Values for two-stage games: Another view of the Shapley axioms

This short study reports an application of the Shapley value axioms to a new concept of two-stage games. In these games, the formation of a coalition in the first stage entitles its members to play a prespecified cooperative game at the second stage. The original Shapley axioms have natural equivalents in the new framework, and we show the existence of (non-unique) values and semivalues for two stage games, analogous to those defined by the corresponding axioms for the conventional (one-stage) games. However, we also prove that all semivalues (hence, perforce, all values) must give patently unacceptable solutions for some two-stage majority games (where the members of a majority coalition play a conventional majority game). Our reservations about these prescribed values are related to Roth's (1980) criticism of Shapley's -transfer value for non-transferable utility (NTU) games. But our analysis has wider scope than Roth's example, and the argument that it offers appears to be more conclusive. The study also indicates how the values and semivalues for two-stage games can be naturally generalized to apply for multi-stage games.Earlier versions of this study were presented at the International Conference on Game Theory and its Applications, organized by Ohio State University in 1987, and at the Workshop on Mathematical Economics and Game Theory at Tel Aviv Unversity. We gratefully acknowledge the valuable comments received on both occasions, especially those of Robert J. Aumann, Roy Gardner, Sergiu Hart, Ehud Kalai, Michael Maschler, Alvin E. Roth, and Lloyd S. Shapley, and also those ofIJGT's anonymous referees. Of course, all responsibility lies with us.     

Feeble forces and gravity

We develop a scenario in which feeble intermediate range forces emerge as an effect resulting from the compactification (à la Kaluza-Klein) of multidimensional theories. These feeble forces compete with gravity and in general permit different bodies to fall to earth with different accelerations. We show that these feeble forces are mediated by vectors (V) and/or scalars (S), whose dimensionless coupling constants are typically of order g_v g_s 10^–10 Under certain plausible assumptions the ranges of these feeble forces are expected to be of order 1 m to 1 km. It is conjectured that the general strategy will prove applicable to realistic multidimensional theories such as the 10-dimensional superstring theories. We speculate that deviations from the standard gravitational force-similar to the ones reported recently as a fifth force-may be interpreted as evidence for higher dimensions.This essay received the second award from the Gravity Research Foundation for the year 1986.Supported by the U.S. Department of Energy under grant # DE-FG03-84ER-40168.     

Anionic polymerization and copolymerization of p-bromostyrene

Monodispersed poly(4-bromostyrenes) (PBs) and their block copolymers with styrene, isoprene, and 3-methylbutene were prepared and characterized by GPC and NMR. Polystyryl and α-methylstyryl carbanions act as effective initiators of the anionic polymerization of Bs in THF. The undersirable side reactions, due to thermally or photochemically induced decomposition of the bromostyryl carbanions, PBs^?, may be eliminated by conducting the reaction at ?78°C and in the dark. Under such conditions, the rate constant of propagation, k_p (?78°), is 1.5 × 10³ M^?1 s^?1. Radical anions, \documentclass{article}\pagestyle{empty}\begin{document}$ {\rm Bs}^{\mathop - \limits_ \cdot} $\end{document}, formed as result of electron transfer from sodium naphthalenide to Bs, may eject spontaneously bromine ions. This step and reactions involving the respective phenyl radicals compete at ?78°C with the addition steps leading to polymer formation. Electron affinity of Bs seems to be much higher than that of styrene or isoprene, and PBs^? carbanions do not add to the latter monomers. Addition of Bs to polyisoprenyl carbanions leads to formation of the BsIBs block copolymers. BsIBs, prepared in THF, may be converted by preferential hydrogenation of the 1–2 adducts into block copolymers of Bs with poly(2-methylbutene) and isoprenyl segments. The effectiveness of Bs as a flame retarding constituent of polymeric systems seems to be much more pronounced when it is incorporated in a “block” than in a random fashion. A considerably larger fraction of PBs is required to achieve the same LOI value when the respective homopolymers are blended. Spatial distribution of the easily charred microdomains in the block copolymers is believed to be responsible for this phenomenon.     

Relaxation time measurements and molecular dynamics of dipalmitoyl-α-lecithin in the solid phase

The relaxation times T₁ and T_D were measured for dipalmitoyl-L-α-lecithin and dipalmitoyl-DL-α-lecithin at various temperatures. It was possible to separate the contribution of the slow motions to the relaxation rates from the fast ones. For some types of motions the activation energies could be estimated.     

Perturbations of molecular extravalence excitations by rare-gas fluids

In this paper we report the results of an experimental study of the vacuum ultraviolet absorption spectra of molecular impurity states of methyl iodide in Ar (density range ? = 0–1.4 g cm^?3) and in Kr (? = 0–2.3 g cm^?3), of carbon disulphide in Ar (? = 0–1.4 g cm^?3) and of formaldehyde in Ar (? = 0–1.25 g cm^?3). The experimental results provide new information regarding medium perturbations of intravalenc transitions, of the lowest extravalence transitions and of transitions to mixed valence—Rydberg configurations, which serve as a diagnostic tool to distinguish between different types of electronic excitations. All the lowest extravalence molecular excitations exhibit appreciable blue spectral shifts at moderate and at high fluid densities, intravalence transitions are practically insensitive to medium effects, while excitations to mixed valence—Rydberg configurations are characterized by a moderate blue spectral shift. New information has been obtained concerning the energetics of molecular ionization processes in a dense fluid. The high n = 2–5 Rydberg states of CH₃l exhibit a large red shift at moderate (? = 0–0.5 cm^?3) Ar densities. The ionization potential E_g and the effective Rydberg constant G for CH₃I in Ar was found to decrease from G = 13.6 eV and E_g = 9.55 eV at ? = 0 and E_g = 9.08 eV and constant G for CH₃l in Ar was found to decrease from G = 13.6 eV and E_g = 9.55eV at ? = 0 and E_g = 9.08 eV and G ≈ 7.15 eV at ? = 0.5 g cm^?3. Experimental evidence was obtained for the identification of n = 2 molecular Wannier impurity states of CH₃I and of CH₂O in liquid Ar. These spectroscopic data result in E_g ≈ 8.6 eV for CH₃I in liquid Ar and E_g ≈ 10.2 eV for CH₂O in liquid Ar.     

Is there a unique consistent theory of quantum gravity?

Superstrings have been proposed as a quantum-theoretical framework for unifying all the fundamental forces, including gravity. We consider the question of whether there might be more general supersymmetric possibilities, based on higher extended objects such as membranes, jellies, etc. We argue that all the possible extended objects in all possible spacetime dimensions are quantummechanically inconsistent except for the 10-dimensional superstring and the 11-dimensional supermembrane. These are also the only two such theories that contain massless gravitons and, thus, that can describe gravity at low energies. It is remarkable that the range of possibilities can be narrowed down to this extent. Whether these can be further narrowed down to just one consistent theory remains open to further research.This essay received the first award from the Gravity Research Foundation for the year 1988.-Ed.     

Global games

Global games are real-valued functions defined on partitions (rather than subsets) of the set of players. They capture “public good” aspects of cooperation, i.e., situations where the payoff is naturally defined for all players (“the globe”) together, as is the case with issues of environmental clean-up, medical research, and so forth. We analyze the more general concept of lattice functions and apply it to partition functions, set functions and the interrelation between the two. We then use this analysis to define and characterize the Shapley value and the core of global games.     

Extension of the static shakedown theorems to softening materials

A generalization of the static shakedown theorems for elastic plastic hardening solids with isotropic [Mech. Res. Commun. 29 (2002) 159] and anisotropic [Acta Mechanica, 2004] damage accounting for the possibility of material softening is proposed.