The Kitty Genovese Equation

Someone's in trouble. You can hear them from your apartment, but you can't tell if any of your neighbors are already rushing down, or already calling the police. It's time sensitive, and you've got to decide now: is it worth spending those precious minutes, or not?

Let's define our variables:

Cost to victim of nobody helping: $C$

cost to each bystander of intervening: $k<C$

Number of bystanders: $N>=2.$ (Since $k<C$, for $N=1$ it's always right to intervene.)

Analysis:

Suppose the bystanders all simultaneously decide whether to intervene or not, with probability p. Then expected world-utility is $U_{C,k,N}\left(p\right)=$$-C(1-p{)}^N-kpN$

Utility is maximized when $0=dU/dp=NC(1-p{)}^{N-1}-kN$ ; In other words, when $(1-p{)}^{N-1}=\frac{k}{C}$. Let $\alpha =\frac{k}{C}$ . Then we have the optimal probability of not helping, $(1-p)={\alpha }^{\frac{1}{N-1}}$ .

One interesting implication of our solution is that the probability that the victim isn't helped, $(1-p{)}^N$, equals ${\alpha }^{\frac{N}{N-1}}$. Since $\alpha <1$, this means P(not helped) starts small at ${\alpha }^2$ for $N=2$ and rapidly rises to $\alpha$.

Examples:

Suppose intervening would cost a minute, and the victim would live 2 years longer on average if you intervened. Then $k/C$ is about one in a million, ${10}^{-6}$. Once you get to seven bystanders, it's optimal to not intervene 10% of the time. $2^{20}$ is about a million, so with 21 bystanders it's optimal for each to take a 50-50 shot at helping.

If $k/C$ is a mere ${10}^{-1}$, you get there six times as fast: a 10% chance to not help at N=2, 50% around N = 4-5, and a whopping 75% chance around N=9.

Application:

This was inspired by friends' varied willingness to intervene in public disputes, and my own experience worrying about how to respond to potential crises around me. Of course, in real life we have a lot of uncertainty around $\alpha$ and around other people's $p$, and we can often wait and observe if someone goes to help. For situations where decisions are pretty simultaneous, though, it would be interesting to see how well people's responses line up with the ${\alpha }^{\frac{1}{N+1}}$ curve.