Chapter I
Two Commissioners and a Bad Surprise
In 1770 Jean-Charles de Borda pointed out to the Académie des Sciences that its elections could go wrong in a specific way. If three candidates stand and the vote splits, the winner on first preferences may be the candidate whom most voters place last. His remedy was to score candidates by position on every ballot and sum.
The Marquis de Condorcet objected that the proper criterion is simpler: a candidate should win if they would beat each rival in a head-to-head majority vote. Then he found that this criterion can fail to pick anybody. For some distributions of preferences, A beats B, B beats C, and C beats A, all by majorities. Collective preference, built from perfectly transitive individual preferences, need not be transitive itself.
This is not a paradox in the sense of a puzzle to be dissolved. It is a fact about majority aggregation, and it means that "what the electorate wants" may not name anything.
Chapter II
Impossibility, Twice
Kenneth Arrow turned the eighteenth-century objections into a theorem by asking what any acceptable aggregation rule must satisfy, and then showing the requirements are jointly unsatisfiable. His conditions are modest individually. The rule must handle every possible profile of individual rankings, since a procedure that fails on some electorates is no procedure. If everyone prefers A to B, so must the social ranking. The social ranking of A against B must depend only on how individuals rank A against B, and not on where some third option sits. And no individual's preference may prevail regardless of everyone else's.
No rule satisfies all four. The proof is a page of combinatorics, and it has generated seventy years of argument about which condition to give up rather than about whether the theorem is right. Independence is the usual casualty, because insisting on it discards all information about how strongly options are preferred — which is why the Borda count, which uses the whole ranking, violates it, and why rules that satisfy it end up relying on pairwise comparisons that can cycle. The theorem's lasting effect was to turn the question from "which voting rule is correct?" into "which failure is acceptable here?".
Amartya Sen added a second impossibility in 1970 with a different moral. Grant each person decisiveness over at least one matter that is plainly their own business — which book they read, which way up they sleep — and also require the Pareto criterion, that a unanimously preferred outcome be chosen. Sen constructs preferences, involving two people who each care what the other reads, for which no social ranking satisfies both. Minimal individual rights and unanimity are logically incompatible once people have preferences about each other's private affairs, which is not a defect of any procedure but a property of the two principles.
Chapter III
A Closer Look: Three Rules, One Set of Ballots, Three Winners
Take an electorate of 100 voters with these preferences:
| Voters | Ranking |
|---|---|
| 40 | A > C > B |
| 35 | B > C > A |
| 25 | C > B > A |
Plurality. Count first preferences only: A has 40, B has 35, C has 25. A wins.
Pairwise majorities. Compare each pair across all ballots.
- A vs B: A is preferred by the 40; B by the 35 and the 25, so 40 to 60. B beats A.
- A vs C: A by the 40; C by the 35 and the 25, so 40 to 60. C beats A.
- B vs C: B by the 35; C by the 40 and the 25, so 35 to 65. C beats B.
C beats both rivals head to head, so C is the Condorcet winner — with the fewest first preferences. And A, the plurality winner, loses to each of the other two by 60 to 40. Sixty per cent of the electorate prefers anyone to the person plurality elects.
Borda count. Award 2 points for a first place, 1 for second, 0 for third.
The totals sum to 300, as they must with 100 voters and 3 points each. C wins, agreeing with Condorcet here but not in general.
The three rules are all defensible and they do not agree, and nothing in the ballots adjudicates between them. Note also what happens to the Borda count if a fourth candidate D, whom everyone ranks last, is added: nothing. But if D is inserted in the middle of some voters' rankings, the gaps between A, B and C change, and the Borda winner can flip without a single voter altering their opinion about A, B or C. That is a violation of independence of irrelevant alternatives, the condition Arrow's theorem says cannot be kept alongside the other three — and here it is, in a worked ballot count, rather than as an abstraction.
Chapter IV
Honesty as a Design Problem
Allan Gibbard and Mark Satterthwaite showed the strategic counterpart to Arrow's result. Any deterministic rule that can select among three or more outcomes, and is not a dictatorship, can be manipulated: there is some situation in which a voter does better by submitting a ranking that misstates their preferences. Tactical voting is therefore structural. The familiar advice not to "waste" a vote on a third candidate is not a failure of civic virtue but a correct response to a feature of the rule.
Faced with two impossibilities, the field turned the question around. Instead of asking which rule is best, ask what is achievable, and design the procedure to make the behaviour you want into each participant's self-interest. William Vickrey gave the founding example in 1961, and it is worth stating exactly because the argument is three lines.
In a sealed-bid auction, let the highest bidder win and pay the second-highest bid. Suppose your value is 100. If you bid 100 and the next bid is 80, you win and pay 80, gaining 20. Could you do better by bidding 90? Only if that changes the outcome — and it changes it only when the highest rival bid is between 90 and 100, in which case you now lose an auction you would have won at a profit. Could you gain by bidding 110? Only when the highest rival bid is between 100 and 110, in which case you win and pay more than the item is worth to you. In every other case your bid makes no difference, because the price is set by someone else. Truthful bidding is a dominant strategy: optimal whatever anyone else does, with no need to guess at their values at all.
Vickrey also proved the first revenue equivalence result: the familiar formats raise the same expected revenue when values are drawn independently from the same distribution. With two bidders whose values are uniform on , the second-price auction collects the lower of two draws, whose expectation is . In a first-price auction the equilibrium is to bid half one's value, so the seller receives half the higher draw, and the higher of two uniform draws averages — giving again. The strategic behaviour differs completely and the revenue is identical.
Roger Myerson completed the framework in 1981 with two results. The revelation principle says that anything achievable by any mechanism is achievable by one in which participants simply tell the truth, which collapses an unmanageable search over procedures into an optimisation over truthful ones. And the revenue-maximising auction for a single item turns out to be the second-price auction with a reserve price — the reserve chosen exactly as a monopolist facing one buyer would choose a price, ignoring competition entirely.
That clean answer has resisted every attempt to extend it to more than one item, which is the open problem above, and the reason real multi-item auctions — spectrum, advertising, electricity — are designed by judgement and simulation rather than derived. The further constraint, that a mechanism must also be computable when there are exponentially many possible allocations, is where this field meets computational complexity, in algorithmic game theory.