Game Theory and the Bystander Effect

Perhaps the most famous illustration of the Bystander Effect is the story of New Yorker Kitty Genovese who was brutally killed in Kew Gardens, Queens. She screamed through the attack for more than half an hour and no one came to her aid or called the police. Later investigations showed that 38 of her neighbours had watched the episode from their windows and did nothing.

Social psychologists offer 3 explanations for the bystander effect:
1) Ambiguity: Witnesses reports revealed some were not sure what was happening.
2) Pluralisitc Ignorance: Everyone was uncertain whether there really was an emergency and looked to others for clues as to how they should react, but all felt the same way, so nobody took action.
3) Diffusion of responsibility: Each observer knew that others could respond to the emergency, and that he/she need not be the one to do something.

In Games of Strategy (2nd ed.), Dixit and Skeath offer a game-theoretic explanation for the bystander effect that is remarkably similar to the diffusion of responsibility. They make a reasonable assumption that everyone would gain personal pleasure if Kitty Genovese was rescued, but each must balance that pleasure against the cost of getting involved. That is, each would like to free-ride on another's effort.

Their explanation is as follows:
There are N pple in the group, action brings each of them benefit B, and the person who take action has to bear the cost C. It is assumed that B>C.
So we can see that the person who takes action gets (B-C) while those who don't get B payoff.
If N=1, we don't have a problem since B-C > 0 and therefore he will help.
But if N>1, there is no pure strategy Nash equilibrium of everyone helping or no one helping bcos each would do better to free-ride or help respectively.

Instead, they propose that there is a mixed strategy equilibrium:
Let P be the prob that any one person will not act. To mix strategies, the person must be indifferent between helping or not helping.

Ok, now it will get really messy if i carry on to write the equations. Basically, find the expected payoff of the one person when he does not act and equate that to the payoff when he does act i.e. B-C (known as indifference principle). Next make P the subject of the formula, and by increasing N from 2 to infinity, P increases from C/B to 1. The probability that any one person will act (1-P) therefore falls from (1- C/B) to 0.

Next find the probablity that not even one person helps, which is P power N. Plugging in the previous result for P, and doing same thing again, we can see that the probablity that at least one person helps decreases as N increases.

Whew, this is really tiring, but I think this is a good example of how economics or more specifically game theory, helps to sharpen the social psychological theory of the bystander effect.
I think of this method as a mathematical proof of the diffusion of responsiblity!

I've tried my best to make this as simple as possible. For a much more succint and neater version, pls refer to Games of Strategy 2nd ed. pp 414-418.