The Paradox of Redistribution: Some Theoretical and Empirical Results

It is widely known among game theorists that the voting weight a member has in a voting body is not a good proxy for his voting power or influence within the body. This fact is easily illustrated. For instance, consider the voting body V = (50 %; 35 %; 45 %; 20 %) where 50 % is the voting rule used in the organization — if any members form a coalition with 50 % or more of the votes, they are winning and can make decisions that are binding on the entire body — and 35 %, 45 % and 20 % are the voting weights of voters 1,2 and 3, respectively. In this body then, although the weights are not distributed equally, it is easy to see that all the members are indeed equally powerful under the 50 % majority rule, since any voter cannot win a vote by himself and any coalition of two or more voters is winning. Consequently, it is clear that voting weights are not perfect proxies for voting power. The question that arises, however, is how bad a proxy are they? In other words, if we were to increase (or decrease) a voter’s voting weight within a voting body would his resulting influence within the organization always increase (or decrease), although possibly not in proportion to the increase in his weight, or could his influence within the organization actually decrease (increase)?