Here’s another lovely Saturday night spent at the Chicago Symphony Orchestra: There’s a Ravel in the crowd yelling Bravi — can you spot him? Tonight’s program is splendid — the orchestra begins with Camille Pépin’s Les Eaux célestes. Anticipating upcoming Ravel, I was not surprised to catch a sense of oriental chords within the surreal, dreamy, beauuuuuutiful music. Turns out, the piece it was inspired by the ancient Chinese legend 牵牛织女鹊桥会 (a love story, read CSO’s official comment). ...

In China, business deals are often made over dinners, where the essential informal/implicit consensus are reached. Drinking alcohol (especially, BaiJiu) in formal business dinner is not only for enjoyment, but more of a cultural etiquette—drinking with the the party is a way to show respect and commitment. Here’s a really interesting behavioral economics paper about the effect of alcohol intoxication affecting people strike a deal in business: The idea is golden — and the paper is only 15 pages! Well, John Nash’s PhD thesis was also just 26 pages… ...

The 2025 Econ Camp is taught by Professor Emir Kamenica for incoming PhD students at Booth. This year, first-years from Griffin and Harris also joined, making the room a cross-department gala of future scholars craving for free coffee. The camp runs for an intensive week, with three hours of lectures every morning. At the Econ Department… Thanks to Professor Kamenica’s exceptionally clean logical organization, these notes preserve the blackboard style. ...

Comparisons of Distributions We study random variables on $\mathbb{R}$ (aka lotteries). Each lottery can be identified with its cumulative distribution function (cdf): $$ F(z) = \Pr(Z \leq z). $$ First-Order Stochastic Dominance (FOSD) Definition. For two distributions characterized by their cdf $F, G$, $$ F \succeq_\text{FOSD} G \iff \int u(x) dF(x) \geq \int u(x) dG(x), \quad \forall \text{increasing } u. $$ Theorem (Characterization of FOSD). $$ F \succeq_\text{FOSD} G \iff F(x) \leq G(x) \quad \forall x. $$ ...

People don’t cherish the things that they obtained without a cost. Here’s an interesting research from the 2025 AI in Social Science Conference, by Jason Sockin from Cornell University. The paper’s not online yet, but an abstract can be found at NBER Interviews (which seems to be a tentative title…?) Elliott Ash, ETH Zurich. Soumitra Shukla, Harvard University. Jason A. Sockin, Cornell University Interviews allow employers to learn about workers, but do they also enable workers to learn about firms? Using data on hundreds of thousands of interview reports from Glassdoor, we find that candidates for high-paying jobs are 9 percent more likely to reject a job offer if they believe the interview was easy. ...

Expected Utility Theory Suppose the space of alternatives has a little more structure: $$ X = X_1 \times X_2 \times \cdots \times X_n. $$ We define Lottery Space on this $X$: Let $\mathcal{L} = \Delta(X)$ = set of lotteries on $X$ (probability distributions). Expected Utility Representation Definition. A preference $\succeq$ on lottery space $\mathcal{L}$ admits an expected utility form if $\exists u:X\to\mathbb{R}$ such that $\forall L,L’\in\mathcal{L}$: $U(L) = \sum_{x\in X} p^L_x u(x)$, $L\succeq L’ \iff U(L)\geq U(L’)$. The von Neumann Morgenstein Theorem Theorem Consider the space of lotteries $\mathcal{L} = \Delta(X)$ over a finite set of outcomes $X$. For any preference relation $\succeq$ on $\mathcal{L}$, iff. it satisfies: ...

Here’s a very interesting matrix completion method and result, which is a direct corollary from Koltchinskii, Lounici and Tsybakov (2011) paper. Consider the following matrix estimation problem: for input matrix $A\in \mathbb R^{m_1\times m_2}$, assume $m_1 < m_2$ and $\text{rank}(A) = r \ll m_2$. $n$ entries of $A$ are observed at uniformly at random, with independent noise $\epsilon_{ij}$. Denote as $\Omega$ the index set of observed entries of $A$, define the scaled observation matrix $Y = [y_{ij}]$ as $$ y_{ij} = \frac{m_1 m_2}n \begin{cases} a_{ij} +\epsilon_{ij} & (i, j) \in \Omega\cr 0 & (i, j)\notin \Omega \end{cases},\quad (i, j)\in [m_1]\times [m_2]. $$ Let $\lambda = C \bar y \sqrt{\frac{\log(m_1+m_2)}{m_1n}}$, where $C$ is a large enough constant depend only on attributes of the random noise $\epsilon_{ij}$. Threshold $Y$’s singular values and obtain an estimated $\hat A$ by ...

Before we begin, it’s helpful to have this framework in mind: $$ \text{Utility Functions }u(\cdot) \Leftrightarrow\text{Preferences } a \succeq b \Leftrightarrow\text{Choices (aka Data) }(\beta, C) $$ Economists care about how to make the above concepts consistent (i.e. under what assumptions would the above $\Leftrightarrow$ hold mathematically). Choices and Choice Structures Given a finite set of mutually exclusive alternatives $X$: Definition. A choice structure is a pair $(\beta, C)$ with $\beta \subseteq \mathcal{P}(X)$ a family of budget sets, $C : \beta \Rightarrow X$ a choice correspondence such that $C(B) \subseteq B$ for all $B \in \beta$. Example. $X=\lbrace a,b,c\rbrace $, $\beta=\lbrace \lbrace a,b\rbrace ,\lbrace a,b,c\rbrace \rbrace $, $C(\lbrace a,b\rbrace )=\lbrace a\rbrace , ; C(\lbrace a,b,c\rbrace )=\lbrace a,c\rbrace $. ...

Given two matrices $A, B$, finding an orthogonal matrix $\Omega\in O(n)$ which most closely maps $A$ to $B$: $$ \min_{\Omega\in O(n)}\Vert \Omega A - B\Vert_F $$ Note: $O(n)$ means the set of n*n orthogonal matrices. The name Procrustes refers to a bandit from Greek mythology who made his victims fit his bed by either stretching their limbs or cutting them off. (Wikipedia) TL;DR: the optimal solution $\Omega^\star = UV^T$, where $U, V$ are given by taking SVD of $BA^T = U \Sigma V^T$. ...

bound SVD outcome by their og matrices