On the resolution of misspecification in stochastic optimization, variational inequality, and game-theoretic problems

Traditionally, much of the research in the field of optimization algorithms has assumed that problem parameters are correctly specified. Recent efforts under the robust optimization framework have relaxed this assumption by allowing unknown parameters to vary in a prescribed uncertainty set and by subsequently solving for a worst-case solution. This dissertation considers a rather different approach in which the unknown or misspecified parameter is a solution to a suitably defined (stochastic) learning problem based on having access to a set of samples. Practical approaches in resolving such a set of coupled problems have been either sequential or direct variational approaches. In the case of the former, this entails the following steps: (i) a solution to the learning problem for parameters is first obtained; and (ii) a solution is obtained to the associated parametrized computational problem by using (i). Such avenues prove difficult to adopt particularly since the learning process has to be terminated finitely and consequently, in large-scale or stochastic instances, sequential approaches may often be corrupted by error. On the other hand, a variational approach requires that the problem may be recast as a possibly non-monotone stochastic variational inequality problem; but there are no known first-order (stochastic) schemes currently available for the solution of such problems. Motivated by these challenges, this thesis focuses on studying joint schemes of optimization and learning in three settings: (i) misspecified stochastic optimization and variational inequality problems, (ii) misspecified stochastic Nash games, (iii) misspecified Markov decision processes. In the first part of this thesis, we present a coupled stochastic approximation scheme which simultaneously solves both the optimization and the learning problems. The obtained schemes are shown to be equipped with almost sure convergence properties in regimes when the function $f$ is either strongly convex as well as merely convex. Importantly, the scheme displays the optimal rate for strongly convex problems while in merely convex regimes, through an averaging approach, we quantify the degradation associated with learning by noting that the error in function value after $K$ steps is $O(\sqrt{\ln(K)/K})$, rather than $O(\sqrt{1/K})$ when $\theta^*$ is available. Notably, when the averaging window is modified suitably, it can be see that the original rate of $O(\sqrt{1/K})$ is recovered. Additionally, we consider an online counterpart of the misspecified optimization problem and provide a non-asymptotic bound on the average regret with respect to an offline counterpart. We also extend these statements to a class of stochastic variational inequality problems, an object that unifies stochastic convex optimization problems and a range of stochastic equilibrium problems. Analogous almost-sure convergence statements are provided in strongly monotone and merely monotone regimes, the latter facilitated by using an iterative Tikhonov regularization. In the merely monotone regime, under a weak-sharpness requirement, we quantify the degradation associated with learning and show that expected error associated with $dist(x_k,X^*)$ is $O(\sqrt{\ln(K)/K})$. In the second part of this thesis, we present schemes for computing equilibria to two classes of convex stochastic Nash games complicated by a parametric misspecification, a natural concern in the control of large- scale networked engineered system. In both schemes, players learn the equilibrium strategy while resolving the misspecification: (1) Stochastic Nash games: We present a set of coupled stochastic approximation distributed schemes distributed across agents in which the first scheme updates each agent’s strategy via a projected (stochastic) gradient step while the second scheme updates every agent’s belief regarding its misspecified parameter using an independently specified learning problem. We proceed to show that the produced sequences converge to the true equilibrium strategy and the true parameter in an almost sure sense. Surprisingly, convergence in the equilibrium strategy achieves the optimal rate of convergence in a mean-squared sense with a quantifiable degradation in the rate constant; (2) Stochastic Nash-Cournot games with unobservable aggregate output: We refine (1) to a Cournot setting where we assume that the tuple of strategies is unobservable while payoff functions and strategy sets are public knowledge through a common knowledge assumption. By utilizing observations of noise-corrupted prices, iterative fixed-point schemes are developed, allowing for simultaneously learning the equilibrium strategies and the misspecified parameter in an almost-sure sense. In the third part of this thesis, we consider the solution of a finite-state infinite horizon Markov Decision Process (MDP) in which both the transition matrix and the cost function are misspecified, the latter in a parametric sense. We consider a data-driven regime in which the learning problem is a stochastic convex optimization problem that resolves misspecification. Via such a framework, we make the following contributions: (1) We first show that a misspecified value iteration scheme converges almost surely to its true counterpart and the mean-squared error after $K$ iterations is $O(\sqrt{1/K})$; (2) An analogous asymptotic almost-sure convergence statement is provided for misspecified policy iteration; and (3) Finally, we present a constant steplength misspecified Q-learning scheme and show that a suitable error metric is $O(\sqrt{1/K})$ + $O(\sqrt{δ})$ after K iterations where δ is a bound on the steplength

The emergence of nonlinear dynamics in three different economic models

A partire dai contributi seminali di Goodwin (1947), Hicks (1950) e Day (1982), nei modelli economici si \ue8 iniziato ad investigare il possibile insorgere di dinamiche non lineari. Uno dei progressi in quest'area di ricerca \ue8 stato quello di capire se tali dinamiche non lineari possono verificarsi quando si considerano comportamenti espliciti degli agenti, ovvero quando il processo decisionale di questi ultimi \ue8 definito attraverso ottimizzazione o un\u2019euristica decisionale. Lo scopo di questa tesi \ue8 quello di mostrare come l'emergere di dinamiche non lineari possa avvenire in diversi contesti economici. Nel secondo capitolo, (i) esaminiamo la letteratura sulla coevoluzione tra variabili economiche e ambiente analizzando sia modelli dinamici cruciali della teoria economica, come il modello Solow, il modello Ramsey-Cass-Koopmans, il lavoro di Day (1982) sia alcuni importanti modelli dinamici a tempo discreto; (ii) analizziamo un modello a generazioni sovrapposte in cui l'attivit\ue0 economica dipende dallo sfruttamento di una risorsa naturale ad accesso libero e nel quale inoltre si assume la presenza di una spesa pubblica per la manutenzione ambientale. Caratterizzando alcune propriet\ue0 della mappa ed eseguendo simulazioni numeriche, vengono analizzate le conseguenze dell'interazione tra spesa pubblica ambientale e settore privato e, pi\uf9 in dettaglio, discussi i diversi scenari nei quali sia equilibri multipli che dinamiche complesse (regimi caotici) possono apparire. Nel terzo capitolo, ricordando il lavoro pionieristico di Cournot (1838), si analizza un modello dinamico di duopolio in cui le imprese producono beni differenziati, i costi marginali sono assunti costanti e le funzioni di domanda microfondate. Inoltre, si suppone che le imprese adottino meccanismi decisionali diversi, basati su un grado ridotto di razionalit\ue0. In particolare, si considera che un'impresa adotti l'approccio Local Monopolistic Approximation (LMA) (Bischi et al., 2007), mentre il rivale adegua il proprio livello di produzione secondo la regola del gradiente (Bischi et al., 1999). Nell\u2019analisi vengono studiate le condizioni per la stabilit\ue0 dell'equilibrio di Nash ed alcuni scenari di biforcazione al variare dei parametri cruciali del modello. Inoltre, viene mostrato come sia un livello alto che basso di differenziazione di prodotto pu\uf2 avere un ruolo destabilizzante nel sistema. Infine, nell'ultimo capitolo, si utilizza un approccio evolutivo, come in Bischi et al. (2009), al fine di studiare l\u2019evoluzione nel lungo periodo delle decisioni delle donne riguardo l\u2019allocazione del loro tempo tra lavoro e famiglia. In particolare, si assume una popolazione composta da due sottogruppi: donne orientate alla famiglia (o family-oriented) e donne orientate alla carriera (o career-oriented). Le preferenze di entrambe le tipologie di donne sono influenzate da benefici estrinseci (ad esempio, un contratto di lavoro basato sulle performance), da costi intrinseci (cio\ue8 la loro innata propensione a trascorrere il tempo sul lavoro o con la famiglia) e dalle norme sociali. Secondo la word of mouth dynamics (Dawid, 1999), si assume inoltre che le donne interagiscano socialmente e confrontino le loro diverse posizioni, imparando su possibili differenziali di payoff. L'interazione sociale scatena quindi l'evoluzione della distribuzione dei tipi di donne (e dei comportamenti corrispondenti) nella popolazione. L'analisi permette di dimostrare come (i) siano ottenibili sia scenari in cui i due tipi di donne coesistono, sia scenari in cui uno dei due sottogruppi tende a scomparire dalla popolazione; e (ii) a causa del ruolo destabilizzante del parametro dell'intensit\ue0 di scelta (assunto come parametro esogeno), possano emergere sia cicli periodici che regimi caotici

Price competition, mergers and structural estimation in oligopoly.

This thesis examines the exercise of market power by oligopolistic firms. The first part deals with a phenomenon that has important implications for market power: horizontal mergers. I seek to uncover why the pattern of equilibria in sequential merger games of a certain type is similar across a fairly wide class of models studied in the literature. By developing general conditions characterising each element of the set of possible equilibria, I show that the solution to models that satisfy a certain sufficient condition will be restricted to the same subset of equilibria. This result is of empirical relevance in that the pattern of equilibria obtained for this class of models is associated with mergers happening, not in isolation, but rather in bunches. I extend the results to the analysis of cross-border mergers, studying two standard models that satisfy my sufficient condition: Sutton's (1991) vertically-differentiated oligopoly and Perry and Porter's (1985) fixed-supply-of-capital model. The second part is concerned with the structural inference of market power, a central theme in empirical Industrial Organisation. I demonstrate that when an industry faces potential entry and this threat of entry constrains pre-entry prices, cost and conduct cannot be identified from the comparative statics of equilibrium. In such a setting, the identifying assumption behind the well-established technique of relying on exogenous demand perturbations to distinguish empirically between alternative hypotheses of conduct is shown to fail. The Brazilian cement industry, where the threat of imports restrains market outcomes, provides an empirical illustration. In particular, price-cost margins estimated using this established technique are biased heavily downwards, underestimating the degree of market power. I propose a test of conduct, adapted to this constrained setting, which suggests that outcomes in the industry are collusive and characterised by market division. Robustness of this result is verified along several dimensions: by considering simple dynamic multimarket games which in equilibrium give rise to market division; by reviewing the spatial competition literature; and by resorting to a gravity model to statistically analyse shipments

On the role of heuristics: Experimental evidence on inflation dynamics

We carry out an experiment on a macroeconomic price setting game where prices are complements. Despite relevant information being common knowledge and price flexibility we observe significant deviation from equilibrium prices and history dependence. In a first treatment we observe that equilibrium values were obtained in the long run but at the cost of a very slow adjustment and thus history dependence. By reporting a business indicator in a simpler form, subjects were given the chance to coordinate their prices by help of a heuristic in a second treatment. This option was widely taken, bringing about excess volatility and a deviation from equilibrium even in the long run. In a third treatment with staggered pricing we observe, contrary to theoretical predictions, the one-round ahead (publicly known) shock is significant, but future inflation is not. Our findings cast light on price dynamics when subjects have limited computational capacities. --Inflation Persistence,Staggered Prices,Sticky Reasoning,New Keynesian Phillips Curve

Planning capacity expansion under oligopoly: the case of the petrochemical industry

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