Operations Research Games: A Survey

This paper surveys the research area of cooperative games associated with several types of operations research problems in which various decision makers (players) are involved.Cooperating players not only face a joint optimisation problem in trying, e.g., to minimise total joint costs, but also face an additional allocation problem in how to distribute these joint costs back to the individual players.This interplay between optimisation and allocation is the main subject of the area of operations research games.It is surveyed on the basis of a distinction between the nature of the underlying optimisation problem: connection, routing, scheduling, production and inventory.cooperative games;operational research

Allocation of fixed costs and the weighted Shapley value

The weighted value was introduced by Shapley in 1953 as an asymmetric version of his value. Since then several approximations have been proposed including one by Shapley in 1981 specifically addressed to cost allocation, a context in which weights appear naturally. It was at the occasion of a comment in which he only stated the axioms. The present paper offers a proof of Shapley's statement as well as an alternative set of axioms. It is shown that the value is the unique rule that allocates additional fixed costs fairly: only the players who are concerned contribute to the fixed cost and they contribute in proportion to their weights. A particular attention is given to the case where some players are assigned a zero weight.cost allocation, Shapley value, fixed cost

Allocation of fixed costs: characterization of the (dual) weighted Shapley value.

The weighted value was introduced by Shapley in 1953 as an asymmetric version of his value. Since then several axiomatizations have been proposed including one by Shapley in 1981 specifically addressed to cost allocation, a context in which weights appear naturally. It was at the occasion of a comment in which he only stated the axioms. The present paper offers a proof of Shapley's statement as well as an alternative set of axioms. It is shown that the value is the unique rule that allocates additional fixed costs fairly: only the players who are concerned contribute to the fixed cost and they contribute in proportion to their weights. A particular attention is given to the case where some players are assigned a zero weight.cost allocation, Shapley value, fixed cost.

Tree-connected Peer Group Situations and Peer Group Games

A class of cooperative games is introduced which arises from situations in which a set of agents is hierarchically structured and where potential individual economic abilities interfere with the behavioristic rules induced by the organization structure.These games form a cone generated by a specific class of unanimity games, namely those based on coalitions called peer groups. Different economic situations like auctions, communication situations, sequencing situations and flow situations are related to peer group games.For peer group games classical solution concepts have nice properties.auctions;cooperative games;peer groups

Complementary cooperation, minimal winning coalitions, and power indices

We introduce a new simple game, which is referred to as the complementary weighted multiple majority game (C-WMMG for short). C-WMMG models a basic cooperation rule, the complementary cooperation rule, and can be taken as a sister model of the famous weighted majority game (WMG for short). In this paper, we concentrate on the two dimensional C-WMMG. An interesting property of this case is that there are at most $n+1$ minimal winning coalitions (MWC for short), and they can be enumerated in time $O(n\log n)$, where $n$ is the number of players. This property guarantees that the two dimensional C-WMMG is more handleable than WMG. In particular, we prove that the main power indices, i.e. the Shapley-Shubik index, the Penrose-Banzhaf index, the Holler-Packel index, and the Deegan-Packel index, are all polynomially computable. To make a comparison with WMG, we know that it may have exponentially many MWCs, and none of the four power indices is polynomially computable (unless P=NP). Still for the two dimensional case, we show that local monotonicity holds for all of the four power indices. In WMG, this property is possessed by the Shapley-Shubik index and the Penrose-Banzhaf index, but not by the Holler-Packel index or the Deegan-Packel index. Since our model fits very well the cooperation and competition in team sports, we hope that it can be potentially applied in measuring the values of players in team sports, say help people give more objective ranking of NBA players and select MVPs, and consequently bring new insights into contest theory and the more general field of sports economics. It may also provide some interesting enlightenments into the design of non-additive voting mechanisms. Last but not least, the threshold version of C-WMMG is a generalization of WMG, and natural variants of it are closely related with the famous airport game and the stable marriage/roommates problem.Comment: 60 page

Sharing the cost of risky projects

Users share the cost of unreliable non-rival projects (items). For instance, industry partners pay today for R&D that may or may not deliver a cure to some viruses, agents pay for the edges of a network that will cover their connectivity needs, but the edges may fail, etc. Each user has a binary inelastic need that is served if and only if certain subsets of items are actually functioning. We ask how should the cost be divided when individual needs are heterogenous. We impose three powerful separability properties: Independence of Timing ensures that the cost shares computed ex ante are the expectation, over the random realization of the projects, of shares computed ex post. Cost Additivity together with Separability Across Projects ensure that the cost shares of an item depend only upon the service provided by that item for a given realization of all other items. Combining these with fair bounds on the liability of agents with more or less flexible needs, and of agents for whom an item is either indispensable or useless, we characterize two rules: the Ex Post Service rule is the expectation of the equal division of costs between the agents who end up served; the Needs Priority rule splits the cost first between those agents for whom an item is critical ex post, or if there are no such agents between those who end up being served

Investment Uncertainty, and Production Games

This paper explores some cooperative aspects of investments in uncertain, real options. Key production factors are assumed transferable. They may reflect property or user rights. Emission of pollutants and harvest of renewable resources are cases in point. Of particular interest are alternative projects or technologies that provide inferior but anti-correlated returns. Any such project stabilizes the aggregate proceeds. Therefore, given widespread risk exposure and aversion, that project’s worth may embody an extra bonus. The setting is formalized as a stochastic production game. Granted no economies of scale such games are quite tractable in analysis, computation, and realization. A core imputation comes in terms of contingent shadow prices that equilibrate competitive, endogenous markets. The said prices emerge as optimal dual solutions to coordinated production programs, featuring pooled resources - and also via adaptive procedures. Extra value - or an insurance premium - adds to any project whose yield is negatively associated with the aggregate.investment, risk attitudes, insurance, covariance-pricing, cooperative games, core, stochastic optimization

On the core of m-attribute games

We study a special class of cooperative games with transferable utility (TU), called (Formula presented.) -attribute games. Every player in an (Formula presented.) -attribute game is endowed with a vector of (Formula presented.) attributes that can be combined in an additive fashion; that is, if players form a coalition, the attribute vector of this coalition is obtained by adding the attributes of its members. Another fundamental feature of (Formula presented.) -attribute games is that their characteristic function is defined by a continuous attribute function (Formula presented.) —the value of a coalition depends only on evaluation of (Formula presented.) on the attribute vector possessed by the coalition, and not on the identity of coalition members. This class of games encompasses many well-known examples, such as queueing games and economic lot-sizing games. We believe that by studying attribute function (Formula presented.) and its properties, instead of specific examples of games, we are able to develop a common platform for studying different situations and obtain more general results with wider applicability. In this paper, we first show the relationship between nonemptiness of the core and identification of attribute prices that can be used to calculate core allocations. We then derive necessary and sufficient conditions under which every (Formula presented.) -attribute game embedded in attribute function (Formula presented.) has a nonempty core, and a set of necessary and sufficient conditions that (Formula presented.) should satisfy for the embedded game to be convex. We also develop several sufficient conditions for nonemptiness of the core of (Formula presented.) -attribute games, which are easier to check, and show how to find a core allocation when these conditions hold. Finally, we establish natural connections between TU games and (Formula presented.) -attribute games.</p