Exploiting Weak Supermodularity for Coalition-Proof Mechanisms

TL;DR

This paper introduces the concepts of supermodularity ratio and weak supermodularity to analyze and bound the profitability of collusion and shill bidding in auctions that are not fully supermodular, such as electricity markets.

Contribution

It defines new measures for weak supermodularity and provides tight bounds on collusion profitability in non-supermodular auctions.

Findings

Supermodularity ratio bounds collusion gains.

Weak supermodularity characterizes near-supermodular functions.

Case studies validate theoretical bounds on IEEE test systems.

Abstract

Under the incentive-compatible Vickrey-Clarke-Groves mechanism, coalitions of participants can influence the auction outcome to obtain higher collective profit. These manipulations were proven to be eliminated if and only if the market objective is supermodular. Nevertheless, several auctions do not satisfy the stringent conditions for supermodularity. These auctions include electricity markets, which are the main motivation of our study. To characterize nonsupermodular functions, we introduce the supermodularity ratio and the weak supermodularity. We show that these concepts provide us with tight bounds on the profitability of collusion and shill bidding. We then derive an analytical lower bound on the supermodularity ratio. Our results are verified with case studies based on the IEEE test systems.