I couldn’t help but delve into a couple of the episodes of the new series, and despite the constant outpouring of emotion, the economist in me did start thinking about game theory and optimal strategy. First things first, the new prizes in each box are listed below…
| £0.01 | £1,000 |
| £0.10 | £2,000 |
| £0.50 | £3,000 |
| £1 | £4,000 |
| £5 | £5,000 |
| £10 | £7,500 |
| £50 | £10,000 |
| £100 | £25,000 |
| £250 | £50,000 |
| £500 | £75,000 |
| £750 | £100,000 |
22 boxes, ranging from 1p to £100,000. This gives us an expected value at the start of the game of £12,917. It could be assumed therefore, that a good, or fortunate game, would bring home more than £12,917. Looking at the distribution of the numbers however, there is a 4 in 22 chance of beating the average, and an 18 in 22 chance of coming home with less than the expected value. That’s just an 18% chance of “beating the banker” at the inception of the game. But let’s delve in to see how a game develops. Given the availability on YouTube, I have picked 24th November with Sylvia. On the first go, contestants will remove 5 boxes, in Sylvia’s case as follows…
| £0.01 | |
| £0.10 | £2,000 |
| £0.50 | £3,000 |
| £1 | |
| £5,000 | |
| £10 | £7,500 |
| £50 | £10,000 |
| £25,000 | |
| £250 | £50,000 |
| £500 | £75,000 |
| £750 |
Unlucky first round taking out the biggest prize, but aside from that I note that every other box is below the expected value. It’s funny hearing the audience’s reaction to losing the £1,000 and £4,000, given this increases the average, and increases the chances of “beating the banker.” Now the average box is worth £10,533 (compared to £12,917) and there is a 3/17 chance of beating the original average. This is still an 18% chance.
Bankers offer: £1,300. Not much decision to be made here, an offer 12% of the current expected value. Though it should be noted, there are 9 boxes lower, and 8 boxes higher, so a 47% chance of beating the £1,300 offer at this stage. Given the massive disparity between the average and offer however, easy no deal.
Next round sees the contestant remove 3 boxes. In Sylvia’s case, £3,000, £750 and £500. All way below the average, so a good round.
| £0.01 | |
| £0.10 | £2,000 |
| £0.50 | |
| £1 | |
| £5,000 | |
| £10 | £7,500 |
| £50 | £10,000 |
| £25,000 | |
| £250 | £50,000 |
| £75,000 | |
Average has now increased to £12,487. So despite losing the biggest box, the average is very close to the inception of the game. Bankers offer: £2,500. This is now 20% of the average, a large jump in comparison to the first round. The chance of beating this offer has slightly dropped to 43%. Overall, this offer is much better from a mathematic point of view, however still far from equal value, so an easy no deal.
3 more boxes, £50,000, £25,000 and £5,000. Ouch, or so someone would think…
| £0.10 | £2,000 |
| £0.50 | |
| £1 | |
| £10 | £7,500 |
| £50 | £10,000 |
| £250 | |
| £75,000 | |
A bad round, two big boxes gone, and no very low ones. That reduces the average to £8,619. Only one box beating the original average, and the average from last round (9% chance of beating it). A bad round on all counts and therefore a large, expected reduction in the bankers offer.
Bankers offer: £3,000. OK, that was entirely unexpected. The offer is now 35% of the average, up from 20%, a big jump. Highly inconsistent and makes me question if the show is deliberately designed with improving offers to entice contestants to keep playing for the audiences’ enjoyment. Think how boring it would be if they accepted the first offer! 3 of the remaining boxes will beat the current offer, giving a 27% chance of this happening. Optimal strategy suggests No Deal due to the expected return, but there is no doubt some argument to be made for a deal here. Sylvia takes the gamble and plays No Deal, certainly reasonable, especially given the increasing percentage of the average. I would have done the same.
Next round, 10p, 50p and 1p. A fantastic round by all accounts, taking out 3 of the very lowest boxes, and no doubt massively increasing the average.
| £2,000 | |
| £1 | |
| £10 | £7,500 |
| £50 | £10,000 |
| £250 | |
| £75,000 | |
This jumps the average up to £11,851. Bankers offer, £5,300. This represents 45% of the average value, a further jump in the percentage. There is a 37.5% chance of beating this offer with the 3 remaining higher boxes.
Worth noting that the offer has gone from 12% to 20% to 35% to 45% of the average of the remaining boxes. It’s not unreasonable to assume the bankers offer in the next round would be anywhere in the 50% to 60% range. So lets test a few theories, assuming now £2k is considered a low box…
1. Losing the £75k, and 2 low boxes. Average would be around £4k. Offer expected to fall into £2k to £2,400 range. 16% likelihood of happening.
2. Losing the £10k, £7.5k and a low box. Average would be around £15k, offer expected to be £7,500 to £9,000 range. 8.5% likelihood of happening.
3. Losing all low boxes. Average up to £19k. Expected offer expected to be £9,500 to £11,400. 18% likelihood of happening.
4. Any scenario that loses the £75k, maximum described as point 1. 43% likelihood of this happening.
So, there is a 43% likelihood that the offer will be up to £2,400 and a 57% likelihood that the offer will be in the £7,500 to £11,400 range. On the lower of this range, the average outcome is £5,300, almost exactly the offer, on the upper range £7,500.
So, whilst optimal strategy clearly dictates taking the offer is almost always a bad idea, given the probabilities of what could happen, it was certainly not unreasonable for Sylvia to take the offer at this stage. In order to get an offer of £11,400 in the next round, 3 low boxes would be needed at 18% likelihood, alongside a 60% expected value offer from the banker. Worth noting, that we are comparing this offer to the expected next offer, not the expected return overall. We are asking the question, should we go a further round?
So here is what happened…
Ouch. £7,500, £10 and £50. One of the 57% outcomes happened, and we know the offer will increase, however, a good time to test by above logic.
| £2,000 | |
| £1 | |
| £10,000 | |
| £250 | |
| £75,000 | |
Bankers offer: £11,000. So I wasn’t far out with my logic, and still maintain that a deal at the last round stage wasn’t a bad statistical call. The average is now £17,450, with the offer accounting for 63% of this expected value. A bit of a higher percentage than I expected, but not by much. It’s easy to get caught up by what happened, but we know full well that there was a 43% chance of this round being a disaster, and from one offer to the next, the expected utility wasn’t changing much, and that is essentially the nature of the gamble. I think Sylvia’s decision was still sound unless we knew she was committing to going the whole way.
Hypothetically if we’re still playing at this stage, here are the main scenarios (2 boxes picked)…
1. Removing the £75k. This is pretty much the only way the offer is going down, and there is a 45% likelihood of this happening. This will essentially kill the offer. At best an average of £4k remaining, with an offer at 70% giving us £2,800.
2. Not removing the £75k, with a 55% likelihood. Worst case within this scenario, taking out the next two highest boxes would give an average of £25k, and following our earlier logic of say a 70% of the average offer, that would give an offer of £17,500. Taking out the two lowest boxes, £29k average and expected offer around £20,000. Lets go down the middle, £27k and an offer of £19k.
Combining this, (£2,800 * 0.45) + (£19,000 * 0.55) = £11,710.
So overall we don’t move much forward from offer to offer. I think a deal here is highly appropriate on a round to round basis. There isn’t much mathematical advantage to going forward by 1 round, and then in order to beat the offer it would have to be the £75k, at just a 20% likelihood.
Committing to going the whole way however plays very differently. 20% chance of winning 6.8x of the cash is a great return by itself. On top of that, there is a further 20% chance of returning with pretty much the cash offer by hitting the £10k. 60% likelihood of going home with much below the current offer. Certainly an excellent price to gamble and keep going to the end, as will always be the case in Deal or No Deal.
Next round…
£10k and £75k taken out. Offer dropped to £560. I don’t think anyone cares if I break this down further, if they ever did during this article!
But key points to be taken from this…
Bankers offer is almost always below the expected value of the box, so on the face of it, No deal is always a good option. However, from round to round, there are certainly instances where playing on offers no additional expected utility. Thinking about what the next offer will be should be a factor when making the decision, and not solely the expected value of the box. Bankers offer on a percentage basis always seems to increase too, how much would you be willing to gamble?
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