ER... Over to you guys!

Simple, just plug in the 5 possible outcomes into the Deal-Eye formula and it should only take it on if the amount on offer is less than the Benchmark. By using goal-seek, I've calculated that the cut-off point that Deal-Eye would take the risk on is £5,140.92. So if the amount at risk is less than that, Deal-eye should go for it, but if it's greater Deal-eye shouldn't take it on.

I'd be more interested in what the 'typical' approach to Box 23 was, and I gather a lot of people would risk a lot more than that on it...

But Deal-Eye isn't supposed to be a human; it's supposed to be a machine free of emotions and targets (especially a £20,000 one that influence some to go for it).

Of course, it's not a human, but in some semblance should represent statistically what WE would consider a logical approach; £5,140.92 is, for me, undoubtedly too cautious and inconsistent with its main gameplay (especially as it shows far more aggressive tendencies on 2-box games at 5-box, for example). £20,000, the inflection point of EV (despite what Excel is telling me due to the half-pence glitch...) is probably too high for the majority of players...

Halfway between those values would seem a perfect compromise... if we could legitimise it!

I think Deal Eye will have the same type of problem with box 23 than it does with the 1-box at a time twist when it is a 1-box game (in that in most cases, there is only a 20% chance of total disaster).

I'm not sure there is a solution now for Deal Eye (other than maybe just ignoring box 23 exists and carrying on with a 22-box scenario)

From a pure EV point of view DE should gamble when it deals less than £20,000 and stick when it deals above £20,000.


From a gameplay perspective I don't think Box 23 makes much difference except that winning a blue now has the potential to turn into £10,000. As it is though DE will generally hit more blues than players so I'm not sure anything needs to or should be changed here.

£10,750 or less

In essence of course, you are right - mathematically, wins of £20k or less should be traded in for Box 23... But the previous nuances of Deal-Eye have not relied on the pure expected value of a situation, largely to create a more realistic picture that's tried to be reflective of how some humans might play the scenarios. We have to imagine Deal-Eye is playing the game ONCE just like the player, and just as you'd reject the average if you had multiple gos, you wouldn't (probably) trade in £20k when there's no 'added value'; if it goes wrong, there's no way back, and you can only expect to get your money back. That just sounds a bit irrational for Deal-Eye purposes...

I think I wouldn't personally gamble more than about £15,000 with it, although it's interesting as the region from £10k-£15k might well be dependent on how the game panned out for me...

I'd argue £10,000+ should not be gambled.

At £10,000 EV is only +£1,000 and it carries the NOTHING risk. £10k->£20k probably isn't as significant as £10k>£0 for most people.

Something around £5,000 seems a sensible maximum to me. Most can live with £5,000 dropping to £0 with £15,000 being a big enough jump to be worth the risk. An EV of +£1,500 represents a 30% increase in value as opposed to the 10% at £10,000.

I reckon depending on the game, I'd definitely deal Box 23 on 4 figure values and less and probably low 5-figure sums (up to about around £14k).

(I reserve the right to change my mind though)

Oh, I agree on the point that the first £10k is more important than the second utility-wise. Every scenario is effectively equal, so let's just imagine they are boxes on a regular game at 5-box, and you've been told the last offer will be a swap (or could be a swap, if that's stretching it a bit).

So, you have a board of £0/£2,500/£5,000/£10,000/£15,000, would you decline £5k...? ACTUALLY, you're starting to convince me about this argument now!

EDIT: Except it's not quite that simple for me for some reason...

My proposal here - have the DE benchmark be the higher of the offer and the FD of the Box 23 possible outcomes. (Not the benchmark of them, as ACB suggested - that's based around the 2-box finish, and that's not appropriate for Box 23!)

The Box 23 FD exceeds the flat offer for values below £3,953.46, incidentally...

Hmm... I suppose this is borrowing some of the the median value/ FD calculations that Deal-Eye is based on, in a way. I don't know... it just feels a bit low for a statistical approach (although we can all have valid arguments for our utility curves all day long!).

Been looking at a more 'fun' (albeit more manipulated) way of arriving at a figure that's pretty much the average of what most have been saying so far:

1. Take the "Expected Value" of the scenarios
2. Take the FDs of the scenarios
3. Find the "Median value" between the two
4. If THAT exceeds the stake, Deal-Eye takes box 23.

Effectively, this creates a cut-off point at £8,085.32, which I find very much more in line with Deal-Eye strategy, and it's about as close as I can get to the general view here so far, based around *some* formula not too dissimiliar from the existing one...

HOWEVER! I think we must also consider something else; Box 23 should lead to more generous offers just by being there, and if DE is judging a offer solely on the board, that doesn't seem quite right now... I'd like it to be a tad more aggressive in the main game. Of course, this will make it even more aggressive in 2-box games at 5-box etc., but there is probably a need to slightly upgrade its existing strategy. I'd expect offers to be 5-10% greater overall with Box 23, but we'll monitor this over the next few weeks/months...

What I would expect to happen in the long run is we see less 3 figure offers as it opens up the £10,000 chance. The banker will want to where possible pitch offers just high enough to stop people taking 23 but as I've said I think when we get to 5 figure offer territory it shouldn't make too much difference as there is little player advantage to 23.

But for the next few weeks at least I think we can expect to see some higher offers.


As a side note I think we are going to get some situations where there is nothing logical for the banker to offer. E.g.

1p vs £1 - Both boxes are essentially worth £2k as the players will gamble for the 1/5 chance of £10k. But the banker can't offer over what the boxes are worth as that pushes the EV up more. Unless we find a situation where the banker views the player as so cautious they'd deal £1,000 and NOT gamble.

1p vs £3,000 - Normally you'd see about £500-£750 here. But who is really gonna deal that? But at the same time it doesn't make sense to increase the offer either as this will only increase the players EV.

Time for me to whip out excel and graph some of this as I'm starting to confuse myself lol

I've managed to get it clear in my head what should be done, and without going into the maths too much here is what the banker should be doing:

Low finishes (£5,000 or less) - Offer the mean, or just slightly less. Or don't bother making an offer at all as box 23 is pretty much getting opened whatever happens so most players won't bother with a deal.

Medium finishes (low and £10k-£20k) - Offer whatever he does now really. Offering the mean here is essentially an AMO for the player so stick to say the £3,000 a blue/£10k would generally get.

High finishes (anything with £35k+) - No change. Box 23 is -EV for the player so if anything try lowballing and getting them to gamble.


In short for the banker... Change nothing, except a slight amount more generosity in low stakes games for the look of the thing.

Very interesting, thanks - nothing particularly needs to be changed on all-blue finishes for Deal-Eye as essentially, in the long term, on average it'll get as many 'pedantic' victories as it loses.

Blue/5k clearly should represent a significant difference for the banker now; whereas before you might've dealt £1,500 with a small blue, now with box 23 you should be looking at close to the average of £2,500... (or perhaps, the swap offer would be better there thinking about it?)

Of course, the implication of all this is... as Deal-Eye is basing part of its calculation on final pairs, and you've identified that some of the goal-posts of these have been moved mathematically, perhaps a slight 'hack' is needed at the spreadsheet...

This is gonna take some thinking about!

Actually... I'm not sure about that, as I think the banker would be more likely to offer the swap (especially with a low blue to nullify double effect). So, at least if that happened, Deal-Eye would remain consistent and FD benchmarks could stay...

What DE should do I think is...

First decide the threshold that it will take box 23. I suggest around £5,000, maybe as high as £7,500.

And then when considering all values under that assume they're the box 23 EV. E.g. an on offer of £2,000 has a box 23 EV of £3,800. Therefore if the benchmark is below £3,800 DE should be dealing £2,000 with a view to gambling it.

This assumes they are going to show what 23 contains even if its not taken.

The more I think about it, the £8,085.32 I came up with earlier is just about perfect, as it follows some of the previous DE methodology, and also utilty-wise makes sense to me - I could certainly handle losing half of it, the utility jumps to £18k & £16k make sense to me, and should I lose it all, I'd get over it!

NOW, I quite like your idea of thinking of offers below the threshold as being in the box 23 EV equation. So, if the banker offers £2k, the benchmark is £3,800, and the EV (clearly) says £3,800, Deal-Eye would ACTUALLY take the £2k at that point - is that what you're saying?