: The paper presents the probabilistic interpretation of ShapleyShubik index and Banzhaf value and the concept of partial homogeneity structure. Probabilities of being critical can be calculated based on this structure. These probabilities constitute the basis for calculating hybrid power indices, ShapleyShubik index and Banzhaf index being the extreme cases. The paper presents the idea of reconstruction of partial homogeneity structure based on maximum likelihood estimation. This method is based on the same assumptions as hybrid power indices. The method is illustrated by simple heuristic examples and results of analysis of clubs’ voting in 7th term of Sejm – the lower house of the Polish parliament.

In fair distribution of a set of indivisible goods it is problematic to provide basic equality if the goods differ in value. One of the most popular solutions to the problems are lotteries. The paper presents seven selected probabilistic procedures: random distribution, lexicographic procedure of equal chances of satisfaction, random serial dictatorship, core from random endowments, probabilistic serial, top trading cycles from equal division and
random priority with infi nite k factor. Some of these procedures always lead to the
same result as some other procedure, therefore these procedures are equivalent. The formal features of the procedures are also analysed in the paper: ordinal efficiency, ex post efficiency, and both strong and week version of: envy-freeness, proportionality, equitability, and individual strategy-proofness.

The paper presents modifi cation of four nonprobabilistic procedures of fair division of a set of indivisible goods with money: Original Knaster, Adjusted Knaster, Equal Shares and Second Prices. The purpose of modification is to adjust the aforementioned methods to the problem of division of a set of goods in a situation of unequal entitlements. Extended versions of six formal properties, namely: strong Pareto optimality, proportionality, envyfreeness, equitability, strategy-proofness and anonymity are also presented. Moreover,proofs of procedures having the same formal properties in a situation of unequal entitlements as in a situation of equal entitlements are shown.