§2, p.8 — all favored-agent mechanisms are feasible in (MDP)

A principal allocates one good among $I\ge2$ agents $i\in\mathcal I$. Agent $i$ has a type $t_i\in\mathcal T_i=[\underline t_i,\overline t_i]$ with $0\le\underline t_i<\overline t_i$, and inspecting agent $i$ costs $c_i>0$. Write $\mathcal T=\prod_i\mathcal T_i$ and $\mathcal X$ for the feasible mechanisms of (MDP).

Let $(p,q)$ be a favored-agent mechanism with any favored agent $i^\star$ and any threshold value $\nu^\star\in\mathbb R$, whose ties $\max_{i\ne i^\star}(t_i-c_i)=\nu^\star$ are resolved by a rule of type (i) or of type (ii), and whose allocation functions are Borel measurable. Then

that is, $(p,q)$ satisfies (FC) and (IC). In particular, no agent can gain by misreporting his type: the winner gets the good with probability one, a loser who overstates his type wins only in case (ii), where he is inspected and the lie is detected.

This is the feasibility half of every optimality claim about favored-agent mechanisms in the paper (Theorems 3 and 4).

Differences from the source:

• encoding: agents I = {1,…,I} with I ≥ 2 → a finite type ι with 2 ≤ Fintype.card ι; reason: relabelling of the agents; same instances
• encoding: mechanisms with p_i, q_i ∈ L(𝒯,[0,1]) (bounded Borel functions on 𝒯) → p q : ι → (ι → ℝ) → ℝ, Borel measurable on ℝ^ι, with [0,1] bounds required only on 𝒯; reason: every Borel function on the closed box 𝒯 extends to a Borel function on ℝ^ι; all notions depend only on the values on 𝒯
• encoding: max_{i≠i⋆}(tᵢ − cᵢ) compared with ν⋆ → ∀ i ≠ i⋆ (for < and ≤) or ∃ i ≠ i⋆ (for > and ≥) on tᵢ − cᵢ; reason: equivalent for a nonempty finite index set, which I ≥ 2 guarantees
• generalization: case (ii) winner i′ = min argmax_i (tᵢ − cᵢ) (lexicographic rule, p.8) → any maximizer of tᵢ − cᵢ, chosen separately in each scenario, with the mechanism assumed Borel measurable; reason: the lexicographic mechanism is one such mechanism (it is Borel measurable), so the statement specializes to the paper’s
• reading: Definition 3: at max_{i≠i⋆}(tᵢ − cᵢ) = ν⋆ "we are free to define (p(t),q(t)) either as in (i) or as in (ii)" → ties resolved by a type (i) rule or a type (ii) rule (p.8), IsFavoredAgent = type (i) ∨ type (ii); reason: p.8 fixes the tie-breaking rule to type (i) or (ii); a scenario-by-scenario choice is false for the claim: two agents, 𝒯 = [0,1]², c = (1,1), i⋆ = 1, ν⋆ = −1/2, t₂ = 1/2, tie resolved as (ii) at t₁ = 0 (agent 2 wins) and as (i) at t₁ = 1; agent 1 of type 0 gains by reporting 1, violating (IC)