Jeroen Suijs

On the balancedness of multiple machine sequencing games

This paper studies m-sequencing games that arise from sequencing situations with m parallel and identical machines. These m-sequencing games, which involve n players, give rise to m-machine games, which involve m players. Here, n corresponds to the number of jobs in an m-sequencing situation, and m corresponds to the number of machines in the same m-sequencing situation. We prove that an m-sequening game is balanced if and only if the corresponding m-machine game is balanced. Furthermore, it is shown that m-sequencing games are balanced ifm 2 f1;2g. Finally, ifm 3, balancedness is established for two special classes of m-sequencing games.

Cooperative Decision-Making under Risk

Acknowledgements. Notations. 1. Introduction. 2. Cooperative Game Theory. 3. Stochastic Cooperative Games. 4. The Core, Superadditivity, and Convexity. 5. Nucleoli for Stochastic Cooperative Games. 6. Risk Sharing and Insurance. 7. Price Uncertainty in Linear Production Situations. A Probability Theory. References. Index.

Stochastic Cooperative Games

For TU-games, the payoff of a coalition is assumed to be known with certainty. In many cases though, the payoffs to coalitions can be uncertain. This would not raise a problem if the agents can await the realizations of the payoffs before deciding which coalitions to form and which allocations to settle on. But if the formation of coalitions and allocations has to take place before the payoffs are realized, the theory of TU-games does no longer apply.

Linear transformation of products: games and economies

In this paper, we introduce situations involving the linear transformation of products (LTP). LTP situations are production situations where each producer has a single linear transformation technique. First, we approach LTP situations from a (cooperative) game theoretical point of view. We show that the corresponding LTP games are totally balanced. By extending an LTP situation to one where a producer may have more than one linear transformation technique, we derive a new characterization of (nonnegative) totally balanced games: each totally balanced game with nonnegative values is a game corresponding to such an extended LTP situation. The second approach to LTP situations is based on a more economic point of view. We relate (standard) LTP situations to economies in two ways and we prove that the economies are standard exchange economies (with production). Relations between the equilibria of these economies and the cores of cooperative LTP games are investigated.

Communication and Cooperation in Public Network Situations

This paper focuses on sharing the costs and revenues of maintaining a public network communication structure. Revenues are assumed to be bilateral and communication links are publicly available but costly. It is assumed that agents are located at the vertices of an undirected graph in which the edges represent all possible communication links. We take the approach from cooperative game theory and focus on the corresponding network game in coalitional form which relates any coalition of agents to its highest possible net benefit, i.e., the net benefit corresponding to an optimal operative network. Although finding an optimal network in general is a difficult problem, it is shown that corresponding network games are (totally) balanced. In the proof of this result a specific relaxation, duality and techniques of linear production games with committee control play a role. Sufficient conditions for convexity of network games are derived. Possible extensions of the model and its results are discussed.

Stable profit sharing in cooperative investments

Abstract.This paper examines profit sharing in cooperative investments where investors bundle their capital endowments to meet the capital requirements of long term investment projects. Furthermore, investors may reinvest intertemporal gains from existing projects into new projects. Focus is on stable allocation schemes as stability is necessary to sustain the long term cooperation of investors. The paper presents sufficient conditions for the existence of stable profit sharing schemes using linear programming techniques.

Cooperation in Capital Deposits

Abstract. The rate of return earned on a deposit can depend on its term, the amount of money invested in it, or both. Most banks, for example, offer a higher interest rate for longer term deposits. This implies that if one individual has capital available for investment now, but needs it in the next period, whereas the opposite holds for another individual, then they can both benefit from cooperation since it allows them to invest in a longer term deposit. A similar situation arises when the rate of return on a deposit depends on the amount of capital invested in it. Although the benefits of such cooperative behavior may seem obvious to all individuals, the actual participation of an individual depends on what part of the revenues he eventually receives. The allocation of the jointly earned benefits to the investors thus plays an important part in the stability of the cooperation. This paper provides a game theoretical analysis of this allocation problem. Several classes of corresponding deposit games are introduced. For each class, necessary conditions for a nonempty core are provided, and allocation rules that yield core-allocations are examined.Zusammenfassung. Die Verzinsung einer Geldanlage kann von der Fristigkeit der Anlage, von der Höhe der Anlagesumme oder von beiden Parametern abhängig sein. Die meisten Banken zum Beispiel bieten eine höhere Verzinsung für längerfristige Anlagen an. Daraus folgt, dass, wenn ein Wirtschaftssubjekt jetzt über Kapital verfügt, das es erst in der nächsten Periode benötigt, während für ein anderes Wirtschaftssubjekt das Gegentei gilt, beide einen Vorteil aus einer Kooperation ziehen, da sie ihnen erlaubt, in eine längerfristige Anlage zu investieren. Eine ähnliche Situation liegt vor, wenn die Verzinsung der Anlage von der Anlagesumme abhängt. Obwohl die Vorteile solch kooperativen Verhaltens allen Wirtschaftssubjekten offensichtlich erscheinen, hängt die tatsächliche Partizipation eines Wirtschaftssubjekts von dem Anteil ab, den es letztendlich vom Gesamtergebnis erhalten wird. Die Aufteilung der gemeinsam erreichten Vorteile auf die Investoren spielt also eine wichtige Rolle für die Stabilität der Kooperation. Diese Arbeit stellt eine spieltheoretische Analyse dieses Allokationsproblems vor. Einige Klassen von entsprechenden Anlagespielen werden eingeführt. Für jede Klasse werden die notwendigen Bedingungen für das Vorhandensein eines nicht leeren Kerns aufgezeigt und Allokationsregeln für die Aufteilung des Kerns untersucht.

Nucleoli for Stochastic Cooperative Games

The nucleolus, a solution concept for TU-games, originates from Schmeidler (1969). This solution concept yields an allocation that minimizes the excesses of the coalitions in a lexicographical way. The excess describes how dissatisfied a coalition is with the proposed allocation. The larger the excess of a particular allocation, the more a coalition is dissatisfied with this allocation. For Schmeidler’s nucleolus the excess is defined as the difference between the payoff a coalition can obtain when cooperating on its own and the payoff received by the proposed allocation. So, when less is allocated to a coalition, the excess of this coalition increases and the other way round.

Cooperative Game Theory

This chapter provides the reader with a brief introduction into cooperative game theory, making the usual distinction between games with transferable utility (TU-games) and games with non-transferable utility (NTU-games). We start with defining a mathematical framework to describe cooperative decision making problems in general. Section 2.2 then shows under which conditions such a problem gives rise to an NTU-game and a TU-game, respectively. By means of an example we try to explain when utility is transferable and when not, and what the consequences are for describing the benefits. Subsequently, we state the formal definition of transferable utility. Section 2.3 provides a brief survey of TU-games. We discuss properties like superadditivity and convexity and define the solution concepts core, Shapley value, and nucleolus. Furthermore, we state the main results concerning these concepts. The succeeding section then shows to what extent these results carry over to the class of NTU-games. Note, however, that both surveys are far from complete. They only provide the concepts necessary for understanding the remainder of this monograph. The chapter ends with a summary of chance-constrained games, in which the benefits from cooperation are stochastic variables. This summary is based on Chames and Granot (1973), (1976), (1977), and Granot (1977) and states the definitions of the so-called prior core, prior Shapley value, prior nucleolus, and the two-stage nucleolus. Furthermore, we compare these definitions with the corresponding ones for TU-games.

Price Uncertainty in Linear Production Situations

This paper analyzes linear production situations with price uncertainty, and shows that the corrresponding stochastic linear production games are totally balanced. It also shows that investment funds, where investors pool their individual capital for joint investments in financial assets, fit into this framework. For this subclass, the paper provides a procedure to construct an optimal investment portfolio. Furthermore it provides necessary and sufficient conditions for the proportional rule to result in a core-allocation.