Holger I. Meinhardt

Convexity of oligopoly games without transferable technologies

We present sufficient conditions involving the inverse demand function and the cost functions to establish the convexity of oligopoly TU-games without transferable technologies. For convex TU-games it is well known that the core is relatively large and that it is generically nonempty. The former property provides us with an answer about the stability of cartels, the latter property gives us an indication about the incentive to found a cartel. Furthermore, for convex games the kernel is a singleton in the core and the Shapley value is located in the center of gravity of the core, thus, there are natural solutions available to split the benefits of a cartel agreement.

Common Pool Games Are Convex Games

For the class of cooperative common pool games the paper focuses on the question of how, during the preplay negotiation process, the ability of coalitions to enforce their claims imposes externalities on the opposition by having an impact on the jointly produced resource. One of our main results is that common pool games are clear games. Based on this result we are able to derive sufficient conditions for the convexity of the characteristic function, which establishes the second main result in the paper, namely that cooperative common pool games are characterized by increasing returns with respect to the coalition size. Copyright 1999 by Blackwell Publishing Inc.

An LP approach to compute the pre-kernel for cooperative games

We present an algorithm to compute the (pre)-kernel of a TU-game (N, v) with a system of (n 2) + 1 linear programming problems. In contrast to the algorithms using convergence methods to compute a point of the (pre)- kernel the emphasis of the chosen method lies not on efficiency and guessing good starting points but on computing large parts or in good cases the whole (pre)-kernel of a game. The chosen algorithm computes on a first step by relying on linear programming the (n 2) largest bi-symmetrical amounts δije which can be transferred from player i to j while remaining in the strong e-core. The associated payoff vector is a midpoint of the e-core segment in i-j direction and is therefore a candidate that satisfies the bisection property. From these results we can determine in a sophisticated pattern-matching procedure the constraints which are needed to construct the final linear programming problem for computing at least a (pre)-kernel point of the game. From the derived final linear program large parts or the whole (pre)-kernel can be easily calculated. Finally, the program checks if the computed (pre)-kernel candidate belongs to the (pre)-kernel. In cases where the candidate does not pass the (pre)-kernel check, the function is called a further time with additional informations about the game. A further call could be necessary if the intersection set of the (n 2) solution sets is empty and no correction of the final LP is applied for, in this case, for at least one distinct pair of players the largest bi-symmetrical transfer is of no importance to compute a (pre)-kernel point, that is, no candidate of the final linear problem satisfies the bisection property. This implies that at least one largest bi-symmetrical transfer δije is greater than the maximal transfer in i-j direction that is possible at a (pre)-kernel point ⇀ y while remaining in the core, that is, δije > δije ( ⇀ y), with ⇀ y ∈ k* (Γ). Hence, if the solution intersection set is non-empty, then all payoff vectors in the intersection set possess the bisection property and are therefore (pre)- kernel elements. The (pre)-kernel of a TU-game with an empty core can be computed, for instance, by providing the epsilon value for the least-core as an additional information.

(AVERAGE-) CONVEXITY OF COMMON POOL AND OLIGOPOLY TU-GAMES

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ON THE SUPERMODULARITY OF HOMOGENEOUS OLIGOPOLY GAMES

The main purpose is to prove the supermodularity (convexity) property of a cooperative game arising from an economical situation. The underlying oligopoly situation is based on a linear inverse demand function as well as linear cost functions for the participating firms. The characteristic function of the so-called oligopoly game is determined by maximizing, for any cartel of firms, the net profit function over the feasible production levels of the firms in the cartel, taking into account their individual capacities of production and production technologies. The (rather effective) proof of the supermodularity of the characteristic function of the oligopoly game relies on the use of maximizers for the relevant maximization problems. A similar proof technique will be reviewed for a related cooperative oligopoly game arising from a slightly modified oligopoly situation where the production technology of the cartel is determined by the most efficient member firm.

Toward an Analysis of Cooperation and Fairness That Includes Concepts of Cooperative Game Theory

In a situation that can be classified as a social dilemma, people involved ought to consider how to overcome the problem of hurting common interests when acting upon individualistic standards; one has to question how to realize gains through cooperation. Human history provides a multitude of examples demonstrating institutional solutions for social dilemmas. Some of these examples have both attained success and survived for a long time (cf. Ostrom, 1990). With respect to the (material) incentive structure, we can distinguish between two different kinds of institutions: first, institutions that change incentives (e.g., introduction of control and sanctions, introduction of binding contracts, etc.); and second, institutions that do not change incentives. Communication belongs to the second. In this chapter, we deal with such a situation. People may talk to each other before deciding on their (individual and hidden) actions. Some economists label such communication “cheap talk.” Even if a negotiation takes place and an agreement is reached, people are not bound to fulfill the agreement by acting accordingly. Drawing on the theory of pure individual rationality, actions are assumed to be determined only by individual incentives and not by the previous discussion of the possible prospects by reaching an agreement. In contrast, there is multiple evidence from field and laboratory studies that agreements are successfully established even in the absence of institutions guaranteeing the fulfillment of these agreements (Dawes et al., 1977; Ostrom, 1998; Sally, 1995). Since we deal with non-binding agreements (called “agreements” from here on), the results may not conform to the agreements when people have carried out their decisions. This might be caused by the fact that people have deviated from the once-agreed-upon outcome in pursuing their own interests. Thus, the question of compliance becomes crucial: Under which conditions can we expect compliance, and in which conditions will the agreement be implemented? The extent of compliance may depend on both the incentive structure and the course of negotiation. A popular hypothesis states that people reduce compliance if they feel they are treated unfairly. If this holds true, it becomes an important issue to examine the

Non-binding agreements and fairness in commons dilemma games

Usually, common pool games are analyzed without taking into account the cooperative features of the game, even when communication and non-binding agreements are involved. Whereas equilibria are inefficient, negotiations may induce some cooperation and may enhance efficiency. In the paper, we propose to use tools of cooperative game theory to advance the understanding of results in dilemma situations that allow for communication. By doing so, we present a short review of earlier experimental evidence given by Hackett, Schlager, and Walker 1994 (HSW) for the conditional stability of non-binding agreements established in face-to-face multilateral negotiations. For an experimental test, we reanalyze the HSW data set in a game-theoretical analysis of cooperative versions of social dilemma games. The results of cooperative game theory that are most important for the application are explained and interpreted with respect to their meaning for negotiation behavior. Then, theorems are discussed that cooperative social dilemma games are clear (alpha- and beta-values coincide) and that they are convex (it follows that the core is “large”): The main focus is on how arguments of power and fairness can be based on the structure of the game. A second item is how fairness and stability properties of a negotiated (non-binding) agreement can be judged. The use of cheap talk in evaluating experiments reveals that besides the relation of non-cooperative and cooperative solutions, say of equilibria and core, the relation of alpha-, beta- and gamma-values are of importance for the availability of attractive solutions and the stability of the such agreements. In the special case of the HSW scenario, the game shows properties favorable for stable and efficient solutions. Nevertheless, the realized agreements are less efficient than expected. The realized (and stable) agreements can be located between the equilibrium, the egalitarian solution and some fairness solutions. In order to represent the extent to which the subjects obey efficiency and fairness, we present and discuss patterns of the corresponding excess vectors.

The Pre-Kernel as a Fair Division Rule for Some Cooperative Game Models

Rather than considering fairness as some private property or a subjective feeling of an individual, we study fairness on a set of principles (axioms) which describes the pre-kernel. Apart from its appealing axiomatic foundation, the pre-kernel also qualifies in accordance with the recent findings of Meinhardt (The pre-kernel as a tractable solution for cooperative games: an exercise in algorithmic game theory. Springer, Berlin, 2013b) as an attractive fair division rule due to its ease of computation by solving iteratively systems of linear equations. To advance our understanding of compliance on non-binding agreements, we start our analysis with a Cournot situation to derive four cooperative game models well introduced in the literature, where each of it represents different aspiration levels of partners involved in a negotiation process of splitting the monopoly proceeds. In this respect, we demonstrate the bargaining difficulties that might arise when agents are not acting self-constraint, and what consequences this impose on the stability of a fair agreement.

Cooperative Game Theory

In preparation for the subsequent chapters we provide the reader in this chapter with some game properties and solution concepts from cooperative game theory with transferable Utility. We conflne ourself in discussing cooperative game theory to the part where the cooperative Output of a coalition can be measured by a numeraire good like money and therefore can be transfered among the players via side-payments. The purpose of this chapter is not to give a comprehensive survey of cooperative game theory. We will just discuss these parts which are valuable to understand the remaining parts of the monograph where we rely to a large extent on cooperative game theory to analyze the incentives for cooperative decision making in common pool situations. In our investigation of a common pool environment we are interested in the feasible gains which are realizable through mutual Cooperation and to the issue whether individuals are better off through Cooperation than acting alone. The formal aspect for Coming up with an answer will be captured through cooperative game theory.

A Pre-Kernel Characterization and Orthogonal Projection

We have established that a quadratic and convex function can be attained from each payoff equivalence class. After that, we proved that the objective function, from which a pre-kernel element can be pursued, is composed of a finite collection of quadratic and convex functions. In addition, from each payoff equivalence class a linear transformation can be derived that maps payoff vectors into the space of unbalanced excess configurations. The resultant column vectors of the linear mapping constitutes a spanning system of a vector space of balanced excesses. Similar to payoff vectors, any vector of unbalanced excesses is mapped by an orthogonal projection on a m-dimensional flat of balanced excesses, whereas m ≤ n. Moreover, each payoff set determines the dimension and location of a particularly balanced excess flat in the vector space of unbalanced excesses. Since, a spanning system or basis of a flat is not unique, we can derive a set of transition matrices where each transition matrix constitutes a change of basis. This basis change has a natural interpretation, which transforms a bargaining situation into another equivalent bargaining situation. It is established that the transition matrices belong to the positive general linear group \({\text{GL}}^{+}(m; \mathbb{R})\). As a consequence, a group action can be identified on the set of all ordered bases of a flat of balanced excesses. Any induced payoff equivalence class of a TU game can be associated with a specific basis or bargaining situation. Finally, a first pre-kernel result with regard to the orthogonal projection method is given.