Suppose that a group of people is asked to rank policies A, B and C, in a decreasing order of preference. There are six possible ways of ranking A, B and C, namely, [A>B>C], [A>C>B], [B>C>A], [B>A>C], [C>A>B] or [C>B>A].
A social welfare function is a function which takes as input the preferences of all the members and returns as output an aggregated ranking of the alternatives.
There are various established social welfare functions used in practice. To describe some of them, consider, for example, the vote count of preferences of the three policies A, B and C shown in the topmost table.
According to the plurality voting function, the output is the outcome which is preferred by most people; in our example, this would be [A>B>C].
In the Condorcet Method, a head-to-head race is conducted between each pair of policies. The bottom table shows that A is preferred over B and over C, and C is preferred over B, so the output would be [A>C>B] using this method.
The Borda Count is a system in which each policy receives a score which is used to rank the policies. Every vote assigns two points to the first preference, one point to the second and none to the third. In our example, the points awarded to A, B and C would be 49, 36 and 41, respectively, leading to the output [A>C>B].
Is there an “ideal” social welfare function?
According to the plurality voting function, the output is the outcome which is preferred by most people
According to a 1950 result by Kenneth Arrow, the answer is “no” – if by “ideal” is meant a preferential voting method that satisfies the following “reasonable” criteria:
• No Dictators (ND): the output should not always be identical to the ranking of one special person (dictator).
• Pareto Efficiency (PE): if every voter prefers Policy A to Policy B, then the outcome should rank Policy A above Policy B.
• Independence of Irrelevant Alternatives (IIA): the outcome’s relative ranking of policies A and B should not change if voters change the ranking of other policies but do not change their relative rankings of A and B.
• Arrow’s Impossibility Theorem says the following: for elections with three or more policies, there is no social welfare function that satisfies ND, PE and IIA.
Emanuel Chetcuti is a professor of mathematics at the Department of Mathematics within the Faculty of Science of the University of Malta. In 2006, he was awarded the Birkhoff-von Neumann prize by the International Quantum Structures Association for his scientific achievements in the field of quantum structures.
Sound Bites
• A model for quantifying power in political systems is the Shapley-Shubik power index, which often reveals surprising power distribution. Suppose you have a decision-making body of 100 people, consisting of group A (50 people), group B (49 people), and group C (1 person). To pass a bill, 51 people are needed but the groups vote as a bloc. One way to quantify the influence is to imagine the voting groups entering a room in some order and forming a growing coalition; when the coalition is just large enough to pass a bill, the voting group that just entered is called the pivotal group.
The Shapley-Shubik index of a voting group is the fraction of orderings for which that group is pivotal. It has been formulated by Lloyd Shapley and Martin Shubik in ‘A Method for Evaluating the Distribution of Power in a Committee System’, American Political Science Review (1954), 48 (3), 787–792. In our example, there are six orderings, namely, ABC, ACB, BAC, BCA, CAB, CBA. In each case, the pivotal group is the one underlined, resulting in a Shapley-Shubik index of 4/6 to A, and 1/6 to B and to C. According to this measure, the one person in group C wields the same power as the 49 people combined in group B!
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DID YOU KNOW?
• For his celebrated Arrow’s Impossibility Theorem, which was, essentially, a mathematical result, Kenneth Arrow received the Nobel prize in economics in 1972.
• Have you watched the movie A Beautiful Mind? Mathematician John Nash also received a Nobel prize in economics in 1994.
• The Shapley-Shubik power index (see Sound Bites) has been applied to the analysis of voting in the Council of the European Union.
For more trivia, see: www.um.edu.mt/think.