the assignment game

The assignment game I: The core

• Published: December 1971
• Volume 1 , pages 111–130, ( 1971 )

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• L. S. Shapley 1 &
• M. Shubik 2  

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The assignment game is a model for a two-sided market in which a product that comes in large, indivisible units (e.g., houses, cars, etc.) is exchanged for money, and in which each participant either supplies or demands exactly one unit. The units need not be alike, and the same unit may have different values to different participants. It is shown here that the outcomes in the core of such a game — i.e., those that cannot be improved upon by any subset of players — are the solutions of a certain linear programming problem dual to the optimal assignment problem, and that these outcomes correspond exactly to the price-lists that competitively balance supply and demand. The geometric structure of the core is then described and interpreted in economic terms, with explicit attention given to the special case (familiar in the classic literature) in which there is no product differentiation — i.e., in which the units are interchangeable. Finally, a critique of the core solution reveals an insensitivity to some of the bargaining possibilities inherent in the situation, and indicates that further analysis would be desirable using other game-theoretic solution concepts.

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Research supported in part by the Cowles Foundation for Research in Economics under Contract Nonr-3055(01) with the Office of Naval Research, in part by the United States Air Force under Project RAND-Contract F 44620-67-C-0045, and in part by The Rand Corporation.

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Shapley, L.S., Shubik, M. The assignment game I: The core. Int J Game Theory 1 , 111–130 (1971). https://doi.org/10.1007/BF01753437

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Received : 15 October 1971

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Remove a code repository from this paper, mark the official implementation from paper authors, add a new evaluation result row, remove a task, add a method, remove a method, edit datasets, the assignment game: new mechanisms for equitable core imputations.

18 Feb 2024  ·  Vijay V. Vazirani · Edit social preview

The set of core imputations of the assignment game forms a (non-finite) distributive lattice. So far, efficient algorithms were known for computing only its two extreme imputations; however, each of them maximally favors one side and disfavors the other side of the bipartition, leading to inequitable profit sharing. Another issue is that a sub-coalition consisting of one player (or a set of players from the same side of the bipartition) can make zero profit, therefore a core imputation is not obliged to give them any profit. Hence core imputations make no fairness guarantee at the level of individual agents. This raises the question of computing {\em more equitable core imputations}. In this paper, we give combinatorial (i.e., the mechanism does not invoke an LP-solver) polynomial time mechanisms for computing the leximin and leximax core imputations for the assignment game. These imputations achieve ``fairness'' in different ways: whereas leximin tries to make poor agents more rich, leximax tries to make rich agents less rich. In general, the two imputations are different. Our mechanisms were derived by a suitable adaptation of the classical primal-dual paradigm from combinatorial optimization. The ``engine'' driving them involves recent insights, obtained via complementarity, into core imputations \cite{Va.New-characterizations} and the pristine combinatorial structure of matching. We have identified other natural games which could benefit from our approach.

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The assignment game I: The core

1971, International Journal of Game Theory

Related Papers

Carles Rafels

On the domain of balanced multi-sided assignment games (Quint, 1991), the core is characterized as the unique non-empty solution satisfying derived consistency and projection consistency. As a consequence, a new characterization of the core of two-sided assignment games (Shapley and Shubik, 1972) is provided by using simultaneously the aforementioned consistency axioms. We also characterize the core on the whole domain of multi-sided assignment games in terms of singleness best, individual anti-monotonicity and derived consistency. Again, as a particular case we obtain a new axiomatization for the bilateral case without making use of the non-emptiness axiom.

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In the one-sided assignment game, any two agents can form a trade; they can decide to form a partnership and agree on how to share the surplus created. Contrary to the two-sided assignment game, stable outcomes often fail to exist in the one-sided assignment game. Hence the core, which coincides with the set of stable payoffs, may be empty. We introduce the idea of tradewise-stable (t-stable) outcomes: they are individually rational outcomes where all trades are stable; that is, no matched agent can form a blocking pair with any other agent, neither matched nor unmatched. We propose the set of constrained-optimal (optimal) t-stable outcomes, the set of the maximal elements of the set of t-stable outcomes, as a natural solution concept for this game. We prove that this set is non-empty, it coincides with the set of stable outcomes when the core is non-empty, and it satisfies similar properties to the set of stable outcomes even when the core is empty. We propose a partnership formati...

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Experience survival horror from a new perspective with The Evil Within: The Assignment. This is the first of a two-part, story driven experience where players take on the role of detective Juli Kidman, Sebastian Castellanos’ mysterious partner, in a concurrent story that looks to answer some of the questions surrounding her whereabouts during The Evil Within. Along the way she will encounter disturbing new enemies, discover new mysteries, and uncover new horrors. The Assignment gives fans of The Evil Within the perfect opportunity to discover detective Kidman’s hidden motivations and her mysterious connection to Mobius, the shadowy group thought to be behind the gruesome events of The Evil Within.

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Guardians rally for 3 runs in 10th inning to stun yankees 8-7 in wild series finale with wild finish.

CLEVELAND -- — Andrés Giménez's sacrifice fly capped Cleveland's wild, three-run rally in the 10th inning as the Guardians salvaged the finale of a three-game series with a stunning 8-7 win over the New York Yankees on Sunday.

Down 7-5, the Guardians scored three in the 10th off Caleb Ferguson (0-2) on a forceout, fielder's choice and sacrifice fly.

“We’re not going to quit,” first-year Cleveland manager Stephen Vogt said. “And that’s one of the things I heard the boys yelling on the way back up to the clubhouse. We don’t quit and that’s what this group’s about.”

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Computer Science > Computer Science and Game Theory

Title: new characterizations of core imputations of matching and $b$-matching games.

Abstract: 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.

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Max Scherzer injury: Rangers ace could begin rehab assignment this week, says he's 'ramping up like normal'

Scherzer has not yet pitched since back surgery in december.

At 11-11, the Texas Rangers currently sit atop the AL West, not that they've been dominant early this season. Up until this weekend, the offense had been humming along nicely, though the pitching staff has been questionable. The Rangers rank 22nd in ERA (4.34) and 25th in pitcher WAR (1.1). The bullpen (0.1 WAR) deserves most of the blame.

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"I want to remove all that 'ahead of schedule,' 'early,' — all of that lingo," he said. "No, I have not been experiencing the symptoms that most people experience, outside of the surgery. I have felt normal. I turned the corner here a couple of weeks ago. Everything I'm doing, in terms of throwing, I'm recovering from it normally. My body feels good. So I'm just ramping up like normal right now. This is not me pushing anything. I'm not trying to cut a single corner."  

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