Inspired by the recent question threads, here are a list of problems that focus on the value of different boards and scenarios. You have a choice in each question, and have to decide without knowing what the offers (if any) will be. What would you do?

1. You have a choice of two non-standard boards, and there are no offers. When you choose a board, you win one of the amounts at random.

Board 1:
£1
£10,000
£15,000
£20,000
£35,000
£250,000

Board 2:
£1
£75,000
£100,000

2. After picking one of the three boards, you'll receive an offer instantly, and in addition for boards 2 and 3, an offer at 2-box.

Board 1:
£1
£100,000

Board 2:
£1
£50,000
£50,000

Board 3:
£1
£35,000
£35,000
£35,000

3. An offer will be made after choosing a board. The second board contains a negative amount, that could cost you money. To choose this board, you would need to secure £5,000 against your assets, taking out a loan etc, if you declined the offer. (If you would struggle to do this in real life, select Board 1.)

Board 1:
£5,000
£75,000

Board 2:
-£5,000
£250,000

4. There's a choice of two boards, and you receive an offer after choosing. The ????? represents one of 16 amounts from £1 to £50,000 that will be selected randomly. The amount will be shown if you choose the second board, followed by an offer.

Edit: Just to clarify, the 16 amounts are all from the DoND board: 8 blues from £1 to £750, and 8 reds from £1,000 to £50,000.

Board 1:
£1
£100,000

Board 2:
?????
£75,000

5. In the first round, you take out 1p, 10p, 50p, £1 and £250,000. You are offered the chance to restart the game.

6. In the first round, you take out £1,000, £3,000, £5,000, £10,000 and £15,000. You are offered the chance to restart the game.

7. There are two boards to choose from. You'll receive an offer at 2-box.

Board 1:
£1,000
£3,000
£5,000
£10,000
£250,000

Board 2:
£5,000
£10,000
£15,000
£20,000
£100,000

8. Two more boards to choose from, and on both boards, you'll receive an offer at 2-box.

Board 1:
£1
£250,000

Board 2:
£1
£15,000
£250,000

A bit complicated, but oh well, here goes.

1) I pick board one...
2) I picked board one.
3) Board 1, £5,000 is still a lot of money
4) Board 2.
5) No, I would continue. There's still 4 power fives.
6) I would.
7) Board 1.
Board 2.

1. Board 1 because there's a greater chance of winning something of worth
2. Board 3 because of the strength of the reds
3. Board 1 because the banker could make a stingy 2-box offer to force you on and I wouldn't want to risk giving £5k to Endemol!
4. Board 2 because if the ????? is a red then the offer has a higher chance of being decent than board 1
5. Restart the game
6. Don't restart the game
7. Board 1 due to the presence of the £250k and you can still win a red even if it goes wrong
8. Board 1 - the £15k/£250k offer probably wouldn't be that different to £1k/£250k. It's not worth the risk of having a £1/£15k climax just to find out!

1. Board 1 because there's a high chance of winning a big amount.
2. Board 3 because there are more reds.
3. Board 1 because I don't have £5000.
4. Board 1 because the '????' is slightly more likely to be a blue than a red.
5. I wouldn't restart.
6. I wouldn't restart.
7. Board 2 because all of the reds except £100,000 are higher than in Board 1.
8. Board 2 because there's a high chance of taking out £1 or £15,000.

Oh now this is brilliant. Absolutely brilliant.

1. Let's start with the maths. Board 2 has an EV of just shy of £60k; Board 1 has an EV of... six amounts, yes... £55k. With that little in the EV, I'm going with the lower chance of £1 on Board 1, but you could replace £35k with £5k and I'm going for Board 2. And if I lived in Liverpool, where £75k gets you a house outright, I'm probably going for Board 2!

2. This depends on prevailing offer generosities!!! Going on something close to long-term averages, I'm anticipating blue/£100k at £22k, blue/£50k at £14k, and blue/£35k at £10k. I'll then assume that I gamble until there's one red left, more as a simplifying assumption than anything, and now Board 2 is actually a £14k/£14k/£50k board (EV £26k) and Board 3 a £10k/£10k/£50k/£50k board (EV £30k). I'll go for Board 3, just.

3. I have enough savings to take on Board 2, so I'm doing that.

4. Depends entirely on the distribution of the random amounts, but probably Board 2. If I am cynical and assume that ????? will be heavily skewed downwards if I am not told about its distribution, I will pick Board 1.

5. Oh that's a really tough decision. This board will probably throw up more near-mean offers, so I will carry on.

6. Definitely playing on - although the board is more dangerous, telling the Banker that I am willing to embrace that danger will induce higher offers.

7. This boils down to two 60-40 splits: a 60% chance of holiday money vs a 40% chance of about £75k, or a 60% chance of family-car money vs a 40% chance of about £40k. I'm just about going for Board 1.

8. Board 1 for certain. I can't see it being less than £50k, and if it is then I'm going for a 50% shot at the jackpot!

1. Baord 1 as you're far more likely to win something decent.
2. Board 2 I think
3. Board 1 for definite as I doubt I could lose £5k of my own money if the banker offered a joke sum!
4. Board 2 as I always think there's little difference between offers on £100k/blue and £75k/blue anyway, and you end up with red back-up with the £75k.
5. I would not restart the game with the ultimate stable board.
6. Probably play on but nowhere near as easy a decision as #5.
7. Actually I fancy the more stable board 2.
8. Keep board 1 for definite, even if I took board 2 and ended up with £250k/£15k, chances are the offer would be low enough to make you ND anyway as they'd want to produce a quarter-millionaire!

1) Board 1 - despite the lower average (£55k to £58.3k), there's a high er chance of winning a red.

2) Board 1 is an instant offer of £26k.
Board 2 has potential 2 box offers of £50k, £15k, £15k (average £25k) so the instant offer won't be higher than board 1.
Board 3 has potential 2 box offers of £35k, £35k, £11k, £11k (average £23k) so again instant offer won't be higher than board 1.

I think I'd take board 1 and deal the 2 box offer.

3) Board 2 will probably give the higher offer and I don't think I'd gamble either so will take board 2.

4) I'd want to know the specific amounts for board 2. If not, then I'm taking the safety of board 1.

5) Carry on with the £100k board.

6) Carry on but not because of the myth of the banker's perfect round but because the average of the board went up and a random first round probably would take out more money on the board.

7) Board 1 has the higher EV at 2 box so will take that.

8. Board 1 - 2 box offer probably around £70k,
Board 2 - 2 box offers £90k, £70k, £5k.

So Board 1

Thanks for the answers, guys. This has helped me to developed a system which I think has an uncanny accuracy to what players feel is a good offer for the board - more on that later!

I've used this system in part in developing these questions to find out whether two (or more) unrelated boards can be considered equivalent, based on their perceived value (PV). It's clear now though that what's perceived equal by some is definitely not by others, though is a good thought exercise.

1. Interesting that there appears to be a consensus on board 1; I expected it to be far closer than that. Though the average is lower on board 1, not one person said that they picked it to win the £250k, opting for the 5/6 chance of winning a reasonable amount; even when there's a 2/3 chance of winning at least £75k - a situation that most people would ordinarily like.

While board 2 has a higher average, board 1 has a slightly higher perceived value. That and the additional risk on board 2 may have swayed things. Personally, though, board 2 would be my choice.

2. KP nailed it on the head: it depends on how generous the banker tends to be! We only know his offer range on the first board: £17,500 to £30,000. Board 3 averages £26,250, so while appears to be safe, could put you at risk if the banker decides that he would like to see you win £1 regardless.

Board 2 is my choice, as even if the offer on £1 and £50k is £12,500, (£50k+£12.5k+£12.5k)/3 = £26,666, and he may well offer more. Didn't think anyone would choose this, but Simon F surprised me!

3. This question didn't quite work out as planned. What I was aiming for, was a scenario where the offer on board 2 would be very likely higher than board 1; but even the perceived risk of losing £5k of your own money would persuade you to choose the first option.

The problem is that adding a dividing line on who has the savings and who doesn't, excludes people from being able to play on. One of the game's best qualities is that anyone can play it from any background, and adding risk further than winning nothing at all undermines that.

If I redid the question, it would more likely be £3,000 and £75,000 vs -£3,000 and £250,000, or even £1,000 and £100,000 vs -£1,000 and £250,000.

While like KP, I could afford the risk, I would probably go all the way on board 1, no dealing what would probably be a £30k offer. Board 2's offer depends on whether the banker would be scared of me choosing it; after all, to him, it's a £5,000 and -£250,000 board! To paraphrase James1978, you don't know whether the banker would want that though!

4. Should have clarified that the ????? amounts are from the DoND board. So, £1, £5, £10 ... £35,000, £50,000. If the banker knows that I pick board 1 instead of 2, then he would probably offer towards the higher range of £17,500 to £30,000. That said, based on a rough calculation, board 2's offers average around £25-30,000, and that's without no dealing any of them.

The average of the 16 amounts is £8,791.62. Guessing that the banker wouldn't want to risk making too low an offer on board 2, I think that would be my choice. But, there's no saying that the banker would make the right offer on a blue - £75k board, or even blue - £100k board for that matter.

5. Just done a quick calc, and the PV of continuing is £11,840. Restarting gives you an EV of the average winnings of the show, around £16,000. Is the £250k worth four low blues?

Now, here's a funny thing: If the banker knows I have a choice of two boards, then I go with restarting. If he doesn't know that I had to choose from one of two boards, then I wouldn't restart. I believe he would make more generous offers if I did restart; with him perceiving me as a gambler, I'm more likely to receive higher offers. But he wouldn't do that if I started from scratch and he wasn't aware of it.

6. Originally, I planned to have the amounts as £3k to £20k, but was persuaded to go lower after hearing feedback. Knew there would be few responses saying deal, even with honest no deals
didn't expect

The PV of this board is £11,040, but is higher than what people would deal at, because the larger amounts are undervalued on extreme boards, so is a more difficult decision than question 5.

7. This is a tricky one. Board 2 is safer, but 1 offers more opportunities.

It depends on how much the banker offers on any-£250k boards. At present, he tends to offer more than £50k in these situations; more than would be produced on board 2. Board 1 is a great bluffing board, in that when the £250k does survive to 2-box, the banker knowing that board was picked above board 2 would worry him slightly.

Because the £5k and £10k are shared between boards, I would just, and only just, pick board 1. If he offers less on any-£250k scenarios, I'm going with board 2.

8. A true test of confidence, this one. There are two ways of looking at this question: one with the banker knowing you choice of board, the other not.

If he knows that you picked £1-£250k, then he will be somewhat concerned that you are considering going all the way. That said, knowing that offering £50k works in these situations, his risk is somewhat limited, and may be tempted to offer less if he thinks you would deal it.

On board 2, I would consider that a gamblers board, in that you're exchanging the reduced average of the board with increased equity when the £15k-£250k scenario works out. You would have to no deal at least £100,000 to make it worthwhile in this case.

That said, when it doesn't work out, and you arrive again at £1-£250k, which you turned down board 1 for, would he punish you for that, knowing that's not what you wanted? My heart says yes, but my head says no. There's still the risk that after choosing board 2, you could still be inclined to continue, and pushing you too far to no deal would be counterproductive.

Being honest, I would pick board 1, if only because I couldn't bear to think about losing £50k on the 1/3 chance of having £1-£15k on board 2, and then 1/2 chance of going the wrong way.

If the banker was offering less on blue-£250k boards in general, there's a small band of offers where I would accept less in that situation, and value board 2 more. But it is small, in that if the offer's too low, I would take board 1 and go all the way.

Last two questions were the reason this post took so long to appear; didn't know exactly what to type until now!