Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 42 / 1

Table of contents
- Solution 1

Solution

Use the usual independent private values model, quasilinear utility, and voluntary participation with zero outside utility. These assumptions matter: without individual rationality, arbitrary type-independent entry charges make revenue unbounded, and correlated types cannot in general be described by their marginal priors alone.

The revelation principle lets us optimize over direct revelation mechanisms satisfying Bayesian incentive compatibility. Write

x (v) \in [0, 1]

for the common project allocation,

X_{i} (t) = E_{V_{- i}} x (t, V_{- i})

for player

i

's interim allocation, and

P_{i} (t)

for its interim payment. The interim payment identity gives

P_{i} (t) = t X_{i} (t) - \int_{\underline{v}_{i}}^{t} X_{i} (s) d s - U_{i} (\underline{v}_{i}) .

(1)

Since interim individual rationality requires

U_{i} (\underline{v}_{i}) \geq 0

, the virtual-surplus revenue identity bounds expected revenue by

E \sum_{i} P_{i} (V_{i}) \leq E [x (V) \sum_{i} ϕ_{i} (V_{i})], ϕ_{i} (t) = t - \frac{1 - F _{i} ( t )}{f _{i} ( t )} .

(2)

The best feasible common allocation at each valuation profile therefore provides the project when total virtual surplus is nonnegative:

x^{*} (v) = 1_{{\sum_{i} ϕ_{i} (v_{i}) \geq 0}} .

(3)

Because each regular prior has a nondecreasing virtual valuation, this allocation is a nondecreasing function of each player's report. Hold

v_{- i}

fixed and charge the critical-value payment

p_{i}^{*} (v) = v_{i} x^{*} (v) - \int_{\underline{v}_{i}}^{v_{i}} x^{*} (t, v_{- i}) d t .

(4)

This is the winning threshold when it lies in the support, the lowest allowed value if every type wins, and zero if the player loses. A truthful winner never pays more than its value; a losing type cannot profit by crossing the threshold. Thus the mechanism has dominant-strategy incentive compatibility and ex post individual rationality, with zero utility at every lowest type. It attains the revenue bound, proving optimality even among mechanisms requiring only Bayesian incentive compatibility. At a zero-virtual-surplus tie, choose any fixed rule that preserves monotonicity.

For independent uniform distributions on