Economic and Game Theory
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ďA boy was hit by a car and is lying on the road, hurt. The driver ran away and 5 people gathered around the boy. The boy needs immediate medical care but for that to happen it is necessary that someone calls 911. Simultaneously and independently, each of the individuals decides if he/she will or not call for help.
If no doctor comes to rescue the boy, each of the individuals will feel guilt which will represent to him/her a utility of -15. Someone who asks for help will pay a cost that represents a utility of -5.
a) This game has a Nash equilibrium with pure-strategies? In case of an affirmative answer, show it/them.
b) This game has any Nash equilibrium with mixed-strategies in which every person has the same probability of calling 911? In case of an affirmative answer, show it. (Suggestion: define p as the probability of someone NOT calling 911).Ē
Concerning the first question, I find it very easy to answer IF there were only 2 bystanders and not 5. Iíll present the normal-form representation of the game for 2 players:
S1= (Call; NotCall)
S2= (Call; NotCall)
Payoff Function of Player 1:
U1 (Call,Call) = -5
U1 (Call, NotCall) = -5
U1 (NotCall, Call) = 0
U1 (NotCall, NotCall) = -15
Payoff Function of Player 2:
U2 (Call,Call) = -5
U2 (Call, NotCall) = 0
U2 (NotCall, Call) = -5
U2 (NotCall, NotCall) = -15
Please note that I considered the possibility of both players call 911 at the same time, therefore both pay the cost. I donít know if thatís correct.
Is there a Nash equilibrium with pure strategies? I believe there isnít (in this 2-players game). This how I reached this conclusion:
If Player 1 decides to Call, Player 2 decides to NotCall.
If Player 1 decides to NotCall, Player 2 decides to Call.
Therefore, there isnít any best strategy from Player 2 to Player 1ís strategies. Which means that there isnít a Nash equilibrium with pure strategies.
Of course, since I assumed there were only 2 players, the outcome of the game with 5 players might be completely opposite to what I believe.
Now the second question gets trickier. Iíve already tried to solve it by using the initial assumption (that there are only 2 players) but it doesnít seem to work out:
(1-p, p) is the mixed strategy in which Player 1 chooses to Call with the probability of 1-p and (1-r, r) is the mixed strategy in which Player 2 chooses to Call with the probability of 1-r.
If Player 1 plays (1-p, p) then Player 2ís expected payoffs are (1-p)(-5) + p(-5) = -5
Now the problem is that the result should be a function of p. Shouldnít it be?