Cooperative Bargaining Games Without Utilities: Mediated Solutions from Direction Oracles
Kushagra Gupta, Surya Murthy, Mustafa O. Karabag, Ufuk Topcu, David Fridovich-Keil
TL;DR
This paper addresses cooperative bargaining when a mediator lacks access to agents' utilities and can only observe each agent's most preferred direction in the decision space. It introduces Direction-based Bargaining Solution (DiBS), an iterative method with updates $\mathbf{x}_{k+1} = \mathbf{x}_k + \alpha_k\sum_{i=1}^N \|\mathbf{x}_k-\mathbf{x}^{*,i}\|_2\,\mathcal{O}^{\mathcal{D}, i}_{\boldsymbol{\ell}}(\mathbf{x}_k)$ that advances toward Pareto-stationary outcomes using only direction oracles. Theoretical results prove that fixed points of DiBS are Pareto stationary, that, under strong convexity and smoothness, the method globally converges to the fixed-point set, and that DiBS enjoys invariance to strictly increasing monotone nonaffine transformations and symmetry (with independence of irrelevant alternatives under stronger conditions). Empirically, the authors validate DiBS on nonconvex multi-agent formation and mediated stock portfolio allocation, showing balanced outcomes and robustness to direction-oracle estimation via comparisons, with code available at the provided repository.
Abstract
Cooperative bargaining games are widely used to model resource allocation and conflict resolution. Traditional solutions assume the mediator can access agents utility function values and gradients. However, there is an increasing number of settings, such as human AI interactions, where utility values may be inaccessible or incomparable due to unknown, nonaffine transformations. To model such settings, we consider that the mediator has access only to agents most preferred directions, i.e., normalized utility gradients in the decision space. To this end, we propose a cooperative bargaining algorithm where a mediator has access to only the direction oracle of each agent. We prove that unlike popular approaches such as the Nash and Kalai Smorodinsky bargaining solutions, our approach is invariant to monotonic nonaffine transformations, and that under strong convexity and smoothness assumptions, this approach enjoys global asymptotic convergence to Pareto stationary solutions. Moreover, we show that the bargaining solutions found by our algorithm also satisfy the axioms of symmetry and (under slightly stronger conditions) independence of irrelevant alternatives, which are popular in the literature. Finally, we conduct experiments in two domains, multi agent formation assignment and mediated stock portfolio allocation, which validate these theoretic results. All code for our experiments can be found at https://github.com/suryakmurthy/dibs_bargaining.