Cost-sensitive, envy-resistant consensus in group decision making

Consensus-reaching processes (CRPs) inherently demand compromise, requiring each decision maker (DM) to adjust their initial opinions—a process that often entails both tangible cost and disutility. Traditional optimisation-based mechanisms in CRPs focus primarily on minimising adjustment cost while overlooking DMs’ discontent. However, repeated and excessive discontent can threaten the sustainability of CRPs, especially when certain DMs are consistently disadvantaged (i.e., required to make significant adjustments) compared to others, leading to envy and disengagement. To address this, we develop a new optimisation-based mechanism in CRPs that balances minimising both cost and envy. Incorporating an envy objective introduces significant computational challenges, as envy is captured by a non-convex function, even when the disutility is convex. Fortunately, many non-convex optimisation problems, such as bilinear and mixed-integer linear programs, can now be solved efficiently using off-the-shelf solvers. Leveraging this capability, we enhance the generalisability of our cost-sensitive, envy-resistant mechanism for consensus by assuming that cost and disutility are representable as polyhedral loss functions, allowing us to reformulate the proposed model as a bilinear program or approximate it with a priori error bound. We demonstrate our model using real data from RateMyProfessors.com, which includes student ratings on the quality and difficulty of courses taught by professors.