Ordinal Bayesian incentive compatible representations of committees

We consider the problem of strategic manipulation for decision schemes that provide an adequate representation (in some sense) of the distribution of power within a committee. Strategy-proof representation is very restrictive: it implies that the committee contains exactly one minimal winning coalition. So we introduce the weaker concept of Ordinally Bayesian Incentive Compatible representation and prove the existence of such representations for weak games under some conditions. Finally, constructing examples, we show first how necessary these conditions are—including the use of chance in the voting procedure — and second that we cannot avoid Condorcet's paradox.