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The equivalence between von Neumann's Minimax Theorem for zero-sum games and the LP Duality Theorem connects cornerstone problems of the two fields of game theory and optimization, respectively, and has been the subject of intense scrutiny for seven decades. Yet, as observed in this paper, the proof of the difficult direction of this equivalence is unsatisfactory: It does not assign distinct roles to the two players of the game, as is natural from the definition of a zero-sum game. In retrospect, a partial resolution to this predicament was provided in another brilliant paper of von Neumann (1953), which reduced the assignment problem to zero-sum games. However, the underlying LP is highly specialized; all entries of its objective function vector are strictly positive and all entries of the constraint matrix and right hand side vector are equal to one. We generalize von Neumann's result along two directions, each allowing negative entries in certain parts of the LP. Our reductions make explicit the roles of the two players of the reduced game, namely their maximin strategies are to play optimal solutions to the primal and dual LPs. Furthermore, unlike previous reductions, the value of the reduced game reveals the value of the given LP. Our generalizations encompass several basic economic scenarios. We end by discussing evidence that von Neumann possessed an understanding of the notion of polynomial-time solvability.

The stable and fair division of profits/costs is a central concern in economics. The core, which ensures stability, has long been the gold standard for profit/cost sharing in cooperative games. Shapley and Shubik([SS71])'s classic work on the assignment game revealed that core imputations can be disproportionately favoring certain agents. Recent work ([Vaz24]) gave leximin and leximax core imputations for this game, achieving better fairness properties. We explore these fairness notions for the cores of three cooperative games: the max-flow game, the minimum spanning tree (MST) game, and the bipartite $b$-matching game. For all three games we give examples to show that an arbitrary core imputation can be excessively unfair to certain agents. Leximin and leximax core imputations are natural extensions of the widely used max-min and min-max fairness notions. We show that finding such imputations in the core is NP-hard for the max-flow and MST games, and likely so for $b$-matching as well. To address this, we introduce the concept of Dual-Consistent Core (DCC) imputations, which are characterized by solutions to the dual linear programs. We give polynomial time algorithms for computing leximin and leximax DCC imputations for all three games. These games have numerous applications and these imputations will provide a more fair way of distributing profit among agents for them., Comment: 50 pages

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}., Comment: 36 pages

Unlike ordinal-utility matching markets, which are well-developed from the viewpoint of both theory and practice, recent insights from a computer science perspective have left cardinal-utility matching markets in a state of flux. The celebrated pricing-based mechanism for one-sided cardinal-utility matching markets due to Hylland and Zeckhauser, which had long eluded efficient algorithms, was finally shown to be intractable; the problem of computing an approximate equilibrium is PPAD-complete. This led us to ask the question: is there an alternative, polynomial time, mechanism for one-sided cardinal-utility matching markets which achieves the desirable properties of HZ, i.e. (ex-ante) envy-freeness (EF) and Pareto-optimality (PO)? We show that the problem of finding an EF+PO lottery in a one-sided cardinal-utility matching market is by itself already PPAD-complete. However, a $(2 + \epsilon)$-approximately envy-free and (exactly) Pareto-optimal lottery can be found in polynomial time using the Nash-bargaining-based mechanism of Hosseini and Vazirani. Moreover, the mechanism is also $(2 + \epsilon)$-approximately incentive compatible. We also present several results on two-sided cardinal-utility matching markets, including non-existence of EF+PO lotteries as well as existence of justified-envy-free and weak Pareto-optimal lotteries.

The Hylland-Zeckhauser gave a classic pricing-based mechanism (HZ) for a one-sided matching market; it yields allocations satisfying Pareto optimality and envy-freeness (Hylland and Zeckhauser, 1979), and the mechanism is incentive compatible in the large (He et al., 2018). They also studied the exchange extension of HZ and gave an example showing that it may not even admit an equilibrium. In this paper, we consider two models of two sided matching markets: when utility functions are symmetric and when they are non-symmetric. We ask if these models always admit allocations satisfying the two basic properties of Pareto efficiency and envy freeness. Our results are negative. A corollary of the former result is a negative result for non-bipartite matching markets as well.

[MV18] introduced a fundamental new algorithmic question on stable matching, namely finding a matching that is stable under two ``nearby'' instances, where ``nearby'' meant that in going from instance $A$ to $B$, only one agent changes its preference list. By first establishing a sequence of structural results on the lattices of $A$ and $B$, [MV18] and [GMRV22] settled all algorithmic questions related to this case. The current paper essentially settles the general case. Assume that instance $B$ is obtained from $A$, both on $n$ workers and $n$ firms, via changes in the preferences of $p$ workers and $q$ firms. If so, we will denote the change by $(p, q)$. Thus [MV18] and [GMRV22] settled the case $(0, 1)$, since they adopt the convention that one firm changes its preferences. Let $\mathcal{M}_A$ and $\mathcal{M}_B$ be the sets of stable matchings of instances $A$ and $B$, and let $\mathcal{L}_A$ and $\mathcal{L}_B$ be their lattices. Our results are: 1. For $(0, n)$, $\mathcal{M}_A \cap \mathcal{M}_B$ is a sublattice of $\mathcal{L}_A$ and of $\mathcal{L}_B$. We can efficiently obtain the worker-optimal and firm-optimal stable matchings in $\mathcal{M}_A \cap \mathcal{M}_B$. We also obtain the associated partial order, as promised by Birkhoff's Representation Theorem, and use it to enumerate these matchings with polynomial delay. 2. For $(1, n)$, the only missing results are the partial order and enumeration. 3. We give an example with $(2, 2)$ for which $\mathcal{M}_A \cap \mathcal{M}_B$ fails to be a sublattice of $\mathcal{L}_A$. In light of the fact that for $(n, n)$, determining if $(\mathcal{M}_A \cap \mathcal{M}_B) = \emptyset$ is NP-hard [MO19], a number of open questions arise; in particular, closing the gap between $(2, 2)$ and $(n, n)$., Comment: 28 pages and 12 figures

LP-duality theory has played a central role in the study of the core, right from its early days to the present time. However, despite the extensive nature of this work, basic gaps still remain. We address these gaps using the following building blocks from LP-duality theory: 1. Total unimodularity (TUM). 2. Complementary slackness conditions and strict complementarity. Our exploration of TUM leads to defining new games, characterizing their cores and giving novel ways of using core imputations to enforce constraints that arise naturally in applications of these games. The latter include: 1. Efficient algorithms for finding min-max fair, max-min fair and equitable core imputations. 2. Encouraging diversity and avoiding over-representation in a generalization of the assignment game. Complementarity enables us to prove new properties of core imputations of the assignment game and its generalizations., Comment: 46 pages. arXiv admin note: text overlap with arXiv:2202.00619

The core is a dominant solution concept in economics and cooperative game theory; it is predominantly used for profit, equivalently cost or utility, sharing. This paper demonstrates the versatility of this notion by proposing a completely different use: in a so-called investment management game, which is a game against nature rather than a cooperative game. This game has only one agent whose strategy set is all possible ways of distributing her money among investment firms. The agent wants to pick a strategy such that in each of exponentially many future scenarios, sufficient money is available in the right firms so she can buy an optimal investment for that scenario. Such a strategy constitutes a core imputation under a broad interpretation, though traditional formal framework, of the core. Our game is defined on perfect graphs, since the maximum stable set problem can be solved in polynomial time for such graphs. We completely characterize the core of this game, analogous to Shapley and Shubik characterization of the core of the assignment game. A key difference is the following technical novelty: whereas their characterization follows from total unimodularity, ours follows from total dual integrality, Comment: 16 pages. arXiv admin note: text overlap with arXiv:2209.04903

LP-duality theory has played a central role in the study of cores of games, right from the early days of this notion to the present time. The classic paper of Shapley and Shubik \cite{Shapley1971assignment} introduced the "right" way of exploiting the power of this theory, namely picking problems whose LP-relaxations support polyhedra having integral vertices. So far, the latter fact was established by showing that the constraint matrix of the underlying linear system is {\em totally unimodular}. We attempt to take this methodology to its logical next step -- {\em using total dual integrality} -- thereby addressing new classes of games which have their origins in two major theories within combinatorial optimization, namely perfect graphs and polymatroids. In the former, we address the stable set and clique games and in the latter, we address the matroid independent set game. For each of these games, we prove that the set of core imputations is precisely the set of optimal solutions to the dual LPs. Another novelty is the way the worth of the game is allocated among sub-coalitions. Previous works follow the {\em bottom-up process} of allocating to individual agents; the allocation to a sub-coalition is simply the sum of the allocations to its agents. The {\em natural process for our games is top-down}. The optimal dual allocates to "objects" in the grand coalition; a sub-coalition inherits the allocation of each object with which it has non-empty intersection., Comment: 14 pages. arXiv admin note: text overlap with arXiv:2202.00619

We give new characterizations of core imputations for the following games: * The assignment game. * Concurrent games, i.e., general graph matching games having non-empty core. * The unconstrained bipartite $b$-matching game (edges can be matched multiple times). * The constrained bipartite $b$-matching game (edges can be matched at most once). The classic paper of Shapley and Shubik \cite{Shapley1971assignment} showed that core imputations of the assignment game are precisely optimal solutions to the dual of the LP-relaxation of the game. Building on this, Deng et al. \cite{Deng1999algorithms} gave a general framework which yields analogous characterizations for several fundamental combinatorial games. Interestingly enough, their framework does not apply to the last two games stated above. In turn, we show that some of the core imputations of these games correspond to optimal dual solutions and others do not. This leads to the tantalizing question of understanding the origins of the latter. We also present new characterizations of the profits accrued by agents and teams in core imputations of the first two games. Our characterization for the first game is stronger than that for the second; the underlying reason is that the characterization of vertices of the Birkhoff polytope is stronger than that of the Balinski polytope., Comment: 32 pages

Motivated by recent insights into the online bipartite matching problem (\textsc{OBM}), our goal was to extend the optimal algorithm for it, namely \textsc{Ranking}, all the way to the special case of adwords problem, called \textsc{Small}, in which bids are small compared to budgets; the latter has been of considerable practical significance in ad auctions \cite{MSVV}. The attractive feature of our approach was that it would yield a {\em budget-oblivious algorithm}, i.e., the algorithm would not need to know budgets of advertisers and therefore could be used in autobidding platforms. We were successful in obtaining an optimal, budget-oblivious algorithm for \textsc{Single-Valued}, under which each advertiser can make bids of one value only. However, our next extension, to \textsc{Small}, failed because of a fundamental reason, namely failure of the {\em No-Surpassing Property}. Since the probabilistic ideas underlying our algorithm are quite substantial, we have stated them formally, after assuming the No-Surpassing Property, and we leave the open problem of removing this assumption. With the help of two undergrads, we conducted extensive experiments on our algorithm on randomly generated instances. Our findings are that the No-Surpassing Property fails less than $2\%$ of the time and that the performance of our algorithms for \textsc{Single-Valued} and \textsc{Small} are comparable to that of \cite{MSVV}. If further experiments confirm this, our algorithm may be useful as such in practice, especially because of its budget-obliviousness., Comment: 27 pages

In the area of matching-based market design, existing models using cardinal utilities suffer from two deficiencies: First, the Hylland-Zeckhauser (HZ) mechanism, which has remained a classic in economics for one-sided matching markets, is intractable; computation of even an approximate equilibrium is PPAD-complete. Second, there is an extreme paucity of such models. This led Hosseini and Vazirani (2022) to define a rich collection of Nash-bargaining-based models for one-sided and two-sided matching markets, in both Fisher and Arrow-Debreu settings, together with very fast implementations using available solvers and very encouraging experimental results. In this paper, we give fast algorithms with proven running times for the models introduced by Hosseini and Vazirani, using the techniques of multiplicative weights update (MWU) and conditional gradient descent (CGD). Additionally, we make the following contributions: (1) By Tr\"obst and Vazirani (2024), a linear one-sided Nash-bargaining-based matching market satisfies envy-freeness within factor two. We show that the other models satisfy approximate equal-share fairness, where the exact factor depends on the utility function being used in the particular model. (2) We define a Nash-bargaining-based model for non-bipartite matching markets and give fast algorithms for it using conditional gradient descent., Comment: 43 pages

This paper addresses two deficiencies of models in the area of matching-based market design. The first arises from the recent realization that the most prominent solution that uses cardinal utilities, namely the Hylland-Zeckhauser (HZ) mechanism, is intractable; computation of even an approximate equilibrium is PPAD-complete. The second is the extreme paucity of models that use cardinal utilities. Our paper addresses both these issues by proposing Nash-bargaining-based matching market models. Since the Nash bargaining solution is captured by a convex program, efficiency follows. In addition, it possesses several desirable game-theoretic properties. Our approach yields a rich collection of models: for one-sided as well as two-sided markets, for Fisher as well as Arrow-Debreu settings, and for a wide range of utility functions, all the way from linear to Leontief. We give very fast implementations for these models using Frank-Wolfe and Cutting Plane algorithms. These help solve large instances with several thousand agents and goods in a matter of minutes on a PC, even for a one-sided matching market under piecewise-linear concave utility functions and a two-sided matching market under linear utility functions. In contrast, using HZ, going beyond even $n = 10$ is prohibitive. Several new ideas were needed, beyond the standard methods, to obtain these implementations. In particular, we present several lower bounding schemes, which not only help improve the convergence of our solution methods but also shed light on fairness properties of the Nash-bargaining-based models., Comment: 45 pages

The classic paper of Shapley and Shubik \cite{Shapley1971assignment} characterized the core of the assignment game using ideas from matching theory and LP-duality theory and their highly non-trivial interplay. Whereas the core of this game is always non-empty, that of the general graph matching game can be empty. This paper salvages the situation by giving an imputation in the $2/3$-approximate core for the latter. This bound is best possible, since it is the integrality gap of the natural underlying LP. Our profit allocation method goes further: the multiplier on the profit of an agent is often better than ${2 \over 3}$ and lies in the interval $[{2 \over 3}, 1]$, depending on how severely constrained the agent is. Next, we provide new insights showing how discerning core imputations of an assignment games are by studying them via the lens of complementary slackness. We present a relationship between the competitiveness of individuals and teams of agents and the amount of profit they accrue in imputations that lie in the core, where by {\em competitiveness} we mean whether an individual or a team is matched in every/some/no maximum matching. This also sheds light on the phenomenon of degeneracy in assignment games, i.e., when the maximum weight matching is not unique. The core is a quintessential solution concept in cooperative game theory. It contains all ways of distributing the total worth of a game among agents in such a way that no sub-coalition has incentive to secede from the grand coalition. Our imputation, in the $2/3$-approximate core, implies that a sub-coalition will gain at most a $3/2$ factor by seceding, and less in typical cases., Comment: 10 pages

The Micali-Vazirani (MV) algorithm for maximum cardinality matching in general graphs, which was published in 1980 \cite{MV}, remains to this day the most efficient known algorithm for the problem. This paper gives the first complete and correct proof of this algorithm. Central to our proof are some purely graph-theoretic facts, capturing properties of minimum length alternating paths; these may be of independent interest. An attempt is made to render the algorithm easier to comprehend., Comment: 43 pages. arXiv admin note: text overlap with arXiv:1210.4594

We prove that a fractional perfect matching in a non-bipartite graph can be written, in polynomial time, as a convex combination of perfect matchings. This extends the Birkhoff-von Neumann Theorem from bipartite to non-bipartite graphs. The algorithm of Birkhoff and von Neumann is greedy; it starts with the given fractional perfect matching and successively "removes" from it perfect matchings, with appropriate coefficients. This fails in non-bipartite graphs -- removing perfect matchings arbitrarily can lead to a graph that is non-empty but has no perfect matchings. Using odd cuts appropriately saves the day., Comment: 16 pages

The Arrow-Debreu extension of the classic Hylland-Zeckhauser scheme for a one-sided matching market -- called ADHZ in this paper -- has natural applications but has instances which do not admit equilibria. By introducing approximation, we define the $\epsilon$-approximate ADHZ model, and we give the following results. * Existence of equilibrium under linear utility functions. We prove that the equilibrium satisfies Pareto optimality, approximate envy-freeness, and approximate weak core stability. * A combinatorial polynomial-time algorithm for an $\epsilon$-approximate ADHZ equilibrium for the case of dichotomous, and more generally bi-valued, utilities. * An instance of ADHZ, with dichotomous utilities and a strongly connected demand graph, which does not admit an equilibrium. Since computing an equilibrium for HZ is likely to be highly intractable and because of the difficulty of extending HZ to more general utility functions, Hosseini and Vazirani proposed (a rich collection of) Nash-bargaining-based matching market models. For the dichotomous-utilities case of their model linear Arrow-Debreu Nash bargaining one-sided matching market (1LAD), we give a combinatorial, strongly polynomial-time algorithm and show that it admits a rational convex program., Comment: 30 pages, 2 figures, added new results

In 1979, Hylland and Zeckhauser \cite{hylland} gave a simple and general scheme for implementing a one-sided matching market using the power of a pricing mechanism. Their method has nice properties -- it is incentive compatible in the large and produces an allocation that is Pareto optimal -- and hence it provides an attractive, off-the-shelf method for running an application involving such a market. With matching markets becoming ever more prevalant and impactful, it is imperative to finally settle the computational complexity of this scheme. We present the following partial resolution: 1. A combinatorial, strongly polynomial time algorithm for the special case of $0/1$ utilities. 2. An example that has only irrational equilibria, hence proving that this problem is not in PPAD. Furthermore, its equilibria are disconnected, hence showing that the problem does not admit a convex programming formulation. 3. A proof of membership of the problem in the class FIXP. We leave open the (difficult) question of determining if the problem is FIXP-hard. Settling the status of the special case when utilities are in the set $\{0, {\frac 1 2}, 1 \}$ appears to be even more difficult., Comment: 22 pages