ECN 614 An Introduction to Game Theory Problem Set 1
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ECN 614 An Introduction to Game Theory
Problem Set 1 (Due on February 24 (Friday), 11:59 pm)
You are encouraged to work in group on homework problems, but you must work through and write up the assignments entirely on your own, after destroying all the written records of the group discussion.
Please scan and send your solutions to [email protected] in a SIN- GLE PDF file. Make sure your scanned homework is in a good quality.
Problem 1. Consider the following extensive form game. Notice that player 2 has two information sets.
(a) By converting the game into normal form game, find all Nash equilibria in pure strategies.
(b) Does player 2 have a pure strategy that is strictly dominated by a mixed strategy?
Problem 2. Consider the given strategic form game.
(a) For player 1, can her strategy D be a best response against a mixed strategy σ2 of player 2?
(b) Is M2 strictly dominated by a mixed strategy for player 2?
(c) What would the result of the iterative elimination of strictly dominated strategies?
Problem 3 (Watson 10.9). Consider the strategic voting game discussed at the end of this chapter, where we saw that the strategy profile (Bustamante, Schwarzenegger, Schwarzenegger) is a Nash equilibrium of the game. Show that (Bustamante, Schwarzeneg- ger, Schwarzenegger) is, in fact , the only rationalizable strategy profile. Do this by first considering the dominated strategies of player L . (Basically, the question is asking you to find the outcome of the iterative elimination of strictly dominated strategies) .
Problem 4. Consider the location game we covered in Lecture 3. Now assume there are three players (vendors) . As we assumed in the lecture, consumers in each area choose the closest vendor and if there are multiple closest vendors then these vendors receive equal share of consumers in the area. Notice Si = {1, 2, 3, ...., 9} for i = 1, 2, 3 . Here are some examples of payoffs: u1 (1, 1, 1) = 3,u1 (1, 1, 9) = u2 (1, 1, 9) = 2.25,u3 (1, 1, 9) = 4.5,u1 (1, 5, 9) = u3 (1, 5, 9) = 2.5 and u2 (1, 5, 9) = 4 .
(a) Is s1(/) = 1 strictly dominated by s1(//) = 2 for player 1?
(b) Is s1(/) = 1 weakly dominated by s1(//) = 2 for player 1?
(c) Can you find a Nash equilibrium in pure strategies?
Problem 5 (Watson 10.2 a-c). Consider a more general Bertrand model than the one presented in this chapter. Suppose there are n firms that simultaneously and inde- pendently select their prices, p1 ,p2 , ...,pn in a market. These prices are greater than or
equal to zero . The lowest price offered in the market is defined as p = min{p1 ,p2 , ...,pn } .
Consumers observe these prices and purchase only from the firm (or firms) charging p, according to the demand curve Q = a - p . That is, the firm with lowest price gets all of the sales. If the lowest prices is offered by more than one firm, then these firms equally share the quantity demanded. Assume that firms must supply the quantities demanded of them and that production takes at a cost of c per unit. That is, a firm producing qi units pays a cost cqi . Assume a > c > 0 .
(a) Represent this game in the normal form by describing the strategy spaces and payoff (profit) functions.
(b) Find the Nash equilibrium of this market game.
(c) Is the notion of a best response well defined for every belief that a firm could hold? Explain.
Problem 6. Consider the following game.
(a) Find all Nash equilibrium in mixed strategies by constructing the best response graphs. Find the Nash equilibrium payoffs.
(b) Can you find a mixed strategy profile (σ1 ,σ2 ) such that u1 (σ1 ,σ2 ) = 3 and u2 (σ1 ,σ2 ) = 3?
2023-02-25