Game theory: The prisoner's dilemma
A lot of our decisions, although we may not realise it, rely on maths. Game Theory is the study of mathematical models of conflict and cooperation between intelligent rational decision makers. It can be applied to almost any subject including economics, political science, psychology, and even biology. The last one is of particular interest to Sergey Gavrilets, a mathematician at the University of Tennessee, who uses game theory to model early human behaviour and he spoke to Tom Crawford...
Sergey - In our models we included two different types of games. One is coming under the name of “Us versus Nature” games. If you think of a situation where group members have to go on a hunting trip, or you can think of a situation where you have to clear part of the forest to be able to plant some plants there.
The other type of games is us versus them games. That would be a conflict with a neighbouring group or territory over some amazing opportunities.
Tom - So would I be right in thinking the idea of us versus nature - say there are eight people in your group and you need to hunt a mammoth for food thinking hunter/gatherer style, provided four people turned up we would kill the mammoth? But then, the idea is that only means half of the group need to give effort because the whole group will be fed, for the whole group to get the reward. Is that the kind of game theory thing where people are thinking should I commit and put the effort in or can I be lazy and free ride through - is that the idea here?
Sergey - Exactly. That the idea. Perhaps we would need more than four people to kill mammoth. Maybe more like ten but, indeed, that’s the idea.These type of games is also coming under the rubric of the volunteer’s dilemma. Just imagine a situation like you and your family are watching a movie and then there is a phone call. The phone is in the next room and you don’t want to go there but it’s so annoying. So this is a situation where everybody would benefit from somebody going and answering it but nobody wants to do it. This is a typical volunteer’s dilemma.
Tom - That ringing phone example occurs pretty much daily at my house. Sergey Gavrilets there from the University of Tennessee.
Georgia - Now that we have a better grasp of how game theory works - Tom, I understand we’re going to do a little experiment of our own in the studio...
Tom - Indeed we are Georgia - it’s called the prisoner's dilemma. And as our resident mathematician I’m going to be running the show, with yourself and Peter Cowley, who’s still with us, our willing volunteers…
Georgia - How does this work?
Tom - This is a hypothetical scenario where both of you have been caught committing a crime.
Georgia - Oh no! What did we do?
Tom - Say you’ve robbed somebody - you’ve done a bit of pickpocketing. You’ve been caught by the police and the police say to you we have enough evidence to put you in jail for one year. They say this to both of you but you’re seperate and you can’t communicate. They also say to you but if you snitch on your friend and cooperate with us, then we’ll let you go free and your friend will go to prison for 7 years.
So you can be a bit naughty and snitch on your mate but, of course, you don’t know what your friends going to do or what you’re going to do. You’ve both been given these two options - you can either cooperate with the police which means you go free and your friend gets 7 years in prison. You can just say absolutely nothing and get one year in prison. And what will happen if you both cooperate? Then, of course, the police are like you’re both just making this up if you both blaming the other person so you’ll each get 3 years if that happens.
So each of you has to decide whether or not you will stay silent and accept your 1 year. Or whether you will cooperate with the police and snitch on your friend. So I’d like you to both to write down whether you would cooperate with the police and snitch or stay completely silent. Of course, you’re not allowed to communicate here. You can’t discuss with each other what’s going on.
Georgia - I was going to say I want Peter to come back on the show so…
Tom - Hypothetical - bear that in mind.
Peter - Yeah. We’re sitting far enough apart we can’t see each other. We can see each other but not our bits of paper. I’ve written something down…
Georgia - I’ve written down something too…
Peter - Are we supposed to show it up like a game at Christmas or something?
Tom - I tell you what. We can do the maths here and work out what’s best. So if we do this first then we can see how you both thought about this problem.
Let’s start with Georgia and say let’s assume that Georgia has decided to cooperate with the police. So, from Peter’s point of view, Georgia has cooperated and snitched. So, at the moment, if Peter stays silent he is getting 7 years in jail because Georgia’s snitched. But if Peter cooperates with the police as well, his 7 years goes down to 3. So if Georgia snitched, Peter should also snitch. But if Georgia stayed silent, then that means at the moment Peter has 1 year but if he snitches he goes down from 1 to 0, whereas if he cooperates he goes up to 3.
Peter - And vice versa?
Tom - Yeah and vice versa. So if you think about it from an individual point of view you’re always reducing your jail time by snitching by cooperating with the police. But, if you think about it from a team perspective and you add up the total number of years in jail, the best option is for you both to stay silent because if you both stay silent you get 1 year each, a total of 2 years. Whereas if you both snitch, which is what you should do to maximise your own personal benefit here, your total time in jail is actually 6 years.
So this is basically how game theory works in a nutshell. It’s considering the individual benefit and individual decision as opposed to the team benefit and the team decision. So, let’s see what you actually have said…
Peter has decided to stay silent.
Georgia - Oh… I’m sorry, I snitched!
Tom - So Peter’s clearly far too nice…
Georgia - Please come back on the show. I’m sorry…
Peter - So the final result of that is how many years?
Tom - The final result would be Georgia would go free and unfortunately, Peter, you would be in jail for 7 years.
Peter - See - I’m a team player and Georgia…
Georgia - I just can’t go to prison.
Tom - But as you said, this is game theory. It’s the team dynamic versus the individual benefit.
Georgia - How do we use this? Who was right essentially? I’ve betrayed Peter massively but how do we use this? If I’m ever in this situation should I be using game theory in this way?
Tom - It’s not so much that you should be using it, it’s more a way of modeling a decision making process. So, if you want to think about it purely from an individual level, then you’re using it to maximise your own individual benefit. Because by always snitching you’re always going to benefit yourself on an individual level but then, on a team level of course, you're not. It’s not so much that you can use it to inform your decision making, it’s perhaps just being aware of what’s going on and it’s a way of modeling how we make decisions as rational decision makers.
Georgia - I see. So it’s you look at what you’ve done and this is a way to understand why. I know in nature game theory is used all the time to explain why animals do certain things the way the do so I guess it’s a really useful tool in that way.
Tom - Exactly. It’s just the idea of thinking things on an individual level and on a team level, there are different options there.
Georgia - Thank you, Tom for showing us the delights of game theory then. And thank you Peter - I’m very sorry I’ve put you in prison for 7 years.