Eco514-Game Theory Problem Set 1: Due Friday, September 30 NOTE: On the“ ethics” of problem sets Some of the theoretical exercise I will assign are actually well-known results; in other cases you may be able to find the answer in the literature. This is certainly the case for the current My position on this issue is that, basically, if you look up the answer somewhere it's your problem. After all, you can buy answer keys to most textbooks. The fact is, you will not have access to such, ehm, supporting material when you take your generals, or, in a more long-term perspective, when you work on your own research. You just cannot learn this stuff without (re)doing the key proofs yourselves and spending considerable time working out actual problems. It is not enough to come to class and do the readings The only enforcement mechanism I will use is that, regardless of your, ehm, "external" sources, I will ask you to turn in your own individual write-up. It's OK to work with your colleagues, of course, as long as each of you turns in a separate homework Sorry to bug you with this sort of things, but, as we say in Italy, patti chiari, amicizia lunga (roughly speaking, "if we agree on the rules beforehand, we shall be friends for a long time:”) 1. The best reply property and strict Dominance Prove that, in a finite game G=(N, (Ai, wieN ), an action ai is a best reply to some possibly correlated) probability distribution a-i∈△(A-) iff there is no a;∈△(A)such that ui(ai, a-i)>ui(ai, a-i)for all a-i E a-i NOTE: this is Lemma 60.1 in OR. It is also proved in the notes for Lecture 2, using LP lowever, I would like to ask you to prove it using the separating hyperplane theorem. A good reference is A. Takayama, Mathematical Economics, Cambridge University Press, pp 39-49, but you can find other sources, too. Please state the exact version of the theorem you are using: you must be careful with strict vs. weak inequalities, closed vs. open sets, and stuff like that. Do some detective workEco514—Game Theory Problem Set 1: Due Friday, September 30 NOTE: On the “ethics” of problem sets Some of the theoretical exercise I will assign are actually well-known results; in other cases, you may be able to find the answer in the literature. This is certainly the case for the current problem set. My position on this issue is that, basically, if you look up the answer somewhere, it’s your problem. After all, you can buy answer keys to most textbooks... The fact is, you will not have access to such, ehm, supporting material when you take your generals, or, in a more long-term perspective, when you work on your own research. You just cannot learn this stuff without (re)doing the key proofs yourselves and spending considerable time working out actual problems. It is not enough to come to class and do the readings. The only enforcement mechanism I will use is that, regardless of your, ehm, “external” sources, I will ask you to turn in your own individual write-up. It’s OK to work with your colleagues, of course, as long as each of you turns in a separate homework. Sorry to bug you with this sort of things, but, as we say in Italy, patti chiari, amicizia lunga (roughly speaking, “if we agree on the rules beforehand, we shall be friends for a long time.”) 1. The Best Reply Property and Strict Dominance Prove that, in a finite game G = (N,(Ai , ui)i∈N ), an action ai is a best reply to some (possibly correlated) probability distribution α−i ∈ ∆(A−i) iff there is no αi ∈ ∆(Ai) such that ui(αi , a−i) > ui(ai , a−i) for all a−i ∈ a−i . NOTE: this is Lemma 60.1 in OR. It is also proved in the notes for Lecture 2, using LP. However, I would like to ask you to prove it using the separating hyperplane theorem. A good reference is A. Takayama, Mathematical Economics, Cambridge University Press, pp. 39-49, but you can find other sources, too. Please state the exact version of the theorem you are using: you must be careful with strict vs. weak inequalities, closed vs. open sets, and stuff like that. Do some detective work! 1