Lottery Preferences: Buying the Payoff Rather Than the Asset

A lottery preference is a willingness to pay more for the shape of a payoff than its average is worth. Small stated chances get used above face value, so a small chance of a very large gain carries a premium the arithmetic does not support. Used that way on a single tail, one small chance produces speculative demand and explains why the demand lands on a few holdings rather than all of them.

Almost everybody who meets this subject reaches for a mood to explain it. Somebody got excited, somebody got greedy, somebody was chasing a thrill. Set the mood aside, and a single measured fact about arithmetic carries the explanation a long way instead. The fact is that people do not use a stated chance at face value, and the direction of the distortion depends on how small the chance is. Take that seriously and a premium for a small chance of a very large gain falls out of it, with nobody having to be foolish, excited or in any particular state of mind at all.

What is a lottery preference, stated as a shape rather than a mood?

A distinction the word lottery usually hides comes first. A lottery ticket is not attractive because of what it costs, and it is not attractive because of what it averages. A ticket is attractive because of the way its value is arranged across the possible outcomes: nearly all of that value sits in one outcome which nearly never happens, and the rest of the outcomes are flat and empty. The spread of value across the outcomes is the payoff shapeHow a return is spread across the possible outcomes, as against what it averages out to., and a lottery preferencePaying more for a small chance of a large gain than its average is worth. is a preference over that arrangement rather than over the average it works out to.

The same preference appears outside money, and it is worth meeting there first. A school fair sells raffle tickets at Rs 100/- each, two hundred tickets in all, one prize of Rs 20,000/-. The stall beside it sells a discount book at Rs 100/- that saves an ordinary buyer about Rs 100/- across the year. The raffle works out like this: Rs 20,000/- shared over two hundred tickets is Rs 100/- a ticket on average, so the two stalls are offering exactly the same average. The queue is always longer at one of them, and it is never the discount book. Nothing separates the two stalls except how the value is arranged, so whatever is driving the longer queue is a preference about arrangement.

The same thing stated in payoffs gives the numbers the rest of this guide works on. Two payoffs both average Rs 12,000/-. The first pays Rs 24,000/- half the time and nothing the other half, and half of Rs 24,000/- is Rs 12,000/-. The second pays Rs 2,40,000/- one time in twenty and nothing the other nineteen times, and one twentieth of Rs 2,40,000/- is also Rs 12,000/-. Every average matches. Every shape differs. Neither payoff is drawn from the Palash decision log, an invented record of what sixty people decided over eight quarters, and nothing in that log has the second shape in it at all. Payoffs arranged that way are rare, and the rarity is the point the whole argument turns on.

Which part of prospect theory does the premium actually come from?

Prospect theory, set out by Daniel Kahneman and Amos Tversky in Econometrica in 1979, carries three separable components, and only the third is doing any work here. The first is the reference point. Gains and losses are measured from wherever a person started rather than from zero. The second is the shape of the value curve, steeper for losses than for gains. The measured coefficient of 2.2 in the Palash cohort comes from that steepness. The third is probability weightingUsing a stated chance at more or less than its face value when a decision is worked out.: a stated chance is not used at face value when a person works out what something is worth to them. Only the third component does any work in the premium.

Two things make that borrowing narrow enough to be safe. First, both payoffs above have exactly one outcome that pays and one that pays nothing, so exactly one chance has to be weighted and there is nothing to argue about regarding the rest. Second, the outcome being weighted is a gain, so the reference point and the steepness of the loss side never enter. Prospect theory stripped down to its third component and applied to a single tail supplies every ingredient the premium needs and not one ingredient more. The overweightingTreating a small stated chance as though it were larger than it is. of a small chance is the whole of the mechanism from here on.

How far above face value is a small chance actually used?

A number has to be put on the distortion, and the number has to come from somewhere honest. The curve drawn here was chosen to show the shape clearly rather than measured from any market, survey or paper, and no reading taken off it is anybody's estimate of the truth. The shape, though, is not controversial: small stated chances get used above face value, large ones get used below, and there is a crossingThe chance at which weighting turns from over to under, so a stated chance is used at exactly face value. somewhere in between.

Read the four anchor points off the curve and then divide. A true 1 per cent is used as 5.5 per cent, or 5.50 times face value. A true 5 per cent is used as 13.0, or 2.60 times. A true 10 per cent is used as 19.0, or 1.90 times. And a true 50 per cent is used as 42.0, or 0.84 times, and 0.84 is below one. The multiple is not a constant lift of a few points; it collapses from more than five times to less than one as the chance grows, and it passes through one on the way. The passing through one is the whole of the mechanism, and everything else in this guide is a consequence of it.

The multiple moves faster than the curve makes obvious, so look at it on its own for a moment. Between a true 1 per cent and a true 10 per cent the used chance only climbs from 5.5 to 19.0, and the climb does not look dramatic. The multiple over the same stretch falls from 5.50 times to 1.90 times, and a fall of that size is a collapse. The premium available for a payoff shape is largest exactly where the chance is smallest, and it drains away long before the chance becomes ordinary. An asset bought for its payoff shape therefore has to have a genuinely remote large outcome in it, not merely an uncertain one.

Why is the premium paid rather than merely felt?

A feeling leaves no record. Somebody can be thrilled by a payoff, talk about it all evening and put no money anywhere, and nothing measurable has happened. A price is different in kind. When a buyer hands over more than a payoff averages, the excess is a completed transaction: it has a date, an amount, a counterparty and a receipt. A completed transaction is the whole reason the mechanism can be studied at all rather than merely asserted, and it is the reason the premium can be stated in rupees.

The concentrated payoff worked through the curve gives the number. The true chance is 5 per cent and the curve uses it as 13.0 per cent, so a buyer running on that used chance values the payoff at 0.130 multiplied by Rs 2,40,000/-, or Rs 31,200/-. The payoff averages Rs 12,000/-. The difference is Rs 31,200/- less Rs 12,000/-, or Rs 19,200/-. The premium is that Rs 19,200/-, a payment rather than a mood, and a payment is something a researcher can count. Nicholas Barberis and Ming Huang, in Stocks as Lotteries in the American Economic Review in 2008, built asset demand out of precisely this step, and Alok Kumar, in Who Gambles in the Stock Market in the Journal of Finance in 2009, went looking for where the preference showed up in what people actually held.

What is Speculative Demand, and what is being demanded?

Speculative demandDemand for a holding that comes from the shape of its payoff rather than from the cash it is expected to produce. is demand for a holding that comes from the shape of what it might pay rather than from the cash it is expected to produce. The word is often used loosely to mean enthusiasm. Enthusiasm is just a restatement of the fact that somebody bought, so used that way the word explains nothing. Used precisely, it names a component of a price, and a component of a price is something that can be added up, subtracted out and argued about with numbers.

Split into two parts, a price stops being vague. One part is the value of the expected cash, here the average of Rs 12,000/-. The other part is the value carried by how the outcomes are spread, over and above the average, here Rs 19,200/-. Together they give the Rs 31,200/- a buyer running on the used chance would pay. The same split runs on the spread payoff. Its true chance of 50 per cent is used as 42.0, so the buyer values it at 0.420 multiplied by Rs 24,000/-, or Rs 10,080/- against an average of Rs 12,000/-. The shape component is positive on the payoff whose value sits in an unlikely outcome and negative on the payoff whose value sits in a likely one. Speculative demand is therefore a statement about arrangement and not about mood. A concentrated payoffA return whose value sits mostly in one unlikely outcome rather than being spread across likely ones. collects a premium; a spread one collects a discount of Rs 1,920/-.

Hold the average fixed at Rs 12,000/- and change nothing but the concentration, and the buyer's value moves a long way. A 1 per cent chance of Rs 12,00,000/- averages Rs 12,000/- and is valued at Rs 66,000/-. A 5 per cent chance of Rs 2,40,000/- averages the same and is valued at Rs 31,200/-. A 10 per cent chance of Rs 1,20,000/- comes to Rs 22,800/-, and the 50 per cent chance of Rs 24,000/- comes to Rs 10,080/-, below the average it started from.

Why does the demand concentrate in a few holdings rather than spreading?

One observation decides whether a separate mechanism is worth naming at all. If a premium simply appears wherever people are keen, it would show up almost everywhere at once and would be another word for a good mood. The premium instead lands hard in a few places and leaves everything else alone, and the landing is what needs explaining. The weighting account answers that in one line: the premium attaches to the shape, and very few things have the shape. Most payoffs are the boring kind, spread across outcomes that are individually quite likely, and on those the same curve produces a discount rather than a premium.

Take it to a street first. Ten stalls at a fair sell ten sensible things and one sells raffle tickets. The queue does not lengthen at all eleven stalls on a cheerful afternoon in proportion to the cheer; it lengthens at the one stall whose product has the arrangement in it. A mood would have to lift everything, and a shape can only lift the few things that have it. Concentration is therefore the observable signature of a shape-based mechanism. A signature like that is a prediction that can be checked rather than a story that can only be agreed with.

One caution belongs here, and the Palash decision log supplies it. Of the 96 buys in that invented log, 41 followed a media mention within three days, being 42.7 per cent, against roughly 11.0 per cent of the eligible list being mentioned in a given week at all. A crowd around a mentioned holding is also a concentration, and it has nothing to do with payoff shape: it is a concentration of attention, a separate mechanism with its own measurement. Two different causes can leave the same footprint of a few holdings attracting a crowd, and the only honest thing to do is to keep them apart rather than let one borrow the other's evidence.

What does the premium cost, and who ends up paying it?

The premium is paid by whoever buys the shape and received by whoever sells it, and the arithmetic of what happens afterwards is short. A buyer who pays Rs 31,200/- for the concentrated payoff is buying something that returns Rs 12,000/- on average, and Rs 12,000/- divided by Rs 31,200/- is 38.5 per cent. Repeat that purchase many times and the average outcome is thirty eight and a half paise in the rupee. Run the same division on the spread payoff bought at its discount: Rs 12,000/- divided by Rs 10,080/- is 119.0 per cent, so the average outcome there is a rupee and nineteen paise in the rupee.

Those two numbers are easy to over-read, so read them carefully. A price above an average produces an average outcome below the price. Arithmetic alone gives that result, and it is not a discovery about anybody. The two numbers do not say the buyer made a mistake. Nobody has measured the true curve in any market, and both numbers are worked on a curve drawn to show a shape. Whether paying Rs 31,200/- was sound depends on something the arithmetic cannot see: what the buyer wanted the payoff for. Whether a middling outcome is any use to that buyer settles it, and the point is a real one rather than a polite one.

Set it out over twenty rounds and the transfer becomes concrete. Twenty purchases at Rs 31,200/- is Rs 6,24,000/- paid out, and one payoff of Rs 2,40,000/- is what comes back, so Rs 3,84,000/- has moved across the twenty rounds. The transfer is Rs 19,200/- a round, exactly the premium calculated earlier.

Where is paying for a payoff shape actually correct?

There is a condition under which paying above the average is exactly right, and the condition is narrow. Paying more than the average for a concentrated payoff is correct whenever a middling outcome is genuinely as useless to the buyer as nothing at all. If that condition holds, only one of the two payoffs can produce the outcome the buyer actually needs. The two are then not equivalent at their averages, whatever the arithmetic says. The premium then buys something real rather than nothing, and calling the buyer confused is the confused position.

Make it concrete with a household. A deposit of Rs 2,40,000/- has to be produced in one payment by a fixed date, and no part payment is accepted. The requirement is not a sum to be reduced but a threshold to be cleared, so Rs 12,000/- in hand does not make it smaller in any way that matters. The spread payoff reliably produces something in the region of Rs 12,000/-, and for that household it does nothing at all. The concentrated payoff produces the required outcome one time in twenty and nothing the other nineteen. Where the useful outcomes are a threshold rather than a total, the shape of a payoff is a real property of it and paying for that shape is buying a real thing.

The limit on that argument matters as much as the argument. The condition is a statement about the buyer's situation and never about the holding, so the same payoff at the same price can be sound for one person and pointless for the person beside them, and nothing about the payoff itself decides which. No general account can settle which of those two people a given buyer is, and nobody selling a payoff shape can settle it either. The only place the answer exists is in what the money was for.

How is any of this used by somebody deciding on behalf of other people?

The answer lives in the buyer's situation rather than in the holding, so Devika Rao, the adviser at the invented Palash Advisory Services Private Limited, does not use any of this to talk anybody out of anything, and could not do so honestly even if she wanted to. She uses it for a split instead. Before a purchase goes through she asks in writing whether the buyer is paying for the cash the holding is expected to produce or for the way its possible payments are spread, so the answer exists afterwards.

Devika Rao, Palash Advisory Services Private Limited, the Palash decision log and the Palash 100 index are invented.
Educational material. Not advice on any investment, tax, budget or market position.