If you’re a fan of crime dramas you will have come across the following scene: the police have arrested two partners in crime, but are lacking evidence. They need a confession. The police try to turn them against each other: they tell the suspects that if they confess and give away their partner they can make a deal to reduce their sentence.
In the show, one of the suspects eventually confesses and takes the deal but is this how it goes in real life? This may seem more like a question for psychologists but if we assume that both suspects are completely rational, we can use a branch of maths called game theory to answer this question.
Consider a situation where both suspect A and suspect B can be arrested for a smaller crime, where they will serve 1 year in prison each (pay-off: −1 years). If they both confess to the bigger crime, they will serve 5 years each (payoff: −5). However if A confesses and B doesn’t then A goes free (payoff: 0) while B gets a 20 year sentence (payoff: −20). The same deal is presented if B confesses and A doesn’t. This is an example of a prisoner’s dilemma. Drawing a table of all possible actions and their payoffs in the form (payoff of A’s action, payoff of B’s action), we get:
|suspect A||confess||(−5, −5)||(0, −20)|
|don’t confess||(−20, 0)||(−1, −1)|
Looking from suspect A’s perspective, no matter what B does, A will get a higher payoff if they confess (−5 > −20 if B confesses and 0 > −1 if B doesn’t confess). So we say that not confessing is strictly dominated by confessing. This means that rationally A will always confess. Knowing that A is rational, B will have to either choose to confess and get a payoff of −5 or not confess and get payoff of −20. Therefore B will also choose to confess. We get the same results if we start from suspect B’s perspective. Therefore the only rational solution in this case is for both to confess.
Unfortunate for them but lucky for the detectives: another case closed!
More from Chalkdust
- Twitch streaming, abacuses and Pascal's secrets feature in our AW23 issue! Plus all your favourite puzzles & columns.
- Can you solve it?
- Today's weather: seasonal, highs of π/18.
- And more importantly, are you on the A-level formula sheet?
- Can you solve it?
- The undisputed chart of the best mathematical games this season