Algebraic Economics

Game theory is an area that has historically benefited greatly from outside ideas. In 1950, Nash published a very influential two-page paper proving the existence of Nash equilibria for any finite game. The proof uses an elegant application of the Kakutani fixed-point theorem from the field of topology. This opened a new horizon not only in game theory, but also in areas such as economics, computer science, evolutionary biology, and social sciences. In this talk, we model different notions of equilibria in terms of undirected graphical models.The vertices of the underlying graph of the graphical model represent the players of the game and the dependencies of the choices of the players are depicted with an edge in the graph. This approach brings game theory in contact with the field of algebraic statistics for the first time, which offers a strong foundation for utilizing algebro-geometric tools to solve interesting problems in game theory. This is joint work with Javier Sendra-Arranz and Bernd Sturmfels.

Supply Network Formation and Fragility (with Matt Elliot and M. V. Leduc); American Economic Review 2022, 112(8): 2701–2747  arXiv  SSRN   Appendix  Slides

We model the production of complex goods in a large supply network. Each firm sources several essential inputs through relationships with other firms. Individual supply relationships are at risk of idiosyncratic failure, which threatens to disrupt production. To protect against this, firms multisource inputs and strategically invest to make relationships stronger, trading off the cost of investment against the benefits of increased robustness. A supply network is called fragile if aggregate output is very sensitive to small aggregate shocks. We show that supply networks of intermediate productivity are fragile in equilibrium, even though this is always inefficient. The endogenous configuration of supply networks provides a new channel for the powerful amplification of shocks.

I’ll also talk more broadly about topics in the following survey: Networks and Economic Fragility (with Matt Elliott); Annual Review of Economics, 14:665-696, August 2022

We introduce the neighbourhood lattice decomposition of a distribution, which is a compact, non-graphical representation of conditional independence that is valid in the absence of a faithful graphical representation. The idea is to view the set of neighbourhoods of a variable as a subset lattice, and partition this lattice into convex sublattices, each of which directly encodes a collection of conditional independence relations. We show that this decomposition exists in any compositional graphoid and can be computed efficiently and consistently in high-dimensions. In particular, this gives a way to encode all of independence relations implied by a distribution that satisfies the composition axiom, which is strictly weaker than the faithfulness assumption. Time permitting, we will discuss applications to interpreting large language models via partial orthogonality in embedding space.

Causal discovery procedures aim to deduce causal relationships among variables in a multivariate dataset. While various methods have been proposed for estimating a single causal model or a single equivalence class of models, less attention has been given to quantifying uncertainty in causal discovery in terms of confidence statements. The primary challenge in causal discovery is determining a causal ordering among the variables. Our research offers a framework for constructing confidence sets of causal orderings that the data do not rule out. Our methodology applies to structural equation models and is based on a residual bootstrap procedure to test the goodness-of-fit of causal orderings. We demonstrate the asymptotic validity of the confidence set constructed using this goodness-of-fit test and explain how the confidence set may be used to form sub/supersets of ancestral relationships as well as confidence intervals for causal effects that incorporate model uncertainty.

Joint work with Sam Wang and Mathias Drton.

https://arxiv.org/abs/2305.14506

In this talk, we present Bayesian and frequentist versions of finite-sample goodness-of-fit tests for three different variants of the stochastic blockmodel for network data. We make use of the fact that when the block memberships are known, stochastic blockmodels are equivalent to log-linear models. The tests for the latent block model versions combine a block membership estimator with the algebraic statistics machinery for log-linear models. We describe sufficient statistics, markov bases and marginal polytopes of these models. The general testing methodology extends to any finite mixture of log-linear models on discrete data, and as such is the first application of the algebraic statistics machinery for latent-variable models.

We consider the problem of the identification of stationary solutions to linear rational expectations models from the second moments of observable data. Observational equivalence is characterized and necessary and sufficient conditions are provided for: (i) identification under affine restrictions, (ii) generic identification under affine restrictions of analytically parametrized models, and (iii) local identification under non-linear restrictions. The results strongly resemble the classical theory for VARMA models although significant points of departure are also documented.

Social disruption occurs when a policy creates or destroys many network connections between agents. It is a costly side effect of many interventions and so a growing empirical literature recommends measuring and accounting for social disruption when evaluating the welfare impact of a policy. However, there is currently little work characterizing what can actually be learned about social disruption from data in practice. In this paper, we consider the problem of identifying social disruption in a research design that is popular in the literature. We provide two sets of identification results. First, we show that social disruption is not generally point identified, but informative bounds can be constructed using the eigenvalues of the network adjacency matrices observed by the researcher. Second, we show that point identification follows from a theoretically motivated monotonicity condition, and we derive a closed form representation. We apply our methods in two empirical illustrations and find large policy effects that otherwise might be missed by alternatives in the literature.

We propose a class of identifiable deep generative models for very flexible data types. The key features of the proposed models include (a) discrete latent layers and (b) a shrinking pyramid- or ladder-shaped deep architecture. We establish model identifiability by developing transparent conditions on the sparsity structure of the deep generative graph. The proposed identifiability conditions can ensure estimation consistency in both the Bayesian and frequentist senses. As an illustration, we consider the two-latent-layer model and propose shrinkage estimation methods to recover the latent structure and model parameters. Simulation results corroborate the identifiability of the model, and also demonstrates the excellent empirical performance of our estimation algorithm. Applications of the methodology to a DNA nucleotide sequence dataset and an educational assessment response time dataset both give interpretable results. The proposed framework provides a recipe for identifiable, interpretable, and reliable deep generative modeling

In this talk we discuss the correlated equilibrium polytope P of a game G from a combinatorial point of view.  We introduce the region of full-dimensionality for this class of polytopes, and prove that it is a semialgebraic set for any game. Through the use of the oriented matroid strata, we propose a structured method for describing the possible combinatorial types of P, and show that for (2×n)-games, the algebraic boundary of each stratum is the union of coordinate hyperplanes and binomial hypersurfaces. Finally, we provide a computational proof that there exists a unique combinatorial type of maximal dimension for (2×3)-games. This is joint work with Marie-Charlotte Brandenburg and Irem Portakal.