(EC.1) — the net benefit of testing as a quadratic in the risk

Let $\mathbf s$ be feasible and $i$ untested in $\mathbf s$. Then

This is the one-person net benefit of testing, the source of the two-sided form of the rule.

Differences from the source:

• reading: a set of genes $\mathcal J$ in the model of §2; Assumption 4, Assumption 5 and Theorem 1's proof are single-gene → one gene: $g_i\in\{0,1,2\}$; reason: Theorem 1 compares $\mathbb P_i(\mathbf s)$ with two real thresholds, which needs a scalar; its proof, Assumption 4 and §4.3 are single-gene (pp.15, ec2), and no multi-gene threshold statement is printed
• generalization: the Mendelian prior (5) → an arbitrary mass function on $\{0,1,2\}^{\mathcal I}$; reason: the argument does not use the pedigree and p.11 says priors incorporate all prior information; (5) is the special case given by the theorem eq_5_mendelian_prior
• encoding: a state as a partition $\{s_0,s_1,s_2\}$ of the tested individuals, state space $\mathcal S=\{\mathbf s:\exists g\in\mathcal G(\mathbf s)\}$ → State I := I → Option (Fin 3), with Feasible meaning $\mathbb P(G\in\mathcal G(\mathbf s))>0$; reason: same objects when $\mathcal G$ is the support of the prior; the family is a Fintype, possibly empty
• encoding: $r_i(\mathbb P_i(\mathbf S'))$ with Definition 3 and Assumption 1 → testedReward: $r_i^0(0)$ at result $0$, $r_i^1(1)$ at results $1,2$; reason: after a test the belief is the true carrier status, which is what Assumption 1 and Definition 3 give
• generalization: the reward assumptions of §4 (Assumptions 1–4) → arbitrary real rewards and costs; reason: the identity is algebraic in the data and does not use them; every instance of the source is covered