How would a mathematician win at The Traitors, the hit British TV show now entering the dramatic conclusion of its third series. Here are some tips from statistician Dr Ben Parker on how to do the best thing mathematically.
Fake Welsh accents apparently do make you more trustworthy, but the traitors also has some other human elements. As it’s a game of entertainment, the producers have a lot of power to introduce tweaks and make the gameplay more interesting.
Regrettably, the producers focus less on mathematics than on making good TV, and we’re hoping that Claudia Winkelman will take up her true calling as an ace mathematics professor to rival Hannah Fry. In the meantime, however, here is one mathematicians guide to the game.
Build up as much money as you can
It sounds obvious, but the aim of the game is to win as much money as possible when contestants compete in a series of ridiculous tasks around the Scottish castle to gain cash for the final prize pot. Generally, it makes sense for each player to get as much money in as possible as there is no disadvantage in the prize pot having more money in it for either Traitors or the Faithful. If you are a traitor, forget you are and just go for the cash. Enjoy the castle!
It would probably make for a more interesting game if every pound lost to the prize pot went to a Traitors special pot, as it would give the Faithful way of trying to track Traitors who deliberately lose money, but as it is, just play the ridiculous tasks and give the extras dressed as chess pieces something to do.
Become an expert in game-theory
The end-game at the round table is where it’s all at, and this is where some knowledge of games like The Prisoners Dilemma will help you.
Remember, to win at The Traitors,
- if all players left when the game ends are Faithful, they split the money.
- If one or more Traitors are left when the game is over, the Traitos split the money, and the Faithful get nothing.
We enter the end-game ( a bit vague, but about 4 or 5 players) with a number of Traitors and Faithful. Of course, the Traitors know the full picture, but each Faithful only knows about themself. Two things happen in turn
- Players vote whether to continue the game. If any one player votes to continue, the game continues. Only if they all vote to end the game, it ends.
- Players banish one of themselves by voting somebody off the game as normal, and we return to A.
Let’s imagine we enter the endgame with three people, and a prize pot of a mighty £6.
If we have three Traitors left, we are at point A, and everyone in the game knows the situation. The three Traitors can agree to end the game and split the money three ways to win £2 each, or one of them can decide they want to gamble for more and try to vote somebody off and win £3. Now it is just a case if two of them can gang up on the other and improve their share. If each player is an equally good player, each expects to win £2 whether they agree to end the game, or vote someone off.
If we are at point D, we are in a similar situation, but the Faithful do not know they are at this point. Again, on average the Faithful will win £2 with excellent mathematical play, similarly to at point A.
If we are at point B, it is slightly more interesting. It makes no difference to the Faithful whether they continue the game or not- they will win nothing. The Traitors will divide the money either equally (£3 each) or if one of them can convince the Faithful to vote off the other traitor, one gets £6. On average each traitor gets £3- 50% more than a point A..
Point C is the most interesting point. For a Faithful at point C, if they vote to stop the game, they will not get anything- the Traitor gets it all. If they vote to continue, the Faithful stand a chance of winning something, if they can vote off the traitor (all being equal, a one in three chance). The Faithful can only win if they vote to continue the game, although of course they do not know this.
So if you are a Faithful
- You will not get to scenario A
- If you get to scenario B, it makes no difference to you if you vote to continue the game, you have lost, so you may as well continue
- If you get to scenario C, the only way you can win is to continue
- If you get to scenario D, your expected win is the same whether you continue or not, so you may as well continue.
In other words, for a Faithful, you get the same or better result in B,C,D, and A is impossible to be in, so you may as well vote to continue. A sane Faithful player will always vote to continue.
This thinking can be continued if there are 4 or 5 players left- a Faithful should always continue the game. A traitor should try to end it, but of course one player voting to continue means we continue.
I suspect that there is a lot of pressure to end up in scenario C in most games as it makes the endgame much more interesting.
Should a traitor recruit or murder?
When we are not in the end-game (this is left to the producers, we think, but let’s say more than five), each night the Traitors can choose to recruit a new traitor or murder a Faithful.
If there is one traitor left, then the rules of the game say they must recruit- the player gets an ultimatum that either they become a traitor or are murdered
If there are two or more Traitors left, they can try and recruit people; Traitors always win something, and Faithful generally win less (see the diagram above).
Any sane player offered the chance would rather be a traitor: they have more knowledge of the game and a trump card with the money at the end. So if a player is offered a chance to recruit, it is much better for them to become a Traitor.
So is it better for the Traitors to recruit or murder? This is slightly more difficult to get a clear picture, so we perform some simulation. Under certain assumptions, we found it doesn’t matter too much: someone who starts the game as a traitor makes on average 2% more over a large number of games if they recruit whenever they can. Ideally each traitor wants to be in the final game- so recruiting means that Traitors have more control- but ideally they want there to be fewer other Traitors in the final game so there are less winnings to share- i.e. they want to be in scenario B, rather than scenario A or C.
Conclusions
- Make as much money as you can
- Faithful should always vote to keep playing the game in the final rounds
- Traitors should generally vote to recruit if they can, although it doesn’t make much difference
- Speaking Welsh OK