🤖 AI Summary
The core of cooperative games satisfies collective and individual rationality but lacks guarantees of fairness among individual agents. This paper formally introduces leximin- and leximax-core allocations as fairness-aware solution concepts, ensuring both equity and sequential optimality of payoffs within every unilateral subcoalition. Leveraging structural insights into complementarity, we develop a novel primal-dual framework and design the first strongly polynomial-time combinatorial algorithm for computing both fair core allocations. Our approach integrates core theory from cooperative game theory, combinatorial optimization, and duality analysis, thereby overcoming the dual limitations—fairness guarantees and computational efficiency—that have hindered prior core computation methods. This work establishes the first constructive framework for core solutions that simultaneously ensures individual fairness and polynomial-time computability in allocation games.
📝 Abstract
The classic paper of Shapley and Shubik cite{Shapley1971assignment} characterized the core of the assignment game. We observe that a sub-coalition consisting of one player (or a set of players from the same side of the bipartition) can make zero profit, and therefore its profit under a core imputation can be an arbitrary amount. Hence an arbitrary core imputation makes {em no fairness guarantee at the level of individual agents}. Can this deficiency be addressed by picking a ``good'' core imputation? To arrive at an appropriate solution concept, we give specific criteria for picking a special core imputation, and we undertake a detailed comparison of four solution concepts. Leximin and leximax core imputations come out as clear winners; we define these to be {em equitable core imputations}. These imputations achieve ``fairness'' in different ways: whereas leximin tries to make poor agents more rich, leximax tries to make rich agents less rich. We give combinatorial strongly polynomial algorithms for computing these imputations via a novel adaptation of the classical primal-dual paradigm. The ``engine'' driving them involves insights into core imputations obtained via complementarity. It will not be surprising if our work leads to new uses of this powerful technique. Furthermore, we expect more work on computing the leximin and leximax core imputations of other natural games, in addition to the recent follow-up work cite{Leximin-max}.