• Consider a random variable ˜θ that is drawn from a distribution F, which support is [θ, θ]
with 0 ≤ θ < θ ≤ 1. • Other parameters: ω ∈ [0, 1], ∆ > 0, and σ ∈ (0, ∆). (Remark: without loss, we can set
∆ = 1).
• We want to compare the profits (defined below) from two possible cases.
• Case 1:
– For a given θP ∈ [θ, θ], we can compute η =
R θ
θP
(1−θ)dF(θ)
R θ
θP
θdF(θ)
, and we can find θA from
the following equation:
σ
∆
Z θ
θA
θ dF(θ) = θP
1 − θP
ω
Z θ
θP
(1 − θ) dF(θ) + (1 − ω)
Z θ
θP
θ dF(θ).
(Note the LHS is monotonic in θA, so there is at most one solution of θA. If for a θP
there is no solution of θA, we do not define profit for this θP .)
– For a given θP , profit is σηθA(1 − F(θA)). So we can write profit as a function of θP
because η and θA are functions of θP .
– We want to find the θP that achieves the maximal profit subject to the constraint that
θP < θA.
• Case 2:
– For a given θP ∈ [θ, θ], we can compute η =
R θ
θP
(1−θ)dF(θ)
R θ
θP
θdF(θ)
, and we define
p = θP
(ω∆ + σ) ·
R θ
θP
(1 − θ)dF(θ)
R θ
θP
θdF(θ)
− (1 − θP ) (−(1 − ω)∆ + σ).
– For a given θP , profit is p(1 − F(θP )). So we can write profit as a function of θP .
– We want to find θP that achieves the maximal profit subject to the constraint that
p < σηθP
Goal: Prove/disprove that Case 1 always dominates Case 2, i.e., the maximal profit in case 1
is greater than or equal to the maximal profit in case 2.
• Ideally, we can prove/disprove the statement analytically.
• You might want to try numerical analysis first.