the group is then acting as if $p_a(\theta)$ were known to be $p_{s\lambda}(\theta)$. If $p_a(\theta)$ were known, it would be possible to calculate $u\lbrack d_{s\lambda} \mid p_a(\theta)\rbrack$ and $u\lbrack d_{si} \mid p_a(\theta)\rbrack$, where $d_{si}$ is the $d$ maximising $u\lbrack d \mid p_{si}(\theta)\rbrack, i = 1, \cdots, k$ and then to use these quantities to assess the effect of adopting the Opinion Pool for any given choice of $\lambda_1, \cdots, \lambda_k$. 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 $p_a(\theta)$ (or, equivalently, on $p_{s1}(\theta)$ and $p_{s2}(\theta)$); Theorem 3.1 puts conditions on $D$ and $u(d, \theta)$ instead.