Weaver's Week 2006-04-30
'Iain Weaver reviews the latest happenings in UK Game Show Land.'
Contents |
Stat and More Stat
30 April 2006
A warm welcome to everyone who has joined us from the write-up in The Times yesterday. Readers are warned that this edition of the Week contains some relatively complex mathematical discussion, and is more demanding than usual. Do stay tuned, as the usual snarking at Paxman and Humphrys will follow shortly.
Deal or No Deal
Much of this article was written in early March, immediately before we heard about the sequenced games. It was intended as a follow-up to the re-review of 26 February. All statistics are accurate up to and including last Thursday's episode, 27 April.
1. Does the distribution of prizes follow a mathematical model?
In particular, is there a statistical model we can use to predict the likelihood of a given outcome? Each day's show is, we can assume, an independent random event, and the final awards vary from one new penny to a six-figure sum. The mean over the shows so far was £16,200 or so, but half the contestants left with £13,000 or less.
This is evidence to suggest an Exponential distribution is appropriate, at least as a starting point. A pure Exponential distribution is "memoryless" - the chance of someone winning less than £5000 is the same as the chance of someone winning £20,000 but not £25,000. This doesn't hold for Deal, where 20% of contestants leave with less than £5000, but 45% of those reaching £20,000 don't make it to £25,000.
In practice, this idea doesn't quite hold water. A fifth of all contestants left with less than £5000; of those who won £20,000, almost half won less than £25,000.
This isn't enough evidence to dismiss the Exponential claim entirely. We mentioned in February that the bank's best bet is to estimate a person's utility curve - the enjoyment they'll get from a certain amount of money. For many people, it will rise steeply through the first few thousands, then begin to flatten out. People will still prefer £250,000 to £200,000, but not as strongly as they'd prefer £60,000 over £10,000.
Reviewing the evidence further, we find that the people who won at least £10,000 had a median win of £20,000, compared to £27,000 for those who won at least £20,000. This is more consistent with an Exponential function - people won't notice the difference between £18,000 and £19,000 as much as they would between £2000 and £3000. There's too little evidence to look at larger winners - only a handful of people have won £30,000 or more, a number so small as to be more anecdote than data.
The preference away from small prizes suggests that the true distribution may be a Gamma distribution with an alpha value slightly above 1, or an Exponential distribution about some (logarithmic?) transformation of the prize. The working hypothesis is that for "large" winnings (and the definition of "large" is unclear), the prize distribution follows an Exponential distribution.
Using this hypothesis, we can test producer Glenn Hugill's assertion (reported 5 March) that slightly more money was given away during the 43 games affected by sequenced boxes than elsewhere in the series. There were 66 games before the new, faulty algorithm began, and Thursday's was the 43rd game after it ended. We are perhaps best served by taking the 43 games immediately before as our control group, as there were obvious differences in bank behaviour during the first few weeks of transmission.
The "average" prize figure given here assumes an Exponential distribution, which allows for an estimate of the mean from the median and quartiles. It is a weighted average; the sample mean has weight 8, the median and upper quartile each have weight 5, and the lower quartile - most affected by face-saving deals - weight 2. Considering the mean alone alters the average prize, but not the conclusions.
The 43 shows before the sequenced games had an "average" prize of £15,240. The sequenced games went up to £19,860, and the games since came to £17,360. It's possible to run some regression techniques to find if these samples come from the same distribution; these results suggest there's a 10% chance the pre-sequenced and sequenced games came from the same set, but a 30% chance that the pre-sequenced and sequenced games followed the same distribution as the post-sequenced games. None of these results are statistically significant.
Mr Hugill's assertion was correct, more money had been given away during the sequenced games; this was most likely the effect of random chance. Though they were not utterly fair games, the group of contestants involved have neither lost nor gained significantly.
2. Is there a mathematical model to solve the game outright?
On inspection, it seems that Deal or No Deal could be solved by a matrix from the Minimax theorem, based on von Neumann's research in the 1920s. For every possible strategy of his own, the bank must consider all the possible responses of the player, and the maximum loss that he could derive. He then plays out the strategy that will result in the minimisation of this maximum loss.
For the bank, a loss is simply having to pay out any money at all. Minimax requires the creation of a matrix giving the pay-off for each move by each player at each stage; this would be a tremendously huge task in the early stages, but something a determined player could do in their head by the time five boxes remain.
Even though we may not know the exact mathematical functions to solve the Minimax problem - and that information should be available to the bank, if it wished to use it - the insight is simple enough for any potential banker to understand: you've got to minimise your maximum pay-out. A sensible bank would make offers that converge asymptotically to the mean, and a sensible player would deal at the point where there is a bigger utility gap between the lowest sum and the offer than the gap between the offer and the mean.
In the pre-sequenced games, the bank appeared to be using this overall strategy, or a close variation thereof. Very early in the sequenced games, a new strategy emerged, one that was based on emotion, not logic. Often, we find the bank making offers in the same general area as the last one, whether in cash terms, or in percentage of the mean, or whatever metric you care to use. This is bad play from the bank, and when repeated as often as it is, makes for irritating television.
Usefully, very usefully, von Neumann's insight is that the optimal strategy for the bank is also the optimal strategy for the player, but only for the player maximising his minimum win. It's not the same as maximising his final win, it's somewhat more risk-averse, and that is where the drama of the game should lie. The random selection of boxes will ensure that the game remains interesting. And if it doesn't, then there's always the host to bring some of his talents to the show. There is no need to go so far as to annoy the viewer.
3. Further research is needed.
A purely von Neumann approach treats the game purely in terms of cash, and we've suggested in the past that Deal or No Deal is more complex than that. Dempster-Shafer theory may be of use here, not so much in the raw probability, but in measuring a person's approach to what they want to believe. This column does not claim to understand Dempster-Shafer in any meaningful way, and throws this idea out in the hope that someone more knowledgeable will be able to perform the research.
Another idea to throw out there. Contestants typically spend twenty or so shows on the benches aside the studio, opening boxes for other people. Part of this is to introduce them to the audience at home, part is so the bank can get to know them better. Could part of the objective be to coach them into the ways of the bank, to give them an understanding of the current going rate in particular situations? We've mentioned before that Noel Edmonds is a proven master at directing people in certain directions, and could he be employed to lead the contestants to conclusions they wouldn't draw if they came cold to the game? There is no proof of this idea, and this column fully expects to have it shot down in very short order.
This column is not entirely sure how much more mileage there is in Deal or No Deal. The ratings are beginning to drop off, this could have something to do with the better weather outside. The game is beginning to become predictable - it's using catchphrases as though it were I'm Sorry I Haven't a Clue. Humph and the teams are funny, the host here is getting to be irritating. Worst of all, the bank's vindictiveness towards some players but not others is verging on the offensive; Wednesday's programme, where someone that many people would love to call "Gran" was on the receiving end of poor offers before she dealt, and hopelessly overstated ones after. Ultimately, Deal or No Deal is feel-bad negative entertainment writ large, and that just leaves a sour taste in the mouth. This feeling is magnified by a quirk of scheduling: the More4 repeat is opposite CBBC's programme of a genuinely positive atmosphere, Raven.
University Challenge
Fourth Quarter-Final: Liverpool v Hertfordshire
Liverpool got here by defeating Hull by a margin, and Trinity Cambridge by a nose. Hertfordshire beat Lucy Cavendish Cambridge by a bit, and Lampeter by just five. Liverpool has outscored their opponents by 460-305 (and not the lower score we erroneously gave recently), and look slight favourites to progress.
So, of course, it's Hertfordshire that gets the first starter. Kudos to the question setter who followed a starter on "turquoise" with a set on blue plaques. And for this buzz:
- Q: Felicity Kendal in 1981, Suzi Quatro in 19...
- Adrian Lewis, Hertfordshire: Rear of the Year.
- '(Thumper chortles)' Yes, spot on! Are you able to recite them all?
Not a terrific number of bonuses tonight, and when Hertfordshire get the visual bonuses - on British gothic architecture - they have a 40-25 lead. It doesn't last; Liverpool knows their musical birds.
A long run of incorrect starters, including one where no one knows the name of the jester in Twelfth Night, much to Thumper's disdain. Liverpool has slightly the better of the exchanges, and though Hertfordshire pick up the audio round - on actors portraying musicians on film - Liverpool's lead is 80-55.
Thumper asks a question about the smallest Westminster constituency by population. Anyone would think he couldn't pronounce Comhairle nan Eilean Siar. Hertfordshire pull to within ten, but Liverpool ensure they retain the lead. Neither side knows the successor to the British Athletics Federation, guessing general sports bodies. That's the pressure of the studio. The second visual round is on electrical circuit components, and Liverpool's lead is a not insurmountable 125-105.
Though they pick up a missignal, Hertfordshire draws level with 5 and a half minutes to play. Then they get another missignal, Liverpool gets one starter, Herts zig on the letter associated with "rhotic" when they should have zagged, it's R and not H, and Liverpool's lead is 40.
And climbing - Thumper is trying to break the studio speed limit of 35 points per minute, and only Liverpool's consideration of the bonuses is slowing him down. Starters are being dropped all over the floor, and when the lead reaches 75, it's got to be game over. The obligatory BBC plug (for children's program Dr Who) slips in moments before the gong, and Liverpool has the win, 195-140.
Adam McNamara-Jones was the best buzzer for Liverpool, making 101 points; the side had 18/33 bonuses and one missignal. Hertfordshire's leader was Adrian Lewis on 97 - the side had 11/30 bonuses and five missignals. Mr Lewis made 194 over the three games, just beating team-mate Bob Chapman (181). Key statistic for the semi-final: Liverpool is ahead of Trinity Hall Cambridge on starters correct and bonus conversion. Trinity Hall have made 800 points, against Liverpool's 665.
The best buzzers amongst the quarter-finalists, and our Team of the Year:
- Gareth Aubrey, Manchester, 295
- Darren Khodaverdi, Gonville & Caius Cambridge, 266
- Chris Smyth, Trinity Hall Cambridge, 258
- Adam McNamara-Jones, Liverpool, 228
Five other players - Ben Hardy and Iain Mathieson of Trinity Hall Cambridge, Richard Hutchinson of Imperial College of Medicine, Graham Ruston of SOAS, and Joseph Meagher of Manchester - have buzzed for 200 points or more. Anthony Lerman is London Business's best buzzer, worth 175.
Next match, 8 May: London Business v Manchester.
Mastermind
Heat 4, 20 April
Gillian Ledwick has swotted up on "The Marlow Family" novels by Antonia Forest. She knows her stuff about these books, scoring 14 (0).
Koushik Chattergee will be telling us about Test Cricket between India and Pakistan. Historically, this series has usually ended in a draw, and this round lives up to the history, ending one-one. Sorry, it ends on 11 (0)
Chris Hardman talks on British Dreadnoughts, 1906-1957. These were large and imposing ships, and this is a large and imposing score. 13 (1) looks promising.
Daniel Longdon will fire on the first three Star Wars films. This round goes completely over our heads, and the contender scores 5 (5). His general knowledge round picks up speed towards the end, finishing on 11 (10).
Mr Chattergee talks about the Indian version of Mastermind, on which he has not yet appeared. He finishes on 15 (6). Mr Hardman also falls into pass hell, ending his round on 18 (9).
Mrs Ledwick needs just five correct answers to progress, or four and a few wrong answers. Mr Humphrys intimates that all books being for children should be happy, revealing the paucity of his literary knowledge. Anyway, she gets two right straight away, eventually a third, and a fourth. Her final score is 22 (4).
This Week And Next
Ratings for the seven days to 16 April, including most of the Easter weekend, and 6.4 million for Jet Set keeps it as the UK's most-watched game show. Dance Fever had just 3.8 million, and falls behind The Apprentice, with 4.6m. The University Challenge double bill had 2.5m and 2.9m, Link 2.6m, Buzzcocks 2.3m, Eggheads 2.2m, and Mastermind 1.9m. On the commercial channels, 4.8m saw Millionaire, but Deal or No Deal (3.9m) was ousted as C4's most popular programme by Grand Designs. Countdown had 1.8m, which really shows the fall of Mastermind. On the digital channels, 806,000 for the Apprentice Aftermath interview, even ahead of BBC3's Dr Who spin-off. Pop Idle US had 600,000, but will have lost out by being on a public holiday. 324,000 for More 4's Deal repeats, 209,000 for Test Em.
Changes to Challenge's daytime schedule this week include the welcome return of Fifteen-to-One in the 4.30 slot. It'll be a slightly shorter Week next week, when we expect to review Challenge's Bullseye and UK Gold's Sitcom Showdown.
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