This one is my favourite:
"As the number of witnesses increases, the probability Kitty survives decreases".
But who is Kitty?
True Story:
Kitty Genovese (a 29-year old girl) was brutally murdered in New York in March 1964. The half hour murder was witnessed by 36 people, yet according to newspapers, no one responded to her screams for help or called the police.
Science-fiction author Harlan Ellison, referred to the witnesses as "thirty-six motherfuckers" and stated that they "stood by and watched" Genovese "get knifed to death right in front of them, and wouldn't make a move"
Kitty Genovese
Mathematical Scenario:
Let's assume there are n witnesses = players to a crime.
Suppose that if one of them reports the crime then the gain for each player is g. The cost in reporting the crime though is c. In other words, the payoff for a player is
- g if someone else reports the crime,
- g - c if he or she reports the crime,
- 0 otherwise.
We assume that g > c > 0.
Applying Symmetric mixed strategy game theory model:
We assume that the i-th player picks a probability p for NOT reporting the crime. We seek a Nash equilibrium where all these probabilities are the same, say equal to p.
For those who don't know what's Nash Equilibrium,
A Nash equilibrium of the game, is a collection of n actions (one for each player),where no player can do better by unilaterally changing their action.
Here, We interpret the Nash equilibrium value of p to be the probability
a witness does not report the crime.
Continuing the calculations,
Let's compute player i 's expected payos for reporting and not reporting.
Report: payo = g- c.
Don't report: payo = 0 x Prob(No one else reports) + g x Prob(Someone else reports) = g(1 - p^n-1)
The Nash equilibrium occurs when
g - c = g(1- p^n-1) ,
p = (c/g)^1/(n-1).
We fix c, g and see what happens as we vary n.
p = (c/g)^1/(n-1) increases as n increases i.e,
The more witnesses there are, the less likely each witness is to report the crime.
OR,
The more witnesses there are, the less likely the crime is reported.
OR,
"As the number of witnesses increases, the probability Kitty survives decreases".
Sources:
Game Theory 101: The Murder of Kitty Genovese, William Spaniel - Amazon.com