In this paper, we address the selfish bin covering problem, which is greatly related both to the bin covering problem, and to the weighted majority game. What we are mainly concerned with is how much the lack of central coordination harms social welfare. Besides the standard PoA and PoS, which are based on Nash equilibrium, we also take into account the strong Nash equilibrium, and several new equilibrium concepts. For each equilibrium concept, the corresponding PoA and PoS are given, and the problems of computing an arbitrary equilibrium, as well as approximating the best one, are also considered.
 A quick attach apparatus for end loaders or the like such as tractor loaders or the like. The tractor loader includes a pair of booms operatively pivotally secured at one end thereof to the tractor. A hydraulic cylinder is operatively pivotally secured to each of the booms and is positioned over the other end thereof. A hook-up bracket is pivotally secured to the said other end of each boom and the hydraulic cylinder and includes a channel-shaped portion extending forwardly therefrom. Each of the various attachments for the loader such as buckets, forks, blades, etc. have a pair of channel-shaped pockets secured to the rearward end thereof which are adapted to receive a hook-up bracket therein. A locking apparatus is provided on each of the brackets to detachably maintain the hook-up brackets in their respective pockets. The locking apparatus includes means for yieldably maintaining the locking apparatus in an unlocked condition and means for automatically locking the locking apparatus after the hook-up bracket has been properly received within its respective pockets. In the following paper we offer a method for the a priori evaluation of the division of power among the various bodies and members of a legislature or committee system. The method is based on a technique of the mathematical theory of games, applied to what are known there as “simple games” and “weighted majority games.” We apply it here to a number of illustrative cases, including the United States Congress, and discuss some of its formal properties. The designing of the size and type of a legislative body is a process that may continue for many years, with frequent revisions and modifications aimed at reflecting changes in the social structure of the country; we may cite the role of the House of Lords in England as an example. The effect of a revision usually cannot be gauged in advance except in the roughest terms; it can easily happen that the mathematical structure of a voting system conceals a bias in power distribution unsuspected and unintended by the authors of the revision. How, for example, is one to predict the degree of protection which a proposed system affords to minority interests? Can a consistent criterion for “fair representation” be found? It is difficult even to describe the net effect of a double representation system such as is found in the U. S. Congress (i.e., by states and by population), without attempting to deduce it a priori . The method of measuring “power” which we present in this paper is intended as a first step in the attack on these problems. Measures of (a priori) power play a useful role in assessing the character of interpersonal interaction found in collective decision making bodies. We propose and axiomatically characterize an alternative power index to the familiarShapley Shubik andBanzhaf indices which can be used for such purposes. The index presented is shown to be unique for the class of simplen-person games. By subsequent generalization of the index and its axioms to the class ofn-person games in characteristic function form we obtain an analog to theShapley value. We have pointed out the theoretical drawbacks of the traditional indices for measuring a priori voting power inasmuch as they are implied in considering the coalition value a private good. This criticism caused us to view the coalition outcome as a public good. From this aspect and additional considerations with respect to power, luck, and decisiveness, we obtained a “story” describing the characteristics of an adequate measure of a priori voting power. These characteristics were found to be fulfilled by an index presented by Holler (1978). Through the above analysis this index has received its theoretical justification. An independent view of this index was then provided by means of an axiomatic characterization. This characterization makes possible abstract comparison of the index with previously established “private good” indices.