[Game Theory] A pedestrian is hit by a car. How many people will help?

[Dostre]

Homework Statement

Consider the following social problem. A pedestrian is hit by a car and lies injured on the road. There are $n$ people in the vicinity of the accident. The injured pedestrian requires immediate medical attention, which will be forthcoming if at least one of the $n$ people call for help. Simultaneously and independently, each of the $n$ bystanders decides whether or not to call for help (by dialing 911 on a cell phone or pay phone). Each bystander obtains $v$ units of utility if someone (anyone) calls for help. Those who call for help pay a personal cost of $c$ . That is, if person $i$ calls for help, then he obtains the payoff $v-c$. If person $i$ does not call but at least one other person calls, then person $i$ gets $v$. Finally, if none of the $n$ people calls for help, then person $i$ obtains 0. Assume $v>c$.

1. The purpose of this question is to find the symmetric Nash equilibrium of this $n$-player game. This equilibrium is in mixed strategies, i.e. such that each person is indifferent between his/her two possible strategies: to call or not to call. Therefore, each player’s payoff must be equal when he/she calls and when he/she does not call.

a. We already know that player $i$’s payoff is $v-c$ when he/she calls. Write the payoff of player $i$ when he/she does not call, letting $p$ be the probability that a person does not call for help. Hint: there are $n-1$ players others than player $i$. Therefore, with probability $p^{n-1}$, no one of the other players will call, and with probability $1-p^{n-1}$ at least one of the other players will call.​

b. By setting player $i$’s payoff equal when he/she calls and does not call, find the probability that a person does not call $p$ in equilibrium (Hint: this will be a function of $c/v$ and $n$).
​

2. Compute the probability that at least one person calls for help in equilibrium $1-p^n$.
How does this depend on n? Can you comment? (Hint: to answer the second part of the question you need to differentiate it with respect to $n$).

Homework Equations

The Attempt at a Solution

All I need is to figure out what the payoffs are, and I will be able to solve the rest. For part 1.a the payoff I came up with is $(1-p^{n-1})v+p^{n-1}(v-c)$, but I am leaning towards the fact that it is wrong. Help appreciated.

 

[D H]

Yep. It's wrong. There's a p^n-1 probability that no one else calls. What is the payoff in this event?

 

[Dostre]

If no one calls it is zero. Then, the payoff of the ith player is $p^{n-1}*0+(1-p^{n-1})v$ . So for part b would it be correct to say that $v-c=(1-p^{n-1})v$ ?

 

[D H]

Correct. That's your Nash equilibrium.

 

[BhargavS]

Hmm... he died in a car crash, how ironic.