[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).
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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.