Stable bargaining outcomes in patent licensing: A cooperative game approach without side payments

By formulating negotiations about licensing payments as cooperative games without side payments, we investigate stable bargaining outcomes in licensing a cost-reducing technology of an external patent holder to oligopolistic firms producing a homogeneous product under two policies: fee and royalty. The final bargaining outcome in fee licensing is uniquely determined, because the bargaining set for a coalition structure in which the patent holder can gain the maximum profit is a singleton. Under the royalty policy, the non-empty core for a coalition structure suggests that the patent holder should license his patented technology to all firms. Moreover, royalty licensing may be superior to fee licensing for the patent holder, when licensing is carried out through bargaining.     

Bargaining sets of cooperative games without side payments

In this paper an analogue of the bargaining setM ₁ ⁱ is defined for cooperative games without side payments. An existence theorem is proved for games of pairs, while it is shown by an example that no general existence theorem holds.     

Bargaining set theory and stability in coalition governments

Various bargaining set theories are compared as predictors of coalition government portfolio distribution. While the kernel and B₁-bargaining set are known to exist in voting games with side payments, it is argued that the kernel, and thus B₁, are poor predictors. The B₂-bargaining set, a subset of B₁, when it exists is shown to be a good payoff predictor in a fractionalized and depolarized parliamentary situation (Finland: 1945ndash;1971). Moreover this predictor provides some explanation for the formation of surplus (winning but not minimal) coalitions.     

On coalition formation: a game-theoretical approach

This paper deals with the question of coalition formation inn-person cooperative games. Two abstract game models of coalition formation are proposed. We then study the core and the dynamic solution of these abstract games. These models assume that there is a rule governing the allocation of payoffs to each player in each coalition structure called a payoff solution concept. The predictions of these models are characterized for the special case of games with side payments using various payoff solution concepts such as the individually rational payoffs, the core, the Shapley value and the bargaining set M₁ ⁽ⁱ⁾. Some modifications of these models are also discussed.     

A three-person cooperative game formulation of the world oil market

The world oil market is modelled as a three-person cooperative game in characteristic function form with and without side payments. The three players are the oil importing countries, the group in OPEC led by Iran and the group in OPEC led by Saudi Arabia. Various solutions of the games are studied such as the core, the Shapley value, the bargaining set and the nucleolus.     

Some recent results inn-person game theory

The discussion will center mainly on some work on two solution concepts: the core for gaines without side payments and the nucleolus for games with side payments (characteristic funtion games). The core has become an important equilibrium concept in mathematical economics. The nucleolus is related to the theory of bargaining sets.This paper was presented at the 7th Mathematical Programming Symposium 1970, The Hague, The Netherlands.This work was supported by the National Science Foundation under grant GK-4795.     

Values of games without side payments

A value forn-person games without side payments is given which coincides with theShapley value for games with side payments, and with theNash value for two-person games.     

A characterization of polyhedral market games

The class of games without side payments obtainable from markets having finitely many commodities and continuous concave utility functions is considered. It is first shown that each of these so-called market games is totally balanced, for a reasonable generalization of the idea of a balanced side payment game. It is then shown that among polyhedral games (i.e., games for which each (V(S) is a polyhedron), this property characterizes the market games.     

The reactive bargaining set: Structure, dynamics and extension to NTU games

The reactive bargaining set (Granot [1994]) is the set of outcomes for which no justified objection exists. Here, in a justified objection the objector first watches how the target tries to act (if he has such an option), and then reacts by making a profit and ruining the target's attempt to maintain his share. In this paper we explore properties of the reactive bargaining set, set up the system of inequalities that defines it, and construct a dynamic system in the sense of Stearns' transfer scheme that leads the players to this set. We also extend the definition of the reactive bargaining set toNTU games in a way that keeps it nonempty. To shed light on its nature and its relative ease of computation, we compute the reactive bargaining set for games that played important role in the game theory literature.     

On ε-cores of balanced games

The present note considers a balanced game without side payments with a continuum of players. The nonemptiness of theε-core for some classes of such games is established.     

ψ-Stability of a cooperative game without side payments

In the present paper we defineψ-stability for theAumann-Peleg theory of cooperative games without side payments, and we prove some theorems which are analogous to the core theorem byAumann andBurger. These theorems provide foundations of the theory ofψ-stability for cooperative games without side payments in addition to being of interest for their own sake. We also consider the composition of two admissible functionsψ ₁ andψ ₂.     

Stationary perfect equilibria of an n-person noncooperative bargaining game and cooperative solution concepts

This paper characterizes the stationary (subgame) perfect equilibria of an n-person noncooperative bargaining model with characteristic functions, and provides strategic foundations of some cooperative solution concepts such as the core, the bargaining set and the kernel. The contribution of this paper is twofold. First, we show that a linear programming formulation successfully characterizes the stationary (subgame) perfect equilibria of our bargaining game. We suggest a linear programming formulation as an algorithm for the stationary (subgame) perfect equilibria of a class of n-person noncooperative games. Second, utilizing the linear programming formulation, we show that stationary (subgame) perfect equilibria of n-person noncooperative games provide strategic foundations for the bargaining set and the kernel.     

The reactive bargaining set for cooperative games

We introduce a new bargaining set for cooperative games in characteristic function form, and investigate its structure and properties. We prove that the new bargaining set is not empty. In fact, we show that it contains the kernel and is contained in the classical bargaining set ${\mathcal{M}^i_1}$ , and we further prove that it consists of the unique symmetric vector for the class of simple majority games.     

The semireactive bargaining set of a cooperative game

The semireactive bargaining set, a solution for cooperative games, is introduced. This solution is in general a subsolution of the bargaining set and a supersolution of the reactive bargaining set. However, on various classes of transferable utility games the semireactive and the reactive bargaining set coincide. The semireactive prebargaining set on TU games can be axiomatized by one-person rationality, the reduced game property, a weak version of the converse reduced game property with respect to subgrand coalitions, and subgrand stability. Furthermore, it is shown that there is a suitable weakening of subgrand stability, which allows to characterize the prebargaining set. Replacing the reduced game by the imputation saving reduced game and employing individual rationality as an additional axiom yields characterizations of both, the bargaining set and the semireactive bargaining set. Received September 2000/Revised version June 2001     

Cores in noncooperative games

Noncooperative games in normal form and in characteristic function form are considered. The supergame of the noncooperative game is defined as an infinite sequence of plays of the original game. The notions of strong Pareto equilibrium point (s.p.e.p.) and essential core are introduced. A relationship between the essential core of a noncooperative game and the set of s.p.e.p. of its supergame is asserted. This result is similar to that ofAumann for cooperative games without side payments.     

Bargaining in cooperative games

This paper introduces a new solution concept for cooperative games without sidepayments, called the aspiration bargaining set, and proves that this set is nonempty for a wide, and economically interesting, class of games.     

A note on a theorem of Aumann and Dreze

Aumann and Dreze (1974) examine sections of cores of side payments games via cores of reduced games. We study sections of cores of games without side payments.A preliminary version appeared as discussion paper no. 392-89 at the University of Mannheim.     

Generalised bargaining sets for cooperative games

Although theM ₁ -bargaining set for games with side payments is known to exist, it frequently contains payoffs which are highly inequitable. For this reason the more restrictedM ₂-bargaining set is of interest. SinceM ₂ is not known to exist in general, this paper introduces anM _*-bargaining set, contained inM ₁ and containingM ₂, and presents an existence theorem. For the class of symmetric, simple games with decreasing returns, theM ₂ -bargaining set is shown to exist, and a fairly severe restriction on payoffs satisfyingM ₂ -stability is obtained.     

Structure of equilibria inN-person non-cooperative games

Here we study the structure of Nash equilibrium points forN-person games. For two-person games we observe that exchangeability and convexity of the set of equilibrium points are synonymous. This is shown to be false even for three-person games. For completely mixed games we get the necessary inequality constraints on the number of pure strategies for the players. Whereas the equilibrium point is unique for completely mixed two-person games, we show that it is not true for three-person completely mixed game without some side conditions such as convexity on the equilibrium set. It is a curious fact that for the special three-person completely mixed game with two pure strategies for each player, the equilibrium point is unique; the proof of this involves some combinatorial arguments.     

An impossibility result concerningn-person bargaining games

In this note we show that a solution proposed byRaiffa for two-person bargaining games, which has recently been axiomatized byKalai/Smorodinsky, does not generalize in a straightforward manner to generaln-person bargaining games. Specifically, the solution is not Pareto optimal on the class of alln-person bargaining games, and no solution which is can possess the other properties which characterizeRaiffa's solution in the two-person case.