SHOULD I PHONE A FRIEND?
How to make the big decisions in game shows
TV executives are always looking for new game shows to fill the schedules. Among the crucial ingredients for a hit show is the creation of some tension building up to a big climax. One way of building tension takes the form of an offer: ‘Do you want to take the money, or gamble?’ It is similar to the decision ‘Take another card, or stick?’ in many card games, including the game show Play Your Cards Right. This is the $64,000 question, and, when it comes to those crunch moments in game shows, a bit of mathematics called decision theory can help.
Decision theory is used all over the place. It is particularly popular with government and management consultants. This chapter could just as easily have been devoted to deciding whether or not to build a new airport in London, or whether to invest in Asia or America, or countless other decisions. But for sheer human interest, you can’t beat TV game shows for investigating the maths behind decisions.
Of all the many game shows, this chapter will concentrate mainly on two. In fact they are two of the most successful game formats of all time.
Imagine you are sitting in the chair of the quiz show Who Wants To Be a Millionaire? So far, you have done really well. You currently have £75,000. Your next question, if you get it right, will win you £150,000. If you get it wrong, however, you lose £25,000. Here it is:
Not sure of the answer? You do have one lifeline left, which is ‘50-50’. This will reduce your choice to two and let’s suppose you decide to take this lifeline.
Now you are left with two choices. They are: (B) the New York Herald Tribune, and (C) The London Times.
Make your choice. You can offer an answer, or you can go away with the £75,000 you have already won. Only you can make this choice. And the answer? Find out later in the chapter.
Who Wants to be a Millionaire? stormed to international success in 1998. Like many successful quiz shows, Millionaire builds the tension by offering contestants the chance to gamble with important sums of money. The contestant has a choice of being greedy or playing safe. There is usually a hint of moral pressure on contestants to gamble, since this makes for much more exciting TV.
For those who have somehow escaped the mechanics of the prize money, here is how it is structured. Contestants start with nothing, and each time they answer a question correctly they move to the next prize level. In the revised prize money table that was introduced in 2007, the prizes from £1000 upwards are as follows:
Contestants who get a question wrong drop down to the safety net below their current level, so a wrong answer at, say, £10,000 means the contestant leaves with £1,000, while a wrong question at £500,000 means the contestant leaves with £50,000.
Decisions on whether to gamble or not are largely based on probability. You were just asked a question about Karl Marx, a man about whom you may or may not know very much. Your intuition will tell you how confident you are with the Karl Marx question, and you can convert that confidence into a probability. The simplest case is when you have no idea what the answer is. That way, you know that you are making a complete guess, and with a choice of two answers you have a 50 per cent chance of getting it right. The decision tree for a guess looks like this:
The branches of the tree represent the different outcomes, and along each branch is the chance of reaching that outcome.
Heads or tails – terminology quiz
What is the chance of tossing a head on a coin?
(a) 1⁄2
(b) 50-50
(c) 50%
(d) 0.5
The answer is … all four. Millionaire has popularised the expression 50-50 but probability experts use all four terms interchangeably.
However, things become more complicated if you feel you know something about the answer. For example, you might reason: ‘I know that Karl Marx lived for a time in London’. This may push you 75 per cent towards this answer, allowing a 25 per cent chance that you might be wrong. What this probability means, incidentally, is that in situations where you feel 75 per cent confident, you might expect to get the answer right three times out of four.
The decision tree would now look like this:
These probabilities are only gut feeling. There is no way of proving that the chance really is 75 per cent that The London Times is the right answer. After all, we know it is either 100 per cent right or 100 per cent wrong! In any case, your own value will be different, depending on what insights you have on the question. But in decision trees, you have to work with the information you have, and if all you can do is make an informed estimate then so be it.
Since you want to maximise your chances of getting the question right, you would in this example opt for the answer The London Times, because that is your hunch. But should you take the risk?
All you know so far are the relative chances of being right or wrong, but there is nothing to tell you whether to opt to have a go. To work this out, you need to put a value against the possible outcome of your decision. Decision trees can help you to do this. We already know the value of the outcomes of the Marx question:
• Get it right-£150,000
• Get it wrong – £50,000
Let’s go back to the easiest case, where you really haven’t got a clue which answer to go for. You might as well toss a coin to decide. You have a 50 per cent (or 0.5) chance of making the right choice. What does the decision tree tell you to do?
You will win £50,000 or £150,0001 after this question, but on average your earnings will be somewhere in the middle. You can calculate what reward to expect by simply multiplying the money values on both branches of the tree by their probability. In this case (0.5 x £150,000) + (0.5 x £50,000) = £100,000.
What to do if you are a millionaire ‘no-hoper’
The hardest part of Millionaire is actually getting to be a contestant in the first place. The odds are heavily stacked against you being invited to the studio, but, even when that happens, you are still competing with nine other participants to get to the Millionaire chair. And if you reckon the rest of the participants are far more knowledgeable than you, then you are in trouble.
But a bit of maths can help you. To get to the chair, you have to rank four items in order as quickly as possible. For example, place the following in order from most westerly to most easterly: (A) Paris, (B) London, (C) Norwich, (D) Brighton. Tricky, isn’t it? Chances are that you are up against somebody who knows the right answer. Your only way to get to the chair is to get the question right, and submit your answer before everyone else. Trouble is, there are lots of possible permutations. It could be ABCD or ACDB, or any one of 22 other options.
If you think you are up against strong competition, your best tactic is actually just to bash the four letters in completely random order as quickly as possible, in less than two seconds if you can do it. That way, you can be confident that at least your answer was quickest. Your chance of getting the right answer is 1 in 24 – about 4 per cent – but, if you miss out first time, there will almost certainly be a second chance, and possibly even a third during the show. In shows where three new contestants are chosen, your chance of getting the choice right and therefore being selected is in fact 1 – (23/24)3, or about 1 in 8. Not fantastic, but a darned sight better than the very remote odds you had when you started.
The value of playing the game is £100,000. To decide whether the game is worth playing, you have to compare this value with the alternative, which is keeping the money. If you do that, you go away with £75,000. Since £100,000 is more than £75,000, this simple decision model is telling you that if you have a 50-50 chance you should always gamble at the £75,000 level of the game. In fact, exactly the same maths says that it is worth gambling on 50-50 at every level in Who Wants To Be a Millionaire?, even when on £500,000 and going for the million.
But is this really what you would do? If you were offered half a million pounds, how certain would you have to be before you would risk losing it to go for £1 million? It all depends on whether you think £1 million is twice as valuable as £500,000. To most people, £1 million is more than they have ever dreamed of having. Then again, so is £500,000. The two prizes have very similar value or, to use the economist’s term, utility, to most of us. Even £50,000 can seem a fortune to most people. To a very wealthy person, on the other hand, what’s another £50,000? Its utility is relatively low. But another £1 million – now you’re talking!
In other words, the value of the rewards in decision trees will be distorted when the prizes are big, and, unless you are looking purely at money for its own sake, you need to substitute utility values for money values.
To see how utilities can vary depending on the person, here are three stereotypical contestants:
• Angie: In debt, £8,000 would be life-changing
• Brian: Comfortably off, but £50,000 would pay off the mortgage
• Clarissa: Wealthy, but £1 million would deliver that longedfor yacht in the Bahamas
Maybe you can see yourself as being at one of these three levels.
Let’s suppose utility is measured on a scale of 0 to 100. The utility graphs of the three might look like this:
By the time that Angie gets above £50,000, the value of the prize begins to be almost irrelevant, because whatever it is it’s enough to change her life. For Clarissa, on the other hand, prizes up to £50,000 are mere pocket money and worth almost nothing, but rapidly become more valuable as they move into six figures.
One model, rather too involved to reproduce here, that takes account of the different utilities at different levels of the game, comes up with the following suggestions on tactics, depending on whether you are type A, B or C.
The table shows how confident each contestant should be before they ‘go for it’ at each level:
If every question in Millionaire meant you doubled your money or lost everything, then the players would have to be more and more confident as the stakes became higher to bother taking the risk. However, there is a safety net at £50,000. A contestant who gets a question wrong above that level is still guaranteed to take home £50,000. This question is therefore a crunch point for all of the players, and has a big distorting effect on players who are in a good position to gamble.
For Angie, every increment of money in the early stages is important, and so it would be foolish to throw away any prize unless she was very confident. Beyond £50,000, however, Angie can relax. The debts are all paid off. She can gamble a bit more.
For Brian and Clarissa, however, the big time doesn’t really begin until the £50,000 level. Smaller prizes won’t hugely affect their lives. It’s even worth their gambling with odds of less than 50-50. Because the £50,000 level is a gateway to £75,000 and beyond, it’s actually worth Brian gambling to get there even if his odds are worse than 50-50. And, because £20,000 is small fry to Clarissa, she gambles on the £50,000 question even if she is making a complete guess out of the four choices.
Curiously, according to this model anyway, it might be that by the time they reach £150,000, Brian is more cautious about risk than Angie. This is Brian’s life-changing level, whereas Angie is well beyond that level, and will still be overjoyed to end up at £50,000. Clarissa will probably be a bigger risk-taker than either of them. Rich people will typically gamble more than those who are less well off at all levels in the game.
At the start of the chapter, you were posed the £150,000 question about Karl Marx. Did you keep the money (that probably makes you a Brian-type person) or did you gamble? The correct answer was: the New York Herald Tribune (a newspaper that had some left-wing leanings). If you got it wrong, you will be relieved this wasn’t real money.
Millionaire’s successor as the biggest talking point in the gameshow world was The Weakest Link. Although the dynamics of this game are very different, in that contestants are competing with each other for a single prize, the mechanisms for winning money in the two shows have a lot in common.
The standard Weakest Link game starts with nine contestants, each of whom is asked a question in turn. If they get the question right, the prize money won by the team moves up one level. Before being asked a question, a contestant can also shout ‘bank’. This puts the money won so far into a communal pot, and the value of the next question goes back to the start level.
The prize money of the original UK version of the show is relatively low. In fact the structure of the prize money in each round is as follows:
Anyone watching this show must at some time have asked themselves: ‘When is the best time to bank?’ One theory is to keep going when you have got a question right, because the increments get bigger each time. On the other hand, if the group gets five questions right in a row and then the sixth one wrong, that is £300 thrown away. Maybe it is better to play safe and bank at low levels.
One way to analyse the game is to look at the prize money you expect to win by following different strategies. The analysis is complicated, because the decision trees become very involved. You can begin to see how you might figure it out, however, by looking at the expected winnings in the early stages of the game.
The most basic strategy is to bank money every time you get a question right, which earns £20 a time. Suppose you expect to get questions right 50 per cent of the time. Here is how your winnings would look after one round:
The average, or expected, winnings after one turn are (0.5 x £20) + (0.5 x £0)= £10.
What about after two questions?
There are four possible paths through the decision tree, correct/correct, correct/incorrect, incorrect/correct and incorrect /incorrect. To work out the value of this strategy, multiply the prize of each option by all of the probabilities along its branches, and then add together the values of each option. In each case, this is 0.5 x 0.5 x the banked money, which works out at an expected return of £20 after two rounds. In fact, with this strategy of always banking £20, and a 50 per cent chance of getting a question right, the expected winnings in the game will be £10 per question. After 25 questions, you would expect on average to have banked £250.
How does this compare with the next simplest strategy, which is to bank only when you reach £50? We are still assuming that you always have a 50 per cent chance of getting questions right. You might expect that, since £50 is more than double £20, this strategy will make more money.
After two questions, the possible outcomes are:
The expected return is (0.5 x 0.5 x £50) + (0.5 x 0.5 x £0) + (0.5 x 0.5 x £20) + (0.5 x 0.5 x £0) = £17.50.
Only £17.50? Remember, with the £20 banking strategy you expected to have £20 after two rounds. Surprisingly, after two questions, the £20 banking strategy is bringing in more money than the £50 banking strategy. And in fact, this continues to be the case no matter how many rounds you go through.
What happens if you expect to get, say, 70 per cent of the questions right. Does this affect which strategy is the best one? Yes, it does. Look at the decision trees after two rounds for the £20 and £50 strategies:
The expected winnings after two questions are (0.7 x 0.7 x £40) + (0.7 x 0.3 x £20) + (0.3 x 0.7 x £20) + (0.3 x 0.3 x £0) = £28
The expected winnings after two questions are (0.7 x 0.7 x £50) + (0.7 x 0.3 x £0) + (0.3 x 0.7 x £20) + (0.3 x 0.3 x £0) = £28.70
So after two rounds, and with a 70 per cent chance of getting questions right, the £50 banking strategy emerges as margin -ally better. And this turns out to be true for longer runs of questions, too.
What also turns out to be true, however, is that, at this 70 per cent success level, always banking at £100 is even better than banking at £50. And banking at £200 is better still. In fact there is no success level at which banking at £50 or £100 is the best strategy, unless you are in the very final seconds, when you should always bank whatever you have.
Although the best banking level moves up in steps as the level of skill of the team goes up, crudely you can reduce the tactics for the basic form of Weakest Link to three very simple rules, as shown in the box overleaf.
Recommended group tactics for The Weakest Link
If you expect to get only half the questions right always BANK AT £20.
If you expect to get about two-thirds of the questions right BANK AT £200, and not before.
If you expect to get over 90% of questions right, aim for £1,000 and DO NOT BANK.
This is actually a practical strategy that you can use to ‘play along’ at home. You can usually assess a group as being a 50 per cent group, a two-thirds group or a 90 per cent group after just one round, and if you adopt the tactics used in the table when watching the programme, ignoring their ‘Bank!’s and using your own instead, you will usually end up with more money than they do. This is because weak groups push their luck instead of banking at £20, and because some strong groups tend to bank too early. In the final few rounds, even strong groups degenerate to the 50 per cent level.
If you were actually a contestant on the show, you would need to adapt your behaviour to reflect your personal circumstances. So, if you, personally, are 90 per cent confident of getting a question right, never bank. If, on the other hand, the people before you have built up to £450, but you only give yourself a 50 per cent chance of getting your next question right, bank immediately.
For higher levels of prize money, especially the US version of the show where the prizes are huge, the complications of ‘utility’ come into play just as they did in the analysis of Millionaire. And there are other psychological factors you can’t ignore, too. For example, it’s all very well taking a question at £450 without banking, but if you do happen to get it wrong then you are almost certain to be voted the Weakest Link, with all the humiliation that that entails. On top of this, there is the issue of not wanting to appear too clever. A contestant who excels in his answers is invariably voted off by the other two when the game is down to the last three players.
A three-card trick – and a tricky decision
There is one game show decision that has become famous not so much for the TV programme, but for the arguments about the right answer. It all centres on the finale of a popular American show from the 1960s called Let’s Make a Deal, which was hosted by Monty Hall. At the end of the show, a contestant was shown three doors, and was asked to choose one of them. Behind one of the doors was a special prize, a car for example, and behind the others there was something of little value, a dustbin, say. Now imagine you are the contestant, and you choose one of the doors (Door 3). Before it is opened, the host opens one of the other two doors (Door 2) to reveal a dustbin. There are now two doors, and behind one of them is the car. You are asked if you would like to swap your choice of door. Do you stick with Door 3 or would you like to swap to Door 1?
Ninety-nine per cent of people when presented with this challenge stick with the door they first chose. The reasoning goes that it is a 50-50 choice, so why swap? But here is where it gets extremely tricky. It is not a 50-50 choice, because the host, being a showman, opened a door behind which he knew there was a bin in order to build up the tension. Your chance of picking the door with the car behind it was 1 in 3. The chance that it was behind one of the other two doors was 2 in 3. After the host has opened the door with the bin behind it, the chance that the other will have the car is still 2 in 3.
Now, experience says that the above short explanation is not nearly sufficient to convince the sceptics that door-swapping is a good idea. By far the best way to understand the problem is by experimenting. Here is one way to do it.
Ask a friend (let’s call him Ralph) to deal three cards, including an ace, face down. Ralph should check so that he knows which one is the ace. Your challenge is to pick the ace, which represents the car. Choose a card, and then Ralph should turn over one card that he knows is not the ace, and then ask you if you want to swap or stick. OK, pick a card…
Here is the card you chose, still face down:
Ralph checks the other cards, then turns one over that he knows is not the Ace, like this:
He then asks you if you want to swap your card. If you decide to stick, you will win this game about one-third of the time. If you decide to swap you will win about two-thirds of the time, and certainly more than half the time unless you are very, very unlucky. Play this game at least ten times and you will begin to see why it isn’t a 50-50 choice at all.2
Confusing? You bet! But this is yet another example where a bit of mathematics can make a big difference to your rewards. Then again, in the heat of a game show, which of us could truly keep our cool to figure it out?
1. Although this is now almost universally known as the Monty Hall problem, it actually dates from the 1930s or even earlier. Indeed it seems very unlikely that the door-swapping situation described ever actually took place on Monty Hall’s show. According to Monty Hall himself, ‘On the show, I did indeed reveal what was behind one of the doors not chosen, but I do not recall giving the contestant an opportunity to trade her selected door for the one remaining. I asked members of my staff if they could recollect my ever doing so, and all but one of them said no.’
2. Getting the right answer is actually worth more than £150,000, because the game doesn’t finish there. If you reach the £150,000 level, you have the chance to play for £250,000 and beyond. For strong contestants, a better estimate of the financial value of getting the £150,000 question right is more like £200,000.