Ulrich Faigle

The Shapley Value for Cooperative Games under Precedence Constraints

Cooperative games are considered where only those coalitions of players are feasible that respect a given precedence structure on the set of players. Strengthening the classical symmetry axiom, we obtain three axioms that give rise to a unique Shapley value in this model. The Shapley value is seen to reflect the expected marginal contribution of a player to a feasible random coalition, which allows us to evaluate the Shapley value nondeterministically. We show that every exact algorithm for the Shapley value requires an exponential number of operations already in the classical case and that even restriction to simple games is #P-hard in general. Furthermore, we outline how the multi-choice cooperative games of Hsiao and Raghavan can be treated in our context, which leads to a Shapley value that does not depend on pre-assigned weights. Finally, the relationship between the Shapley value and the permission value of Gilles, Owen and van den Brink is discussed. Both refer to formally similar models of cooperative games but reflect complementary interpretations of the precedence constraints and thus give rise to fundamentally different solution concepts.

Sensory gating revisited: Relation between brain oscillations and auditory evoked potentials in schizophrenia

Disturbances of auditory information processing have repeatedly been shown in schizophrenia. To contribute to a better understanding of the neurophysiological underpinnings of habituation in auditory processing and its disturbance in schizophrenia we used three different approaches to analyze auditory evoked responses, namely phase-locking (PL) analyses, single trial amplitudes, and averaged event-related potentials (P50 and N100). Given that brain oscillations reflect the neuronal correlates of information processing we hypothesized that PL and amplitudes reflect even more essential parts of auditory processing than the averaged ERP responses. In 32 schizophrenia patients and 32 matched controls EEG was continuously recorded using an auditory paired click paradigm. PL of the lower frequency bands (alpha and theta) was significantly reduced in patients whereas no significant differences were present in higher frequencies (gamma and beta). Alpha and theta PL and amplitudes showed a marked increase after the first click and to a minor degree after the second one. This habituation was more prominent in controls whereas in schizophrenia patients the response to both clicks differed only slightly. N100 suppression was significantly reduced in schizophrenia patients whereas no group differences were present with respect to the P50. This corresponded to the finding that gamma mostly contributed to the prediction of the P50 response and theta mostly to the N100 response. Our data showed that analyzing phase and amplitude in single trials provides more information on auditory information processing and reflects differences between schizophrenia patients and controls better than analyzing the averaged ERP responses.

Some Convergence Results for Probabilistic Tabu Search

During recent years, much work has gone into the exploration of general fundamental principles underlying local search strategies for combinatorial optimization. Many of these strategies can be subsumed under the general framework of Tabu Search, which introduces mechanisms of guidance and control based on flexible memory processes, broadening the range of strategic possibilities beyond those incorporated in memoryless search heuristics such as Simulated Annealing. We consider some examples of such memory based strategies for modifying both the generation and acceptance probabilities and investigate their impact on convergence results. It turns out that several Tabu Search ideas can be subjected to mathematical analyses similar to those applied to Simulated Annealing, making it possible to establish corresponding convergence properties based on a broader foundation. INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Note on scheduling intervals on-line

An optimal on-line algorithm is presented for the following optimization problem, which constitutes the special case of the k-track assignment problem with identical time windows. Intervals arrive at times ti and demand service time equal to their length. An interval is considered lost if it is not assigned to one of k identical service stations immediately or if its service is interrupted. Minimizing the losses amounts to coloring a maximal set of intervals in the associated interval graph properly with at most k colors. Optimality of the on-line algorithm is proved by showing that it performs as well as the optimal greedy k-coloring algorithm due to Faigle and Nawijn and, independently, to Carlisle and Lloyd for the same problem under full a priori information.

Cores of games with restricted cooperation

Games with restricted cooperation are cooperativeN-person games with sidepayments, where the collection of feasible coalitions need not comprise all subsets of players and thus is restricted. We study balanced and completely balanced games in this context and derive the corresponding core theorems from a sandwich theorem for set functions within the setting of linear programming. In particular, we discuss general convex games, which Edmonds and Giles (1977) have shown to be of particular importance also in combinatorial optimization.ZusammenfassungSpiele mit beschränkter Kooperation sind kooperativeN-Personenspiele mit Nebenzahlungen, wobei nicht jede Teilmenge von Spielern zulässig zu sein braucht. In diesem Sinn sind die Kooperationsmöglichkeiten beschränkt. Balancierte und vollständig balancierte Spiele werden in diesem Zusammenhang untersucht. Die entsprechenden Sätze über die Existenz von Kernen werden von einem Sandwichsatz über Mengenfunktionen im Rahmen der linearen Programmierung abgeleitet. Insbesondere werden allgemeine konvexe Spiele diskutiert, deren Bedeutung auch für die kombinatorische Optimierung Edmonds and Giles (1977) aufgezeigt haben.

On the computation of the nucleolus of a cooperative game

Abstract. We consider classes of cooperative games. We show that we can efficiently compute an allocation in the intersection of the prekernel and the least core of the game if we can efficiently compute the minimum excess for any given allocation. In the case where the prekernel of the game contains exactly one core vector, our algorithm computes the nucleolus of the game. This generalizes both a recent result by Kuipers on the computation of the nucleolus for convex games and a classical result by Megiddo on the nucleolus of standard tree games to classes of more general minimum cost spanning tree games. Our algorithm is based on the ellipsoid method and Maschlers scheme for approximating the prekernel.

On some approximately balanced combinatorial cooperative games

A model of taxation for cooperativen-person games is introduced where proper coalitions Are taxed proportionally to their value. Games with non-empty core under taxation at rateɛ-balanced. Sharp bounds onɛ in matching games (not necessarily bipartite) graphs are estabLished. Upper and lower bounds on the smallestɛ in bin packing games are derived and euclidean random TSP games are seen to be, with high probability,ɛ-balanced forɛ≈0.06.

Computing the nucleolus of min-cost spanning tree games is NP -hard

Abstract. We prove that computing the nucleolus of minimum cost spanning tree games is in general NP-hard. The proof uses a reduction from minimum cover problems.

Geometries on partially ordered sets

Abstract Geometries on finite partially ordered sets extend the concept of matroids on finite sets to partially ordered sets. Geometries are defined in terms of closure operators on partially ordered sets. The lattice of closed sets is semimodular, and every finite semimodular lattice is isomorphic to the lattice of closed sets of some geometry. A distinction between geometries and pregeometries is made, and deletion and contraction are discussed as constructions on pregeometries. Strong geometries and strong semimodular lattices are introduced, and strong geometries are characterized as those geometries for which every contraction results in a geometry. Cryptomorphic descriptions of pregeometries are given in terms of flats, hyperplanes, rank function, independent, and B-independent sets. It is shown that among the geometries precisely the strong geometries possess the Kuros-Ore exchange property, and a general marriage theorem for pregeometries is proved. To each pregeometry on a partially ordered set a dual on the dual partial order is associated as an operation of period 2. Deletion and contraction are seen to commute under duality, and it is shown that the dual of a geometry is, in general, only a pregeometry.

On approximately fair cost allocation in Euclidean TSP games

We consider the problem of allocating the cost of an optimal traveling salesman tour in a fair way among the nodes visited; in particular, we focus on the case where the distance matrix of the underlying TSP problem satisfies the triangle inequality. We thereby use the model of TSP games in the sense of cooperative game theory. We give examples showing that the core of such games may be empty, even for the case of Euclidean distances. On the positive, we develop an LP-based allocation rule guaranteeing that no coalition pays more thanα times its own cost, whereα is the ratio between the optimal TSP-tour and the optimal value of its Held-Karp relaxation, which is also known as the solution over the “subtour polytope”. A well-known conjecture states thatα≤4/3. We also exhibit examples showing that this ratio cannot be improved below 4/3.SummaryWir betrachten die Aufgabe, die Kosten einer optimalen Traveling-Salesman-Tour fair unter den besuchten Knoten zu verteilen; insbesondere untersuchen wir den Fall, daß die Kostenmatrix des zugrundeliegenden TSP-Problems die Dreiecksungleichung erfüllt. Dazu wird das Modell von TSP-Spielen im Sinne der kooperativen Spieltheorie benutzt. Wir zeigen anhand eines Beispiels, daß der Core eines solchen Spiels leer sein kann, selbst im Falle euklidischer Distanzen. Andererseits geben wir eine LP-basierte Verteilungsregel an, die garantiert, daß keine Koalition mehr als dasα-fache ihrer eigenen Kosten bezahlen muß, wobeiα das Verhältnis zwischen den Kosten einer optimalen TSP-Tour und dem Optimum der Held-Karp-Relaxation ist, die auch als Lösung über dem “subtour polytope” bekannt ist. Es wird allgemein vermutet, daßα≤4/3. Abschließend geben wir eine Klasse von Beispielen an, die beweist, daß keine allgemeine Verteilungsregel für das TSP-game ein generell besseres Verhältnis als 4/3 zwischen der Belastung einer Koalition und ihren Kosten garantieren kann.