Class Of Transferable Utility Games Research Articles

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.

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.

Risk allocation under liquidity constraints

Risk allocation games are cooperative games that are used to attribute the risk of a financial entity to its divisions. In this paper, we extend the literature on risk allocation games by incorporating liquidity considerations. A liquidity policy specifies state-dependent liquidity requirements that a portfolio should obey. To comply with the liquidity policy, a financial entity may have to liquidate part of its assets, which is costly.The definition of a risk allocation game under liquidity constraints is not straightforward, since the presence of a liquidity policy leads to externalities. We argue that the standard worst case approach should not be used here and present an alternative definition. We show that the resulting class of transferable utility games coincides with the class of totally balanced games. It follows from our results that also when taking liquidity considerations into account there is always a stable way to allocate risk.

Risk Allocation Under Liquidity Constraints

Risk allocation games are cooperative games that are used to attribute the risk of a financial entity to its divisions. In this paper, we extend the literature on risk allocation games by incorporating liquidity considerations. A liquidity policy specifies state-dependent liquidity requirements that a portfolio should obey. To comply with the liquidity policy, a financial entity may have to liquidate part of its assets, which is costly. The definition of a risk allocation game under liquidity constraints is not straight- forward, since the presence of a liquidity policy leads to externalities. We argue that the standard worst case approach should not be used here and present an alternative definition. We show that the resulting class of transferable utility games coincides with the class of totally balanced games. It follows from our results that also when taking liquidity considerations into account there is always a stable way to allocate risk.

Resource allocation games: a compromise stable extension of bankruptcy games

This paper presents an extension of the traditional bankruptcy problem. In a resource allocation problem there is a common-pool resource, which needs to be divided among agents. Each agent is characterized by a claim on this pool and an individual linear monetary reward function for assigned resources. Analyzing these problems a new class of transferable utility games is introduced, called resource allocation games. These games are based on the bankruptcy model, as introduced by O’Neill (Math Soc Sci 2:345–371, 1982). It is shown that the properties of totally balancedness and compromise stability can be extended to resource allocation games, although the property of convexity is not maintained in general. Moreover, an explicit expression for the nucleolus of these games is provided.

A mesh-based notion of differential for TU games

We introduce the notion of Burkill–Cesari (BC) differentiability for transferable utility (TU) games and compare it with some other analogous established notions existing in cooperative game theoretical literature. We also apply our notion to the study of the core of a new class of TU games.

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 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.

A Compromise Stable Extension of Bankruptcy Games: Multipurpose Resource Allocation

This paper considers situations characterized by a common-pool resource, which needs to be divided among agents. Each of the agents has some claim on this pool and an individual reward function for assigned resources. This paper analyzes not only the problem of max- imizing the total joint reward, but also the allocation of these rewards among the agents. Analyzing these situations a new class of transferable utility games is introduced, called multipurpose resource games. These games are based on the bankruptcy model, as intro- duced by O'Neill (1982). It is shown that every multipurpose resource game is compromise stable. Moreover, an explicit expression for the nucleolus of these games is provided.

Multi-issue allocation situations

This paper introduces a new class of transferable utility games, called multi-issue allocation games. These games arise from various allocation situations and are based on the concepts underlying the bankruptcy model, as introduced by O'Neill [Math. Social Sci. 2 (1982) 345]. In this model, a perfectly divisible good (estate) has to be divided amongst a given set of agents, each of whom has some claim on the estate. Contrary to the standard bankruptcy model, the current model deals with situations in which the agents' claims are multi-dimensional, where the dimensions correspond to various issues. It is shown that the class of multi-issue allocation games coincides with the class of (non-negative) exact games. The run-to-the-bank rule is introduced as a solution for multi-issue allocation situations and turns out to be Shapley value of the corresponding game. Finally, this run-to-the-bank rule is characterised by means of a consistency property in the spirit of O'Neill.

Bargaining property of nucleolus and τ-value in a class of TU-games

First, we introduce pairwise-bargained consistency with a reference point, and use as reference points the maxmin and the minmax value within pure strategies of a certain constant-sum bimatrix game, and also the game value within mixed strategies of it. Second, we show that the pairwise-bargained consistency with reference point being the maxmin or the minmax value determines the nucleolus in some class of transferable utility games. (This result is known in the bankruptcy games and the pseudoconcave games with respect to supersets of the managers.) This class of games whose element we call a pseudoconcave game with respect to essential coalitions, of course, includes the bankruptcy games and the pseudoconcave games with respect to supersets of the managers. It is proved that this class of games is exactly the same as the class of games which have a nonempty core that is determined only by one-person and ( n − 1)-person coalition constraints. And we give a sufficient condition which guarantees that the bargaining set coincides with the core in this class of games. Third, we interpret the τ-value of a quasibalanced transferable utility game by the pairwise-bargained consistency with reference point being the game value. Finally, by combining the second and the third results, if a transferable utility game in this class is also semiconvex, then the nucleolus and the τ-value are characterized by the pairwise-bargained consistency with different reference points which are given by the associated bimatrix game.

A shapley value for games with restricted coalitions

A “restriction” is a monotonic projection assigning to each coalition of a finite player setN a subcoalition. On the class of transferable utility games with player setN, a Shapley value is associated with each restriction by replacing, in the familiar probabilistic formula, each coalition by the subcoalition assigned to it. Alternatively, such a Shapley value can be characterized by restricted dividends. This method generalizes several other approaches known in literature. The main result is an axiomatic characterization with the property that the restriction is determined endogenously by the axioms.

An axiomatization of the weighted NTU value

Non-symmetric generalizations of the non-transferable utility (NTU) are defined and characterized axiomatically. The first of these is a weighted NTU value that is identical to the (symmetric) NTU value when players have the same weights. On the class of transferable utility games, this weighted NTU value coincides with the weighted Shapley value and on the pure bargaining games it coincides with the non-symmetric Nash bargaining solution. A further extension, the random order NTU value, is also defined and axiomatized and its relationship to the core is discussed.