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 ...

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

The Chicago Symphony Orchestra’s celebratory fundraising concert took place on September 20, opening the 2025–26 season with a sense of homecoming. Symphony Center was filled to the brim, the audience bubbling with anticipation and festive spirits, rather like the champagne glasses at the pre-concert reception. The program was elegant and lighthearted curated. We began with Weber’s Oberon Overture, followed by Mendelssohn’s A Midsummer Night’s Dream — the Scherzo and the famous Wedding March. ...

I recently attended the Joffrey Ballet’s production of Carmen, and while I had the privilege of watching from a perfect front-row, center seat (thanks to a remarkable $30 ticket), I left the theater somewhat unsettled. The dancers’ technique was solid, the orchestra gave a nice performance, and yet — something about the production went fundamentally astray. what went wrong in my pov: Carmen Herself The greatest shortcoming, in my view, lies in the characterization of Carmen. She was not the radiant, magnetic figure we expect — the embodiment of confidence, independence, and dangerous charm. Instead, this Carmen seemed curiously subdued, at times even mournful. When she rejected Don José, she appeared more sorrowful than defiant, as though regret had clouded her resolve. ...

Two fascinating cases of market design for grad students at UChicago. Bidding for classes At Booth, students enroll in courses through an auction system. The process runs across four stages — each a separate round of auction, with waitlist and drop-refund rules layered in. The bidding schedule for Fall 2025 quarter At the start, students are endowed with virtual currency (e.g. 8,000 Bid Points for new PhDs; returning students carry over 2,000 points per course completed). They can see the total seat supply, remaining spots, and the clearing prices from all past rounds — including the current quarter. ...

There’s a carillon on the top of the Rockefeller Chapel at University of Chicago. The Rockefella Chapel viewed from the 7th floor of the David Rubenstein Forum building. It’s kinda like a piano, but instead of strings or pipes (like an organ), Carillons use bronze bells of different sizes, each producing a distinct pitch. The Rockefeller one operates pure mechanically. ...

Some fun algebraic manipulations.