Primer in Social Choice Theory by Gaertner Wulf;

Author:Gaertner, Wulf;
Language: eng
Format: epub
Publisher: Oxford University Press
Published: 2009-04-11T04:00:00+00:00

According to the broad Borda rule, the choice set is hBB(w, S) = {z}. For S′ = {x, y, v}, the result is hBB (w, S′) = {y}.

Let us modify the above profile to profile w′ in the following way:

According to the broad Borda rule, the choice set now is hBB (w′, S) = {x}. For S′ = {x, y, v}, the outcome is hBB (w S′) ={x}. So far, so good. Note, however, that the two profiles w and w′ are exactly the same over the set S′, viz.

We clearly obtain hBB(w, S′) = hBB(w′, S′), where w = w′ over S′. This means that the Arrovian independence condition is violated, appropriately redefined for rank-order rules or scoring functions. Obviously, the broad Borda method has no problems with contraction consistency (note that the Borda rule actually generates an ordering over all alternatives). Since the scores from the superset are used for each possible subset, an alternative that scored highest in the superset will always be chosen in a subset, as long as it belongs to this subset. This characteristic does not apply to the narrow Borda rule, as the following example will show for S = {x, y, z} and profile w′′ for three individuals:

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