Bribeproof mechanisms for two-values domains

Schummer (Journal of Economic Theory 2000) introduced the concept of bribeproof mechanism which, in a context where monetary transfer between agents is possible, requires that manipulations through bribes are ruled out. Unfortunately, in many domains, the only bribeproof mechanisms are the trivial ones which return a fixed outcome. This work presents one of the few constructions of non-trivial bribeproof mechanisms for these quasi-linear environments. Though the suggested construction applies to rather restricted domains, the results obtained are tight: For several natural problems, the method yields the only possible bribeproof mechanism and no such mechanism is possible on more general domains.

💡 Research Summary

The paper tackles the problem of designing mechanisms that are immune not only to strategic misreporting (strategy‑proofness) but also to manipulations through side‑payments (bribe‑proofness) in quasi‑linear environments where agents can transfer money among themselves. While previous work (notably Schummer 2000) showed that in unrestricted or sufficiently rich domains the only bribe‑proof mechanisms are trivial ones that always output a fixed outcome, this paper demonstrates that non‑trivial bribe‑proof mechanisms do exist when the agents’ private cost‑per‑unit types are restricted to two possible values.

Model and Definitions
There are n ≥ 2 agents, each receiving an allocation a ∈ ℝⁿ₊ of “work”. Agent i’s cost for allocation a is a_i·θ_i, where the private type θ_i belongs to a publicly known domain Θ_i. In a two‑values domain each Θ_i = {L, H} with L < H. A mechanism (A, p) maps reported type profiles θ̂ to an allocation A(θ̂) and a payment vector p(θ̂). Bribe‑proofness requires that for any pair of agents i, j and any deviation of i’s reported type, the sum of utilities of i and j cannot increase; formally inequality (1) in the paper. Setting i = j recovers ordinary strategy‑proofness. Strong bribe‑proofness extends the condition to simultaneous deviations by both i and j.

λ‑Linear Mechanisms
The authors propose a simple family of mechanisms called λ‑linear mechanisms. For a fixed λ ∈