Problem Set 1 1. Games in Strategic Form: Do questions 1-4 and ...

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So consider the game below. If the row player used b this round, the column player’sbest-response would have been z for a payoff of 3. If the column player’s strategies aredecided by best-response dynamics, then, we’d predict he should play z in the next round.x y za 3,3 1,2 1,0b 0,1 2,2 3,3c 1,2 -1,-1 4,4(a). Find all Nash equilibria.(b). Show that if you start at (b,y), play eventually reaches (c,z) and stays there permanently.Is this a Nash equilibrium?(c). Show that if you start at (a,z), the learning process never reaches a Nash equilibrium.(d). Is there any starting point that converges to the Nash equilibrium at (a,x)? Why isthis a “fragile” Nash equilibrium under best-response dynamics, especially if players sometimesmake mistakes?(e). Show that if each player uses the same strategy twice, then their strategies form apure-strategy Nash equilibrium, and they will play those strategies in all future periods.(f). Is this a good model of learning or not? Explain your answer (particularly taking youranswer to part (v) into account).i. Find all Nash equilibria. Show that if you start at (b,y), play converges to (c,z).Is this a Nash equilibrium?The Nash equilibria are underlined.Here is a map of how the game moves:(a,x) → (a,x)(a,y) → (b,x)(a,z) → (c,x)(b,x) → (a,z)(b,y) → (b,z)(b,z) → (c,z)(c,x) → (a,z)(c,y) → (b,z)(c,z) → (c,z)Thebestthingtodoisdrawarrowsonthestrategicformshowingtheaboveinformation;this lets you visualize the dynamics.iii. Show that if you start at (a,z), the game never converges to an equilibrium.From the “map” above, (a,z) → (c,x) and (c,x) → (a,z), so the players bounce backand forth between those two profiles forever.iv. Is there any starting point that converges to the Nash equilibrium at (a,x)?Why is this a “fragile” Nash equilibrium under best-response dynamics?Only (a,x). If you look at the map, no profile ends at (a,x) except (a,x) itself. Thismeans that unless the game starts there, the players will never find it, and if eitherplayer makes a mistake and does something else, the game will never return to thisoutcome. Imagine that the dynamics are like a ball rolling on a bowl. If the bowl is6
ight-side up and you drop the ball into the bowl, it will roll to the bottom and staythere. If you tweak the ball, it will return to the bottom. If you flip the bowl upsidedown, you can balance the ball exactly and it will stay there, but if you tweak it, it willroll away. That’s the difference between the Nash equilibrium in the upper-right cornerand the lower-left corner.v. Show that if either player knew the other was using best-response dynamics,that player could “guide” the other player to the Nash equilibrium at (c,z).If the column player uses z twice, the row player will use c on the second roundand they can stay at the “good” Nash equilibrium forever. If the row player uses ctwice, the column player will use z for sure on the second round and they can stay atthe “good” Nash equilibrium forever. This shows that if your opponent is using bestresponsedynamics, you probably don’t have an incentive to use best-responsedynamicsyourself.vi. Is this a good model of learning or not? Explain your answer (particularlytaking your answer to part (v) into account).Yes: It incorporates feedback that players receive through their payoffs to makedecisions about future play. Normal game theory assumes that players “know” whatequilibrium they are using or can reason their way to a unique prediction, which isn’talways a good assumption. Here, the players update over time, and as long as we don’tend up in the cycle they converge fairly quickly to a Nash equilibrium anyway.No: In part iii, iv, and v, we see that players can end up in unrealistic situationsas a result of best-response dynamics, where real players would probably abandon thatapproachandtrysomethingelse. Insteadofusingaforward-looking, deductiveapproachitreliesonabackward-looking, inductiveapproachthatisnotalwaysgoingtoaccuratelypredict future play. Ideally, we’d like something in the middle, but this seems toosimplistic.5b. The purpose of this question is to show that while the tools we have adopted (bestresponses,strategy dominance, iterated deletion of dominated strategies) are useful, thereare other ways to approach the material.Rather than focus on dominance, let’s try another idea, called “rationalizability”. Astrategy is rationalizable if it is a best-response to some profile of strategies your opponentscould use.(a). Write a mathematical definition for a rationalizable strategy similar to the definitionof a dominant strategy, using the u i (s i ,s −i ) notation. It’s OK if you find this hard, or don’tthink your answer is correct; this is just an exercise in thinking in alternative ways aboutplayers, games and behavior.(b). Write out a process similar to iterated deletion of strictly dominated strategies called‘iterated deletion of unrationalizable strategies’. Use it to solve the following game:7
Page 1 and 2: Problem Set 11. Games in Strategic
Page 3 and 4: The Nash equilibrium is (A,A), sinc
Page 5: (1) CL RU 2,2,3 1,1,2B D 1,3,4 1,4,
Page 9 and 10: the strategy d is a best response t
Page 11: The usual profiles (U,L) and (D,R)

Problem Set 1 1. Games in Strategic Form: Do questions 1-4 and ...

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