Question

Two classmates in a game theory class collaborate on a problem set (incidentally, recall that collaboration on this exam is not allowed!). Each student i = 1, 2 invests an amount of effort e; > 0; the cost of effort for student i is ez. Both students receive the same grade: 41 +222 + eje2 (the effort of student 2 is more valuable because she's better at calculus). Hence, the payoff function of student i is given by v;(e1,e2) = 21 +222 +€162 - 6 (a) What's the best-response of student 1 if student 2 invests ez? What's the best- response of student 2 if student 1 invests ei? (b) Find the Nash equilibrium of this game. (c) Professor Manea recommends that the students choose ei = = 2. Does this outcome Pareto dominate the Nash equilibrium (i.e., is it better for both students)?

Answer 1

ule, c)-e, +2e, tee- en do It ez-zero leie- lte 2 Sinnlaly a Ite, 2 soluj e- 1 + {1+ lei-1) 4162 - 1] ule, - 1+ 2+ 1-3 2tite, 4 2 b) =) 4 (-) 3 c) el 2 ge=2, 252) = 2+212) + 4 - 4 Yes, if Ge =2 then both can be made better off but there might be coordination problion. ble of incentive of cheat