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Class Note for ECON 309 at UMass(4)

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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Massachusetts taught by a professor in Fall. Since its upload, it has received 34 views.

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Date Created: 02/06/15
1 The Battle of the Bismarck Sea In 1943 Rear Admiral Kimura has been ordered to transport Japanese troops across the Bismarck Sea to New Guinea and Admiral Kenny wants to bomb the troop transports Kimura must choose between a shorter northern route or a longer southern route to New Guinea and Kenny must decide where to send his planes to look for the Japanese If Kenny sends his planes to the wrong route he can recall them but the number of days of bombing is reduced The payoff table is Kimura North South Kenny North 2 2 2 2 South 1 1 3 3 a Does Kenny have a strictly or weakly dominated strategy No b What about Kimura Kimura has a weakly dominated strategy South c Can you solve the game using the notion of strictly or weakly dominant equilibrium Yes Kenny may delete the column of South and then choose North d Can you solve the game using the notion of Nash equilibrium de ned in terms of best responses I highlighted best replies of each player There is one cell that is a mutual best reply Thus North North is a NE e Can you solve the game using the notion of Nash equilibrium de ned in terms of nodeviation North North is a NE because there is no player who would like to deviate from this North South is not a NE because Kenny would like to deviate South North is not a NE because Kenny has an incentive to deviate South South is not a NE because Kimura has an incentive to deviate 2 The Battle of the Sexes You and your fiance wish to go out together Two concerts are available one of music by Bach and one of music by Stravinsky You prefer Bach and your fiance prefers Stravinsky If you and your fiance go to different concerts each of you is equally unhappy listening to the music of either composer The payoff structure of the game is the following Your fiance Bach Stravinsky You Bach 2 1 0 0 Stravinsky 0 0 1 2 a Does anyone have a dominated strategy No one has a dominated strategy b Can you solve the game using the notion of dominant equilibrium No c Can you solve the game using the notion of Nash equilibrium Yes There are two NE Bach Bach and Stravinsky Stravinsky 3 Hawk Dove Two animals ofthe same species compete for a resource eg food whose value is v gt 0 Each animal can be either aggressive hawkish or passive dovish If both animals are hawkish they ght until one is seriously injured the winner obtains the resource without sustaining any injury whereas the loser suffers a loss of c lt v The two animals are equally likely to win so each one s expected payoffis v c Ifboth animals are dovish then each obtains the resource with probability without a ght Finally if one animal is hawkish while the other is dovish then the hawk obtains the resource without a ght a Construct the oayoff table for this game See below The other animal Hawk Dove One animal Hawk v c v c V 0 Dove 0 v v v b Solve the game using the notion of Nash equilibrium There are two pure strategy Nash equilibria Hawk Hawk and Dove Dove 4 l The policy pair 05 05 is the unique Nash equilibrium and the two parties have 50 chances of winning at the equilibrium It is easy to see that this is a Nash equilibrum Case 1 Suppose party 1 deviates leftward from 05 to 05s where sgt0 given that party 2 keeps 05 Then all voters with policy preferences from 058 2 to 1 vote for party 2 whereas all voters with policy preferences from 0 to 0582 vote for party 1 Party 1 loses for sure by the deviation Case 2 Suppose party 1 deviates rightward from 05 to 05s sgt0 given that party 2 keeps 05 Then all voters with policy preferences from 05s 2 to 1 vote for party 1 whereas all voters with policy preferences from 0 to 05s2 vote for party 2 Party 1 loses for sure by the deviation In like manner one can show that party 2 does not bene t by deviating unilaterally from 05 05 Since no party bene ts from any deviation this is a Nash equilbrium 2 There is no pure strategy NE if there are three Downsian parties Consider any strategy pro le in which each party locates at a separate point Such a strategy pro le is not an equilibrium since the two parties nearest the ends would edge in to squeeze the middle pa1ty s vote share If a strategy pro le has any two parties at the same point the third party would be able to acquire a share of at least 05s by moving next to them and if the third player39s share is that large one of the doubledup parties would deviate by jumping to its other side and capturing its entire vote share Finally a strategy pro le in which all three parties are at the same point is not also a NE Why

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