A few days ago we heard the story of how a waitress rescued an 86 year old lady who been stuck in her bath for four days. The waitress contacted the police after becoming concerned that Doreen had not come in for her usual lunch and wine. A story with a happy ending.
A story with a not so happy ending is the infamous murder of Kitty Genovese in 1964 in New York. This murder caught the public's attention because of the supposed number of witnesses who did nothing to stop the crime. The exact details of what happened are debated. One thing is, however, for certain: Several people must have seen or heard the attack and none of them called the police.
To try and make sense of these conflicting stories let us look at simple game theoretic model. Suppose that there is someone called Doreen that needs rescuing and there are n witnesses who can call the police. If (at least) one person calls the police then Doreen is rescued and all the witnesses feel relief equal to B. Calling the police incurs a cost of c < B. Note that calling the police is a public good because everyone, not just the caller, benefits.
If n = 1 and so there is only one witness then it is a simple decision. The witness should call the police because the benefit of doing so exceeds the cost. So, Doreen is saved!
If n > 1 and so there are multiple witnesses, then things become trickier because of a free-rider problem. In short, each witness would prefer that another witness calls the police. That way they get benefit B without incurring the cost c. If we assume that all of the witnesses think alike then we need to look for a symmetric Nash equilibrium where each witness independently calls the police with some probability p. In equilibrium we require p to be at a level where all the witnesses are indifferent between calling or not calling the police. (To see why we need this indifference, suppose witnesses prefer calling. Then everyone will call. But this cannot be an equilibrium because we only need one person to call. Similarly, if witnesses prefer not calling then nobody calls. But this cannot be an equilibrium because each witness would want to rescue Doreen.) Let us look at the incentives of a typical witness called Sonia.
If Sonia calls then she is guaranteed payoff B - c because Doreen is rescued.
If Sonia does not call the police then she avoids cost c but relies on someone else calling. There are n - 1 other witnesses and so the probability that none of them call is (1 – p)n – 1 . This means the probability that at least one calls is 1 – (1 – p)n – 1 . The expected payoff from not calling is, therefore, B(1 – (1 – p)n – 1).
Equating the payoff from calling with the payoff from not calling we get an equilibrium probability of calling:
Unsurprisingly, this probability is decreasing in n. In other words the more witnesses there are then the lower the probability that any one witness will call. The following graph illustrates what happens when c/B = 0.1.
The really crucial question, though, is what happens to the overall probability of someone calling. What chance does Doreen have of being rescued? The probability that at least one person calls is:
This is also decreasing in n. So, the more witnesses there are the less likely it is that Doreen gets saved! The following graph plots the probability of her being saved when c/B = 0.1.
To many this seems like a counter-intuitive result. It shows, though, the dangers of free-riding. In terms of producing public goods, more is not necessarily better.