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Exam 1ECONS 424: Strategy and Game TheorySpring 2017General Directions: You have 50 minutes to complete the exam. You are allowed towrite on the exam and use provided scratch paper. Please write and circle your answerson the exam clearly. Notes and phones are not allowed. You may use a simple or scientificcalculator, but no graphing calculators. Please ask me for clarification on questions ifyou need any.For questions 1,2 and 3, answer True or False. If the answer is False explainin a short answer why the statement is false.Problem 1.(2 points) (True or False): In simultaneous games a rational player willnot play a strictly dominated strategy, but in sequential games sometimes they will.Solution:False: By definition, a strategy, call it s is strictly dominated by anotherstrategy s0 if ui (s0 ) ui (s) for any strategy choice of the other players. Because a playeralways does strictly better by playing s0 than they would by playing s, a rational playermust always choose s0 over s. Hence, a rational player will never play a strictly dominatedstrategy in any type of game. Weakly dominated strategies are a different story.Problem 2.(2 points) (True or False): A strategy si for player i in a game G, willspecify a single action for every one of player i’s information sets that are actually reachedwhen s i is the strategy profile of the other players.Solution:False: By definition, a strategy is a complete contingent plan for a playerwhich describes exactly how they will act at ALL of their information sets, regardless ofwhether any player’s actions prevent certain information sets from being reached. Thismeans it must specify one of the available actions at every single information set a playerhas.Problem 3.(2 points) (True or False): In symmetric games network effects can leadto symmetric pure strategy Nash equilibria and congestion can lead to asymmetric purestrategy Nash equilibria.Solution: True.1

Problem 4.(2 points)(Circle the correct choice) Nash equilibrium requires that therationality of the players be common knowledge so that players choose their strategies tomaximize their payoffs given their beliefs about the strategies chosen by the other playersAND it requires those beliefs to beA.) rationalizableB.) fairC.) correctD.) subjectiveE.) none of the aboveProblem 5.(2 points)(Circle the correct choice) Consider the game below played byBob and Alice.AliceFormalCasualFormal2, 31, 1Casual0, 03, 2BobThe game facing Alice and Bob is a game ofA.) pure conflictB.) constant sum conflictC.) zero sum conflictD.) coordinationE.) anti-coordinationF.) pursuitG.) none of the aboveProblem 6.(2 points)(Circle all that are correct) A strategic pre-commitment musthave the properties:A.) irreversibleB.) result in a higher security level for the playerC.) known and believed by other playersD.) established before other players make decisionsE.) A best response2

Problem 7. (8 points) Study the extensive form game below and answer the followingquestions or problems in the space provided below. (Note in the game assume that forspace reasons W W ar and N W N o war)A.) Is this a game of perfect or imperfection information?B.) How many players are in this game?C.) Write down two pure strategies for each player in the game.D.) How many pure strategies does each player in the game have?E.) How many pure strategy profiles are there in the game?F.) How many subgames are there in the game?Solution:A.) All information sets are not singletons, so this is a game of imperfect information.B.) The game has 3 players: Iraq, UN, and US.C.) Iraq has 3 information sets, each a singleton. Therefore each of Iraq’s strategies willspecify exactly 3 actions, one for each information set. The UN has only a singleinformation set with 2 actions so a strategy will specify just a single action (for the 1information set). Finally, the US has a total of 4 information sets and so any strategy3

specify a total of 4 actions. Below are two strategies from each player’s strategy set.Iraq : W M D/Deny/Deny, W M D/Deny/Allow&hideUN : Inspect, Do not inspectUS : W ar/W ar/W ar/N o W ar, N o W ar/W ar/N o W ar/N o W arD.) We can use the number of information sets for each player and the number of actionsat each information set to determine the total number of strategies for each player.Iraq’s strategies are a list of 3 actions, one for each information set and there are 2possible actions at two of the information sets and 3 available actions at the last.Hence, Iraq has 2 2 3 12 possible strategies. The UN has only 2 possible strategies as it has only 1 information set with 2 actions. Finally, the US has 4 informationsets, with available number of actions equal to 2 at each of those information sets.Hence, there are 2 2 2 2 16 pure strategies for the US. In summaryNumber ofStrategies12216PlayerIraqUNUSE.) Now that we know the total number of strategies for each player we can calculate thetotal number of possible pure strategy profiles for this game. Remember, a strategyprofile is a list of strategies, exactly 1 for each player. In this case it will be a triple(sIraq , sU N , sU S ). The first component can take 12 different values, the second 2possible values and finally the 3rd can take 16 different values. This means that wehave 12 2 16 384 strategy profiles.F.) The smallest proper subgame belongs to the US and is the node that is marked withIV. After this, however we see that there are no other proper subgames as theywill either fail to have a unique initial node or they will “break” an informationset. Hence, the only other subgame is the whole game itself. Therefore there are 2subgames.4

Problem 8. (8 points) In the game below determine the following:A.) The total number of proper subgames.B.) The total number of pure strategies for each player.C.) The total number of pure strategy profiles.Solution:A.) The smallest proper subgames are those where player X gets to move. Traversing upthe tree we see that the only other valid subgame is the whole game itself. However,because the whole game, while a subgame, is not a proper subgame so we do notcount it. Therefore the game has 4 proper subgames.B.) Every player in the game has exactly 2 available actions at each of their informationsets.Player InfosetsEH1JF1X4Y4Z45Strategies222 2 2 2 162 2 2 2 162 2 2 2 16

C.) A pure strategy profile for the game will contain exactly 1 strategy for each player.In this case it will be a list of 5 strategies ( a 5-tuple), (sEH , sJF , sX , sY , sZ ). Since wehave calculated the total number of strategies each players have we can count the totalnumber of possible combinations of those strategies as 2 2 16 16 16 16, 384.6

Problem 9.(14 points) For the game below, find all pure and mixed strategy Nashequilibria.Solution: The first step is to realize that this is a simultaneous game since both playersmake their strategy choices without knowledge of the other player’s strategy. We convertto a strategic form game and proceed from there. Each player has 1 information set withfour actions meaning they each have 4 pure strategies and therefore we will have a totalof 4 4 16 possible strategy profiles. Listing player 1’s strategies down the rows andplayer 2’s strategies across the columns we have the strategic form game below.Player 2wxyza 3, 24, 12, 30, 4b 4, 42, 51, 20, 4c1, 33, 13, 14, 2d 5, 13, 12, 31, 4Player 1To reduce the difficulty in searching for both pure and mixed strategy Nash equilibria ofthis game we can take advantage of the assumption of rationality as common knowledgeto perform iterated deletion of strictly dominated strategies (IDSDS) since these are neverbest responses and will not be played by rational players.We note that b is strictly dominated by d for player 1 and that strategy y is strictlydominated by z for player 2. Hence we delete both strategies from the strategic form andwe are left with Because rationality is common knowledge all players know the deleted7

Player 2wxz4, 10, 41, 33, 14, 2d 5, 13, 11, 4a 3, 2Player 1 cstrategies and the fact that strategy x is strictly dominated by z for player 2 and will bedeleted.Player 2wza3, 20, 4Player 1 c1, 34, 2d5, 11, 4Because everyone knows that player 2 will not play x, its deletion causes strategy a tobecome strictly dominated by d for player 1 and we now have the reduced strategic formgame after a couple rounds of IDSDS.Player 2wzc1, 34, 2d5, 11, 4Player 1The remaining 2 2 game cannot be reduced any further by IDSDS since none of thestrategies are strictly dominated. We now proceed to look for pure and mixed strategyNash equilibria.Using the best reply method to seek for pure strategy Nash equilibria we see that thereare not any. We now seek for mixed strategy Nash equilibrium. Let p be the probabilitythat Player 2 plays w and q be the probability that Player 1 plays c. Rational playersare only willing to randomize if and only if they are indifferent between the strategiesover which they can randomize. Player 1 will only randomize if Player 2 behaves such8

Player 2cwz1, 34 ,2Player 1d5 ,11, 4that the expected payoff from c is the same as d.p 4(1 p) 5p (1 p) {z} {z}Eu1 (c)Eu1 (d)p 4 4p 5p p 14 3p 4p 13 7p3p 7Player 2 is indifferent if3q (1 q) 2q 4(1 q) {z} {z}Eu2 (w)Eu2 (z)3q q 1 2q 4q 42q 1 4 2q4q 33q 4Therefore the only Nash equilibrium of the game is in mixed strategies and is 33 14c, d ,w, z4 4779

Problem 10.(14 points) Consider two players, Alice and Bob, who will play theultimatum game with a pot worth 100. You and a friend are talking about how ultimatum games give power to the proposer because in subgame perfect equilibrium play theproposer is able to get the receiver to accept a smaller portion of the pot. Your friendthen remarks,“You know what the problem is? . The proposer doesn’t face any competition.If we add another player and allow the receiver to decide who they want toplay against, then the proposers will compete with each other and erase theirproposer power.”You consider this argument, but remain skeptical and relying on your ECONS 424 trainingyou decide to test the argument with a simplified ultimatum game in which Alice, Boband now Jill will play. Alice will choose whether to player the ultimatum game with Bobas the proposer, or with Jill as the proposer. If Alice rejects the proposer’s offer than theproposer gets 40 and Jill will only receive 20. The remaining 40 of the pot is surrenderedto an outside party. The order of payoffs on the terminal nodes is (Alice, Bob, 0A30070R20040Find all subgame perfect Nash equilibria of the above game and then use the concept ofsequential rationality to make an argument for why your friend is correct or incorrectabout his hypothesis of competition improving the ultimatum problem.Solution:10

0R20040Using backward induction we see there are two pure strategy subgame perfect Nashequilibrium profiles(Bob/A/A/A/A, 70-30, 70-30) (Jill/A/A/A/A, 70-30, 70-30)The payoff to Alice in both SPNE is 30, which is the same as she gets in SPNE playwhen she doesn’t have a choice between proposers. However, note that Alice is indifferentbetween her two SPNE strategies since she receives a payoff of 30 no matter who she picksto play against. Notice that Bob and Jill are not indifferent between the two SPNE. Theleft profile gives Bob 70 and while Jill gets 0. The right profile gives Bob 0 but Jill gets70.Suppose that Alice chooses Bob with probability b. Then his expected payoff from hisSPNE strategy of 70 30 is 70b 0(1 b) 70b. If instead he was to play 50 50 hisexpected payoff would be 50b 0(1 b) 50b. Since 70b 50b for all b, Bob cannot bemade indifferent between his strategies and will not randomize.Similarly, Jill’s expected payoff from her SPNE strategy would be 0b 70(1 b) 70 70band switching to 50 50 would yield 0b 50(1 b). Again since 70(1 b) 50(1 b)for all b, Jill will always want to play 70 30.11