Today we’ll be analysing the latest player-by-player data from the fabulous OneQuestionShootout website, and investigating a single issue: how well can we predict a contestant’s chances of getting back to the desk based solely on the Cash Builder amount that he or she recorded?

METHODOLOGY

To perform this analysis, we’ll use only the following pieces of information:

• The contestant’s Cash Builder amount

• The seat number that he or she was in

• The offer that he or she accepted (Low, Middle, or High, regardless of the £ amount)

In the chart below we’ve shown the proportion of times that contestants have won their head-to-head, given these three pieces of information. The dots record the actual data, and show that, generally speaking:

• Controlling for the offer type accepted and the Cash Builder size, contestants from each seat number perform roughly equally as well as one another

• The higher the offer (given seat number and Cash Builder amount), the lower the chances of winning the head-to-head

• The higher the Cash Builder amount (given the seat number and offer taken), the higher the chances of winning the head-to-head

Note that the size of each dot reflects the number of contestants who have recorded that combination of Cash Builder, seat number, and offer taken, so there is more sampling variability associated with the smaller dots.

The chart also includes green lines that track the fitted values from a binary logit, the best functional form for which was determined to be:

Win Head-to-Head ~ Constant + Cash Builder (in £’000’s) x Seat Number + Offer Chosen

When fitted to the available data, the coefficients shown at right were returned.

These suggest that, consistent with what we described from reviewing the actual data, above:

• After controlling for seat number and offer chosen, a contestant’s chances of winning the head-to-head increase with his or her Cash Builder amount

• After controlling for seat number and Cash Builder amount, a contestant’s chances of winning the head-to-head decrease as the offer type moves from Low to Middle and, from Middle to High

• After controlling for the offer chosen, there’s a complex relationship between the contestant’s chances of winning the head-to-head and his or her seat number, which requires us to include the Cash Builder amount

This model fits the data quite well, as you can see from the earlier chart.

One interesting - if both simple and simplistic - thing you can do with this model is answer the following questions:

• Given a contestant’s seat number and Cash Builder amount, what Low Offer would leave the player with the same expected contribution to the prize pool as what it would be if he or she took the Middle Offer?
  (In other words, what Low Offer makes Pr(Win Head-to-Head | Middle Offer) x Middle Offer = Pr(Win Head-to-Head | Low Offer) x Low Offer)

• Given a contestant’s seat number and Cash Builder amount, what High Offer would leave the player with the same expected contribution to the prize pool as what it would be if he or she took the Middle Offer?
  (In other words, what High Offer makes Pr(Win Head-to-Head | Middle Offer) x Middle Offer = Pr(Win Head-to-Head | High Offer) x High Offer)

The answers to these questions (for Cash Builder amounts between £3,000 and £9,000) appear below:

So, for example, an average contestant from Seat One who recorded a £3,000 Cash Builder should, on average, put £1,700 in the prize pool (by winning the head-to-head and contributing £3,000 about 55% of the time, and by losing the head-to-head and contributing nothing about 45% of the time).

Given that, any Low Offer above £2,300, given his or her estimated chances of winning the head-to-head when taking such an offer, would represent a higher expected value than taking the Middle Offer. Similarly, any High Offer above £4,700 would also represent a higher expected value than taking the Middle Offer.

Contestants obviously consider a lot more than just expected contribution when choosing which offer to take, but it’s interesting to see how relatively small High Offers can be while still representing a higher expected value. That’s because the estimated (and, in some cases, actual) head-to-head win rate for contestants taking the High Offer is only about 20 to 25% lower than if they take the Middle Offer.

We should note here that the size of that difference will be affected by any selection bias in contestant propensity to take the High Offers. If contestants have a better idea about their underlying ability than their Cash Builder amount reveals, it could be that genuinely better contestants are more likely to take the High Offer, and more likely to succeed than the average contestant we’ve been talking about so far. This would tend to increase the minimum High Offer that a truly average contestant would need to feel that it was equivalent to the Middle Offer in terms of expected contribution.

SUMMARY AND CONCLUSION

There is, as I’m sure many of you might have guessed, quite a bit of information about a contestant’s relative ability that we can glean from his or her Cash Builder, and quite a bit we can say about his or her chances of making the Final Chase based on that Cash Builder, his or her seat number, and which offer he or she took.

Contestants who record larger Cash Builders are more likely to progress to the Final Chase than those who record smaller Cash Builders, after we control for the offer they take, and that is true as we move from the smallest recorded Cash Builders to the largest, although empirical data is harder to come by at the extremes.