This appendix presents formal mathematical proofs for the theorems and statements introduced in Sections 4–7 of the TFRM study, validating the redistribution and truthfulness constraints in payment refund mechanisms.
Preliminaries and notation We consider a set of agents N with n agents. Each agent i has a private valuation v_i for the assigned good or outcome. The mechanism M defines an allocation x(v) and payments p_i(v) for each valuation profile v. The utility of an agent i is u_i = v_i(x) - p_i. We call the refund function r_i(v) the amount returned by the mechanism after calculating base payments g_i(v) (e.g., VCG). The truthfulness or incentive compatibility requirement implies that for all i and all v_{-i}, the dominant strategy is to report v_i honestly, i.e., to maximize u_i by reporting v_i^true.
Definition A rebate mechanism is truthful if for all i and all profiles v, for any alternative report v_i', the inequality v_i(x(v_i,v_{-i})) - p_i(v_i,v_{-i}) = v_i(x(v_i',v_{-i})) - p_i(v_i',v_{-i}) holds. We call a mechanism feasible if it satisfies weak budget balance p_sum - r_sum = 0 and rational if u_i = 0 for agents who participate honestly.
Lemma 1 (Local characterization of truthfulness) Consider a mechanism with payments differentiable with respect to each agent's report in continuous spaces. If for all i and profiles v_{-i}, the partial derivative of payment p_i with respect to v_i satisfies dp_i/dv_i = X_i(v) where X_i(v) = sum of marginal allocation probabilities, then the mechanism is truthful. Proof. Considering the first-order condition for an interior maximum of i's utility when reporting r, truthfulness requires that the derivative of u_i with respect to r at r = v_i be zero and the second derivative negative. That is, d/dr[v_i(x(r,v_{-i})) - p_i(r,v_{-i})]_{r=v_i} = 0, which gives d x/dr|_{r=v_i} · v_i - dp_i/dr|_{r=v_i} = 0. Reapplying the definition of X_i as the rate of change of the allocation induced by the report, the equality dp_i/dv_i = X_i ensures that reporting v_i cancels the marginal gain from manipulation, so truth is locally optimal. Convexity or additional monotonicity conditions guarantee global optimality, concluding the proof.
Theorem 1 (Impossibility of total redistribution while maintaining truthfulness and efficiency) Under allocative efficiency and truthfulness constraints, there is no mechanism that fully redistributes all surplus without violating feasibility or individual rationality in general cases. Proof. Consider a profile v in which the sum of externalities generated by the efficient allocation is positive. If one attempts to return all payments to agents while maintaining VCG base payments minus free, then for some agent i the net transfer may exceed its quasilinear utility, violating individual rationality or generating a deficit in alternative valuation scenarios. Formally, we construct two profiles v and v' that differ only in the valuation of an agent j different from i and show that the truthfulness conditions for both profiles create a contradictory inequality if total redistribution were possible. This implies the announced impossibility.
Proposition 1 (Upper bound on anonymous redistribution) For anonymous mechanisms that respect truthfulness and feasibility, there exists a bound R_max such that the expected sum of refunds cannot exceed R_max in the worst case. Proof. Applying worst-case limit techniques, we consider extreme profiles where a single agent has high valuation and the others zero. Truthfulness requires that the payment structure penalize marginal manipulation, limiting how much can be redistributed without inducing incentives to lie. The explicit calculation of R_max is obtained by optimizing the sum of refunds subject to truthfulness and feasibility inequalities, and it is shown that the optimum is achievable by mechanisms that assign refunds proportional to marginal contributions under symmetry.
Construction of an optimal mechanism We present a constructive TFRM mechanism that achieves the redistribution bound R_max under standard conditions. The mechanism calculates VCG-type base payments g_i(v) and applies a refund rule r_i(v) = a · h_i(v) where h_i(v) is a symmetric measure of i's contribution and a is chosen to saturate feasibility without breaking truthfulness. Proof of correctness. We verify that with this choice, the incentive compatibility inequalities are preserved by linearity and by the choice of a within an allowed interval derived from monotonicity conditions. Furthermore, we verify that the budget is conserved and the sum of r_i is maximal by construction.
Robustness property and computational complexity We also show that computing r(v) is polynomial when h_i(v) can be expressed as a linear combination of aggregate statistics (sums, max, order statistics). Proof. If the statistics used are computed in O(n log n) time or better, evaluating r for any profile requires polynomial time in n. Additionally, we demonstrate stability against bounded noise in valuations: small perturbations in v imply limited changes in r due to the continuity of h.
Applications and examples We illustrate the theorems with two examples: (i) an auction of indivisible goods with efficient allocation and proportional refund, where the bound R_max is verified analytically and compared with numerical simulation; (ii) a mechanism for allocating public resources with homogeneous agents that shows the proposed anonymous structure is optimal within the considered class.
Technical conclusion In summary, the formal proofs demonstrate that truthfulness and feasibility constraints impose concrete limits on the amount of redistribution possible in a rebate mechanism. The TFRM family shows that it is possible to approach the optimal bound through symmetric and computationally efficient refund rules, while preserving truthfulness, individual rationality, and stability against perturbations.
About Q2BSTUDIO Q2BSTUDIO is a custom software and application development company specialized in innovative solutions for businesses. We offer custom software, custom applications, and comprehensive services in artificial intelligence, cybersecurity, and aws and azure cloud services. Our team combines experience in business intelligence services with the ability to develop AI agents, AI for businesses, and advanced dashboards in power bi. We design and implement artificial intelligence solutions adapted to business processes, integrating cybersecurity practices and scalable deployments in aws and azure cloud services to ensure continuity and data protection. If your project requires custom software, custom applications, AI agents, or business intelligence services, Q2BSTUDIO brings technical expertise and a commitment to excellence.
Keywords and positioning custom applications custom software artificial intelligence cybersecurity aws and azure cloud services business intelligence services AI for businesses AI agents power bi
Contact and closing For technical questions about the proofs, implementation of the TFRM mechanism, or to request a custom solution with integration of artificial intelligence, cybersecurity, and aws and azure cloud services, please contact the Q2BSTUDIO team. Our offering includes consulting, development, and complete integration for projects requiring custom applications and custom software with advanced artificial intelligence capabilities and power bi analytics.
