A rule that weighs two walkers spares the one serving the worse would starve

Two walkers share a layout drawn for the worse of them put two visitors in front of a projector turned fifteen degrees to its wall, and drew one layout for both at every moment: the layout that gave the worse-served of the two the most detail per degree at their own worst place, or the even picture where nothing did better for them. Walking the same way a metre apart, the walker nearer the picture’s centre kept 99 per cent of what a layout following them alone would give and the other kept 77. Crossing, nobody gained. And at different distances from the wall the rule turned into a transfer: the walker two metres out was served and the one five metres out was charged, below the even picture.

The essay ended on the rule itself. Serving the worse-served is one notion of fairness, the one the even picture is fair to a centred room used for a seated audience, and it does not care what the other viewer loses. A rule that weighed the walkers would care: maximise the sum of their gains, or of their gains as shares of what each could have had alone, or the smaller of those two shares. The question was what such a rule gives up of the worse walker’s detail, what it keeps of the other’s, and how often either is pushed below the detail an eye can resolve.

Four rules over one family of layouts #

The projector, the wall and the budget are the earlier essays’: the even picture’s budget of panel samples, spent column by column across the corrected picture, every layout capped where the panel is coarser. Two viewers walk at a metre a second, a frame every fifth of a second. At every frame the shared layout is chosen from the even picture and a family running from the layout that would follow one walker alone to the layout that would follow the other — the two single-viewer shapes blended in twentieths, each scaled back to the budget. Each rule picks from that family:

• serve the worse-served — the earlier essay’s layout, drawn exactly;
• the most detail in all — the largest sum of the two walkers’ gains over the even picture, in detail per degree at each one’s worst place;
• the most of both shares — the largest sum of each walker’s gain as a share of what following them alone would give;
• the largest smaller share — the layout under which the less-favoured walker keeps the most of their own best.

Each rule is judged three ways: the worse-served walker’s mean gain over the even picture; the share of their own following gain each walker keeps; and the share of frames in which either walker is held below sixty pixels a degree at their worst place when the even picture would have left them at or above it — the line a pixel is not a point draws for what an eye resolves.

Two walkers two and five metres from the wall, the nearer one leading, are the case that motivates the question. Serving the worse-served gives the near walker 11.7 per cent over the even picture on average and takes the far walker below the even picture — to minus 84 per cent of their own gain. Keeping the smaller share largest gives the near walker 6.8 per cent and leaves both walkers at least 61 per cent of their own best. The other two rules sit between: the most detail in all and the most of both shares each give the near walker about 5.5 per cent and keep half of its gain and nearly all of the far walker’s. The rule the earlier essay used is the only one of the four that drives anybody below the even picture.

Walking the same way, the rules barely differ #

For two walkers the same way at one depth, the choice of rule is a small adjustment.

A metre apart, serving the worse-served keeps 99 per cent of the leader’s gain and 78 of the trailer’s; keeping the smaller share largest keeps 91 and 83. Four metres apart, 100 and 44 against 91 and 48. The leader gives up a few points of a gain it was given in full, and the trailer gets a few points back. The worse-served walker’s own gain over the even picture falls a little under the weighing rule: 9.4 per cent against 8.4 a metre apart, 8.8 against 6.2 four metres apart. These walkers are the earlier essay’s easy case, both on the projector’s side for most of the walk wanting nearly the same layout, and any reasonable rule gives them nearly the same answer.

The most detail in all is the exception even here. A metre apart it keeps 79 per cent of the leader’s gain and 95 of the trailer’s; three metres apart, 56 and 91 — it has swung the other way, serving the walker further out on the projector’s side, who stands to gain most from the panel’s surplus because it sees the near edge from furthest. The leader, nearer the centre and worse served, keeps barely half. Summing the gains favours whoever can be helped most cheaply, which is the opposite of whoever needs it most.

Kept, and taken #

The second figure sets out every pair and every rule.

Two rules ever leave a walker with less than nothing, and they do it in different places. Serving the worse-served leaves the far walker at minus 84 per cent of their own gain when the near one leads, and the near walker — now the trailer — at minus 71 when the far one leads: at different depths the nearer walker is the worse-served almost throughout, and the layout drawn for them takes from the other. The most detail in all leaves nobody below nothing on these walks but, crossing, keeps 53 per cent of both walkers’ gains by serving the worse walker below the even picture, as the next section shows. The two share-weighing rules never leave either walker below nothing; the smaller-share rule never leaves either below 45 per cent away from a crossing, where nothing can be given to anyone.

Detail an eye would have seen #

A share of a gain is an abstraction. What a visitor notices is whether the picture in front of them got coarser than they can see.

Serving the worse-served takes visible detail from the other walker in 21 per cent of the frames when the near walker leads and 31 per cent when the far one does: a visitor five metres back who would have seen the picture’s far side at sixty-odd pixels a degree under the even picture sees it at less, for the benefit of a visitor two metres out who sees the near side more finely. The most detail in all does the same in 16 per cent of a crossing. The two share-weighing rules never do it, on any walk.

That is the argument the earlier essay’s question anticipated: a rule that gives up a little detail the worse-served walker cannot see, in order to keep detail the other can see, is the better rule for the people in the room. The measurement adds a qualification it did not anticipate. The worse-served walker here is served below sixty pixels a degree for much of every walk — the even picture gives forty in front of the picture — so the detail the weighing rule gives up is detail they could see. What the weighing rule buys is that nobody is made worse than the even picture to pay for it.

The walk, frame by frame #

The case at different depths is worth following through the walk, because the transfer has a shape.

The near walker, two metres out, sees the picture’s near part squarely and from close and is the worse-served almost throughout. Under the worse-served rule the layout is drawn for it: samples move to the near edge, where it looks hardest, taken from the far edge, where the walker five metres back looks. That walker starts better served than the near one and ends below the even picture for 45 per cent of the walk, by up to sixteen pixels a degree. Under the smaller-share rule the layout sits part way between the two walkers’ own, the near walker gains less, and neither falls below the even picture at any frame.

The walkers here are a metre apart in their walk and three metres apart in depth, and the second difference is the one that matters. A walker is followed only on the projector’s side found the single walker’s gain set by where they stood across the room; two walkers at different depths add a second axis, and on it the further walker always has detail to spare and the nearer one always needs it. A rule that only ever serves the worse-served turns that into a steady transfer. A rule that weighs them limits it.

Crossing, the sum’s arithmetic turns on the worse walker #

Crossing was the earlier essay’s case in which nobody gained: whenever the two stood on opposite sides some way out, the one on the far side was the worse-served and nothing could be given to them, so the shared layout was the even picture.

Three of the four rules agree, keeping one or two per cent of either walker’s gain. The most detail in all does not. It keeps 53 per cent of both walkers’ gains — and serves the worse walker 9 per cent below the even picture on average, pushing a walker out of resolvable detail in 16 per cent of the frames. Summed, the walker on the projector’s side gains more pixels a degree than the walker on the far side loses; the sum goes up, and the worse walker pays. This is the familiar failure of a utilitarian sum, arrived at in pixels: it is indifferent to who gains, so it will always find a way to give more to someone who already has a lot.

What a projector with two visitors should do #

The earlier essay’s policy was to serve whichever visitor was worse-served and to accept that the other paid. The measurement here gives a better one, and it is simple. Choose the shared layout that leaves the less-favoured visitor the largest share of their own best. Walking the same way at one depth it costs the worse-served walker a point or two of gain and returns a few points to the other; crossing it does what any sane rule does, which is nearly nothing; and at different depths it stops the transfer that serving the worse-served makes, at the price of five percentage points of the near walker’s gain. It never makes anybody worse than the even picture, and it never takes visible detail away.

The rule to avoid is the sum. It sounds like the rule that does most good, and it is the rule that most reliably helps whoever is easiest to help — the walker far out on the projector’s side, who sees the near edge from furthest and needs it least — and, crossing, it does so at the worse walker’s expense.

Why every rule is a choice about the near edge #

It helps to see what all four rules are arguing over, because it is less than it looks. An even spend of the panel is an uneven picture found that the even picture throws away the near edge’s surplus of panel samples to match the far edge’s coarseness. Every layout here differs from the even picture only in how much of that surplus it keeps and where on the near half of the picture it puts it; nothing can be added to the far edge, because the panel has nothing finer to give there. A walker whose worst place is on the far half — anyone standing on the far side of the picture’s centre — can only lose under any of the rules, never gain, and the rules differ in how much they let them lose.

That is why the arguments are so lopsided. Serving the worse-served lets the far walker lose without limit, so long as the near walker gains. The sum lets the worse walker lose whenever another gains more. The two share rules count a loss as a negative share, which weighs against it heavily, and on none of these walks did either let anyone fall below what they had. A projector is a camera run backwards, and like a camera it can choose where to be sharp; what it cannot do is be sharper everywhere, and the rule is only ever a decision about who it is soft for.

Where the rules come from #

The family of layouts the rules choose from is narrow on purpose: blends of the two walkers’ own following layouts, and the even picture. A rule allowed any layout at all could do more, or differently, and the worse-served rule is the earlier essay’s exact layout rather than the family’s nearest member, so it is not handicapped by the restriction. What the family captures is the whole of the trade the panel offers, because keystone correction spends the panel unevenly: the only thing any layout has to give is the near edge’s surplus, and the only question is how far towards each walker’s own shape to spend it. The screen sets the distance is the other half of the arithmetic — how much a picture can show an eye depends on where the eye is — and the two halves together fix who can be helped and who can be charged at every pair of positions.

The share-weighing rules have a familiar form outside projection: they are the proportional and the max-min-share rules of dividing a resource between claimants with different capacities to use it, and they behave here as they do there. Serving the worse-served is the max-min rule on absolute outcomes, which is fair to the person in front of the picture and blind to the person behind them. One picture and three people met the same conflict for a curved screen drawn for one seat, where a choice of seat decided who paid; a rule that weighs shares decides it by how much each could have had.

What the comparison assumes #

Each walker cares about their worst place. Every rule scores a walker by the detail at the part of the picture they see least finely. A visitor looking at one part of the picture cares about that part, and a projector that knew where each was looking would weigh differently.

The projector knows where both walkers are, without lag. A rule that weighs shares needs each walker’s own best layout as well as the shared one, which is twice the arithmetic and no extra tracking; a late layout hurts every rule, as it hurt the single walker.

Two walkers. With a crowd, the smaller-share rule protects the least-favoured visitor’s share, which in a crowd is usually somebody on the far side with nothing to gain; it then behaves like the even picture, as the worse-served rule does for the same reason.

The projector is turned fifteen degrees. The gains, and so every share, scale with the panel’s surplus, which grows with the turn.

Still open: whether a rule that knows where each walker is looking can serve both #

Every rule here scored each walker by their worst place — the part of the picture they see least finely — because a projector that tracks bodies and not eyes has nothing better to go on. But a visitor walking past a projected picture looks at one part of it at a time, usually the part nearest them or the part that moves, and the worst place for their position may be a part they are not looking at.

The measurement that settles what knowing the gaze is worth gives each walker a stated point of regard — the picture’s middle, the part straight in front of them, or the part a stated angle ahead of them along their walk — scores each by the detail at that point instead of their worst place, and asks whether the four rules then still disagree, whether the transfer at different depths survives when each walker is scored where they are actually looking, and how much of the near walker’s gain under the worse-served rule went to parts of the picture it was not looking at.