[EM] A simpler geographical example showing that mere ordinal info is inadequate
simmonfo at up.edu
Tue Jun 14 10:12:56 PDT 2005
Jobst recently gave a nice, but rather elaborate geographical example with different winners for different methods.
With a more limited objective in mind (showing the inadequacy of mere ordinal information) I present a simpler example:
This example is in the form of two related scenarios that obviously give rise to the same ordinal information, although it seems that different winners are appropriate in the different scenarios.
Scenario I. A certain family with members in three city states A, B, and C, are trying to decide in which of the three to hold their reunion. Round trip air fares are $1000 between B and either of the other two cities, and $1100 round trip between A and C. All other expenses and amenities are essentially equal. The family has 45%, 12%, and 43% of its members in the respective cities A, B, and C. From this information we can deduce that rational preferences are
Scenario II. Same as the above scenario except that hostilities have broken out between city states A and C, so it is no longer possible for citizens of the two belligerents to obtain visas to enter each others' states. In effect the $1100 cost has increased to infinity. However, the rational preferences retain their order.
In this second scenario we can also deduce the rational approval cutoffs:
Alternative B is both the Approval Winner and the Condorcet Winner, in this scenario. It's pretty obvious that the family reunion should be held in city B.
But what about scenario I ?
The family air fare would total $88,000 for a meeting at B, contrasted with only $59,300 for a meeting at A, or $61,500 for a meeting at C.
It seems that in this case B is not the best result.
IRV would pick A in both cases. That would be the utility maximizer in the first scenario, but the totally unacceptable infinite cost solution for the second scenario.
Condorcet would pick B in both cases, which would be great in the second scenario, but far from optimal in the first scenario.
IRV gives a better answer in the first case, while Condorcet gives a better answer in the second case.
Personally, I would prefer random ballot in the first case. That would lead to an expected airfare of $636.90 per family member, only $43.90 more than the cost minimizer solution, a small cost for the greater diversity. But this preference of mine depends on more than the ordinal information.
Since the ordinal data is the same in both cases, but the most appropriate outcome is not the same, it would appear that ordinal data is generally inadequate to satisfactorally decide elections.
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