Sample Midterm Solutions

# Sample Midterm Solutions -     1A)      1B) ...

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Unformatted text preview:     1A)      1B)    1C)       The only pure strategy NE is (Hide,Take). We will learn later in the quarter how to  find Mixed strategy equilibria.    1D)           1E)    2)            3) Harrington 2.11    Only game (b) satisfies perfect recall. In game (a), consider the  information set for player 1 that includes two nodes. One node is associated with  1 having chosen a and 2 having chosen y. The other is associated with 1 having  chosen b and 2 having chosen x. At this information set, 1 is then unsure whether  she chose a or b. That violates perfect recall. As to game (c), the information set  for player 1, which includes four nodes, captures the property that, when 1 chooses  between actions c and d, she doesn’t know what player 2 chose (which is not in  violation of perfect recall) nor what she originally chose at the initial node (which  is in violation of perfect recall). Game (b) satisfies perfect recall. When 1 chooses  between actions c and d, she cannot discriminate between the nodes in which play  was b x and play was a  y, nor between the nodes in which play was b  x and  play was b  y. The former reflects 1’s uncertainty over 2’s action but knowledge  that she originally chose a. The latter reflects 1’s uncertainty over 2’s action but  knowledge that she originally chose b.    4) Harrington 3.4    a) For player 1, a is strictly dominated by b. Neither b nor c is strictly  dominated. For player 2, z is strictly dominated by x. Player 1 plays either b or c  and player 2 plays either x or y.    b) By the assumption, we can go two rounds of the iterative deletion of  strictly dominated strategies (IDSDS). After eliminating the strictly dominated  strategies, the game matrix has the c row and z column deleted.       X  y  B  3,1  2,2  c  0,2  1,2         In the reduced game, b strictly dominates c for player 1. Neither of player 2’s  strategies is strictly dominated. Thus, player 1 chooses b and player 2 chooses either  x or y.    c) After the first two rounds, the game is as shown above. Now y strictly dominates x  for player 2. Thus, player 1 chooses b and player 2 chooses y.    5) Harrington 3.9      6) Harrington 4.1      There are four symmetric strategy profiles and thus four candidates for  symmetric Nash equilibrium. Note that if a person fails to vote for the person that  everyone else votes for, then no one takes on the task. Thus, at a symmetric strategy  profile, an individual player’s choice is always between the person who the others  are voting for and no one. Consider the symmetric strategy profile in which all vote  for Boromir. This strategy is optimal for both Boromir and Frodo as each would  rather that Boromir take on the task than that no one do so. This is clearly not a  Nash equilibrium, however, as both Legolas and Gimli would prefer to vote for  someone else, and the result would be that no one takes the ring to Mordor. By a  similar argument, itis not a Nash equilibrium for all to vote for Gimli, or for all to  vote for Legolas. Now consider all voting for Frodo. Since each person prefers that  Frodo do it than that no one do it, each player’s strategy is optimal. The unique  symmetric Nash equilibrium is then for all to vote for Frodo.    7) Harrington 4.8    8) Harrington 5.1    a) One class of Nash equilibria has 19,999 people requesting \$100 and the  other 80,001 people requesting \$20. There are then many Nash equilibria, and they  differ only in terms of who are the lucky ones to receive \$100. There is also a class  of Nash equilibria in which at least 20,001 people request \$100, with the remainder  requesting \$20. They all end up with zero.    b) No one submitting a request is the unique Nash equilibrium. Note that  submitting a request for \$20 is strictly dominated by not submitting a request. If  less than 20% request \$100, then a player’s payoff is ‐1.95, and if 20% or more  request \$100, then her payoff is ‐21.95. In contrast, the payoff is 0 by not submitting  a request. Thus, a Nash equilibrium must entail no one submitting a request for  \$20. But if no one submits a request for \$20, then, if anyone participates, they all    must submit a request for \$100. Since the fraction of requests for \$100 exceeds  20%, all participants get zero, which makes their payoff ‐21.95 (the cost of the  magazine). As that is lower then the payoff from not submitting a request, then, in  equilibrium, it must be the case that no one submits a request for \$100 either.    c) Going back to part (a), it is no longer a Nash equilibrium to buy the  magazine when you don’t expect to get some money. Nash equilibria now entail  19,999 people requesting \$100 and the other 80,001 people requesting \$20; with  the former players receiving a payoff of \$80.05 and the latter getting 5 cents.    9) Harrington 5.2     Any strategy profile with a person bidding a positive amount below the highest bid  is not a NE, because it results in a negative payoff, but such a person can get a 0  payoff by bidding zero.    Suppose top bid is b’   ‐Suppose only 1 player bids b’, rest bid 0.      If b’>1, top bidder can increase payoff by lowering bid by 1. Thus b’=1         10) Harrington 6.3  11) Harrington 6.7      ...
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## This note was uploaded on 03/03/2011 for the course ECON 414 taught by Professor Staff during the Spring '08 term at Maryland.

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