Game-Theoretically Secure Distributed Protocols for Fair Allocation in Coalitional Games

TL;DR

The paper tackles distributed fair allocation in coalitional games by approximating the Shapley value under mutual distrust and without a central randomness source. It introduces a maximin-security framework with a violation budget $C$ and studies permutation-sampling protocols (NaivePerm, SeqPerm) to achieve $(1-\epsilon)$-multiplicative guarantees in expectation or high probability, providing upper and lower bounds on the required number of permutation samples $R$ that depend on the max-to-mean ratio $\Gamma$, budget $C$, and error parameters $\epsilon,\delta$. Through formal adversarial models, adaptive strategies, and experimental validation on synthetic and real data (including edge-synergy graphs), the work derives tight sampling bounds for known and unknown budgets and demonstrates the (limited) advantage of stronger permutation-generation schemes under certain conditions. The results offer practical, provable tools for secure distributed allocation that remain robust to adversarial behavior in settings lacking trusted randomness.

Abstract

We consider game-theoretically secure distributed protocols for coalition games that approximate the Shapley value with small multiplicative error. Since all known existing approximation algorithms for the Shapley value are randomized, it is a challenge to design efficient distributed protocols among mutually distrusted players when there is no central authority to generate unbiased randomness. The game-theoretic notion of maximin security has been proposed to offer guarantees to an honest player's reward even if all other players are susceptible to an adversary. Permutation sampling is often used in approximation algorithms for the Shapley value. A previous work in 1994 by Zlotkin et al. proposed a simple constant-round distributed permutation generation protocol based on commitment scheme, but it is vulnerable to rushing attacks. The protocol, however, can detect such attacks. In this work, we model the limited resources of an adversary by a violation budget that determines how many times it can perform such detectable attacks. Therefore, by repeating the number of permutation samples, an honest player's reward can be guaranteed to be close to its Shapley value. We explore both high probability and expected maximin security. We obtain an upper bound on the number of permutation samples for high probability maximin security, even with an unknown violation budget. Furthermore, we establish a matching lower bound for the weaker notion of expected maximin security in specific permutation generation protocols. We have also performed experiments on both synthetic and real data to empirically verify our results.

Figures (2)

• Figure 1: The number $R$ of P-samples to reach $\epsilon$-expected maximin security under different $n$, $C$ and $\epsilon$. The default values are $n=100, C=20, \epsilon=0.01$. (\ref{['exp11_1']}), (\ref{['exp11_2']}) and (\ref{['exp11_3']}) are for the synthetic game; (\ref{['exp21_1']}), (\ref{['exp21_2']}) and (\ref{['exp21_3']}) are for edge synergy game on DBLP.
• Figure 2: Empirical Cumulative Distribution Function of multiplicative error $\widehat{\epsilon}$ over 1000 simulations. (\ref{['exp12']}) is for the synthetic game, and (\ref{['exp22']}) is for the edge synergy game.

Theorems & Definitions (37)

• Theorem 1.1: Known Budget (Informal)
• Theorem 1.2: Unknown Budget (Informal)
• Theorem 1.3: Upper Bound (Informal)
• Theorem 1.4: Lower Bound (Informal)
• Definition 2.1: Maximin Security for Shapley Value
• Lemma 2.2: $\Gamma$ Ratio for Monotone Games
• proof
• Lemma 2.4: Monotonicity of Ranks
• proof
• proof