I do not mean to insult your intelligence and knowledge, but based on last years ex- perience, it's probably helpful to remind you that probabilities are nonnegative. You will (hopefully! see why I am pointing this out 2. Correlated Rationalizability as a Fixpoint Solution Concept Consider a finite game G=(N, (Ai, uiieN). Define the constrained best response corre- spondence cri: 2A-i\10)=24\0 as follows: for any B_i C A-i such that B-i+ 0, ai E cri (B_i)iff there exists a probability distribution a-i E A(A-i) such that:(i) ui(ai,a-i2ui(a, a-i) for all a; E Ai, and(ii)a-i(B-i) Correlated rationalizability can then be defined as follows: for every i E N, let A= Ai then,forn≥1 and for every i∈N,letA=cr;(4x1) (i)[trivial] Prove that there exists N> l such that An= A for all n>N (ii)Prove that an action profile a is in A, i.e. is correlated rationalizable, iff there exists a set b= Ilen Bi C A(i.e. B is a Cartesian product)such that(i)aE B, and (ii)for every z∈N,B1Ccr;(B- Gi) If we modify cri so as to incorporate the restriction that beliefs must be independent probability distributions, the preceding argument obviously goes through and leads to an alternative characterization of (independent)rationalizability. Conclude that any strategy in the support of a Nash equilibrium is rationalizable (iv)Prove that A includes any set b=llen B, with the property that, for every iE N, Bi C cri(B-i) 3. The Beauty Contest Game Consider the following situation: N individuals are asked to(simultaneously) write down an integer ai between 0 and 100. Payoffs are determined as follows: first, the average a of the N numbers is computed; then, the individuals whose number is closest to a are deemed winners; finally, winners share(equally)a prize P>0, while all other individuals receive 0 First, what would you do in this situation, if you could not think about it for more than 30 seconds? You will obviously not be graded on this: I'm just curious Now analyze the game using the notions of correlated rationalizability and Nash equi- librium. How did you do, based on your gut feeling? If you did poorly, don't worry: the overwhelming majority of people do 4. From or:18.2,183,35.1,64.1I do not mean to insult your intelligence and knowledge, but based on last year’s ex￾perience, it’s probably helpful to remind you that probabilities are nonnegative. You will (hopefully!) see why I am pointing this out. 2. Correlated Rationalizability as a Fixpoint Solution Concept Consider a finite game G = (N,(Ai , ui)i∈N ). Define the constrained best response corre￾spondence cr i : 2A−i \ {∅} ⇒ 2 Ai \ {∅} as follows: for any B−i ⊂ A−i such that B−i 6= ∅, ai ∈ cr i(B−i) iff there exists a probability distribution α−i ∈ ∆(A−i) such that: (i) ui(ai , α−i) ≥ ui(a 0 i , α−i) for all a 0 i ∈ Ai , and (ii) α−i(B−i) = 1. Correlated rationalizability can then be defined as follows: for every i ∈ N, let A0 i = Ai ; then, for n ≥ 1 and for every i ∈ N, let An i = cr i(A n−1 −i ). (i) [trivial] Prove that there exists N ≥ 1 such that An i = AN i for all n ≥ N. (ii) Prove that an action profile a is in AN , i.e. is correlated rationalizable, iff there exists a set B = Q i∈N Bi ⊂ A (i.e. B is a Cartesian product) such that (i) a ∈ B, and (ii) for every i ∈ N, Bi ⊂ cr i(B−i). (iii) If we modify cr i so as to incorporate the restriction that beliefs must be independent probability distributions, the preceding argument obviously goes through and leads to an alternative characterization of (independent) rationalizability. Conclude that any strategy in the support of a Nash equilibrium is rationalizable. (iv) Prove that AN includes any set B = Q i∈N Bi with the property that, for every i ∈ N, Bi ⊂ cr i(B−i). 3. The Beauty Contest Game Consider the following situation: N individuals are asked to (simultaneously) write down an integer ai between 0 and 100. Payoffs are determined as follows: first, the average a¯ of the N numbers is computed; then, the individuals whose number is closest to 1 2 a¯ are deemed winners; finally, winners share (equally) a prize P > 0, while all other individuals receive 0. First, what would you do in this situation, if you could not think about it for more than 30 seconds? [You will obviously not be graded on this: I’m just curious!] Now analyze the game using the notions of correlated rationalizability and Nash equi￾librium. How did you do, based on your gut feeling? If you did poorly, don’t worry: the overwhelming majority of people do. 4. From OR: 18.2, 18.3, 35.1, 64.1 2