The Chase UK: Predicting the High and Low Offers During the Course of an Episode
/After you’ve watched The Chase for a while, you become familiar with the patterns of the contest and feel as though you can make reasonable guesses about the Low and High Offers that will be made to a contestant at the start of his or her head-to-head, based on how well they did in their Cash Builder, which seat they came from, and the overall state of the contest.
Last year, we built models to predict the High and Low Offers made to Australian contestants, and today we’ll do the same for offers made to UK contestants. What’ll be different, however, is that we’ll build separate models for each of the four seats, and we’ll take into account the state of the contest and previous offers when making predictions for Seats 2 to 4. So, you can imagine us using these models, sequentially, whilst an episode was in progress, using what we’ve learned from previous offers to update our views about likely future offers.
To do this, we’ll create simple linear regressions, using the playrer-by-player data from the OneQuestionShootout website, excluding the celebrity episodes and any episodes where there were multiple Chasers (which was the case in Episode 1,000).
THE OFFERS FOR SEAT 1
In creating our model for the Seat 1 Low Offer, we’ll assume that the only relevant information we have is:
The identity of the Chaser
The size of the Cash Builder set by Player 1
The model for Seat 1’s Low offer, using these variables, is summarised at right.
Our best estimate of the Low Offer for Seat 1 is equal to:
28% of their Cash Builder
Plus some adjustment for The Chaser’s identity of between -£143 (Jenny Ryan) and +£51 (Shaun Wallace). Note that Anne Hegerty is the “reference category”, so the adjustment for her is £0
Less £392
Let’s work through an example:
Paul Sinha is The Chaser, and the player in Seat 1 amassed a Cash Builder of £6,000..
Our estimate of the Low Offer would be:
28% x £6,000 - £40 - £392 = £1,248
In reality that might mean that he’ll usually offer £1,000 in this situation, but occasionally go above or below this..
There are a couple of things to note about this model:
It fits the data fairly well. The R-squared value tells us that it explains about 53% of the variability in the Low Offers for Seat 1, at least a portion of which will be due to offers generally being rounded
The average Seat 1 Cash Builder is about £5,400, so the average contribution of the Cash Builder term to the expected Low Offer is about £1,500. Every additional question correct adds £280 to the expected Low Offer
(Note that we lose one episode due to it having multiple Chasers, and three more where there was no Low Offer because the player in Seat 1 amassed a £0 Cash Builder. We lose sample in future models for similar reasons)
What about the High Offer for Seat 1? Chronologically, this is set after the Low Offer for Seat 1 is revealed, so we can use that Low Offer as another predictor on the basis that it might give us some clue about the relative level of generosity of the producers in the current episode.
The model appears at left, and our best estimate of the High Offer for Seat 1 is equal to:
200% of their Cash Builder
Less 110% of the Low Offer for Seat 1
Plus some adjustment for The Chaser’s identity of between -£475 (Shaun Wallace) and +£2,127 (Darragh Ennis). Note that Anne Hegerty is here too, as for all models we’ll build, the “reference category”, so the adjustment for her is £0.
Plus £15,205
Let’s work through an example:
Jenny Ryan is The Chaser, and the player in Seat 1 amassed a Cash Builder of £7,000. The Low Offer was £2,000.
Our estimate of the High Offer would be:
200% x £7,000 - 110% x £2,000 + £1,340 + £15,205 = £28,345
So, you’d say her most likely High Offer is £30,000, although she will occasionally go higher or lower than that.
The average Seat 1 HIgh Offer is about £24,900, with a standard deviation of about £5,800.
For this model, we should note:
The model fits the data less well than does the model for the Low Offer, but nonetheless explains about 31% of the variability in High Offers
There is a positive relationship between the Low and High Offers in that the larger was the Low Offer, the smaller we expect the High Offer to be. The effect size, however, is small. At the average Seat 1 Low Offer of about £1,100, the deduction is only £1,200. In only about 1 episode in 30 has the Low Offer for Seat 1 exceeded £2,000.
At the average Seat 1 Cash Builder of about £5,400, the average contribution of the Cash Builder term to the expected High Offer is about $10,800. Every additional question correct adds £2,000 to the expected High Offer
THE OFFERS FOR SEAT 2
As we move to look at the offers for Seat 2, we now have available to us a couple of new variables reflecting the producers’ existing risk: the amount banked by the team so far, and how many contestants are safely back at the desk (just 0 or 1 at this stage of the contest). We also have a sense of whether the producers are in a relatively high offer or low offer mood, based on the Low and High Offers already made to Seat 1.
We include those variables and build a model for Seat 2’s Low Offer, which appears at right.
Our best estimate of the Low Offer for Seat 2 is equal to:
22% of Seat 2’s Cash Builder
Less 6% of the money banked by Player 1 Plus £156 if Seat 1 is back at the desk
Plus 13% of the Low Offer made to Player 1 less 1% of the High Offer made to Player 1
Plus some adjustment for The Chaser’s identity of between -£1 (Mark Labbett) and +£112 (Darragh Ennis)
Plus £61
Let’s work through an example:
Anne Hegerty is The Chaser, and the player in Seat 1 banked £5,000 after receiving a Low Offer of £1,000 and a High Offer of £25,000. Seat 2’s Cash Builder was £8,000.
Our estimate of the Low Offer would be:
22% x £8,000 - 6% x £5,000 + £156 + 13% x £1,000 - 1% x £25,000 + £61 = £1,557.
So, the Low Offer would likely most often be £1,000 or £2,000.
Here, we should note:
The model fits the data less well than does the model for the Low Offer for Seat 1, but nonetheless explains about 41% of the variability in Low Offers
An average Cash Builder of about £5,250 will contribute £1,150 to the expected Low Offer. Every additional question answered correctly will add £220 to the expected Low Offer
The average money banked after Seat 1 is about £4,300. At that value, the MoneyBankedBefore term would, on average, lead to a decline in the expected Low Offer of about £250
The average Seat 1 Low Offer is about £1,100 and the average Seat 1 High Offer is about £25,000. At those levels, the nett contribution of the two Seat 1 Offer terms is about -£100
We move then to the model for the High Offer for Seat 2, and now have the Low Offer for Seat 2 available as an additonal predictor. The model appears below.
Our best estimate of the High Offer for Seat 2 is:
120% of Seat 2’s Cash Builder
Less 11% of the money banked by Seat 1 less £548 if Seat 1 got home
Less 270% of the Low Offer for Seat 1 plus 82% of the High Offer for Seat 1
Plus 71% of the Low Offer for Seat 2
Plus an adjustment for The Chaser ranging from £0 (Anne Hegerty) to +£2,127 (Darragh Ennis)
plus £8,018
Working through an example:
Assume Mark Labbett is The Chaser, and the player in Seat 1 banked £20,000 after a Low Offer of £2,000 and High Offer of £20,000. Seat 2’s Cash Builder was £6,000, and Low Offer was £1,000.
Our estimate of the High Offer would be:
120% x £6,000 - 11% x £20,000 - £548 - 270% x £2,000 + 82% x £20,000 + 71% x £1,000 + £442 + £8,018 = £26,602.
So, our best estimate of the High Offer would be £25,000 or £30,000.
The average Seat 2 HIgh Offer is about £31,800, with a standard deviation of about £7,200.
Here, we should note:
The model fits the data quite well and explains almost half the variability in High Offers to Seat 2s
The High Offer for Seat 2 is positively related to the High Offer for Seat 1, possibly suggesting that the producers’ level of risk is set at different levels and for entire episodes. Numerically, the contribution of the term related to the High Offer for Seat 1 is likely to be high (around £20,000 for a typical Seat 1 High Offer of £25,000)
The contribution attributable to the contestant’s Cash Builder is likely to be the next highest amount (around £6,300 for a typical Seat 2 contestant with a Cash Builder of £5,250). Every additional question correct in the Cash Builder adds £1,200 to the expected High Offer
The nett contribution of the two Low Offer terms will likely be next largest in magnitude. If the Seat 1 Low Offer is at its average of around £1,100, and the Seat 2 Low Offer at its average of around £900, their combined contribution will be about -£2,300
The expected High Offer decreases with the amount of money already banked. Numerically, the effect of this is likely to be quite small unless Seat 1 banked the High Offer in which case it could reduce the expected High Offer for Seat 2 by as much as £3,000.
THE OFFERS FOR SEAT 3
We now have the same variables as we did for Seat 2, plus the offers made to Seat 2. To capture the general level of offers made so far, we’ll use the mean Low Offer and mean High Offer to Seats 1 and 2.
The model for the Low Offers to Seat 3 appears below.
Our best estimate of the Low Offer for Seat 3 is equal to:
18% of Seat 3’s Cash Builder
Less 13% of the money banked by Seats 1 and 2 plus £90 times the number of contestants back at the desk
Plus 29% of the average Low Offers made to Seats 1 and 2 less 2% of the average High Offers made to Seats 1 and 2
Plus some adjustment for The Chaser’s identity of between -£128 (Mark Labbett) and +£192 (Darragh Ennis)
Plus £560
Let’s work through an example:
Anne Hegerty is The Chaser, and the players in Seats 1 and 2 between them banked £5,000 after receiving Low Offers of £1,000 and £2,000, and High Offers of £25,000 and £35,000. Seat 3’s Cash Builder was £6,000.
Our estimate of the Low Offer would be:
18% x £6,000 - 13% x £5,000 + £90 + 29% x £1,500 - 2% x £30,000 + £560 = £915.
So, the Low Offer would likely most often be £1,000.
Here, we should note:
The model fits the data better than any other model so far. It explains about 55% of the variability in Low Offers for Seat 3
An average Seat 3 Cash Builder of about £4,500 will contribute about £800 to the expected Low Offer. Every additional question answered correctly will add £180 to the expected Low Offer
The average money banked after Seat 2 is about £8,800. At that value, the MoneyBankedBefore term would, on average, lead to a decline in the expected Low Offer of about £1,150
The average Seat 1 and Seat 2 Low Offer is about £1,000 and the average Seat 1 and Seat 2 High Offer is about £28,000. At those levels, the nett contribution of the two Seat 1 and 2 Offer terms is about -£300
To the HIgh Offer for Seat 3 model then, which appears below.
Our best estimate of the High Offer for Seat 3 is:
88% of Seat 3’s Cash Builder
Less 19% of the money banked by Seats 1 and 2 less £608 per player home safely
Less 240% of the average Low Offers for Seats 1 and 2 plus 110% of the average High Offers for Seats 1 and 2
Less 18% of the Low Offer for Seat 3
Plus an adjustment for The Chaser ranging from £0 (Anne Hegerty) to +£2,187 (Darragh Ennis)
plus £8,481
Working through an example:
Assume Mark Labbett is The Chaser, and the player in Seat 1 banked £20,000 after a Low Offer of £2,000 and High Offer of £20,000. The player in Seat 2 banked nothing after a Low Offer of £1,000 and High Offer of £40,000. Seat 3’s Cash Builder was £6,000, and Low Offer was £1,000.
Our estimate of the High Offer would be:
88% x £6,000 - 19% x £20,000 - £608 - 240% x £1,500 + 110% x £30,000 - 18% x £1,000 + £671 + £8,481 = £39,244.
So, our best estimate of the High Offer would be £40,000.
The average Seat 3 HIgh Offer is about £39,000, with a standard deviation of about £9,200.
Here, we should note:
The model fits the data quite well and explains just over half the variability in High Offers to Seat 3s
The High Offer for Seat 3 is positively related to the average High Offers to Seats 1 and 2. Numerically, the contribution of the term related to the average High Offers for Seats 1 and 2 is likely to be around £31,400, since the average Seat 1 and 2 High Offer is £28,500
The contribution attributable to the contestant’s Cash Builder is likely to be the next highest amount (around £4,000 for a typical Seat 3 contestant with a Cash Builder of £4,500). Every additional question correct in the Cash Builder adds £880 to the expected High Offer
The nett contribution of the two Low Offer terms will likely be next largest in magnitude. If the average Seat 1 and 2 Low Offer is at its average of around £1,050, and the Seat 3 Low Offer at its average of around £150, their combined contribution will be about -£2,350
The expected High Offer for Seat 3 decreases with the amount of money already banked. On average, teams have banked about £9,000 after Seat 2 is done. At that value, the contribution of the MoneyBankedBefore term will be to reduce the expected Seat 3 High Offer by abot £1,700.
THE OFFERS FOR SEAT 4
We now have information about the offers made to Seats 1 through 3, and create a model for the Low Offers to Seat 4 that appears below.
Our best estimate of the Low Offer for Seat 4 is equal to:
6% of Seat 4’s Cash Builder
Less 14% of the money banked so far less £460 times the number of contestants back at the desk
Plus 110% of the average Low Offers made to Seats 1 through 3 less 5% of the average High Offers made to Seats 1 through 3
Plus some adjustment for The Chaser’s identity of between -£305 (Jenny Ryan) and +£215 (Darragh Ennis)
Plus £1,657
Let’s work through an example:
Anne Hegerty is The Chaser, and the players so far have banked £40,000 after receiving average Low Offers of £1,500 and average High Offers of £32,000. There are two players back at the desk. Seat 4’s Cash Builder was £4,000.
Our estimate of the Low Offer would be:
6% x £4,000 - 14% x £40,000 - 2 x £460 + 110% x £1,500 - 5% x £32,000 + £1,657 = -£4,500.
So, the Low Offer would likely most often be -£4,000 or -£5,000
Here, we should note:
The model fits the data better than any other model so far. It explains about 64% of the variability in Low Offers for Seat 4
An average Seat 4 Cash Builder of about £4,850 will contribute about £290 to the expected Low Offer. Every additional question answered correctly will add only £6 to the expected Low Offer
The average money banked after Seat 3 is about £12,000. At that value, the MoneyBankedBefore term would, on average, lead to a decline in the expected Low Offer of about £1,700
The average Seat 1 to Seat 3 Low Offer is about £740 and the average Seat 1 to Seat 3 High Offer is about £32,000. At those levels, the nett contribution of the two Seat 1 to 3 Offer terms is about -£800
Finallly, to the HIgh Offer model, which appears below.
Our best estimate of the High Offer for Seat 4 is:
21% of Seat 4’s Cash Builder
Less 45% of the money banked by Seats 1 to 3 less £865 per player home safely
Less 100% of the average Low Offers for Seats 1 to 3 plus 140% of the average High Offers for Seats 1 to 3
Less 68% of the Low Offer for Seat 4
Plus an adjustment for The Chaser ranging from -£241 (Shaun Wallace) to +£7,405 (Darragh Ennis)
plus £11,139
Working through an example:
Assume Mark Labbett is The Chaser, and the players have so far banked £12,000 after average Low Offers of £900 and average High Offers of £40,000. Two players are back at the desk. Seat 4’s Cash Builder was £6,000, and Low Offer was -£1,000.
Our estimate of the High Offer would be:
21% x £6,000 - 45% x £12,000 - £865 x 2 - 100% x £900 + 140% x £40,000 - 68% x -£1,000 + £600 + £11,139 = £61,649.
So, our best estimate of the High Offer would be £60,000.
The average Seat 4 HIgh Offer is about £52,000, with a standard deviation of about £14,000.
Here, we should note:
The model fits the data quite well and explains over half the variability in High Offers to Seat 4s
The High Offer for Seat 3 is positively related to the average High Offers to Seats 1 to 3. Numerically, the contribution of the term related to the average High Offers for Seats 1 to 3 is likely to be around £45,000, since the average Seat 1 to 3 High Offer is about £32,000
The contribution attributable to the contestant’s Cash Builder is likely to be around £1,000 for a typical Seat 4 contestant with a Cash Builder of £4,850. Every additional question correct in the Cash Builder adds £210 to the expected High Offer
The nett contribution of the two Low Offer terms will likely be around +£250 if the average Seat 1 to 3 Low Offer is at its average of around £740, and the Seat 4 Low Offer is at its average of around -£1,450.
The expected High Offer for Seat 3 decreases with the amount of money already banked. On average, teams have banked about £12,000 after Seat 3 is done. At that value, the contribution of the MoneyBankedBefore term will be to reduce the expected Seat 4 High Offer by abot £5,400.
SUMMARY AND CONCLUSION
Clearly it is possible to build simple regressions using relatively few variables that explain a significant proportion of the variability in the Low and High Offers made to contestants in the different seats.
Below is a table that summarises all eight of the models that we’ve built.
From this table, it’s interesting to note that:
Darragh tends to make the highest offers, especially High Offers. Jenny’s High Offers are also consistently above the average. Anne - who you’ll recall has zeroes in every model - is consistently below average with her offers
Low and High Offers are positively related to each contestant’s Cash Builder, but the strength of the relationship diminishes as we move from Seat 1 to Seat 4
Conversely, the amount of money banked and the number of contestants safely home has more and more influence on the size of the offers made - especially the High Offers - as we move from Seat 1 to Seat 4
There does appear to be a general “offer generosity” level established for each episode in that there is a positive relationship between the Low Offers made to later Seats and those made to earlier Seats, and a positive relationship between the High Offers made to later Seats and those made to earlier Seats, once we adjust for all other factors in the model. So, once we’ve seen a relatively generous High Offer to Seat 1, the chances are that we’ll see relatively generous High Offers to later seats, as well
Notwithstanding that these models do a reasonable job of fitting historical offer data, there is still a large proportion of variability in Low Offers and in High Offers that is not explained by them - as much as 60% or even 70% for some, and at least 35% for all.
I would be interested to hear your thoughts on what might be missing that could explain more of the variability, or if you think it’s just down to decisions made on the day that don’t follow a pattern.
