How should the ‘Seer’ have used their power in the Traitors
Why the seer's choice was, information theoretically, less than optimal (although dramatically completely brilliant).
***Spoilers*** If you’re not fully caught up with the third series of UK Traitors (what have you been up to) then you might want to hold off from reading this until you’re finished as it contains spoilers about the last couple of episodes. If you’re all caught up, read on.***
Our family just finished the third series of UK Traitors. All four of us in our house are obsessed with it. Because it’s on a bit late for the kids, we’ve been saving it up to binge watch together at the weekends. When we’re not sitting around speculating, I often daydream about how my skills as a mathematician might help me out if I were in Traitor’s castle. I used to develop the puzzles for a mathematical game-show called Dara O Briain’s School of Hard Sums and I’ve written about maths and game-shows before, so I have more than a little interest in the process.
In truth, in practice, I’m not sure how helpful being a mathematician on the Traitors would be, but in theory there are at least a few places where a little practical maths could allow a faithful to glean some insights. One place that really stood out to me was in an element of the game that was new to this year’s series - the “Seer”. But before I explain how maths can come in useful, let me give those unfamiliar with the programme a quick recap. If you’re a Traitors veteran then you can skip the next paragraph.
Each year, twenty or so people volunteer to spend two weeks confined to a beautiful Scottish castle, playing, what is in effect a murder mystery game on a grand scale. Some of the players are selected from the beginning to be “traitors” while the remaining players are “faithfuls”. The traitors’ job is to meet together secretly each night to ‘murder’ a faithful (to kick them off the show) and to avoid detection, while the faithfuls are supposed to pick up on subtle behavioural cues and gather what other scraps of evidence they can to try to figure out who the traitors are. Each evening all the players sit around the “round table” and share their theories until the discussion ends in a banishment – each player votes for the person they think is a traitor and the person with the most votes is “banished” from the castle, revealing, before they go, whether they were a traitor or a faithful. The players left standing at the end of the series take home a share of the prize money they have raised with their fellow competitors throughout the show. The twist is that if a traitor makes it all the way to the end then they take all the cash leaving the remaining faithfuls with nothing.
This series, for the first time, the show’s creators introduced a new feature to the game when there were just five players remaining. The player who won the “seer” ability got to choose one other player and to have their identity revealed – traitor or faithful – in a face-to-face meeting. While the traitors know who each other are and, as a result, who the faithfuls are, this is the first time that a faithful had been offered a chance to know with 100% certainty the status of another player. The “Seer” ability is an opportunity to gain information about the game which could potentially take you all the way to the final, so one way of looking at the problem is to answer the question, “How should you choose the player to maximise the information you gain?”
Fortunately, there is a whole branch of mathematics known as information theory which is dedicated to helping us answer questions like this. We can think of the process of finding out whether someone is a traitor or a faithful as a simple binary read out – 1 for traitor and 0 for faithful. One way of reframing the question is to think about which person will provide us with the most information when we find out their status and to some extent that depends on what we currently think the probability of them being a traitor is.
Imagine for a moment playing Scrabble with a binary alphabet made up of just two characters, 0 and 1. The figure below shows how the information gain – sometimes referred to as entropy - associated with picking out a single tile changes with the frequency of 1s and 0s in the bag. If both characters come up equally often in a given language then the Scrabble bag for that language should contain equal numbers of 0s and 1s. When you pick a tile from the bag at random you are equally surprised if it is a 0 or a 1. Your initial 50-50 uncertainty about the character you will pick out means that, when that uncertainty is resolved by looking at the letter you picked out, you gain one bit of information (see the figure below). A bit is the fundamental unit of information and the word itself comes from a contraction of the words binary and digit.
Conversely, in a language which is 90 percent 0s and only 10 percent 1s, you’re not surprised when you pick a 0 out of the bag – there should be nine times more 0s in the bag than 1s, after all. You should be more surprised when you pick out a 1, but this happens less often, so on average the overall surprise (or more technically the entropy) for that language is less (at approximately 0.47 bits – see the figure above) than the language with equal numbers of 0s and 1s.
Taking this example to its extreme, imagine that you have a language which only uses 0s and never uses 1s. Every tile you pick out of the bag will be a 0, so there is no surprise. There is no uncertainty associated with the event of picking a tile from this bag, so you gain no information (again check out the figure above). You knew what would happen in advance. And this makes sense intuitively. In a language that only has one character, how can you write a message that contains any information? You can’t.
This tells us something important about how to extract the most information from a situation. Imagine you’re playing a round of Guess Who – the game in which you try to pick out the character your opponent has chosen, from a line up in front of you by asking “yes/no” questions about their appearance. By the time you have ruled out all the other cards and are left with just one, you have gained a certain amount of information. It’s the same amount of information whether you asked five questions or 50, but the aim of the game is to ask as few questions as possible. Consequently this means that you should try to gain as much information, on average, from each question, as you can.
It’s always tempting to ask questions which would, in one fell swoop, rule out huge swathes of the cartoon card population. Because there were just three hairless avatars in the version I played as a kid, “Are they bald?” was always a favourite question of mine. But although you gain a lot of information if the answer to the question is ‘yes’, it’s highly unlikely that this obscure question will be answered in the affirmative, so on average you gain less information than is optimal. The most information you can gain, on average, when asking a binary “yes/no” question is one bit. As we saw when picking tiles from the Scrabble bag, you get closest to realising this potential if the ‘yes’ and ‘no’ responses are equally probable. From this perspective asking, “Are they male?” might be a good first question, allowing you to rule out roughly 50% of the unknowns.
Taking this knowledge back to Traitors castle tells us how Frankie should have chosen which player’s status to reveal in order to gain the most information. She should have chosen the person who she was wavering about the most. People who she believed very likely to be faithful (or similarly very likely traitors) wouldn’t typically have surprised her very much – wouldn’t, on average, have given her very much information.
In the end, Frankie chose her close friend Charlotte, who she was extremely confident was a faithful, having established a strong rapport with her from the start. Having her confirmed as a faithful would have put her mind at rest and allowed her to forge an alliance, but seemingly would have gained her relatively little information. As it transpired, however, Charlotte was, in fact, the last traitor left in the castle and Frankie gained more information than she could have hoped to by picking someone she was less certain of, like Alexander. Information theoretically it was the wrong play on average, but on this occasion she learned a huge amount – sometimes you pick a zero out of the scrabble bag with 90 percent 0s and only 10 percent 1s, sometimes your random pick on Guess Who is the right one.
Ultimately, as Frankie discovered the power of the Seer can be a curse rather than a blessing and it’s fair to say that information theory isn’t the only (or probably even the guiding) principle that should be used to play Traitors, but it does provide some interesting insights.
Nice article Kit