Another convenient way to view, compare, and solve Cheap Talk vs. Bayesian Persuasion. Linear-Programming Formulation: Cheap Talk vs. Bayesian Persuasion Consider an information environment $\langle \Omega, A, M, \mu_0, u, v \rangle$ with state space $\Omega$, action space $A$, signal space $M$, prior $\mu_0$, receiver utility $u(\omega,a)$, and sender utility $v(\omega,a)$. (Let every set be finite). Consider direct mechanisms. In general, the sender’s strategy $\sigma:\Omega \to \Delta(A)$ maps each state to a distribution over recommended actions, and the receiver’s strategy $\rho:A \to \Delta(A)$ specifies how she responds to each recommendation. ...

A general continuous-time dynamic programming problem is given by reward function $h(x, u)$ control function $g(x, u)$ and initial state $x_0$: $$ \begin{align*} V^\star (x_0) :=& \max_{\lbrace u_t\rbrace_{t\ge 0}} \int_0^\infty e^{-\rho t} h(x_t, u_t),\text{d}t\cr & \text{s.t. }\dot x_t = g(x_t, u_t). \end{align*} $$ The Hamiltonian functino $H(x, u, \lambda)$, and Maximized Hamiltonian $H^\star(x, \lambda)$ is defined as $$ \begin{align*} & H(x, u, \lambda) = h(x, u) + \lambda^Tg(x, u),\cr & H^\star(x, \lambda) = \max_u H(x, u, \lambda). \end{align*} $$ It’s classical convex analysis result that $H^\star(x, \lambda)$ is convex in $\lambda$. But also, interestingly ...

When I learnt the continuous time and discrete time neoclassical model, it’s feels so tempting to try to put them together in a uniform view. It’s not trivially easy though. And here’s one way to do it. Continuous-Time Neoclassical Growth Model Given $k_0$, $f(\cdot), U(\cdot)$: $$ \begin{align*} & \max_{\lbrace c_t\rbrace_{t\ge 0}} \int_0^\infty e^{-\rho t} U(c_t),\text{d}t\cr & \text{s.t. }\dot k_t = f(k_t) - \delta k_t - c_t \end{align*}\tag{1} $$ Taking (appropriate) derivative of the Hamiltonian function solves this problem — in general, for any continuous-time problem in the following form: $$ \begin{align*} V^\star (x_0) :=& \max_{\lbrace u_t\rbrace_{t\ge 0}} \int_0^\infty e^{-\rho t} h(x_t, u_t),\text{d}t\cr & \text{s.t. }\dot x_t = g(x_t, u_t). \end{align*} $$ then, the following first-order optimality conditions are necessary (under regularity assumptions) and sufficient (under regularity and convexity assumptions) for the path $\lbrace x_t, u_t\rbrace_{t\ge 0}$ to be optimal: $$ \begin{align*} & \text{let } H(x, u, \lambda) = h(x, u) + \lambda^Tg(x, u)\cr & \text{FOCs}:\begin{cases}H_u(x_t, u_t, \lambda_t) = 0\cr \dot \lambda_t = \rho \lambda_t - H_x(x_t, u_t, \lambda_t)\cr \dot x_t = H_\lambda (x_t, u_t, \lambda_t) = g(x_t, u_t). \end{cases} \end{align*} $$ For $(1)$, the first order optimality conditions are $$ (1)’s\text{ FOCs}:\begin{cases} U’(c_t) = \lambda_t\cr \dot \lambda(t) = \rho\lambda_t - \lambda_t(f’(k_t) - \delta)\cr \dot k_t = f(k_t) - \delta k_t - c_t. \end{cases} $$ ...

I thought it was another song Johannes Brahms wrote for Clara Schumann. In fact, Brahms had it dedicated to one of his former lover: The cradle song was dedicated to Brahms’s friend, Bertha Faber, on the occasion of the birth of her second son. Brahms had been in love with her in her youth and constructed the melody of the “Wiegenlied” to suggest, as a hidden counter-melody, a song she used to sing to him. ...

You can but don’t necessarily have to use fancy harmonic to write a masterpiece (Like math for research). Chopin proves this point — you need only two chords to write a masterpiece. Here’s his wonderful Berceuse (Op. 57). It’s a very sophisticated, rich work full of room for imagination. Note: I really likes the following Sumino’s recording where, it sounds like either he’s using the sostenuto pedal or it’s a really soft piano. ...

The musical scale is arguably humanity’s oldest algorithm for beauty. Music theory can seem intimidating (for instance, I probably should have learned tuning theory back in 16 when I was sitting in orchestra). But it’s actually quite straightforward. Imagine you’re a piano designer: you want your instrument to sound nice, but you also don’t want it to have an infinite number of keys. Axiom 1 (Sounding nice) Two notes sound consonant when their frequencies form a ratio of small integers. The simpler the ratio, the more pleasant the sound. For example, a perfectly harmonic octave has a ratio of $2:1$, and a nice fifth has $3:2$. ...

AI can now make decent, impressive music. Check out Tunee and their example projects. Tunee’s interface features a conversational input, a project diagram, and options to preview AI-generated music with lyrics. However, there’s no way to export stems or sheet music yet. Out of curiosity, I impatiently splashed some random prompts on my keyboards into Tunee: the results were decent Bach, whimsical bouncy cool woodwinds, and a few suspiciously catchy ballads—of course, since AI is now quite good at generating lyrics: ...

After surviving a macro midterm where the True/False section felt like coin flipping, I rushed downtown for Dvořák’s Symphony No. 9 with the Chicago Symphony Orchestra under Sir Riccardo Muti (Concert Note digital version) Can you believe that Muti’s 84?! And he probably definitely counts bars better than I do. I sat in the center front row of the Terrace—right behind the stage. That seat feels almost illicit, as if you’ve joined the orchestra. Muti’s expressions are right there in front of you; his every gesture, every cue, goes directly under your eyes. He is astonishingly clear. Every motion carries intention—he shakes for trills, marks beats for the horns, molds phrases with his hands. His precision and humor are inseparable. It’s a masterclass in detail. Honestly, that seat should cost more when Muti’s conducting—as a orchestra flute player myself I’d say he’s the best conductor one could hope for. ...

Another note from info economics class. Recall Aumann’s common knowledge: fix an event $E$ and consider the two agents’ posteriors of event $E$ after agents each conduct their own deterministic partition experiment. (see this post for a recap) Theorem (Aumann 1976) If A’s posterior is $q_A$ and B’s posterior is $q_B$ is common knowledge for A and B, then $q_A = q_B$. But how come $q_A$ and $q_B$ become common knowledge in the first place? Geanakoplos and Polemarchakis’s paper We Can’t Disagree Forever (1982) gives a way that the two people can use (non-strategic) communication to arrive at a common understanding of each other’s posterior: ...

Who understands copyright best? IMSLP would definitely be shortlisted. They have an excellent article explaining general how copyright works: A work that is in the public domain is not protected by copyright and can be freely used for any purpose. Which works are in the public domain varies from country to country. For example, [in Canada] if a copyright lasts for 70 years after an author’s death, and the author who died on March 14, 1955, then the term lasts until December 31, 2025 and the work is in the public domain from 2026 onwards. ...