One of the recent fad topics in econometrics is the use of machine learning methods to estimate high dimensional nuisance parameters, with perhaps the most famous paper along these lines being that of Chernozhukov, Chetverikov, Deimrer, Duflo, Hansen, Newey, and Robins. In the two years since first opening this paper to read, I have been … Continue reading A Sparse Reading of Chernozhukov et. al. (2017)

In a previous post, we used a nonlinear pricing monopoly as means to see the key ideas in mechanism design in action. In this post, I would like to show how the same basic logic can be used to study the formation of partnerships. By studying the first order conditions of the leader of a … Continue reading Mechanism Design and New Institutional Economics

One potentially unsatisfying aspect of the convergence result given in part 4 is that being based on the regret matching strategy, it is not robust against adversarial play. But recall from part 3 that if agents play according to the Blackwell strategy, their strategy will be in the limit a best response (in the sense … Continue reading Rationality from Irrationality Addendum 2: Adversary Robust Learning

In a previous post, we showed how the maximum likelihood estimator can be motivated by the fact that if a distribution belongs to some parametric model $latex \Theta$, the true parameter $latex \theta_0$ can be shown to solve a certain variational problem: $latex \theta_0 = \mathrm{argmin}_{\theta \in \Theta}\, D(f(x | \theta_0) || g(x | \theta)) … Continue reading A difficult inequality and 2 proofs

One of the most elegant corners of economic theory is mechanism design. Its apparent generality combined with the simplicity of its answers makes it an exciting field to study. But many expositions are either not rich enough to convey the beauty of the theory or too technical, relying on the tools of convex analysis, sometimes … Continue reading Mechanism Design Made Simple

It is of potential theoretical interest to know whether or not the results in the previous two posts extend to games with compact action spaces (in particular, it seems to me that these models give us a perspective from which to think about the dynamics of price discovery in oligopoly games). Like in a lot … Continue reading Rationality from Irrationality Addendum 1: Finiteness to Compactness

In the last post, we sketched the proof of a remarkable result: that agents following relatively naive strategies might still play with empirical distributions that are increasingly indistinguishable from some rational equilibrium. But we left off with one complaint: that the agents somehow are able to perfectly reconstruct the counterfactual of what would have happened … Continue reading Rationality from Irrationality Part 4: Creating Our Own Propensity Scores

Suppose we have some random vector $latex (X, Y)^T \in \mathbb R^{m+n}$ following some (non-degenerate in the sense of having positive definite covariance) multivariate Gaussian distribution with mean $latex (\mu_X, \mu_Y)$ and covariance matrix $latex \begin{pmatrix}\Sigma_{xx}& \Sigma_{xy}\\ \Sigma_{xy}^T & \Sigma_{yy}\end{pmatrix}$ It is a well known fact that the conditional distribution of $latex Y | X$ … Continue reading Multivariate Gaussian Conditioning Without Explicit Inversion

Having built up most of the prerequisite mathematical machinery, we are now finally ready to define the setting considered in Hart and Mas-Colell. We will consider games of the form $latex \Gamma = \left(N, (S^i)_{i\in N}, (u^i)_{i\in N}\right)$ where $latex N$ is the (finite) set of players, $latex S^i$ is the (finite) strategy set of … Continue reading Rationality from Irrationality Part 3: From Matching Pennies to Matching Regrets

Last time, we covered some probability tools that will be useful going forward. Today, we will turn our attention to some ideas from convex analysis that are useful for theoretical economics more broadly, but are central to Hart and Mas-Colell in particular. Let's begin with some basic definitions. Definition 1: A set $latex \mathcal C … Continue reading Rationality from Irrationality Part 2: Convex Analysis and Blackwell Approachability