Transferable Utility Games Research Articles

Characterization of TU Games with Stable Cores by Nested Balancedness

A balanced transferable utility game (N, v) has a stable core if its core is externally stable, that is, if each imputation that is not in the core is dominated by some core element. Given two payoff allocations x and y, we say that x outvotes y via some coalition S of a feasible set if x dominates y via S and x allocates at least v(T) to any feasible T that is not contained in S. It turns out that outvoting is transitive and the set M of maximal elements with respect to outvoting coincides with the core if and only if the game has a stable core. By applying the duality theorem of linear programming twice, it is shown that M coincides with the core if and only if a certain nested balancedness condition holds. Thus, it can be checked in finitely many steps whether a balanced game has a stable core. We say that the game has a super-stable core if each payoff vector that allocates less than v(S) to some coalition S is dominated by some core element and prove that core super-stability is equivalent to vital extendability, requiring that each vital coalition is extendable.

On the Internal and External Stability of Coalitions and Application to Group Purchasing Organizations

Two new notions of stability of coalitions, based on the idea of exclusion or integration of players depending on how they affect allocations, are introduced for cooperative transferable utility games. The first one, called internal stability, requires that no coalition member would find that her departure from the coalition would improve her allocation or those of all her partners. The second one, called external stability, requires that coalitions members do not wish to recruit a new partner willing to join the coalition, since her arrival would hurt some of them. As an application of these two notions, we study the stability of Group Purchasing Organizations using the Shapley value to allocate costs between buyers. Our main results suggest that, when all buyers are initially alone, while small buyers will form internally and externally stable Group Purchasing Organizations to benefit from the best price discount, big buyers will be mutually exclusive and may cooperate with only small buyers.

Antiduality in Exact Partition Games

This note shows that the egalitarian Dutta and Ray (1989) solution for transferable utility games is self-antidual on the class of exact partition games. By applying a careful antiduality analysis, we derive several new axiomatic characterizations. Moreover, we point out an error in earlier work on antiduality and repair and strengthen several related characterizations on the class of convex games.

A weighted power distribution mechanism under fuzzy behavior systems

In real situations, players might represent administrative areas of different scales; players might have different activity abilities. Thus, we propose an extension of the Banzhaf-Owen index in the framework of fuzzy transferable-utility games by considering supreme-utilities and weights simultaneously, which we name the weighted fuzzy Banzhaf-Owen index. Here we adopt three existing notions from traditional game theory and reinterpret them in the framework of fuzzy transferable-utility games. The first one is that this weighted index could be represented as an alternative formulation in terms of excess functions. The second is that, based on an reduced game and related consistency, we offer an axiomatic result to present the rationality of this weighted index. Finally, we introduce two dynamic processes to illustrate that this weighted index could be reached by players who start from an arbitrary efficient payoff vector and make successive adjustments.

The Shapley value, the Proper Shapley value, and sharing rules for cooperative ventures

In this note, we discuss two solutions for cooperative transferable utility games, namely the Shapley value and the Proper Shapley value. We characterize positive Proper Shapley values by affine invariance and by an axiom that requires proportional allocation of the surplus according to the individual singleton worths in generalized joint venture games. As a counterpart, we show that affine invariance and an axiom that requires equal allocation of the surplus in generalized joint venture games characterize the Shapley value.

The average covering tree value for directed graph games

We introduce a single-valued solution concept, the so-called average covering tree value, for the class of transferable utility games with limited communication structure represented by a directed graph. The solution is the average of the marginal contribution vectors corresponding to all covering trees of the directed graph. The covering trees of a directed graph are those (rooted) trees on the set of players that preserve the dominance relations between the players prescribed by the directed graph. The average covering tree value is component efficient, and under a particular convexity-type condition it is stable. For transferable utility games with complete communication structure the average covering tree value equals to the Shapley value of the game. If the graph is the directed analog of an undirected graph the average covering tree value coincides with the gravity center solution.

The Core of a Transferable Utility Game as the Solution to a Public Good Market Demand Problem

The core of a monotonic transferable utility (TU) game is shown to be the set of prices that incentivize each individual to demand the grand coalition in a market demand problem in which the goods being demanded are coalitions viewed as public goods. It is also shown that the core is the intersection of superdifferentials evaluated at the grand coalition of the covers of person-specific TU games derived from the original game. These characterizations of the core demonstrate how a market demand approach to the core in the spirit of Baldwin and Klemperer is related to the approach to the core using the cover of a TU game and it superdifferential at the grand coalition developed by Shapley and Shubik, Aubin, and Danilov and Koshevoy.

Novel equal division values based on players’ excess vectors and their applications to logistics enterprise coalitions

Some sub-coalitions can not be formed or fail to satisfy the superadditivity in many realistic cooperative transferable utility (TU) games, which particularly exists in the logistics service industry. To exploit some novel solutions for addressing these TU games, we firstly propose again the equal surplus division value based on the least square method and players’ common excess vector, and then introduce the weighted equal surplus division value. Both of them belong to the family of the least square values. Inspired by the fact that many TU games base the profit distribution strategies not only on the egalitarian principle but also on the utility principle, the equal contribution division value and the weighted equal contribution division value based on the least square method and players’ contribution excess vector are spontaneously generated. An algorithm is described to make the four solutions proposed in this paper satisfy the property of individual rationality. Finally, to show the advantages, the practicability and the rationality of the four solutions, a practical example about the profit distribution strategy of a logistics enterprise coalition is illustrated and the contrastive analysis among them is given.

Nontransferable utility bankruptcy games

In this paper, we analyze bankruptcy problems with nontransferable utility (NTU) from a game theoretical perspective by redefining corresponding NTU-bankruptcy games in a tailor-made way. It is shown that NTU-bankruptcy games are both coalition-merge convex and ordinally convex. Generalizing the notions of core cover and compromise stability for transferable utility (TU) games to NTU-games, we also show that each NTU-bankruptcy game is compromise stable. Thus, NTU-bankruptcy games are shown to retain the two characterizing properties of TU-bankruptcy games: convexity and compromise stability. As a first example of a game theoretical NTU-bankruptcy rule, we analyze the adjusted proportional rule and show that this rule corresponds to the compromise value of NTU-bankruptcy games.

Pay to change lanes: A cooperative lane-changing strategy for connected/automated driving

This paper proposes a cooperative lane changing strategy using a transferable utility games framework. This allows vehicles to engage in transactions where gaps in traffic are created in exchange for monetary compensation. The proposed approach is best suited to discretionary lane change maneuvers. We formulate gains in travel time, referred to as time differences, that result from achieving higher speeds. These time differences, coupled with value of time, are used to formulate a utility function, where utility is transferable. We also allow for games between connected vehicles that do not involve transfer of utility. We apply Nash bargaining theory to solve the latter. A cellular automaton is developed and utilized to perform simulation experiments that explore the impact of such transactions on traffic conditions (travel-time savings, resulting speed-density relations and shock wave formation) and the benefit to vehicles. The results show that lane changing with transferable utility between drivers can help achieve win-win results, improve both individual and social benefits without resulting in any adverse effects on traffic characteristics in general and, in fact, result in slight improvement at traffic densities outside of free-flow and (bumper-to-bumper) jammed traffic.

Fuzzy transferable-utility games: a weighted allocation and related results

By considering the supreme-utilities among fuzzy sets and the weights among participants simultaneously, we introduce the supreme-weighted value on fuzzy transferable-utility games. Further, we provide some equivalent relations to characterize the family of all solutions that admit a potential on weights. We also propose the dividend approach to provide alternative viewpoint for the potential approach. Based on these equivalent relations, several axiomatic results are also proposed to present the rationality for the supreme-weighted value.

Liability games

We analyze the question of how to distribute the asset value of an insolvent firm among its creditors and the firm itself. Compared to standard bankruptcy games as studied in the game-theoretic literature, we treat the firm as a player and define a new class of transferable utility games called liability games. We show that the core of a liability game is empty. We analyze the nucleolus of the game. The firm always gets a positive payment, at most equal to half of the asset value. Creditors with higher liabilities receive higher payments, but also suffer from higher deficiencies. We provide conditions under which the nucleolus coincides with a generalized proportional rule.

Sharing a River with Downstream Externalities

We consider the problem of efficient emission abatement in a multi polluter setting, where agents are located along a river in which net emissions accumulate and induce negative externalities to downstream riparians. Assuming a cooperative transferable utility game, we seek welfare distributions that satisfy all agents’ participation constraints and, in addition, a fairness constraint implying that no coalition of agents should be better off than it were if all non-members of the coalition would not pollute the river at all. We show that the downstream incremental distribution, as introduced by Ambec and Sprumont (2002), is the only welfare distribution satisfying both constraints. In addition, we show that this result holds true for numerous extensions of our model.

Procedural and optimization implementation of the weighted ENSC value

The main purpose of this article is to introduce the weighted ENSC value for cooperative transferable utility games which takes into account players’ selfishness about the payoff allocations. Similar to Shapley’s idea of a one-by-one formation of the grand coalition [Shapley (1953)], we first provide a procedural implementation of the weighted ENSC value depending on players’ selfishness as well as their marginal contributions to the grand coalition. Second, in the spirit of the nucleolus [Schmeidler (1969)], we prove that the weighted ENSC value is obtained by lexicographically minimizing a complaint vector associated with a new complaint criterion relying on players’ selfishness.

Associated Games to Optimize the Core of a Transferable Utility Game

In view of the core optimization, this paper establishes a new associated game starting from one with a nonempty core and proposes a sequence of associated games recursively. We prove that the cores of the associated games are increasingly stable in two aspects. Firstly, the core of each game is contained in the one it precedes. Secondly, any allocation outside the core of the corresponding associated game is indirectly dominated by a certain allocation in it. Therefore, the last one of the nonempty cores in this sequence is the final optimized set. More interestingly, if this sequence does not encounter a game with an empty core, we show that it converges and that the limit game is a constant-sum one by the matrix approach. In this case, we can ideally select a unique point from the core of the original game, which is the core of such a limit game.

The Procedural Egalitarian Solution and Egalitarian Stable Games

This paper identifies the maximal domain of transferable utility games on which aggregate monotonicity (no player is worse o when the worth of the grand coalition increases) and egalitarian core selection (no other core allocation can be obtained by a transfer from a richer to a poorer player) are compatible. On this domain, which includes the class of large core games, we show that these two axioms characterize a unique solution which even satisfies coalitional monotonicity (no member is worse off when the worth of one coalition increases) and strong egalitarian core selection (no other core allocation can be obtained by transfers from richer to poorer players).

Alternative Representation of Semivalues, the Inverse Problem and Coalitional Rationality

The concept of semivalue of a transferable utility game has been introduced by Dubey, Neyman and Weber as weighted sum of marginal contributions. Later, Puente has introduced a particular class of semivalues, called binomial semivalues, where weights are obtained through a recursive procedure. In the present paper, we extend Puente's procedure to obtain an equivalent representation of semivalues that turns out to be useful to solve the inverse problem and the question of coalitional rationality.

On Convexity in Games with Externalities

We introduce new notions of superadditivity and convexity for games with coalitional externalities. We show parallel results to the classic ones for transferable utility games without externalities. In superadditive games the grand coalition is the most efficient organization of agents. The convexity of a game is equivalent to having non decreasing contributions to larger embedded coalitions. We also see that convex games can only have negative externalities.

Decomposing a balanced game: A necessary and sufficient condition for the nonemptiness of the core

The Bondareva–Shapley condition is the most eminent necessary and sufficient condition for the core of a transferable utility game to be nonempty. In this paper, we provide a new necessary and sufficient condition. We show that a game has a nonempty core if and only if the game can be decomposed into some simple games.

Necessary players, Myerson fairness and the equal treatment of equals

This article addresses linear sharing rules on transferable utility games (TU-games) with various structures, namely communication structures and conference structures as defined by Myerson in two papers (Myerson in Mathematics of Operations Research 2:225–229, 1977; Myerson in International Journal of Game Theory 9:169–182, 1980). Here, using matrix expressions, we rewrite those sharing rules. With this presentation we identify the close relationship between the fairness property and an equal treatment of necessary players axiom. Moreover, we show that the latter is implied by the equal treatment of equals, linking the fairness property to the notion of equality.