Re: Can anyone Solve this???

Posted by Amit Goyal on Oct 02, 2013; 8:44am
URL: http://discussion-forum.276.s1.nabble.com/Can-anyone-Solve-this-tp7584050p7584056.html

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.