Social Sciences

Abstract

The mathematical properties of the variety of methods used in different countries and organisations to vote for their legislatures or executives is examined, with copious examples to illustrate the merits and problems that arise. We meet Arrow’s celebrated “Impossibility Theorems”, which show that, when seeking to choose a winner from more than two candidates, there can be no “best” method, and discuss the inevitability of tactical voting under the systems in use. How the USA has chosen its House of Representatives illustrates problems of logic, involving simple arithmetic ideas. We describe Simpson’s Paradox, and point out how failure to appreciate its existence can lead to flawed conclusions. Medical tests for drug use, or the presence of cancers, are not infallible; we look at the balance between false positives, and failure to detect actual cases. The Gini Index is one way of measuring inequality in income or wealth; we note this, and several other ways, and make mathematical comparisons between them, noting that students can expect to see this general idea when they meet metric spaces later in their career.

Appendix

5.1.1 Proofs of Arrow’s ImpossibilityTheorems

Proof

Our logic is to prove the Second Theorem, and then deduce the First. We write $A \succ B$ to mean that A is strictly preferred to B, and $A \succeq B$ to mean that A is ranked level with B, or higher. Assume there are $n \ge 3$ candidates, and that the voting system satisfies conditions (1), (3) and (6). We will show that there is some voter, $v*$, who is a dictator, i.e. that (5) cannot hold. To do this, suppose B is a candidate, and every voter ranks B either first or last. Then if the system does not rank B as either first or last, there is some candidate, A, with $A \succeq B$, and another candidate C with $B \succeq C$.

Suppose now that every voter makes a simple change, if necessary, to move C above A in their own ranking, making no other changes. Since all voters rank B either top or bottom, this does not affect any voter’s preferences between A and B, or B and C; so by property (3), we still have $A \succeq B \succeq C$. However, because every voter ranks C above A, by (6), $C\succ A$ in the voting system—a contradiction. The system must rank B either first or last.

So suppose first all voters rank B last. We have just seen that the system must rank B either first or last—and it cannot be first, by (6), so B must be ranked last. Plainly, if all voters now move B from last to first, (6) shows that the system must rank B at the top. Write $\{v_1,v_2,\ldots ,v_m\}$ as the list of voters who originally all ranked B last, and now change one at a time in that order to make B first. There is some first voter, $v_x$ say, who’s vote change causes B to move from first to last. Call $v_x$ a pivotal voter; we show that $v_x$ is a dictator.

For, let A and C be two candidates other than B, and suppose $v_x$ prefers A to C in some list, L, of rankings by all voters. By property (3), the system’s choice between A and C must be unaffected if either

• (i) $v_x$ moves B to between A and C, or

• (ii) all of $v_1,v_2,\ldots ,v_{x-1}$ move B to the top of their lists, or

• (iii) all of $v_{x+1},\ldots ,v_m$ move B to the bottom of their lists.

So suppose all these changes are made giving a new list of rankings $L'$. In $L'$, $v_1,\ldots ,v_{x-1}$ prefer B to A, while $v_x,\ldots ,v_m$ prefer A to B. But since $v_x$ is the pivotal voter, as only $v_1,\ldots ,v_{x-1}$ rank B first, $L'$ still ranks B last; in particular, $L'$ ranks A above B. But in addition, since B would be ranked top if $v_x$ now moved B to the top from its position of above C but below A, this has no effect on the relative positions of B and C, so $L'$ must prefer B to C. But if $A \succ B$ and $B \succ C$, then $A \succ C$ in $L'$. But as we know that these changes do not alter the overall choice between A and C, then A is ranked above C in the original list L. Because the pivotal voter $v_x$ prefers A to C, so does the system.

This pivotal voter was chosen by reference to a particular candidate B. So let D be some candidate other than B, and let $v_y$ be the corresponding pivotal voter. Let E be someone other than B or D. We have just seen that whatever choice $v_x$ makes between two candidates other than B—in this case, D and E—the system follows the choice made by $v_x$. But, as $v_y$ is pivotal for D, $v_y$ also determines the relative ranking of D and E. So $v_x$ and $v_y$ are the same person—there is a single pivotal voter, $v*$ say, who controls the relative rankings of anyone other than B, anyone other than D, and anyone other than E. $v*$ is a dictator.

To deduce the First Theorem , it is enough to show that conditions (2), (3) and (4) imply condition (6); for then, if all of (1) to (5) hold, so must (6), and we have found that (1), (3), (5) and (6) are incompatible.

Given (2), (3) and (4), suppose all voters prefer candidate A to candidate B in some list L of rankings. By (4), we know that there is some list $L'$ which would lead to $A \succ B$; modify $L'$ if necessary to produce $L''$ in which every voter prefers A to B. By (2), the system prefers A to B under $L''$; L and $L''$ may be quite different, but the relative ranking of A and B is the same for every voter each time so using (3), we see that the system must rank A above B in the original list L. Condition (6) does indeed hold.