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1 4. Game Theory Midterm I Instructions. This is an open book exam; you can use any written material. You have one hour and 0 minutes. Each question is 35 points. Good luck!. Consider the following game in extensive form. L R l r l r,, 0,3 l, r,4 (a) Apply backwards induction in this game. State the rationality/knowledge assumptions necessary for each step in this process. The backwards induction outcome is as below. We first eliminate action r for player, by assuming that player is sequentially rational and hence will not play r, which is conditionally dominated by l.we also eliminate action r for player, assuming that player is sequentially rational. Thisisbecauserisconditionally dominated by l. Second, assuming that player is sequentially rational and that player knows that player is sequentially rational, weeliminater. This is because, knowing that player is sequentially rational, player would know that will not play r, and hence r wouldleadtopayoff of. Being sequentially rational shemustplayl. Finally, assuming that (i) player is sequentially rational, (ii) player knows that player is sequentially rational, and (iii) player knows that player knows that player is sequentially rational, we eliminate R. This is because (ii) and (iii) lead player to conclude that will play l and l, and thus by (i) he plays L.

2 L R l r l r,, 0,3 l, r,4 (b) Write this game in normal-form. Each player has 4 strategies (named by the actions to be chosen). l l l r r l r r Ll,,,, Lr,,,, Rl 0,3, 0,3, Rr 0,3,4 0,3,4 (c) Find all the rationalizable strategies in this game use the normal form. State the rationality/knowledge assumptions necessary for each elimination. First, Rr is strictly dominated by the mixed strategy that puts probability.5 on each of Ll and Rl. Assuming that player is rational, weconcludethathewould not play Rr. We eliminate Rr, so the game is reduced to l l l r r l r r Ll,,,,. Lr,,,, Rl 0,3, 0,3, Now r r is strictly dominated by l l. Hence, assuming that (i) player is rational, andthat(ii)player knows that player is rational, weeliminater r. This is because, by (ii), knows that will not play Rr, and hence by (i) she would not play r r. The game is reduced to l l l r r l Ll,,,. Lr,,, Rl 0,3, 0,3 There is no strictly dominated strategy in the remaining game. Therefore, the all the remaining strategies are rationalizable.

3 (d) Comparing your answers to parts (a) and (c), briefly discuss whether or how the rationality assumptions for backwards induction and rationalizability differ. Backwards induction gives us a much sharper prediction compared to that of rationalizability. This is because the notion of sequential rationality is much stronger than rationality itself. (e) Find all the Nash equilibria in this game. The only Nash equilibria are the strategy profiles in which player mixes between the strategies Ll andlr,andmixesbetweenl l and l r,playingl l with higher probability: NE = {(σ, σ ) σ (Ll)+σ (Lr) =, σ (l l )+σ (l r )=, σ (l r ) /}. (If you found the pure strategy equilibria (namely, (Ll,l l )and(lr,l l )), you will get most of the points.). Consider two players A and B, who own a firm and want to dissolve their partnership. Each owns half of the firm. The value of the firm for players A and B are v A and v B, respectively, where v A >v B > 0. Player A sets a price p forhalfofthefirm. Player B then decides whether to sell his share or to buy A s share at this price, p. If B decides to sell his share, then A owns the firm and pays p to B, yielding playoffs v A p and p for players A and B, respectively. If B decides to buy, then B owns the firm and pays p to A, yielding playoffs p and v B p for players A and B, respectively. All these are common knowledge. Find the subgame-perfect equilibrium of this game. Given any price p, the best response of B will be buy if v B p>p, i.e., if p<v B /; sell if p>v B /; {buy, sell} if p = v B /. In equilibrium, B must be selling at price p = v B /. This is because, if he were buying, then the payoff of A as a function of p would be ½ p if p v B /; v A p if p>v B /, which can be depicted as in Figure. Then, no price could maximize the payoff of A, inconsistent with equilibrium (where A maximizes his payoff given what he anticipates). Hence, the equilibrium strategy of B must be ½ buy if p<vb /; sell if p v B /. In that case, the payoff of A as a function of p would be ½ p if p<v B /; v A p if p v B /, which can be depicted as in Figure. 3

4 U A (p) v B p Figure : U A (p) v B p This function is maximized at p = v B /. A sets the price as p = v B /. 3. Two start ups are competing for leadership in a software market. The leader wins, and the other loses. Each firm can invest some x [0.00, ] unit for research and development by paying cost of x/4. Ifafirm invests x units and the other firm invests y units, the former wins with probability x/ (x + y). Therefore, the payoff of the former start up will be x x + y x/4. All these are common knowledge. (a) Compute all pure strategy Nash equilibria. Call them as Firm and Firm. Firm maximizes x x + y x/4 4

5 over x, and Firm maximizes y x + y y/4 over y. The best response function of Firm as a function of y is given by 0 = µ x x x + y x/4 = µ y x x + y x/4 = y (x + y) /4, i.e., x (y) = y y. Similarly, the best response function of Firm is y (x) = x x. Note that x (y) >ywhenever y<. Therefore, the graphs of x and y intersect each other only at x = y = as shown in the figure below. Therefore, (,) is the only Nash equilibrium. (b) Compute all rationalizable strategies. (,) is the only rationalizable strategy profile. Since y y , then any strategy x<x (y 0 ) is strictly dominated by x = x (y 0 ),andtherefore eliminated. Write also x 0 = y 0 and x = y. Now, the remaining strategy space of each player is [x, ]. Note that x = x (.00) > 0.00 = x 0. Now, similarly, we can eliminate any strategy x<x x (y ). Applying this iteratively, after nth elimination we are left with a strategy space [x n, ] where x n = x n x n and x 0 =.00. It is clear from the figure that x n as n. Hence in the limit we are left with strategy space {}. 5

6 You do not need to do this: More formally, Hence, x n = x n x n > x n = x / n. >x n >x (/)n 0. Of course, as n, (/) n 0, and hence x (/)n 0. Therefore, x n. 6