Is there some sort of formula to work out a good offer for the board?

The best formula that someone has come up with is the Fair Deal for the board.

http://www.bothersbar.co.uk/dealexplanation.htm

There is a webpage that was made in the days of the old Bother's Bar commentaries that will work it out for you.

http://www.bothersbar.co.uk/dornd.htm

Obviously the banker can't use the same offer on every contestant on similar boards, some are willing to gamble unless the offers are extremely generous and some cop out on offers that are fair but weak where there is more of a chance that the contestant will get a better offer next time but can't take the risk. In all or nothing situations such as the £250,000 and four blues left at five box. The banker can offer as much as five times less than the average and still get the contestant to deal because it's an all or nothing situation but on a board where are lots of big reds stacked together towards the end, the banker has to offer about the average to stand any chance of getting the players out of the chair. It also depends how confident he feels, if he feels over-confident that the players will crash then he will do poor offers but however if he likes the players he gives them very good offers.

The average is always a simple enough figure to work out and it gives a good idea, but it is flawed as it doesn't take distribution and volatility of the board into account. I used to write down the percentage of the average that each player was offered while I was on the show. I used to remember this as the game was being filmed and then write them down in the green room. There is definitely a bit of a pattern. The offers seem to be most generous around 5 box, less generous at 8, even less at 11 and at 2 box it was very tough to tell due to the fact that the board is very volatile at this point.

The offers that usually go down pretty well are the ones in the 50-65% bracket and if you are lucky to get offered more than that, it shows the banker is showing you a bit of respect.

One simple way of taking volatility into account is to take the average, then take an average minus the top and bottom figure, then take an average minus the top and bottom 2 (if at 8 box.) Then average the 3 numbers. This is similar to the way that gymnastics are judged, to remove anomalies.

For example, on a board of: -
Blue, Blue, Blue, Blue, £5k, £10k, £100k, £250k

Average is approx: - £46k
Take off the top and bottom, average is: - £19k
Take off the top 2 and bottom 2, average is: - £3.75k

AVERAGE OF THE THREE FIGURES: - £23k

so although the average on the board is £46k, you may expect a good offer would be in the region of £30k, but with the volatility taken into account you should expect probably closer to £20k. I often used to simulate a good, bad and average round, then asses what the board would look like in that circumstance by the time the next offer comes around. Then work out the probability of it happening and seeing if the reward was worth it. That is the EXACT reason for why I dealt at 11 box.

The one thing that was very clear though is that the type of player and your demeanor on the show, in the green room and in your game helps your offers massively. The list I made of the offers proved that the more vocal members got better generosity. I used to drum that into everyone. Never say the immortal words "I've never had that kind of money before", because if you no deal, you aren't ever going to get much more. Be positive and politely decline offers with little sign of regret. poker face is key.

The truth is that no equation can work out a good offer, because the value of an offer is dependent on the player and players are very complex variables, and ones that no-one really can work out until they play their games. The offer is good if it makes someone deal. Simple as. Maths helps, but the player knowing the value of the money on offer is what makes people make good decisions.

The way I decide if an offer is fair or not is quite simple.

Firstly, I look at the mean: anything at or above 50% of the mean has got my attention.

I then look at the volatility, and I may have a slightly different definition of this than others. When looking at this I'm looking at how much of the average is made up from single boxes. The main culprit here is of course the £250,000 - and with this in play we should never be expecting particulary generous offers in relation to the mean, simply because taking it out cripples the mean in most instances. But there are a few more subtle ways in which this can apply. Take the following 8box boards:

6 blues*
£10,000
£100,000

4 blues
£10,000
£15,000
£35,000
£50,000

Both boards have a mean value of £13,750. But the first board is so much more volatile than the second, simply because the £100,000 makes up so much of that figure. If we knock it out then we're in big trouble, and are probably opening our box. The second board is a bit better, we should hope for a slightly higher offer on this board. I'd personally be looking for about 60% of the mean: £8,250. On the first board I'd probably accept a little less (£6,000+) and then bang my head on the table when it turns out £100,000 was in my box all along.


And after we've looked at all of this we must look at future rounds and how that may affect the offer, and what the chances are of these future rounds. Will's game today was a nice example of this when he was offered £20,000 multiple times. I suggested dealing at each one of them (I even took the 14box offer). The reason was because I felt the offer was never really likely to increase enough to make it worth the risk of taking out the big numbers. Once the banker starts hitting the "real money" the generosity of the offers as a % of the mean usually drops, and this is fair, because the gamble is now much bigger. With Will's board today 75k/100k/250k and another low red, I felt that unless he held onto all 3 values til 5box he was unlikely to be offered significantly better than £20,000. And the chance of having a bad round and taking these values out to me made £20,000 a fairly easy deal.

Another nice example of future rounds was David's game where he had (I think) 3 blues, £20,000, £35,000 at 5box and dealt for £6,500. He correctly read that a future offer was unlikely to be any higher, meaning that unless he is prepared to gamble £6,500 and potentially win a blue he was correct to deal.

*I consider all blues to be of equal value: "blue". The only exception to this is late in the game the £750 and maybe £500 may take on some significane. If I was ever on the show though the first thing I would do was tell Noel to shut up about the bloody 1p, I swear he gets so many offers pulled down by how much he goes on about it.

Pretty much what I said. I agree. thats why I like to do the average, then the average without top and bottom and then without top 2 and bottom 2. If you get a big drop between your 3 figures, your board is very volatile, if there is less of a drop between the figures, the board is stable.

This simple equation gives a monetary value that incorporates the mean, but also takes volatility into account and it is easy enough to do on the spot in the game environment..

I tried out your formula Dan on Will's game as I felt that it was interesting in the sense that for me every deal from the 2nd offer onwards was dealable.

Mean-2 is with the top and bottom values removed, Mean-4 is the top and bottom 2 removed. The numbers are all aproximate, blues count as 0.

14box

Mean: £35k
Mean-2: 20k
Mean-4: £14k
£23,000
Actual offer: £20,000
Fair deal: £12,200


11box

Mean: £39k
Mean-2: £20k
Mean-4: £11k
£23,300
Actual offer: £20,000
Fair deal: £12,400

8box

Mean: £44k
Mean-2: £17k
Mean-4: £0.75k
£20,600
Actual offer: £20,000
Fair deal: £13,000

5box

Mean: £50k
Mean-2: 0
Mean-4: 0
£16,666
Actual offer: £16,500
Fair deal: £10,750


It's interesting to note your formula comes to pretty much what the banker offered in all of these cases. Am I correct to assume you would have dealt all of these offers? Would you have dealt £20,000 at 14box in real play in the studio? Out of interest I calculated the numbers for your 14box offer, something which you no dealt at the time:

Dan 14box:

Mean:£18k
Mean-2: £12.5k
Mean-4: £7.6k
£12,700
Actual offer: £12,000
Fair deal: £7,500

How come you no dealt this offer seeing as it comes so close to your magic number? I'm assuming it was because you felt another round had a good chance of improving the quality of the board and hence the offer. But looking back and ignoring the result this seems like a deal to me, as the negatives of taking out the 100k/75k seem to outweigh the positives of keeping them in play and generating a slightly better offer. (I don't want this to seem like i'm needlessly challenging your decisions, indeed I see nothing wrong with them, I just think it makes for some interesting discussion).


My final point is the FD values. These at times seem really low to me, and with the exception of Dan's FD all come in at 33% of the mean or less. Even taking variance into consideration this seems like a very small amount to me, and I think i'd find myself No Dealing all of these amounts.

So yeah, a largely pointless post, but I felt that some of the numbers were interesting

The FD tends to be relatively low, and the established general consensus ended up being that a below-FD Deal is one that's probably not a good move. (That said, two-box is an exception because the FD cannot be less than 1/x of the mean where x is the number of boxes, and half the mean on Power 5/peanuts is plenty enough for me.)

Why did Dan play on at 14-box? The same reason the other 1,000+ second offers have all been rejected. Stigma. End of. If Dot had set the precedent, we'd either have had four or five by now, or we'd have joke 14-box offers all the way, or quite possibly both.

While we're talking about systems, I've been using two since the show started. One's called the 2-box median system, the other the Hi-Lo system.

2-box median works on the averages possible at 2-box. All of the averages are calculated, and then the median of them is found.

On an 8-box board, there are 28 possible 2-box combinations. Here's an example:

50p
£250
£500
£3k
£5k
£20k
£35k
£250k

Looking at the 14th and 15th best board in terms of average, we get £5k and £20k (£12,500), and £3k and £20k (£11,500). The median here is £12,000.

The problem with this system, is that it doesn't take into account volatility of the board, and can only give a limited idea of the strength of the board.

The second system, Hi-Lo, uses two figures. With the 8-box board above, each amount is equally likely to appear at 2-box. Different amounts are more likely to be either the lower or higher of the two amounts in play.

E.g. If £35k remains to 2-box, it is much more likely to be the higher of the two amounts. £250k will never be on the low side, and 50p never on the high side.

The low figure is calculated by adding all of the 'low' amounts, and dividing by the number of combinations. The high figure is the same, but for the 'high' amounts.

Here's the number of times the amounts'll appear as a low amount at 2-box:

50p x7
£250 x6
£500 x5
£3k x4
£5k x3
£20k x2
£35k x1
Total £106,003.50, 28 combinations. Average is £3,785.84.

And for high:

£250 x1
£500 x2
£3k x3
£5k x4
£20k x5
£35k x6
£250k x7
Total £2,090,250, 28 combinations. Average is £74,651.79.

So the low figure is £3,785, high is £74,651, and the average £39,218.

The system gives an idea of the board based on what could happen if things go wrong. On this board, Jean (17/04/09) received an offer of £20,000. If she was forced to play on, and things didn't work out, she could expect to win an average of just under £4,000.

The key here is to assume that the £20,000 offer is the last one of the game. On a board of £4,000 and £75,000, you might not take £20,000. She declined the offer in the game.

While the system may seem complicated, the most important number, the low, can be worked out with addition. Starting with £35k, £40k can be added (£20k x2), to give £75k, then £15k (£5k x3), and so on. The blue amounts usually don't need to be added. A rough use of the combinations (e.g. use 30 instead of 28) can give an approximation of the low figure.

Using the low and the average, a range of statistics can be made about the strength of offers. One use of this is to take the offer, and divide it by the low figure. This is a form of volatility, in that the higher the number, the harder it is for the contestant to continue.

On the above board, it is 5.28, which is high at this stage, due to the influence of the £250k. Since developing the system, I haven't seen anyone take an offer below the low figure (i.e. it's always above 1.00).

Another use is to work out the percentage of the offer, between the low and the average. So if the low is a starting point of £3,785, and the average £39,218, the offer of £20,000 is 45.76% of the way there.

Sometimes the percentage will be close to the percentage of the average, sometimes there is a large difference. It gives an idea of the quality of the offer based on the board. Usually if the percentage is at least 40% is an offer worth considering.

I've uploaded a spreadsheet with the Hi-Lo system, downloadable below. To use, delete the amounts from the a-column when they disappear from the board.

Hi-Lo system spreadsheet for OpenOffice