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Small cases lie — page 2

Patterns that hold for every example anyone would check by hand, and then stop. The cases within reach are not a sample of the cases.
quota = population × 27 ÷ 10000populationexact quotaquotafloorremainderseatsABCDEsum571015417/100015.41715417/10001526707209/10007.2097209/10007650351/2001.7551151/20025301431/10001.4311431/10002440297/2501.188147/2501100002727.00025227rounded up from its floorHamilton: floors, then the largest remainders, over 27 seats and 5 regions of 10000 peoplethe exact quotas sum to 27 and so do the awarded seats; the floors account for 25, leaving 2the 2 spare seats went to the largest remainders: C and D Applied

The seat that vanishes when the house grows

Twenty-seven whole seats have to be divided between five regions whose exact shares are 15.417, 7.209, 1.755, 1.431 and 1.188. Every rule for rounding those five numbers breaks something, and the instance drawn here breaks all three of the classical ways at once.

 before the link A→B existsSABTcost 3flow 3cost 7flow 3cost 7flow 3cost 3flow 33 units each way, every traveller takes 10after the link A→B is addedSABTcost 6flow 6cost 7flow 0cost 7flow 0cost 6flow 6cost 0flow 6all 6 units one way, every traveller takes 12S→A and B→T cost 1 for each unit on them; A→T and S→B cost 7 whatever the trafficaverage travel time per traveller02468101214before the link10after the link12least possible119/12every traveller takes 10 before the zero-cost link exists and 12 after it doesthe least total travel time on the larger network is 119/2, an average of 119/12 = 9.917, so theequilibrium costs 144/119 = 1.210 times the least possiblechecked over every route at both flows, and against all 703 splits of the traffic on a lattice of sixths Applied

The road that makes everyone later

An equilibrium is a state nobody can improve alone, which is a much weaker thing than a state anybody would choose. Adding a link that costs nothing to use makes every traveller in this network strictly slower, and the arithmetic says by exactly how much.

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