Question

5. [20] N 3 team members can contribute efforts to a project. The value of the project depends on everyone's efforts, but each member bears her own cost of effort. In particular, letting e; be the effort of member i (measured in number of hours spent on the project in a day, so that 0 Sei < 24), the payoff of member i is Ti(C1, ..., en) = (e1e2...en) - ez. (a) (10) Find the symmetric Nash equilibrium effort level. What happens to the equilibrium effort level and the payoff as N oo? (b) (10) Find the (symmetric) socially optimal effort level, where social opti- mum is defined as maximizing the sum of the individual payoffs: Tilen...., en).

Answer 1

ñi = (C.e.. .eN) - e2 then foo NE, fin ding BR funch of each in dividual o solving them, BR:: Tre: -2ei %3D TTe ec I product of all e except for = %3D ec = e 2 Nowi Symmetoi c NE?- (e)N- ITej J+ i 2e. thus eN-2 2. to e*= (2)Ñ-2 (R fns So ler = N-2 dim exr Now . (a) eech effost level tends to 1 Ang as N-0,

b.) b-) Social payoff Z = Now Σπ: (9, .eN) - le²+ez²+ eg²t %3D eN) N(G. e2. %3D az 7 N ITej деr 201 NOW POC: =0. as 8ymmetoic Egm, N(ěNH then So 2e - 2-N- Ans 27N-2) es N- 2 %3D