Abstract:
This thesis contributes to the program of numerical characterisation and classification of simple games outlined in the classical monograph of von Neumann and Morgenstern. One of the most fundamental questions of this program is what makes a simple game a weighted majority game. The necessary and sufficient conditions that guarantee weightedness were obtained by Elgot and refined by Taylor and Zwicker. If a simple game does not have weights, then Taylor and Zwicker showed that rough weights may serve as a reasonable substitute. Not all simple games are roughly weighted, and the class of projective games is a prime example. We give necessary and sufficient conditions for a simple game to have rough weights. We define two functions f(n) and g(n) that measure the deviation of a simple game from a weighted majority game and a roughly weighted majority game, respectively. We formulate known results in terms of lower and upper bounds for these functions and improve those bounds. Also we suggest three possible ways to classify simple games beyond the classes of weighted and roughly weighted games. We introduce three hierarchies of games and prove some relationships between their classes. We prove that our hierarchies are true (i.e. infinite) hierarchies. In particular, they are strict in the sense that more of the key "resource" yields the flexibility to capture strictly more games. Simple games has applications in the theory of qualitative probability orders. The concept of qualitative probability takes its origins in attempts of de Finetti to axiomatise probability theory. An initial segment of a qualitative probability order is a simplicial complex dual to a simple game. We initiate the study of abstract simplicial complexes which are initial segments of qualitative probability orders. This is a natural class that contains the threshold complexes and is contained in the shifted complexes, but is equal to neither. In particular we construct a qualitative probability order on 26 atoms that has an initial segment which is not a threshold simplicial complex. Although 26 is probably not the minimal number for which such example exists we provide some evidence that it cannot be much smaller. We prove some necessary conditions for this class and make a conjecture as to a characterization of them. The conjectured characterization relies on some ideas from cooperative game theory.