Eco 290E: Game Theory (Winter, 2014-15)
Midterm Exam: Due on March 9
1. Basic Concepts (10 points)
Explain the difference between “dominant strategy” and “dominated strategy”. 2. Static Game (30 points)
Ann and Bob are in an Italian restaurant, and the owner offers them a free 3- slice pizza under the following condition. Ann and Bob must simultaneously and independently announce how many slice(s) she/he would like: Let a and b be the amount of pizza requested by Ann and Bob, respectively (you can assume that a and b are integer numbers between 1 and 3). If a + b ≤ 3, then each player gets her/his requested demands (and the owner eats any leftover slices). If a + b > 3, then both players get nothing. Assume that each players payoff is equal to the number of slices of pizza; that is, the more the better.
(a) Draw a payoff matrix of the game described by the above story. (b) Is there a dominant strategy? If not, explain the reason.
(c) Derive all the pure-strategy Nash equilibria. 3. Continuous Game (30 points)
Two neighboring homeowners, 1 and 2, simultaneously choose how many hours to spend maintaining a beautiful lawn (denoted by l1 and l2). Since the appearance of one’s property depends in part on the beauty of the surrounding neighborhood, homeowner’s benefit is increasing in the hours that neighbor spends on his own lawn. Suppose that 1’s payoff is expressed by
6 − l1+ l2 2
l1 and 2’s payoff is symmetrically expressed by
6 − l2+l1 2
l2. Then, answer the following questions.
(a) Derive the best response function for homeowner 1, BR1(l2).
Hint: Recall how we have derived the best response functions in Cournot model or Bertrand model with product differentiation. You can solve this question in a similar way.
(b) Graph the best response functions of both players (taking l1 on horizontal axis and l2 on vertical axis), and show the Nash equilibrium on your figure. (c) Compute the Nash equilibrium.
4. Mixed-Strategy Equilibrium (30 points)
Three Firms (1, 2 and 3) put three items on the market and can advertise these products either on morning (= M ) or evening TV (= E). A firm advertises exactly once per day. If more than one firm advertises at the same time, their profits become 0. If exactly one firm advertises in the morning, its profit is 1; if exactly one firm advertises in the evening, its profit is 2. Firms must make their daily advertising decisions simultaneously.
(a) Derive all pure strategy Nash equilibria.
(b) Show that the following type of Nash equilibria does NOT exist: One firm chooses pure strategy M , and other two firms use mixed strategies.
(c) Derive a symmetric mixed strategy Nash equilibria. You may assume that each firm chooses M with probability p and E with probability 1 − p, then calculate an equilibrium probability, p.
5. Focal Point (5 points, bonus!)
Choose one course offered in GRIPS in the winter term, and write down the name (do NOT write more than one names!). If the course you choose becomes the most popular answer, you would get 5 points. Otherwise, you would get 0 point. You need not explain the reason why you choose that course.