Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 42 / 6

The Gibbard-Satterthwaite theorem concerns a deterministic social choice function on all profiles of strict preferences over a finite set of at least three alternatives. If it is onto and strategyproof, it is a dictatorship in social choice: one fixed voter always obtains its top alternative. Equivalently, every onto nondictatorial rule on this unrestricted domain is manipulable. Dictatorships themselves are strategyproof. We prove the implication for two voters.

First derive the rank-raising monotonicity lemma. If changing one report changes the selected alternative from $a$ to $b$, strategyproofness in the old profile requires $a\succ b$ in the old order, while strategyproofness in the new profile requires $b\succ a$ in the new order. Therefore a change that never lowers the selected $a$ relative to any formerly lower alternative cannot change the outcome.

Onto-ness now implies unanimity. For every $a$, some profile selects it. Raise $a$ to first place in each voter's order; the preceding lemma keeps the outcome at $a$. Moreover the outcome cannot be $b$ when both voters rank some $a$ above $b$: starting from such an outcome, raise $b$ to second place directly below $a$ in each order. These changes never lower $b$ relative to anything, so would preserve $b$ at a profile unanimously topping $a$, a contradiction. Thus the rule respects unanimous pairwise preference.

For each pair $a,b$, promote that pair to the first two places in each report, preserving the voter's order between them. Both dominate every outsider unanimously, so the outcome is $a$ or $b$. The result depends only on the two voters' comparisons of $a,b$, not on the lower alternatives: changing a tail cannot reverse the choice between two alternatives whose relative order did not change, since one direction of that change would be a profitable report. These binary choices define a complete strict social relation satisfying pairwise unanimity and independence from other comparisons.

This relation is transitive. To see why, suppose its choices on some triple form a directed cycle. Put those three alternatives first in both reports, preserving their individual relative orders. Unanimity excludes all outsiders, so the rule chooses one of the three, say $a$. Promoting $a$ with either other member to the first two positions does not lower $a$ and therefore preserves its selection. Thus $a$ must win both its binary comparisons, contradicting the cycle. A complete strict relation with no directed three-cycle is transitive. Also the original social choice is its top: for any original selected $a$, promoting $a$ and any $b$ preserves $a$, so it wins every binary comparison.

It remains to prove two-voter dictatorship for this binary relation. Choose any conflict between $a,b$. Suppose voter 1 prefers $a$ to $b$, voter 2 prefers $b$ to $a$, and the social relation follows voter 1. Say voter 1 wins this ordered comparison. For any third alternative $c$, consider the two profile patternswith other alternatives below the triple. In pattern I, the social relation has $a\succ b$ by independence and $b\succ c$ by unanimity, hence $a\succ c$. This proves that voter 1 wins the conflict $a$ against $c$. In pattern II, it has $c\succ a$ by unanimity and $a\succ b$ by independence, hence $c\succ b$, proving that voter 1 also wins $c$ against $b$.

Thus winning $a$ against $b$ implies winning $a$ against every third $c$ and every such $c$ against $b$. Applying the implication to $a$ against $c$ gives $b$ against $c$, and then to $b$ against $c$ gives $b$ against $a$. The implication therefore supplies both directions of all pairs involving the original alternatives and any new one. For an arbitrary ordered pair $x,y$ distinct from a fixed base alternative $a$, use the already obtained win of $x$ against $a$ and the third alternative $y$ to get the win of $x$ against $y$. Hence voter 1 wins every conflict. Unanimous comparisons also follow its preference, so the social relation is always voter 1's full order, and the social choice is its top.

If the initially chosen conflict follows voter 2, swap the voter roles in the same argument. We have therefore proved every onto two-voter strategyproof rule with at least three alternatives is dictatorial. This two-voter dictatorship from binary choice proof establishes the required special case without assuming the general theorem's conclusion.

There is a parity qualification in the printed claim. With the usual strict-majority definition, an even electorate need not have a Condorcet winner. On the axis $a<b<c$, the two orders $a\succ b\succ c$ and $c\succ b\succ a$ are both single-peaked preferences, but every pairwise contest is tied. Thus no alternative strictly defeats every other one. We first prove the intended strict-winner result for odd $n$.

Let $n=2k+1$ and order the peaks along the common axis. Their median $w$ has at least $k+1$ peaks at or to each side. For any $y<w$, the $k+1$ voters whose peaks are at or to the right of $w$ prefer $w$ to $y$ by single-peakedness. For any $y>w$, the corresponding $k+1$ voters on the left prefer $w$. Thus the median peak is the unique Condorcet winner.

The corresponding median voter rule is strategyproof. Fix the other $2k$ peaks $q_1\le \cdots \le q_{2k}$. As one voter reports a peak $p$, the selected median iswhere minimum and maximum refer to the common axis. All attainable outcomes lie between $q_k$ and $q_{k+1}$. If the true peak lies inside this interval, truthful reporting obtains the voter's top. If it lies left of the interval, truthful reporting obtains its left endpoint, which the voter's single-peaked order prefers to every larger attainable outcome. The case right of the interval is symmetric. No report improves the outcome. For one voter the rule simply selects its peak.

There is also a proof that does not depend on knowing the axis. If a voter truly prefers a proposed new winner $z$ to the current strict Condorcet winner $w$, that voter already opposes $w$ in the contest $w$ versus $z$. The strict majority supporting $w$ in that contest therefore consists of other voters and is unaffected by its report. Thus $z$ cannot become a strict Condorcet winner after a profitable misreport. This proves strategyproofness on the domain of profiles for which the selected strict winner exists, including all admissible odd-electorate single-peaked profiles.

For even $n$, a weak Condorcet winner is guaranteed: every alternative between the two middle peaks weakly defeats each other alternative, allowing ties. With a fixed common axis, consistently choosing the lower median or consistently choosing the upper median gives a single-valued strategyproof rule, by the same interval-clamping argument for an order statistic. Arbitrary tie selection is not asserted to have this property. The corrected conclusion is therefore a unique strict winner and its strategyproof selection for odd electorates; a weak winner with a specified median rule for even electorates.