The multiplicative pickup model scales your count by a historical ratio

A security and fire show has gone from 6,900 registrations to 11,600 in four editions. That is about 19 per cent a year. Sixty days out this year the file holds 5,400, and the forecast someone built by averaging the last four editions' absolute gains says 10,781, which is below last year's final on a show that has grown every year since it launched.

Everyone in the room can see the number is wrong. Nobody can say why in one sentence. The multiplicative pickup model is the one sentence: divide the count in hand by the share of the final total your show usually holds at this point, instead of adding an increment measured in registrations.

The division that replaces the subtraction

Take the same four editions of the security show, measured at 60 days before doors.

The 2022 edition held 2,760 registrations at day 60 and finished at 6,900, so it was 40 per cent complete. The 2023 edition held 3,486 and finished at 8,300, which is 42 per cent. The 2024 edition held 4,059 and finished at 9,900, which is 41 per cent. The 2025 edition held 4,872 and finished at 11,600, which is 42 per cent again.

Average the four shares. 0.40 plus 0.42 plus 0.41 plus 0.42 gives 1.65, and dividing by four gives 0.4125.

Divide today's count. 5,400 over 0.4125 is 13,091 registrations at show open.

Now look at what the absolute gains did over the same four editions: 4,140, then 4,814, then 5,841, then 6,728. They grew 63 per cent across the series. Their average of 5,381 describes an edition somewhere in the middle of the show's history, which is why the additive form returns 10,781 and understates by 2,310 registrations. The shares moved by two percentage points over the same period. One of those two quantities is stable on this show and the other is not.

Why does the ratio form suit a show whose volume moved?

Because it carries a proportion across editions and lets the volume come from this year's file.

Both forms make a claim about what past editions tell you. The additive claim is that the number of registrations still to come is about the same as last time in absolute terms. The multiplicative claim is that the fraction of the campaign already completed is about the same as last time. Those two claims agree on a flat show and diverge fast on a growing one.

The divergence has a direction you can predict. On a growing show the additive form reads low, because the increments it averages came from smaller editions. On a shrinking show it reads high, for the same reason in reverse. The multiplicative form has no such bias with respect to size, and it buys that at the cost of a different sensitivity, which is that it multiplies whatever noise sits in the current count.

Talluri and van Ryzin place both forms inside revenue management practice in their 2004 Springer volume, The Theory and Practice of Revenue Management, which runs to 713 pages and gives estimation and forecasting a chapter of its own. The framing there is worth importing. A pickup forecast is a demand estimate feeding a capacity decision, and the point of putting a number on the final registration count is that somebody downstream orders badges, books shuttle buses and decides whether to open the second hall against it.

The chained version, which is what the literature means

The share-of-final calculation above is the compact version. The construction in the reservation forecasting literature builds the ratio interval by interval.

Papayiannis, Johnson and Duck set it out cleanly in their 2016 comparison of pickup methods for airport carpark arrivals, presented at ICORES. Lay the booking history out as a matrix with one row per arrival date and one column per lead time, holding the cumulative bookings on hand. The multiplicative version replaces each cell with the ratio of the cumulative count at one review point to the cumulative count at the review point before it. Each column of that ratio matrix is then a short time series in its own right, and you forecast the unknown cells column by column and multiply along the row to get the total.

The practical difference is granularity. The compact version estimates one number, the share at day 60. The chained version estimates a ratio for every interval between now and doors, so 60, 45, 30, 21, 14, 7 and 0 days out gives you six ratios to estimate and multiply. On a hotel with hundreds of past Tuesdays, the chained version wins because each of those six series has real data behind it. On a show with four past editions, each of the six ratios rests on four numbers, and multiplying six noisy estimates together is a good way to manufacture variance.

Papayiannis and colleagues also separate what they call classical from advanced pickup. Classical uses only fully completed booking curves. Advanced also uses the partially completed curves of instances that have not happened yet. For an annual show the distinction mostly collapses, because there is only one live edition at a time, but it matters for a portfolio running several shows a quarter off a shared audience.

What does the ratio form do at a long horizon?

It inflates, and the mechanism is arithmetic rather than bad luck.

Dividing by a small share multiplies whatever error is in the numerator. At 60 days out on the security show the divisor is 0.4125, so an error of 100 registrations in the current count becomes an error of 242 in the forecast. At 150 days out, when the share might be 0.12, the same 100 registration error becomes 833. Add the fact that the share itself is estimated from four editions and is proportionally less stable early in the campaign, and the early forecast is wide in a way the single number never shows.

The empirical version of this is in the carpark study. Across 28 pickup variations, four carparks and horizons of 7, 14, 28 and 56 days, Papayiannis, Johnson and Duck found the additive variations dominating the top of the accuracy tables, and reported that of the eighty entries making up the best-five lists across all carparks and horizons, only three were multiplicative. Their explanation is the one above: as the horizon lengthens the on-hand figure is low and volatile, and the ratio turns that volatility into an inflated projection.

Their data is carparks with a strong weekly cycle, so the numbers do not transfer to a trade show. The mechanism does. Treat any multiplicative forecast made while the historical share is below about 0.30 as an early indicator instead of a forecast, and say so on the slide.

When the shares themselves are drifting

The four shares on the security show were 0.40, 0.42, 0.41 and 0.42, which is noise around a level. Plenty of shows produce 0.51, 0.47, 0.44 and 0.40 instead, which is a trend, and averaging a trend is the wrong operation.

A drifting share usually means registrations are arriving later in the campaign than they used to. That can be your own doing, if the early bird deadline moved or the launch email went out three weeks later, and it can be a market condition affecting every show in the sector at once. Either way the mean of four shares sits above the current one, and dividing by a number that is too high returns a forecast that is too low.

Two repairs are available and both are cheap. Weight the recent editions more heavily, for instance four parts to the most recent edition, three to the one before, then two and one, which on 0.51, 0.47, 0.44 and 0.40 gives 0.437 instead of the flat mean of 0.455. Or fit a straight line through the four shares against edition number and read off the fifth, which on that series gives about 0.365 and is the more aggressive of the two. Say which you used. Dividing by 0.437 instead of 0.455 raises the forecast by 4 per cent, and dividing by 0.365 raises it by 25 per cent, so the choice between three defensible treatments of the same four numbers is worth a quarter of the answer. A figure that moves that far on a modelling preference should not present itself as a measurement.

The band around it is lopsided

Divide by the edges of the share range as well as the mean. 5,400 over 0.42 is 12,857. 5,400 over 0.40 is 13,500.

The point estimate was 13,091, so the range runs 234 below it and 409 above. The four shares sat within one percentage point either side of the mean, and the implied finals do not. Dividing by a share is a reciprocal, and a reciprocal turns a symmetric input into an asymmetric output. Anyone who takes the point estimate and puts plus or minus 400 around it has quietly moved the low end of the forecast down by 166 registrations.

Where this stops

The multiplicative form fixes the additive form's problem with growth and inherits two of its own.

It assumes the campaign shape held. If registration opened at a different distance from doors this year, the share at day 60 is measuring a different fraction of the campaign, and no ratio computed from history repairs that. Putting every edition on a common days to open axis is the prerequisite, and that is O9's subject.

It also assumes that four shares are enough to estimate a fifth. They are not, in any strong sense. Four observations give you a mean and a very rough spread, and the spread is what the operations lead actually needs. The honest deliverable is the range, the four historical shares printed underneath it, and a note of anything that made one of those editions unusual.

The larger caveat is that neither form is obviously right for your show, and the choice is answerable from your own file rather than from an argument. Compute both sets of residuals at the same days-to-open point and compare their spread, which is the diagnostic O3 works through.

Take four closed editions, find each one's count at the same days-to-open value, divide by its final, and write the four shares in a row. If they sit inside a two point band while the absolute gains behind them differ by more than a thousand registrations, the ratio form is the one your show supports, and you can say that in the meeting with the four numbers on the screen. The rest of the machinery for holding those snapshots comparably sits with the other forecasting methods in this cluster.