https://www.nuffield.ox.ac.uk/users/robertsk/IrrelevantAlternatives.pdf
One notion of IIA relates to the idea that the choice set for some set of available alternatives should change with the set of alternatives only if chosen elements become excluded or alternatives are added which ëdominateíthe chosen alternatives. One way of expressing this condition is
Can an unordered generalised logit model be used to model descriptive models?
The Independence of Irrelevant Alternatives CriterionIf an election is held and a winner is declared, this winning candidate should remain the winner in any recalculation of votes as a result of one or more of the losing candidates dropping out..
As a publicity stunt for its soon-to-be-published cookbook, the Culinary Club of Smallville decided to have a 'Best Pie' contest. The entries were narrowed down to three for the final round of the contest. In this final round, each club member ranked the 3 pies. Each first place vote is worth 3 points, each second place vote is worth 2 points, and each third place vote is worth 1 point. Here is a summary of the results:
27 members ranked Al's Apple Pie first, Chris' Cream Pie second, and Pat's Peach Pie third;
24 members ranked Pat's Peach Pie first, Chris' Cream Pie second, and Al's Apple Pie third;
2 members ranked Chris' Cream Pie first, Pat's Peach Pie second, and Al's Apple Pie third.
Based on the rules above,
Al's Apple Pie got 27 first place votes at 3 points each (81 points) and 26 third place votes at 1 point each (26 points) for a grand total of 107 points.
Chris' Cream Pie got 2 first place votes at 3 points each (6 points) and 51 second place votes at 2 points each (102 points) for a grand total of 108 points.
Pat's Peach Pie got 24 first place votes at 3 points each (72 points), 2 second place votes at 2 points each (4 points), and 27 third place votes at 1 point each (27 points) for a grand total of 103 points
Thus, Chris' Cream Pie gets first place (108 points), Al's Apple Pie gets second place (107 points), and Pat's Peach Pie gets third place (103 points).
Before the results can be publicized, Pat (who is upset about a third place finish) demands that her Peach Pie entry be retroactively withdrawn from the contest. Bowing to her wishes, the club removes the Peach Pie and recalculates points (2 points for each first place, 1 point for each second place).
Since Pat's Peach Pie is now out, the rankings are:
27 members ranked Al's Apple Pie first, Chris' Cream Pie second, andPat's Peach Pie third;
24 members rankedPat's Peach Pie first, Chris' Cream Pie second first, and Al's Apple Pie third second;
2 members ranked Chris' Cream Pie first,Pat's Peach Pie second, and Al's Apple Pie third second.
The new results are,
Al's Apple Pie gets 27 first place votes at 2 points each (54 points) and 26 second place votes at 1 point each (26 points) for a grand total of 80 points;
Chris' Cream Pie gets 26 first place votes at 2 points each (52 points) and 27 second place votes at 1 point each (27 points) for a grand total of 79 points.
Thus, Al's Apple Pie gets first place (80 points), and Chris' Cream Pie gets second place (79 points).
Notice what happened. Because a loser (Pat) dropped out, the winner changed (from Chris to Al). This is a violation of the the Independence of Irrelevant Alternatives Criterion.
27 members ranked Al's Apple Pie first, Chris' Cream Pie second, and Pat's Peach Pie third;
24 members ranked Pat's Peach Pie first, Chris' Cream Pie second, and Al's Apple Pie third;
2 members ranked Chris' Cream Pie first, Pat's Peach Pie second, and Al's Apple Pie third.
Based on the rules above,
Al's Apple Pie got 27 first place votes at 3 points each (81 points) and 26 third place votes at 1 point each (26 points) for a grand total of 107 points.
Chris' Cream Pie got 2 first place votes at 3 points each (6 points) and 51 second place votes at 2 points each (102 points) for a grand total of 108 points.
Pat's Peach Pie got 24 first place votes at 3 points each (72 points), 2 second place votes at 2 points each (4 points), and 27 third place votes at 1 point each (27 points) for a grand total of 103 points
Thus, Chris' Cream Pie gets first place (108 points), Al's Apple Pie gets second place (107 points), and Pat's Peach Pie gets third place (103 points).
Before the results can be publicized, Pat (who is upset about a third place finish) demands that her Peach Pie entry be retroactively withdrawn from the contest. Bowing to her wishes, the club removes the Peach Pie and recalculates points (2 points for each first place, 1 point for each second place).
Since Pat's Peach Pie is now out, the rankings are:
27 members ranked Al's Apple Pie first, Chris' Cream Pie second, and
24 members ranked
2 members ranked Chris' Cream Pie first,
The new results are,
Al's Apple Pie gets 27 first place votes at 2 points each (54 points) and 26 second place votes at 1 point each (26 points) for a grand total of 80 points;
Chris' Cream Pie gets 26 first place votes at 2 points each (52 points) and 27 second place votes at 1 point each (27 points) for a grand total of 79 points.
Thus, Al's Apple Pie gets first place (80 points), and Chris' Cream Pie gets second place (79 points).
Notice what happened. Because a loser (Pat) dropped out, the winner changed (from Chris to Al). This is a violation of the the Independence of Irrelevant Alternatives Criterion.
0
|
I'm trying to do a cross sectional country analysis of various factors (corruption, inequality, region, etc) affecting the standard of living of a country (Using an index of GDP and HDI). I was reading up in generalized logit models, and apparently Independence of irrelevant alternatives is an important assumption of the model. Can we use the model to model this particular theory? Because it seems like if one of the independent variable changes, our countries standard of living might go up or down depending on the values of other variables.
|
No comments:
Post a Comment