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Ladder

Equilibrium — the ladder

2 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. the row chooser mixes00.20.40.60.81-4-20246weight on row Apayoff to the row choosercol Xcol Ycol Z8/15 → 19/15the most the row chooser can guarantee: 19/15the column chooser mixes00.20.40.60.81-4-20246weight on col Xpayoff to the row chooserrow Arow B7/15 → 19/15the least the column chooser can concede: 19/15both sides name 19/15 = 1.267the value is 19/15 = 1.267, reached by the row chooser mixing 8/15 on A and by the column chooser mixing 7/15 on Xthe lower envelope of 3 lines peaks where two cross; the upper envelope of 2 bottoms out at that heightchecked against every pure reply on both sides, and over 61 mixtures on one side and 91 on the other

    The value from both sides

    Two choosers move at the same instant, and each asks the cautious question — how much can be guaranteed, whatever the other does. With pure choices the two answers are usually different numbers; allow a probability and they are forced to be the same one.

    rung 1 · applied
  2. 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

    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.

    rung 2 · applied

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