I need someone to do a favour. It’ll not take up much of their time, but if no one does it, then everyone will suffer. So if I email lots of people, I am bound to get a positive response, right?
It’s a nice idea, but unless the rewards are going to be amazing for everyone, asking one or two people might be your best bet.
We’ll suppose that if *someone* volunteers, then everyone gets £10, but the cost to that individual is £1, meaning that they only get £9.
This means that if I volunteer, I’ll be guaranteed to get £9, but if I don’t, I will either get £10 or £0.
If I am certain no-one else will volunteer, I will be guaranteed to get £0 if I do not volunteer. This means I really should volunteer if I want any reward at all. However, if I am certain someone else will volunteer, volunteering myself will mean I get £1 less than what I should, so I lose £1 for no good reason. Not volunteering is the best solution here.
The problem arises when I am only 90% sure someone else will volunteer, as my expected return is the same whether I volunteer or not.
Here is where the paradox arises, because it turns out that the more people you have in your volunteering pool, the more likely it is that nobody will volunteer.
Let’s suppose, then, that John and I are certain we will get £20 in the pub quiz jackpot, but there is a £1 entry fee, and we only have £1 coins (and a communication issue which means we can’t talk about who puts the money in). As far as John is concerned, there’s a 90% chance I will put the pound in, and I regard John the same way. This means that the probability of neither of us putting the £1 in is 1%, so relatively unlikely.
If John’s girlfriend Sarah joins us and the jackpot is now £30 (to make the rewards the same), it changes slightly- as far as I am concerned there is a 90% chance of neither of them putting the money in and they all regard each other the same way. The probability of any individual not putting the money in is therefore higher at about 31%.
This means that the chances of no one putting the money in rises to 3%. Eventually, the more people that come, the closer this value gets to 10%, and it can’t get higher than that.
But, of course, as I mentioned at the start, if I am less than 90% certain someone else will put the money in, I should step up, but if I am more certain than that I will hold fire.
There are more serious adaptations than these- if lots of people witness a crime, there exists a probability that none of them will call the police (as they will have to spend time giving statements or will be a nuisance to the police etc.).
If you think about it, that’s actually quite a scary thought, because if means that everyone has either been made indifferent to whether or not they call the police, or that it would have been a detriment to them if they had done so.
So, if you want something done, it’s maybe best to ask a small group of people first. Better yet, you can do it yourself!
(Adapted from Game Theory-A Very Short Introduction by Ken Binmore)