## The bystander effect

On March 13^{th}, 1964, 3:00 AM. When New York bar manager Kitty Genovese arrived at her house, she was assaulted by a bandit with a knife. She screamed loudly to beg for help. There are 38 neighbors for her and a lot of people had witnessed this crime through the window. However, they did nothing until one of them call the police after the bandit left away. This incident turned out to be a tragedy: Genovese expired because she could not receive medical treatment in time.

What is the reason that no one give Kitty a hand? It is a common sense that people will blame these Neighbors as cold-blooded man. But there is a theory to explain this kind of circumstance: psychologists Shows that if there are more people present in a dangerous event, there shall be less possibility that one People would come out and provide help. This effect is called “bystander effect”. Next I will use the theory of Nash Equilibrium to prove it.

Let’s say the prerequisite is everyone tries to maximize their profit. And then we suppose that there are n bystanders in the incident. If someone call the police, every bystanders will earn a profit of a. (This could be understand as a relief that the victim is rescued and justice is served). For the people who provide the help, he may lose b points. (This could be understand as the extra time and energy cost when giving a hand, also doing this may put him exposed to the criminal). As a result, the total profit that the helper earned shall be a – b.Based on the information above, we could draw the profit grid:

Others don’t call police Others call police

A don’t call police 0 a

A call police a – b a – b

Suppose the probability that each people will not call the police is p, and then:

The expect of A not call the police shall be : Q1 = 0 * p ^{n-1} + a * ( 1 – p ^{n – 1}).

The expect of A call the police shall be : Q2 = (a – b) * p ^{n-1} + (a – b) * ( 1 – p ^{n – 1}).

As a result, the total expect of A should be : Q(p) = p * Q1 + (1-p) * Q2.

We expand this equation: Q(p) = a*p – a*p ^{n} + (1 – p) * (a – b).

Q(p) = a – b + p*b – a * p ^{n} . To get the maximum of Q(p), we need to know d(Q(p))/dp = 0.

So the equation turns out to be: b – a * n * p ^{n-1} = 0

As a result, the value p should be p = (b/(na)) ^{1/(n – 1)}

Based on this function, we could say that when n is larger than 2, p will grow when n grows.

As a result, the more people exists, p will be larger, which means the probability of lending a hand is smaller. Therefore, there will be less possibility that a person is willing to help when there are more people around.

This is the link to the original essay : http://www.guokr.com/article/6172/.