The Constrained Equal Awards Rule for Bankruptcy Problems with a Priori Unions

In this paper, we provide two extensions of the constrained equal awards rule for bankruptcy situations to the class of bankruptcy situations with a priori unions. We present some characterisations and relations with corresponding games. The two new extensions are illustrated by a specific application.     

A vertex oriented approach to the equal remaining obligations rule for minimum cost spanning tree situations

In this paper, we consider spanning tree situations, where players want to be connected to a source as cheap as possible. These situations involve the construction of a spanning tree with the minimum cost as well as the allocation of the cost of this minimum cost spanning tree among its users in a fair way. Feltkamp, Muto and Tijs 1994 introduced the equal remaining obligations rule to solve the cost allocation problem in these situations. Recently, it has been shown that the equal remaining obligations rule satisfies many appealing properties and can be obtained with different approaches. In this paper, we provide a new approach to obtain the equal remaining obligations rule. Specifically, we show that the equal remaining obligations rule can be obtained as the average of the cost allocations provided by a vertex oriented construct-and-charge procedure for each order of players.     

A new look at the 3:1 rule of combat through Markov Stochastic Lanchester models

The 3:1 rule of combat states that in order that for the attacker to win the battle, his forces should be at least three times the force of the defender. This somewhat vague statement has resulted in numerous interpretations and discussions from historical and military science points of view. In this paper we attempt to examine this rule by utilising a number of Markov Stochastic Lanchester models that correspond to various basic combat situations and to draw some conclusions from their implementations. We identify general combat situations where the 3:1 rule is reasonable as well as situations where the force ratio should be either smaller or larger. Since the analysis is performed in the formal and somewhat ‘sterile’ setting of (pure) mathematical modeling, the results should be appropriately interpreted as reasoning of a certain abstraction of the battlefield.     

排序对策的一个新的收益分配准则

针对一个机器的排序问题,给出了排序问题中成本增加量的表达式,提出了收益分配的不小于成本增加量准则。针对一类特殊的排序问题,给出一个符合不小于成本增加量分配准则的解,并证明了它满足有效性,哑元性和单调性。结合一个算例,对本文的提出的方法进行了分析验证。     

Drop out monotonic rules for sequencing situations

This note introduces a new monotonicity property for sequencing situations. A sequencing rule is called drop out monotonic if no player will be worse off whenever one of the players decides to drop out of the queue before processing starts. This intuitively appealing property turns out to be very strong: we show that there is at most one rule satisfying both stability and drop out monotonicity. For the standard model of linear cost functions, the existence of this rule is established.     

The degree value for games with communication structure

A new value concept, called degree value, is proposed by employing the degree game induced by an original game for hypergraph communication situations (including graph communication situations). We provide an axiomatic characterization of the degree value for arbitrary hypergraph communication situations by applying component efficiency and balanced conference contributions, which is a natural extension of balanced link contributions introduced in Slikker (Int J Game Theory 33:505–514, 2005) for graph communication situations. By comparing the degree value with the position value and the Myerson value, it is verified that the degree value is a new allocation rule that differs from both the Myerson value and the position value, and the degree value highlights the important role of the degree of a player in hypergraph communication situations. Particularly, in a uniform hypergraph communication situation, where every conference contains the same number of players, we show that the degree value coincides with the position value.     

Cooperation indices and coalitional value

Summary We introduce an allocation rule for situations defined by a TUgame, a cooperation index and a coalition structure, and characterize it axiomatically. This rule is an extension of Owen's coalitional value; in fact, a variety of previously studied game situations is shown to be embodied and unified by our model.     

A probabilistic position value

In this article, we generalize the position value, defined by Meessen (Master??s thesis, 1988) for the class of deterministic communication situations, to the class of generalized probabilistic communication situations (Gómez et al. in European Journal of Operational Research 190:539?C556, 2008). We provide two characterizations of this new allocation rule. Following in Slikker??s (International Journal of Game Theory 33:505?C514, 2005a) footsteps, we characterize the probabilistic position value using probabilistic versions of component efficiency and balanced link contributions. Then we generalize the notion of link potential, defined by Slikker (International Game Theory Review 7:473?C489, 2005b) for the class of deterministic communication situations, to the class of generalized probabilistic communication situations, and use it to characterize our allocation rule.     

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.     

Balanced contributions for TU games with awards and applications

Calleja et al. [Calleja, P., Borm, P., Hendrickx, R., 2005. Multi-issue allocation situations. European Journal of Operational Research 164, 730–747] introduced multi-issue allocation situations with awards. In this paper, we extend the classical model of cooperative games with transferable utility to the cooperative games with transferable utility and awards. We define a run-to-the-bank rule for cooperative games with transferable utility and awards and characterise it in terms of a property of balanced contributions. We apply our main result to bankruptcy problems and multi-issue allocation situations with awards.     

Extended proportionality in division problems with multiple references

We consider an extension of the classic problem of division with claims to situations in which the characteristics of the agents are multi-dimensional. The proportional rule is the most prominent in the one-dimensional case, however there is no obvious way to define proportionality for the extended model. In this paper we introduce a property which generalizes proportionality, identify the family of rules satisfying it, and characterize a particular rule within this family on the basis of a property of symmetry.     

On a new stopping rule for stochastic approximation

Summary A new stopping rule for the Robbins-Monro process, based on an F-statistic criterion is proposed and its asymptotic behavior established. On the basis of evidence obtained through experimental sampling, the procedure seems to work well over a wide variety of situations. A two-stage procedure, coupling the new rule with an earlier one proposed by Sielken [1973] is recommended for practical use.     

Algorithms with Adaptive Smoothing for Finite Minimax Problems

We present a new feedback precision-adjustment rule for use with a smoothing technique and standard unconstrained minimization algorithms in the solution of finite minimax problems. Initially, the feedback rule keeps a precision parameter low, but allows it to grow as the number of iterations of the resulting algorithm goes to infinity. Consequently, the ill-conditioning usually associated with large precision parameters is considerably reduced, resulting in more efficient solution of finite minimax problems.The resulting algorithms are very simple to implement, and therefore are particularly suitable for use in situations where one cannot justify the investment of time needed to retrieve a specialized minimax code, install it on one's platform, learn how to use it, and convert data from other formats. Our numerical tests show that the algorithms are robust and quite effective, and that their performance is comparable to or better than that of other algorithms available in the Matlab environment.     

New efficient shortest path simplex algorithm: pseudo permanent labels instead of permanent labels

We introduce a new network simplex pivot rule for the shortest path simplex algorithm. This new pivot rule chooses a subset of non-basic arcs to simultaneously enter into the basis. We call this operation a multiple pivot. We show that a shortest path simplex algorithm with this pivot rule performs O(n) multiple pivots and runs in O(nm) time. Our pivot rule is based on the new concept of a pseudo permanently labeled node, and it can be adapted to design a new label-correcting algorithm that runs in O(nm). Moreover, this concept lets us introduce new rules to identify negative cycles. Finally, we compare the network simplex algorithm with multiple pivots with other previously proposed efficient network simplex algorithm in a computational experiment.     

A characterization of the uniform rule based on new robustness properties

We study the problem of allocating a divisible good among a group of people. Each person’s preferences are single-peaked. We consider situations in which there might be more of the resource to be assigned than was planned, or there might be less of the resource. Two robustness properties are formulated, which we call one-sided composition up and one-sided composition down. We show that only one rule satisfies irrelevance of null agents, the equal-division lower bound, and our robustness properties. This rule is the uniform rule.     

Contractually stable networks

We consider situations where players are part of a network and belong to coalitions in a given coalition structure. We propose the concept of contractual stability to predict the networks that are going to emerge at equilibrium when the consent of coalition partners is needed for adding or deleting links. Two different decision rules for consent are analyzed: simple majority and unanimity. We characterize the coalition structures that make the strongly efficient network contractually stable under the unanimity decision rule and the coalition structures that sustain some critical network as contractually stable under the simple majority decision rule and under any decision rule requiring the consent of any proportion of coalition partners. Requiring the consent of coalition members may help to reconcile stability and efficiency in some classical models of network formation.     

New characterizations of the constrained equal awards rule in multi-issue allocation situations

Multi-issue allocation situations are used to study the problem of having to divide an estate among a group of agents. The claim of each agent is a vector specifying the amount claimed by each agent on each issue. We present several axiomatic characterizations of the constrained equal awards rule generalizing some characterizations of these rules in bankruptcy situations.     

Bargaining with nonanonymous disagreement: Decomposable rules

We analyze bargaining situations where the agents’ payoffs from disagreement depend on who among them breaks down the negotiations. We model such problems as a superset of the standard domain of Nash (1950). We first show that this domain extension creates a very large number of new rules. In particular, decomposable rules (which are extensions of rules from the Nash domain) constitute a nowhere dense subset of all possible rules. For them, we analyze the process through which “good” properties of rules on the Nash domain extend to ours. We then enquire whether the counterparts of some well-known results on the Nash (1950) domain continue to hold for decomposable rules on our extended domain. We first show that an extension of the Kalai-Smorodinsky bargaining rule uniquely satisfies the Kalai and Smorodinsky (1975) properties. This uniqueness result, however, turns out to be an exception. We characterize the uncountably large classes of decomposable rules that survive the Nash (1950), Kalai (1977), and Thomson (1981) properties.     

Using multiple sources of information to recognize and classify objects

In this paper we consider the problem of selecting an object or a course of action from a set of possible alternatives. To give the paper focus, we concentrate initially on an object recognition problem in which the characteristic features of the object are reported by remote sensors. We then extend the method to a more general class of selection problems and consider several different scenarios.

Information is provided by a set of knowledge system reports on a single feature, and the output from these systems is not totally explicit but provides posible values for the observed feature along with a degree of certitude.We use fuzzy sets to represent this vague information. Information from independent sources is combined using the Dempster-Shafer approach adapted to the situation in which the focal elements are fuzzy as in the recent paper by J. Yen [7]. We base our selection rule on the belief and plausibility functions generated by this approach to accessing evidence.

For situations in which the information is too sparse and/or too vague to make a single selection, we construct a preference relationship based on the concept of averaged subsethood for fuzzy sets as discussed by B. Koskoin [4]. We also define an explicit metric upon which to base our selection mechanism for situations in which the Dempster-Shafer rule of combination is inappropriate     

Bargaining with nonanonymous disagreement: Decomposable rules

We analyze bargaining situations where the agents’ payoffs from disagreement depend on who among them breaks down the negotiations. We model such problems as a superset of the standard domain of Nash (1950). We first show that this domain extension creates a very large number of new rules. In particular, decomposable rules (which are extensions of rules from the Nash domain) constitute a nowhere dense subset of all possible rules. For them, we analyze the process through which “good” properties of rules on the Nash domain extend to ours. We then enquire whether the counterparts of some well-known results on the Nash (1950) domain continue to hold for decomposable rules on our extended domain. We first show that an extension of the Kalai–Smorodinsky bargaining rule uniquely satisfies the Kalai and Smorodinsky (1975) properties. This uniqueness result, however, turns out to be an exception. We characterize the uncountably large classes of decomposable rules that survive the Nash (1950), Kalai (1977), and Thomson (1981) properties.