Player 2 H T T -1,1 1, -1

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1 1 1 Question 1 Answer 1.1 Q1.a In a two-player matrix game, the process of iterated elimination of strictly dominated strategies will always lead to a pure-strategy Nash equilibrium. Answer: False, In many games (including games with multiple equilibria, games with mixed-strategy equilibria, and games with no strictly dominated strategies for either player), IESDS will not lead to PSNE. One specific example is matching Pennies - since no strategy for either player is strictly dominated, IESDS does not give any restriction on what strategies can be played. Player 1 Player H T H 1, -1-1, 1 T -1,1 1, Q1.b Every Strict Dominant Strategy Equilibrium is a Nash equilibrium. Answer: True, This can be proved directly or by contradiction. The proof by contradiction goes as follows: Suppose, the statement was false, i.e., there exists a game with a Strict Dominant Strategy Equilibrium that is not a Nash Equilibrium. Let s denote this DSE. Since s is not a Nash Equilibrium, there is some i and some s i such that: U i ( s i, s i ) < U i (s i, s i ) But then s i is not a strictly Dominant Strategy, contradicting the assumption that s i is played as part of a Dominant Strategy Equilibrium.

2 Question Answer You are going to take part in the coming Inter-IIT 017 sports meet as a sprinter. In practice today, you fell down and hurt your ankle. Based on the results of an x-ray, the doctor thinks that its broken with probability 0.. The question is, should you take part in the race in this coming inter-iit games? If you run, you think you ll win with probability 0.1. If your leg is broken and you run on it, then you ll damage it further. So, your utilities are as follows: if you win the race and your leg isn t broken, +100; if you win and your leg is broken, +50; if you lose and your leg isn t broken 0; if you lose and your leg is broken -50. If you don t run, then if your leg is broken your utility is -10, and if it isn t, it s 0. 1.Draw the decision tree for this problem. Answer: Figure 1: Decision tree for the runner s choice. Evaluate the tree, indicating the best action choice and its expected utility. Answer: U(Run)= 0and U(Don t Run)= - You might be able to gather some more information about the state of your

3 3 leg by having more tests. You might also be able to gather more information about whether youll win the race by talking to your coach or the TV sports commentators. 3. Compute the expected value of perfect information about the state of your leg. Answer: Figure : Decision tree given perfect information about the leg Given the perfect information about the leg, we have the tree in figure, so the expected value of the information is E(U info ) E(U no info ) = 6 0 = 6 4. Compute the expected value of perfect information about whether youll win the race. Answer: Given the perfect information about winning, we have the tree in figure 3, so the expected value of the information is E(U info ) E(U no info ) = 7. 0 = 7. 3 Question 3 Answer Compute the pure-strategy Nash equilibria in the following linear Cournot oligopoly game for any number of n firms: each firm has marginal cost c (0, 1)

4 4 Figure 3: Decision tree given perfect information about winning and a fixed cost F > 0, which it needs to incur only if it produces a positive amount; the inverse-demand function is given by p(q) = max1 Q, 0 where Q is the total supply given by Q = m i=1 q i and m n. [Hints: recollect that Cournot oligopoly game is a quantity competition game and in a Nash equilibrium, some firms may not produce anything due to the presence of fixed cost.] Answer: Suppose that m n firms produce some positive quantity and the remaining firms produce 0. For any i with positive production q i * q i * = 1 c j i q j *

5 5 As in the usual Cournot model above, the uniqe solution to this equation system is q i * = 1 c m + 1 for each firm with positive production. The payoff is Π * = ( ) 1 c F m + 1 Of course, the firm also has the option of not producing at all and obtaining zero profit. Hence,Π * 0, i.e., m 1 c F 1 m * Moreover, if m < n the firm that produce 0 must not profit from deviation to positive production: ( ) 1 c mq * F 0 this simplifies to m m* 1. Hence, the set of Nash equilibria is as follows. For any integer m [m m* 1,..., m * ] firms produce (1 c)/(m + 1) each, and the remaining firms produce 0.