Can anyone Solve this???

[vandita24x7]

There are two individuals, 1 and,2. Each individual has an initial endowment of 30. There is a

machine with the following property : Should individuals 1 and 2 provide respectively

endowments x1, and x2, to the machine, the machine first computes the aggregate contribution,

x1 +x2 . This done, the machine responds by providing each of the individuals fresh

endowments equal to 5(x1+ x2)^1/2. Thus the utility of individual i from the contribution profile

{x1, x2) is Ui{x1, x2) = 5(x1 + x2)^1/2 + (30 - xi). Note also that the endowment given to the

machine by individual i, xi. cannot exceed her initial endowment of 30. 

 Which of the following contribution profiles (x1, x2)maximizes the sum of the utilities of

the two individuals, U1(x1 ,x2) +U2(x1,x2) ?

(a) (30, 30) "

(b) (15, 15)

(c) (0, 25)

(d) None of the above

 Suppose the individuals make their . respective contributions, x1 and x2

simultaneously. This means that when an individual chooses her contribution level, she

is unaware of the contribution level. chosen by the other person. For this

simultaneous-move game, which of the following contribution profiles constitute a Nash

equilibrium?

(a) (15, 15)

(b) (0, 25)

' {c) (25 /8, 25 /8)

(d) None of the above

[Prerna Rakheja]

Hi Vandita,

Answer is part (c) for both the questions.

[vandita24x7]

pls solve here and explain

On Wed, Oct 2, 2013 at 12:06 AM, Prerna Rakheja [via Discussion forum] <[hidden email]> wrote:

Hi Vandita,

Answers is part (c) for both the questions.

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[vandita24x7]

Hi Prerna I want to verify my solution wid urs.. kindly explain the solution in detail. 

Regards!

On Wed, Oct 2, 2013 at 9:37 AM, vandita mishra <[hidden email]> wrote:

pls solve here and explain

On Wed, Oct 2, 2013 at 12:06 AM, Prerna Rakheja [via Discussion forum] <[hidden email]> wrote:

Hi Vandita,

Answers is part (c) for both the questions.

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[Amit Goyal]

Hi Vandita,

For the first one, solve the following problem:
Maximize w.r.t x,
10 (x)^(1/2) + 60 - x
subject to
0 ≤ x ≤ 60

where x is the sum of contributions.

Solving the above problem we get, x = 25.

Hence, the contribution vector (0, 25) maximizes the sum of utilities.

For the second one, solve the following problem (taking y as given):
Maximize w.r.t x,
5 (x + y)^(1/2) + 30 - x
subject to
0 ≤ x ≤ 30

where x is the contribution of player 1 and y is the contribution of player 2.

Solving the above problem we get the best response function of player 1, x = max{0, 25/4 - y}.
By symmetry, the best response function of player 2, y = max{0, 25/4 - x}.

Hence, set of all Nash equilibria is {(x, y)| x ≥ 0, y ≥ 0, x + y = 25/4}
Clearly, (25/8, 25/8) is one of the Nash equilibrium.

[vandita24x7]

This is exactly how I solved . however I was confused since in the options explicit values of x1 and x2 is given. So i could not be sure with it since the solution obtained is implicit. 

Thank you :)

On Wed, Oct 2, 2013 at 2:16 PM, Amit Goyal [via Discussion forum] <[hidden email]> wrote:

Hi Vandita,

For the first one, solve the following problem:
Maximize w.r.t x,
10 (x)^(1/2) + 60 - x
subject to
0 ≤ x ≤ 60

where x is the sum of contributions.

Solving the above problem we get, x = 25.

Hence, the contribution vector (0, 25) maximizes the sum of utilities.

For the second one, solve the following problem (taking y as given):
Maximize w.r.t x,
5 (x + y)^(1/2) + 30 - x
subject to
0 ≤ x ≤ 30

where x is the contribution of player 1 and y is the contribution of player 2.

Solving the above problem we get the best response function of player 1, x = max{0, 25/4 - y}.
By symmetry, the best response function of player 2, y = max{0, 25/4 - x}.

Hence, set of all Nash equilibria is {(x, y)| x ≥ 0, y ≥ 0, x + y = 25/4}
Clearly, (25/8, 25/8) is one of the Nash equilibrium.

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