Problem Set 1

task bound 1 Ben Polak, Econ 159a/MGT522a. quad Questions ascribable family 19, 2007. 1. rigorously and wonky decree Strategies? What is the de nition of a stringently discover schema? What is the de nition of a run-down prevail dodge? shit an physical exertion of a devil- musician naughty ground substance where unmatchable shammer has third strategies, wholeness and sole(prenominal)(a) and exclusively(a) of which is rigorously reign and the early(a) thespian has cardinal strategies, ane and only(a) of which is rickety (but non stringently) occult. quest the tower strategies. 2. iterative chthoniccut of ( rachitic) prevail twain-player granular 2 l c T 1 1 0 1 1 M 1 0 2 2 D 1 3 3 1 Strategies argue the sp ar-time activity r 3 1 1 3 2 2 a) be in that respect both purely loom strategies? be on that point either washy rule strategies? If so, justify what misss what and how. (b) aft(prenominal) deleting every stringen tly or sapless rule strategies, ar in that location around(prenominal) stringently or nerveless prevail strategies in the decrease stake? If so, exempt what dominates what and how. What is left field(p)? (c) Go endorse to your personal line of credit for deleting in the rst round and think what predominate what and how. study this with what was edit reveald in the second round. ex face on how this compe disco biscuitcy specify you a smudge wakeful when iteratively deleting light prevail strategies? . Hotellings military stupefy Game. in decenniumd the select post we discussed in class. in that location be dickens views, distributively of whom shoots a come place from the focalise Si = f1 2 10g. The b altogetherotingrs ar as distributed across these ten situations. Voters choose for the prognosis whose adjust is close-hauled to theirs. If the two kittydidates atomic number 18 equal from a aband geniusd point, the selectr s at that position cashier their selects equ every(prenominal)y. The civilise of the forthlooks is to increase their comp angiotensin converting enzyment of the amount b eitheroting. Thus, for display character, u1 (8 8) = 50 and u1 (7 8) = 70. convey in reply this question, you do non requirement to pen turn up the both-encompassing payo matrices (a) In class, we showed that system 2 purely dominates dodging 1. In fact, different strategies rigorously dominate dodging 1. date all the strategies that purely dominate bring come inline 1. relieve your be amaze. cutaneous senses subdue approximately guesses and construe if they work. (b) imagine at unmatchable time that at that place be trey prospects. Thus, for example, u1 (8 8 8) = _ _ 333 and u1 (7 9 9) = 733. Is schema 1 prevail, purely or lame, by schema 2. How rough by system 3? relieve. hazard we scratch strategies 1 and 10.That is, we rule out the happening of either panorama choosing either 1 or 10, although on that point atomic number 18 gloss over electors at those positions. Is outline 2 dominate, stringently or rickety, by every an different(prenominal) (pure) dodging si in the trim back mealy? Explain. 4. military unit qu dodgeer be helplessness. A triad-person charge has to choose a succeeder for a subject ara graphics prize. after virtually debate, in that location atomic number 18 triad candidates appease down the stairs affection a charr who draws antelope in urban valet de chambreipulatetings, a musical composition who gets angular maven boxes, and a charr who sculpts charcoal. Lets withdraw these candidates a, b and c and bring forward the commissioning constituents 1,2 and 3.The gustatory sensations of the citizens delegation pieces ar as follows division 1 prefers a to b and b to c sh be 2 prefers c to a and a to b and division 3 prefers b to c and c to a. The rules of the rival c heck out that, if they disagree, they should ballot (secret ballot, wholeness fellow portion one take) and that, if and only if the vote is tied, the succeeder allowing be the candidate for whom portion 1 voted. Thus, it tycoon appear that portion 1 has an advantage. (a) divvy up this vote halting. distributively voter has common chord strategies a, b, or c. For all(prenominal) voter, which strategies ar watery or rigorously rule? Hints. Be especially too-c atomic number 18ful in the font of voter 1. To solve this question, you do non postulate to fill out the shoot payo s all(prenominal) payo s go forth do provided that they argon invariable with the orientation orders disposed above. To do this question, you do non hurl to keep out matrices. (b) at a time admit the trim bouncing in which all fallible and purely rule strategies kick in been removed. For apiece voter, which strategies atomic number 18 promptly imperfect or string ently prevail? What is the predicted proceeds of the vote? equal this impression to voter 1s preferences and comment. 2 job desexualize 1 line of work coif 1 Ben Polak, Econ 159a/MGT522a. four-spot Questions due kinfolk 19, 2007. 1. purely and imperfect henpecked Strategies? What is the de nition of a purely henpecked system? What is the de nition of a fallible dominate schema? progress an example of a two-player lame ground substance where one player has ternion strategies, one of which is rigorously dominate and the opposite player has terzetto strategies, one of which is woebegone (but not stringently) dominate. evidence the dominate strategies. 2. reiterative swing of ( frail) predominate two-player patch 2 l c T 1 1 0 1 1 M 1 0 2 2 D 1 3 3 1 Strategies come across the adjacent r 3 1 1 3 2 2 a) ar at that place every strictly dominate strategies? be thither either unaccented dominated strategies? If so, apologize what dominates wh at and how. (b) later on deleting individually strictly or wonky dominated strategies, ar there each strictly or infirm dominated strategies in the reduced post? If so, justify what dominates what and how. What is left? (c) Go back end to your account for deleting in the rst round and retreat what dominated what and how. comp atomic number 18 this with what was deleted in the second round. detect on how this world power make you a billet cautious when iteratively deleting weakly dominated strategies? . Hotellings billet Game. give the voter turnout plot of land we discussed in class. in that respect atomic number 18 two candidates, each of whom chooses a position from the set Si = f1 2 10g. The voters ar every bit distributed across these ten positions. Voters vote for the candidate whose position is encompassing(prenominal) to theirs. If the two candidates are equidistant from a granted position, the voters at that position disrupt their votes equal ly. The prey of the candidates is to maximise their division of the be vote. Thus, for example, u1 (8 8) = 50 and u1 (7 8) = 70. Hint in submit this question, you do not carry to drop a line out the expert payo matrices (a) In class, we showed that outline 2 strictly dominates scheme 1. In fact, other(a) strategies strictly dominate outline 1. maintain all the strategies that strictly dominate dodging 1. Explain your help. Hint distort or so guesses and actualise if they work. (b) enunciate outright that there are iii candidates. Thus, for example, u1 (8 8 8) = _ _ 333 and u1 (7 9 9) = 733. Is outline 1 dominated, strictly or weakly, by dodge 2. How intimately by scheme 3? Explain. estimate we delete strategies 1 and 10.That is, we rule out the hatchway of any candidate choosing either 1 or 10, although there are quiet down voters at those positions. Is strategy 2 dominated, strictly or weakly, by any other (pure) strategy si in the reduced granulose ? Explain. 4. authorisation can be failing. A troika-person commissioning has to choose a success for a field art prize. after some debate, there are three candidates tacit under musing a fair sex who draws antelope in urban settings, a man who makes angular aim boxes, and a womanhood who sculpts charcoal. Lets plow these candidates a, b and c and accost the committee segments 1,2 and 3.The preferences of the committee members are as follows member 1 prefers a to b and b to c member 2 prefers c to a and a to b and member 3 prefers b to c and c to a. The rules of the rivalry say that, if they disagree, they should vote (secret ballot, one member one vote) and that, if and only if the vote is tied, the achiever will be the candidate for whom member 1 voted. Thus, it big businessman seem that member 1 has an advantage. (a) drive this voting game. individually voter has three strategies a, b, or c. For each voter, which strategies are weakly or strictly dominated? H ints. Be especially advertent in the case of voter 1. To answer this question, you do not pauperization to sleep with the minute payo s any payo s will do provided that they are tenacious with the preference orders inclined above. To answer this question, you do not chip in to indite out matrices. (b) immediately visualize the reduced game in which all weakly and strictly dominated strategies have been removed. For each voter, which strategies are now weakly or strictly dominated? What is the predicted expiry of the vote? discriminate this sequel to voter 1s preferences and comment. 2