ECON104 Assignment 1 Choice Theory

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Exercies 1.3 The previous example was built starting from the case $x\succ y\succ z\succ x$. Does the same reasoning work if $x\succ y\succsim z\succsim$x? What about $x\succsim y\succsim z\succsim x$? CourseNana.COM

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Exercies 1.5 Show that the strict preference $\succ$ and the indifference relation ~ are transitive. Moreover,show that if $x\succsim y$ and $y\succ z$, then $x\succ z$. CourseNana.COM

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Exercies 1.7 So far we have not talked about how an agent compares an alternative with itself. By assuming economic rationality, what can you say about this question? Does your conclusion make sense? (Hint: the assumption of completeness also applies when the agent is faced to choose between x and x). CourseNana.COM

Exercies 1.9 Can you think of a economically rational preference that is consistent with any observed possible behavior? CourseNana.COM

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Exercies 1.12 Show that if ? is a preference and x∉C?≿(A) then there is y∈A such that y≻x CourseNana.COM

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Exercies 1.13 Let $X=a,b,c$ and? $\succsim$ be such that $a?\succsim a$, $b\succsim ?b$, $c?\succsim c$, $a?\succsim b$, $b?\succsim c$, and $c\succsim a$, are all the relations in $\succsim$ ?. Show that $\mathcal{C}{?}_{\succsim }(\{a,b,c\})=\mathrm{\varnothing }$. Can you explain why this happens, and its relation with the definition of economic rationality? CourseNana.COM

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Exercies 1.15 Suppose CourseNana.COM

$X=a,b,c$ and $\mathcal{C}$ is such that $\mathcal{C}(a,b)=a$, $\mathcal{C}(b,c)=b$, and $\mathcal{C}(a,c)=c$. Show that $\mathcal{C}$ is not rationalizable. CourseNana.COM

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Exercies 1.18 Let $X=\{a,b,c\}$ and suppose $\mathcal{C}$ is rationalizable and satisfies CourseNana.COM

Find $\mathcal{C}(\{a,b,c\})$. CourseNana.COM

Exercies 1.20 In terms of economic rationality, does HA makes sense? Explain (in your own words) why it does or does not. CourseNana.COM

Exercies 1.21 Show that if $\mathcal{C}$ is rationalizable then it satisfies HA. CourseNana.COM

Exercies 1.22 Show that the choice rule in Exercise 1.15 fails HA. [Hint: try different possibilities for
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$\mathcal{C}(\{a,b,c\})$] CourseNana.COM

Exercies 1.25 The Associated Press constructs a weekly national ranking of the 25 best College Football Teams (https://apnews.com/hub/ap-top-25-college-football-poll). This ranking is constructed by a survey completed by a “panel of 63 sports writers and broad- casters from around the country”, where each person submits their own ranking (i.e., they submit their preferences). They aggregate the rankings using the Borda Count, which can be defined in base of the following choice rule. Let
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$P$ be the panel of people completing the survey, and ?${\succsim }_i$ be the preferences reported by person $i\in P$ . The Borda Count defines the choice rule (for any set X, the notation |X| represents the number of elements in X) CourseNana.COM

$C(A)=\{x\in A:|\{i\in P:x\in {\mathcal{C}}_{?{\succsim }_i}(A)\}|\ge |\{i\in P:y\in {\mathcal{C}}_{{\succsim }_i}(A)\}|forally\in A\}.$ CourseNana.COM

In other words, $x\in \mathcal{C}(A)$ if there is no other alternative that is chosen by strictly more individuals than x. CourseNana.COM

Suppose $X=\{a,b,c\}$, $P=\{1,2,3\}$, and the reported preferences are CourseNana.COM

Show that the Borda Count violates Sen’s $\alpha$. CourseNana.COM