Applied — page 3

The product that makes a division fair

Divide goods to make the total happiness as large as possible and the result can be monstrously unfair; make the least happy person as happy as possible and it can waste. Multiply the people's values together and maximise the product instead, and something unexpected happens — nobody envies anybody when goods can be split, and nobody envies by more than one item when they cannot.

The people every stable answer leaves out

Let the lists be short and let one side take several partners. Stable matchings still exist and there can be many of them — but every one leaves out exactly the same people, and a member who is left with an empty place holds exactly the same partners in every one. A three-line count proves it.

A ring that no pairing can break

Put everybody in one pool and a stable pairing may not exist. Allow rings as well as pairs and something stable always exists — and the pairs-only answer fails exactly when that stable arrangement contains a ring of odd length. Two sides make every ring even, which is the whole reason the two-sided theorem holds.

When one of the two numbers is missing

The duality theorem is usually quoted as an equality: a linear program and its dual reach the same number. That is one of four cases. A program can run away to infinity, or have no feasible point at all, and then its dual is forced into a matching failure. Every small program with coefficients from minus one to one has been classified, and the table has exactly four occupied cells out of nine.

The lines the optimum lies under

Change the resources a linear program is given and its best value changes too, tracing a graph. Every solution of the dual is a straight line lying above that graph, and the graph is exactly the lowest of those lines — a bent roof of finitely many planks. Require the answer to be in whole numbers and the roof stays where it was while the graph falls away beneath it in steps, and the space between is the part of the problem no price can see.

The nearest consistent verdict

When a court's majorities contradict each other, one repair is to announce the consistent verdict that disagrees with the judges least. It treats the premises and the conclusion alike, which neither of the two standard procedures does. On the classic case it returns a three-way tie; on five judges, with every question weighted equally, it never returns a single answer on a troubled profile at all — and what breaks the tie is a decision about which question matters more.

Agendas that cannot contradict themselves

A court voting on two unconnected questions never contradicts itself, and neither does one voting on a chain of thresholds. A court voting on two premises and their conjunction sometimes does. What separates them is the size of the smallest sets of judgements that cannot all be true: pairs are harmless, because two majorities always share a judge, and triples are not. The same count says exactly how large a supermajority has to be to stay consistent on any agenda.

What a missing input is worth

A model prices a house at 180 from its size, its garden and its bedrooms, and the question is how much of the price each input is responsible for. Make the inputs the players and the average over orders answers it — once somebody decides what the model says when an input is not known. Three reasonable decisions give bedrooms nothing, nothing, and sixteen, for a model that never reads them.

Cutting a link costs both of its ends the same

Three players, any two of whom can earn 1 together — but only if they are linked. Link all three and each is due a third. Remove one link and the player holding both of the others is due two thirds. Averaging over orders on the game the network allows is the one rule under which breaking any link costs the two players it joined exactly the same, and it pays go-betweens more than their links.

How often the majority goes in a circle

Three voters and three candidates give 216 profiles, and 12 of them are cycles. Count every electorate up to 41 voters exactly and the share climbs towards 8.77%, a number Guilbaud found in 1952 as the solid angle where three half-spaces at the tetrahedral angle overlap. Add candidates and a winner goes missing half the time; let voters share one axis and cycles vanish. The number is always a property of the model of how ballots are drawn.

A majority wiser than its members

Condorcet's other theorem turns voting round: the voters no longer have preferences but judgements about a single fact, each a little more likely right than wrong. Then a simple majority of many of them is almost certainly right — 6,763 voters who are each right 51% of the time make a majority right 95% of the time. The theorem survives voters worse than a coin, if the average is better. It does not survive voters who share their mistakes, and when their skills differ the right rule weighs votes rather than counting them.

One extra person on one side

In a random market of a thousand a side, whoever proposes gets about their seventh choice and whoever receives gets about their hundred-and-fortieth. Add one person to one side and the advantage of proposing all but disappears: the shorter side does well and the longer side badly, whichever side proposes, and most people are left with exactly one stable partner.

The matching in the middle

List every stable matching of a market, give each member their stable partners sorted from best to worst, and hand each the one in the middle. Nothing says the result should even be a matching — two people might pick the same partner — and yet it always is one, it is always stable, and the other side gets its median partners too.

Envy that any single item would cure

Envy-free up to one item lets a person's envy be excused if removing the envied bundle's best item would cure it. The stronger standard asks that removing any item would — even the one that person values least. Every allocation of three people's items can be searched, and an allocation meeting the stronger standard was there every time; for two people cut and choose finds one, for three it took until 2020 to prove, and for four nobody knows.

A rent nobody envies

Three housemates, three rooms that are not alike, one rent. Every way of splitting the rent is a point of a triangle; ask, at each point of a fine grid, which room one housemate would take at those prices, taking turns so that each small triangle has one corner for each of them. Sperner's lemma then promises a small triangle where all three would choose different rooms — and as the grid is refined, the envy at that triangle shrinks to nothing.

Prices the bidders raise

The cheapest assignment is certified by a price on every task, and those prices can be found without anyone in charge. Let each unassigned person bid for the task that suits them best at current prices, raise its price by a little more than it is worth to them over the next best, and wait. The bidding ends, and when the increment is small enough the prices it ends at are a proof of optimality.

The prices nobody can break away from

When houses are sold to buyers who value them differently, there is a whole range of prices at which nobody wants to walk away, and it has a remarkable shape — one corner best for every buyer at once, one best for every seller at once, and the buyers' corner pays each buyer exactly what the market would lose without them.

Prices at every corner

The duality theorem says a linear program's best value equals its dual's, and says nothing about how to find either. The simplex method finds both at once — it walks from corner to corner, and at each one asks the constraints that meet there for prices. A negative price names an edge that climbs; when none is negative, the prices are the proof.

The cube that takes every corner

The simplex method is fast on every program anybody meets in practice. In 1972 Victor Klee and George Minty squashed a cube so that the method, choosing the steepest edge each time, visits all of its corners — 2ⁿ − 1 moves in n variables, with the optimum one edge from the start.

The corners are the orders of arrival

Line the players up, let each join in turn, and pay each what it adds on arrival: every order gives a split. When a newcomer always adds at least as much to a bigger group, those splits are exactly the corners of the core — so the core is never empty, it is the outline of the orders, and the average over all of them lies inside it. For a group worth the square of its size the outline is a hexagon whose corners are the six orderings of 1, 3 and 5.

Each user pays for its own last link

Several users must be connected to a source, and the cheapest network that does it is a tree. Dividing its cost so that no group of users would rather build its own looks like a hard search, and it has a one-line answer: each user pays for the link that joins it to the tree on its way to the source. No group is ever overcharged — while the average over orders of arrival, the rule that settles so much else, can charge a pair more than its own connection costs.

How few voters any majority needs

Any pattern of head-to-head majorities whatever — cycles within cycles, a candidate who beats the winner of every other contest and loses to its loser — can be produced by voters who each rank the candidates sensibly. McGarvey's recipe needs n(n − 1) of them for n candidates. The truth is far fewer: every pattern on five candidates takes three voters at most, a counting argument shows the number must eventually grow, and it grows only like n divided by its logarithm.

The fewest swaps to a winner

When no candidate beats every other head to head, Charles Dodgson proposed in 1876 to elect the one that is closest to doing so — the candidate that the fewest swaps of neighbouring names on the ballots would turn into a winner of every contest. The rule is easy to state and hard to compute: the count needs a search, and deciding the winner is provably among the hardest problems of its kind. A much simpler count, the votes still to be won, usually agrees, more often the larger the electorate.

A bound that may be off by a third

The shortest tour through a set of cities is hard to find, and a linear programme gives a lower bound for it in polynomial time: give every road a weight between nought and one, two at each city, at least two across every division of the map. On random cities the bound is almost always exact. On two triangles joined by three long paths it falls short by nearly a third, and whether a third is the worst it can ever do has been conjectured for decades and never proved.