IMAGE OF ORIGINAL QUESTION ABOVE Recall the setup of the “volunteer’s dilemma” f
ID: 3053493 • Letter: I
Question
IMAGE OF ORIGINAL QUESTION ABOVE
Recall the setup of the “volunteer’s dilemma” from Section 3.4.2 of the textbook.
In this game, players can pay a cost to generate a public good that everyone benefits from. If one or more people pay the cost, everyone receives a value of 1. (They receive 1 regardless of how many people volunteer as long as at least one does.) Those who volunteered also pay a cost c ? ? (0, 1). If no one volunteers, everyone receives 0.
a) Suppose there are n players. Let ?N represent the probability that a player does not volunteer. Further, suppose that ?N is identical for all players. As a function of n, what is any single player’s expected utility for not volunteering?
b) What is any single player’s expected utility for volunteering?
c) As a function of n, for what value of ?N is a player indifferent between volunteering and not volunteering?
d) What happens to the probability of no one volunteering as n increases? If you were the man- ager of a business, what does this say about assigning someone in particular to do a task versus requiring that a group complete the task?
Ignore Call Ignore 0,0 1, 1-c Call 1-c,1 1-c, 1-c Recall the setup of the "volunteer's dilemma" from Section 3.4.2 of the textbook. In this game, players can pay a cost to generate a public good that everyone benefits from. If one or more people pay the cost, everyone receives a value of 1. (They receive 1 regardless of how many people volunteer as long as at least one does.) Those who volunteered also pay a cost c E (0, 1). If no one volunteers, everyone receives 0 Original game Ignore sall lgnoreCall 0,0 1-c, 1 1,1-c 1-c, 1-c a) Suppose there are n players. Let Oy represent the probability that a player does not volunteer. Further, suppose that ?N is identical for all players. As a function of n, what is any single player's expected utility for not volunteering? b) What is any single player's expected utility for volunteering? c) As a function of n, for what value of ?N is a player indifferent between volunteering and not volunteering? d) What happens to the probablity of no one voluntering as n increases? f you were the man- ager of a business, what does this say about assigning someone in particular to do a task versus requiring that a group complete the task?Explanation / Answer
(a)The expected utility can be calculated using the same general formula that is used to calculate expected value. Instead of multiplying probabilities and values, you multiply probabilities and utility amounts. That is, the expected utility (EU) equals probability x amount of utiles derived.
So, the expected utility of a single player for not volunteering is
SigmaN*0 +SigmaN*1 = SigmaN
(b) The expected utility of a single player volunteering is
(1-SigmaN)*(1-c) + (1-SigmaN)(1-c) = 2(1-SigmaN)(1-c)
(c) The player will be indifferent between volunteering and not volunteering when the expected utility from both the strategies will be same.
This happens when SigmaN = = 2(1-SigmaN)(1-c)
SigmaN/(1-SigmaN) = 2(1-c)
(1-SigmaN)/SigmaN = 1/[2(1-c)]
1/SigmaN - 1 = 1/[2(1-c)]
1/SigmaN = 1/[2(1-c)] + 1
1/SigmaN = [1+[2(1-c)]]/[2(1-c)]
SigmaN = [2(1-c)] / {1+[2(1-c)]}
This when the probability of not volunteering is equal to the above expression then the player is indifferent between volunteering and not volunteering. Here, c is the cost of volunteering.
(d) An n increases the probability of not volunteering will also decrease. Since the number of participants increases it will be assumed that some or the other will volunteer However the probability of not volunteering could also increase as the players assume that some or the other will volunteer and so the players will try to avoid the cost of volunteering so as to get a total benefit of 1.
This shows that assignments of a task to a group of people will ensure that the tasks will be completed with a greater probability since there are larger number of participants in a group so it will be assumed that some or the other person will volunteer to do the work.
However there is also a chance that the group might assume that “some other member” will do the work and hence they will try to avoid the efforts of doing the task. This will simply depend on the selection of people in the group.