Supplementary Material: Market Scoring Rules Act As Opinion Pools For Risk-Averse Agents

Here, we present detailed proofs of all theorems in the main paper, as well as some observations and experiments which we had to exclude from the main paper owing to paucity of space. 1 Model and definitions Here, we provide the proof of Lemma 1 from Section 2 of the main paper. Restatement of Lemma 1. For a two-ouctome forecasting problem where an expert’s report can be specified in terms of a single probability p ∈ [0, 1], if f2(r1, r2) and fn−1(q1, q2, . . . , qn−1) are valid opinion pools for two probabilistic reports r1, r2 and n − 1 probabilistic reports q1, q2, . . . , qn−1 respectively, then f(p1, p2, . . . , pn) = f2(fn−1(p1, p2, . . . , pn−1), pn) is also a valid opinion pool for n reports. Proof. Recall from Definition 1 in the main paper that a valid opinion pool p̂ = φ(p1, p2, . . . , pm), where p1, p2, . . . , pm ∈ [0, 1] are reported expert probabilities of occurrence of binary event X , must satisfy 1. Unanimity: If pi = p ∀i = 1, 2, . . . ,m, then p̂ = p. 2. Boundedness: min{p1, p2, . . . , pm} ≤ p̂ ≤ max{p1, p2, . . . , pm}. 3. Monotonicity: p̂ increases monotonically as pi increases, pj being held constant ∀j 6= i, i = 1, 2, . . . ,m, i.e. ∂φ ∂pi > 0 everywhere ∀i. By the condition of the lemma, all the above three properties are possessed by each each of f2 and fn−1, and we need to prove that f has each of these properties, too. To prove the unanimity of f : Let pi = p ∀i = 1, 2, . . . , n. Then, f(p, p, . . . , p) = f2(fn−1(p, p, . . . , p), p) = f2(p, p), by unanimity of fn−1, = p, by unanimity of f2.