Questions:

Partnership Game Two individuals, 1 and 2, are going to supply input to a joint project They share the output (profits) equally O s ers 4: effort level chosen by i=1,2 Profit: 1 = 4 (e1+ez+ b*el*ez), where Os bs 14 b: a constant; indicates the level of complementarity, synergy between the two individuals (extra benefit you get from working with the other person) If they both devote more effort, they are both better off The cost of effort for player i: e Payoffs: UI(e1, ez) = 12 [4(entez+b*er*ez)] - e12 uz(e1, ez) = 12 [4(entez+b*er*ez)] - ezz Best response functions: Player 1: We need to maximize un(e1, ez) with respect to Player 1's choice variable, namely e1. Note that this optimization problem takes Player 2's choice variable, namely ez, as constant. This is because although the payoff to Player 1 depends on ez, from Player 1's point of view ez is exogenously determined (by Player 2's problem), hence Player 1 takes it as given. Max UI(e1, ez) = 12 [4(entez+b*en*ez)] - e12 with respect to e1, given ez FOC: dui(e, ez) /den = 0(1/2 [4(e1+ez+b*en"ez)] - ex?)/de1=0 2(1+b*ez) - 2e1=0 el = 1 + b*ez > BRI(ez) = 1 + bez (Best Response Function of Player 1) The Best Response function of Player 1, BRi(ez), tells us the payoff maximizing level of e, for a given level of ez. (Note that, since the game is a simultaneous move game, the "given levels" will refer to the beliefs of each player about the level of effort that the other player would choose.) Player 2: The arguments for Player 1 apply for Player 2 as well. Hence we get: Max uz(e1, ez) = 1/2 [4(en+ez+b*el*ez)] - ez? with respect to ez, givene, FOC: duz(e1, ez) /0 ez= 0(1/2 [4(e1+ez+b*e1"ez)] - ez?)/dez=0 2(1+b*e1) - 2ez=0 ez = 1 + b*e, > BRz(e1) = 1 + ben (Best Response Function of Player 2) Nash Equilibrium (NE):We know that at the NE, both players will be playing best responses to each other's action. So we need to find the levels of en and ez that solve both best response functions simultaneously. e1 = 1 + b*ez & ez = 1+ b*el In order to solve these two equations simultaneously, plug in the equation for e, into that for ez": ez = 1 + b*el (BR function of Player 2) } ez = 1 + b*(1 + bez) (Plugged in the BR function of Player 1 to "er" in the BR func of Player 2)+ ez = 1 + b + b-ez (1 - b4) *e2 = 1 + b ez = (1 + b) / (1 - b') = (1+ b] / [(1 -b)(1 + b)] ez = 1 / (1- b) By plugging in this value for the level of ez in BR function of Player 1 we get the NE level of ex: e1 = 1 + b*ez = 1 + b*(1/(1 - b]) = 1/ (1-b) Then the NE is given by: (1 / (1 - b), 1 / (1 -b)) NOTE: Since the game is symmetric (players' payoff functions have the same functional form and they have same strategy sets), we know that at the NE players would choose the same levels of effort. We could as well use this observation to substitute e, for ez, and vice versa, in the best response functions and solve for the NE levels. Don't forget: This short-cut method works only if the game is symmetric! Graphically, the NE will be at the intersection of the BR functions. We need to assume a value for b in order to draw the best response functions. Assume b= 14.How do we solve this game? This is a sequential move game with complete information > Backward Induction! But now the strategies are continuous: 0 BR,(q1)= (a-c)/2b - q,/2 .Firm 1's decision problem: max Profit, = q1 x (P - MC) subject to BR2(q1) Profit, = q1 X (a - bq1 - bq2 - c) Profit, = q1 x (a - bq, - b[BR2(q1)]-c) BR2(q1) FOC: a Profit,/2q1 = (a - c)/2 - bq1 = 0 7 qis = (a -c)/2b 7 925 = BR2(q,s) = (a - c)/4b > Firm 1 is producing more > higher profit! 91 "First mover advantage"1. [20 points] In a two-player game, let O be a mixed strategy of Player 1, and R be a mixed strategy of Player 2. Both Q and R may assign (+) probabilities to one or more pure strategies available to Player 1 and Player 2, respectively. Suppose Player 1's strategy set is {A, B, C}, and that of Player 2 is {D, E, F, G, H} . Define Q* = (1/3, 0, 2/3), and R* = (1/5, 0, 0, 2/5, 2/5). Choose the appropriate phrase to fill in the blanks in the following statements or whether (True/False), whichever applies. Briefly explain your reasoning. Answers without explanation will not be evaluated. and will get zero credit. a. If (Q*, R*) is a Nash Equilibrium, the payoff to Player 1 is maximized by playing Q* if Player 2 is playing R*. TRUE FALSE b. If (Q*, R*) is a Nash Equilibrium, whenever Player 2 is playing R*, the payoff to Player 1 from playing the mixed strategy Q'= (3/4, 0, 1/4) would be HIGHER THAN / LOWER THAN/ SAME AS his/her payoff at the NE, (Q*, R*). c. If (Q*, R*) is a Nash Equilibrium, given that Player 1 is playing Q*, Player 2's payoff from the pure strategy D would be HIGHER THAN/ LOWER THAN/ SAME AS her/his payoff from playing R*. d. If (* is a best-response to R*, the pure-strategy A IS / IS NOT a best-response for Player 1 against layer 2 playing R*.2. [30 points] Consider the Partnership Game discussed in class. (Refer to the lecture notes and the additional "Partnership Game" handout provided on Moodle.) Now rather than considering the Partnership Game as a simultaneous-move game, consider the perfect information sequential-move game version of the Partnership Game where one of the players, say Player 1, moves first and sets their effort level. Player 2 observes the choice of Player 1 and then chooses their effort level. The payoff functions and everything else about the interaction remains the same as the simultaneous-move version covered in class. a. Find the Backward Induction equilibrium. b. Compared to the Nash Equilibrium of the simultaneous-move game, is Player 1 better-off, worse- off, or equally well-off? What about Player 2? c. Is there a first-mover advantage or a second-mover advantage, or neither? Explain. 3. [30 points] (Tragedy of the Commons) Consider two fishers who fish in the same lake. They consume their catch and do not engage in any kind of exchange, nor do they make any agreements about how to pursue their economic activities. Yet the activities of each affect the payoff of the other: the more one fishes, the harder it is for the other to catch fish. The preferences the fishers are represented by the following utility functions: Fisher 1: 141 (e1, ez) = (1/2)(1 - 2ez)e, - ex Fisher 2: Uz(e1, ez) = (1/2)(1 - 2e,)ez - ez where e is the amount of time (fraction of a twenty-four-hour day) that fisher / spends fishing, i=1,2. Assume each fisher chooses their strategy, ex, without knowing the strategy chosen by the other. a. Find the best response functions of each fisher. b. Draw the BR functions. c. Find the Nash equilibrium. d. Is the Nash equilibrium Pareto efficient? Explain. 4. [20 points] For a two-player game, the payoff function for player 1 is VI(XI, X2) = XI + 10 X1 X2, and for player 2 is V2(X1, X2) = X2 + 20 X1 X2. The players' strategies are continuous. Player 1's strategy set (i.e. the values that X1 can take) is the interval [0,100], and player 2's strategy set (i.e. the values that x2 can take) is the interval [0,50]. Find all Nash equilibria. Hint: Recall our discussion about the intuition behind using the first-order-condition to get the max value of a function. What would it mean to have the first-order-derivative of the utility function to have a nonzero value