Yes, "The New Fair Deal" sounds like an Iain Duncan-Smith tabloid-baiting programme to make the poor poorer, I know. It's not.

I can't believe that the original Fair Deal, which I devised in week one of the show in November 2005, is still the cornerstone of the statistical analysis here. It's the building block of Deal-Eye, and widely recognised as a first approximation of board value that's at least good for gauging that nebulous concept, "statistical generosity."

It's been criticised for not adjusting to stake level, a quirk of the square root function (it has a constant first derivative, hence it doesn't change any faster as the stakes rise - hence the £10/£100k FD is precisely ten times the £1/£10k FD when most people would take rather less than ten times as much on the former finish as on the latter.) I actually knew this in 2005 - I wanted a logarithmic solution. I've now found it, after re-reading some economic research into DoND which mentioned logarithmic utility functions as a proxy for explaining decisions.

For these functions to work in this context, you have to set a base level of wealth as a comparison context. (This serves to model less financially secure contestants taking less generous offers.) The best approximation seems to be annual household income - for the sake of a formula calculation, I've put £20k, which is close to the post tax average for middle income households in the UK according to ONS research, and a nice round number.

The NFD calculation is like the old FD calculation - transform every board value, take the average of these new values, reverse the transformation. But with logarithms it's more complex:
* Take the natural logarithm of (value+£20k) for every value left;
* Take the (mean) average of these values;
* Calculate the inverse natural logarithm (exponent) of this average;
* Subtract the £20,000. You have the NFD.

The NFD is more generous than the FD on most finishes, especially low value volatile finishes like 1p/£1k. It's notably less generous on £250k finishes - it values £10k/£250k at only £70k, compared to an old FD of £90k (amusingly £70k has been offered and dealt on this finish!) - and although I think it overvalues £100k/blue finishes quite a bit (around £29k), otherwise it deals with the weakness of the old FD while being otherwise fairly comparable.

More on this as I experiment with it, and possibly a hybrid of FD and NFD...

Hmm, interesting... This does at least sound workable within some kind of logarithmic formula as you say, and not arbitrarily calculated on individual utility. That was an area I was keen to steer away from with Deal-Eye, as I wanted its formula to be consistent across the board (which inevitably has its flaws with certain pairings, but equally adds to its quirkyness and 'drama' of ill-gotten points and near-misses etc., I think! ). Also, I didn't really want the system to have MY own utility calculations, at least not entirely anyway.

I quite like the theory of the default setting "Base level of wealth". You may hit some vastly contrasting FD pairs more accurately with this, but I'd be interested in the blues/ mid-reds pairs, as Deal-Eye is often weaker than the average player with the desired offers there. But then it's not generous enough on some all-red finish, where the concept of "chunky sum guaranteed" clearly affects the utility.

In essence, something between Deal-Eye and this idea could work. But let's not forget, I like to think of Deal-Eye as "A computer's interpretation of playing the game", and just like some of the players, it is unpredictable and it changes its strategy according to the scenario, and I like that quirky element to it, flaws and all.

That's just one interpretation, this is another! That's the fascination!

Nice analysis. Logs are awesome.

Glad it's sparked a bit of interest! I see this as almost a DE side project, and I did actually produce a DE sheet based on the NFD for each pair!

That produces a starting benchmark of £12,013.41 - the median pair is a lot higher (£7,110.88 compared to £5,070.96), but the mean pair is actually marginally lower (£16,915.94 compared to £17,070.93) because of the power of the £250k being muted in NFD.

On blue/mid-red, I think the NFD overcompensates - 1p/£15k is valued at £6,457.52, compared to £3,756.13 on the traditional calculation. Even 1p/£35k is as high as £13,166.26 - FD puts it under £9,000. The magnitude of the blue matters a lot less though, £750/£15k comes in at £6,949.03 on NFD (£5,614.55 on FD).

StatsMan also asked about stable finishes - the NFD arguably accounts for that less than FD. For instance, £20k/£75k is valued at £41,644.14 on NFD, compared to £43,114.92 on FD (which is almost £5k lower on the infamous 1p/£75k pair, which NFD values at exactly £23,589!).

With each formula having quirks, I decided to hack them together in a very simple manner - recreate DE with each pair value being the mean of FD and NFD. This puts 1p/£15k just over £5k, for instance. £35k/peanuts comes in around £11k, £50k/peanuts around £15k, and £100k/peanuts just over £27k. The scream finish is at £58,004.86, the Tegen/Morris/Bunney finish £93,389.19, and the Laura finish £67,873.31 on this hybrid value (NFD comes in below £59k, FD almost £77k!).

I'll post a screen shot of all the values for "red" finishes for NFD and the FD/NFD hybrid when I get something other than a phone for browsing the net at home - which, if my landline ADSL speeds are going to be awful, might be a phone as a modem for my laptop anyway!