Mean-field games of speedy information access with observation costs

We investigate a mean-field game (MFG) in which agents can exercise control actions that affect their speed of access to information. The agents can dynamically decide to receive observations with less delay by paying higher observation costs. Agents seek to exploit their active information gathering by making further decisions to influence their state dynamics to maximize rewards. In the mean field equilibrium, each generic agent solves individually a partially observed Markov decision problem in which the way partial observations are obtained is itself also subject of dynamic control actions by the agent. Based on a finite characterisation of the agents' belief states, we show how the mean field game with controlled costly information access can be formulated as an equivalent standard mean field game on a suitably augmented but finite state space.We prove that with sufficient entropy regularisation, a fixed point iteration converges to the unique MFG equilibrium and yields an approximate $\epsilon$-Nash equilibrium for a large but finite population size. We illustrate our MFG by an example from epidemiology, where medical testing results at different speeds and costs can be chosen by the agents.Comment: 33 pages, 4 figure

Large Banks and Systemic Risk: Insights from a Mean-Field Game Model

This paper aims to investigate the impact of large banks on the financial system stability. To achieve this, we employ a linear-quadratic-Gaussian (LQG) mean-field game (MFG) model of an interbank market, which involves one large bank and multiple small banks. Our approach involves utilizing the MFG methodology to derive the optimal trading strategies for each bank, resulting in an equilibrium for the market. Subsequently, we conduct Monte Carlo simulations to explore the role played by the large bank in systemic risk under various scenarios. Our findings indicate that while the major bank, if its size is not too large, can contribute positively to stability, it also has the potential to generate negative spillover effects in the event of default, leading to increased systemic risk. We also discover that as banks become more reliant on the interbank market, the overall system becomes more stable but the probability of a rare systemic failure increases. This risk is further amplified by the presence of a large bank, its size, and the speed of interbank trading. Overall, the results of this study provide important insights into the management of systemic risk

LQG Risk-Sensitive Mean Field Games with a Major Agent: A Variational Approach

Risk sensitivity plays an important role in the study of finance and economics as risk-neutral models cannot capture and justify all economic behaviors observed in reality. Risk-sensitive mean field game theory was developed recently for systems where there exists a large number of indistinguishable, asymptotically negligible and heterogeneous risk-sensitive players, who are coupled via the empirical distribution of state across population. In this work, we extend the theory of Linear Quadratic Gaussian risk-sensitive mean-field games to the setup where there exists one major agent as well as a large number of minor agents. The major agent has a significant impact on each minor agent and its impact does not collapse with the increase in the number of minor agents. Each agent is subject to linear dynamics with an exponential-of-integral quadratic cost functional. Moreover, all agents interact via the average state of minor agents (so-called empirical mean field) and the major agent's state. We develop a variational analysis approach to derive the best response strategies of agents in the limiting case where the number of agents goes to infinity. We establish that the set of obtained best-response strategies yields a Nash equilibrium in the limiting case and an $\varepsilon$-Nash equilibrium in the finite player case. We conclude the paper with an illustrative example

Dynamics of Social Networks: Multi-agent Information Fusion, Anticipatory Decision Making and Polling

This paper surveys mathematical models, structural results and algorithms in controlled sensing with social learning in social networks. Part 1, namely Bayesian Social Learning with Controlled Sensing addresses the following questions: How does risk averse behavior in social learning affect quickest change detection? How can information fusion be priced? How is the convergence rate of state estimation affected by social learning? The aim is to develop and extend structural results in stochastic control and Bayesian estimation to answer these questions. Such structural results yield fundamental bounds on the optimal performance, give insight into what parameters affect the optimal policies, and yield computationally efficient algorithms. Part 2, namely, Multi-agent Information Fusion with Behavioral Economics Constraints generalizes Part 1. The agents exhibit sophisticated decision making in a behavioral economics sense; namely the agents make anticipatory decisions (thus the decision strategies are time inconsistent and interpreted as subgame Bayesian Nash equilibria). Part 3, namely {\em Interactive Sensing in Large Networks}, addresses the following questions: How to track the degree distribution of an infinite random graph with dynamics (via a stochastic approximation on a Hilbert space)? How can the infected degree distribution of a Markov modulated power law network and its mean field dynamics be tracked via Bayesian filtering given incomplete information obtained by sampling the network? We also briefly discuss how the glass ceiling effect emerges in social networks. Part 4, namely \emph{Efficient Network Polling} deals with polling in large scale social networks. In such networks, only a fraction of nodes can be polled to determine their decisions. Which nodes should be polled to achieve a statistically accurate estimates

Essays on strategic trading

This dissertation discusses various aspects of strategic trading using both analytical modeling and numerical methods. Strategic trading, in short, encompasses models of trading, most notably models of optimal execution and portfolio selection, in which one seeks to rigorously consider various---both explicit and implicit---costs stemming from the act of trading itself. The strategic trading approach, rooted in the market microstructure literature, contrasts with many classical finance models in which markets are assumed to be frictionless and traders can, for the most part, take prices as given. Introducing trading costs to dynamic models of financial markets tend to complicate matters. First, the objectives of the traders become more nuanced since now overtrading leads to poor outcomes due to increased trading costs. Second, when trades affect prices and there are multiple traders in the market, the traders start to behave in a more calculated fashion, taking into account both their own objectives and the perceived actions of others. Acknowledging this strategic behavior is especially important when the traders are asymmetrically informed. These new features allow the models discussed to better reflect aspects real-world trading, for instance, intraday trading patterns, and enable one to ask and answer new questions, for instance, related to the interactions between different traders. To efficiently analyze the models put forth, numerical methods must be utilized. This is, as is to be expected, the price one must pay from added complexity. However, it also opens an opportunity to have a closer look at the numerical approaches themselves. This opportunity is capitalized on and various new and novel computational procedures influenced by the growing field of numerical real algebraic geometry are introduced and employed. These procedures are utilizable beyond the scope of this dissertation and enable one to sharpen the analysis of dynamic equilibrium models.Tämä väitöskirja käsittelee strategista kaupankäyntiä hyödyntäen sekä analyyttisiä että numeerisia menetelmiä. Strategisen kaupankäynnin mallit, erityisesti optimaalinen kauppojen toteutus ja portfolion valinta, pyrkivät tarkasti huomioimaan kaupankäynnistä itsestään aiheutuvat eksplisiittiset ja implisiittiset kustannukset. Tämä erottaa strategisen kaupankäynnin mallit klassisista kitkattomista malleista. Kustannusten huomioiminen rahoitusmarkkinoiden dynaamisessa tarkastelussa monimutkaistaa malleja. Ensinnäkin kaupankävijöiden tavoitteet muuttuvat hienovaraisemmiksi, koska liian aktiivinen kaupankäynti johtaa korkeisiin kaupankäyntikuluihin ja heikkoon tuottoon. Toiseksi oletus siitä, että kaupankävijöiden valitsemat toimet vaikuttavat hintoihin, johtaa pelikäyttäytymiseen silloin, kun markkinoilla on useampia kaupankävijöitä. Pelikäyttäytymisen huomioiminen on ensiarvoisen tärkeää, mikäli informaatio kaupankävijöiden kesken on asymmetristä. Näiden piirteiden johdosta tässä väitöskirjassa käsitellyt mallit mahdollistavat abstrahoitujen rahoitusmarkkinoiden aiempaa täsmällisemmän tarkastelun esimerkiksi päivänsisäisen kaupankäynnin osalta. Tämän lisäksi mallien avulla voidaan löytää vastauksia uusiin kysymyksiin, kuten esimerkiksi siihen, millaisia ovat kaupankävijöiden keskinäiset vuorovaikutussuhteet dynaamisilla markkinoilla. Monimutkaisten mallien analysointiin hyödynnetään numeerisia menetelmiä. Tämä avaa mahdollisuuden näiden menetelmien yksityiskohtaisempaan tarkasteluun, ja tätä mahdollisuutta hyödynnetään pohtimalla laskennallisia ratkaisuja tuoreesta numeerista reaalista algebrallista geometriaa hyödyntävästä näkökulmasta. Väitöskirjassa esitellyt uudet laskennalliset ratkaisut ovat laajalti hyödynnettävissä, ja niiden avulla on mahdollista terävöittää dynaamisten tasapainomallien analysointia

A two-player price impact game

We study the competition of two strategic agents for liquidity in the benchmark portfolio tracking setup of Bank, Soner, Voss (2017), both facing common aggregated temporary and permanent price impact \`a la Almgren and Chriss (2001). The resulting stochastic linear quadratic differential game with terminal state constraints allows for an explicitly available open-loop Nash equilibrium in feedback form. Our results reveal how the equilibrium strategies of the two players take into account the other agent's trading targets: either in an exploitative intent or by providing liquidity to the competitor, depending on the ratio between temporary and permanent price impact. As a consequence, different behavioral patterns can emerge as optimal in equilibrium. These insights complement existing studies in the literature on predatory trading models examined in the context of optimal portfolio liquidation problems.Comment: 41 pages, 6 figure

The variational structure and time-periodic solutions for mean-field games systems

Here, we observe that mean-field game (MFG) systems admit a two-player infinite-dimensional general-sum differential game formulation. We show that particular regimes of this game reduce to previously known variational principles. Furthermore, based on the game-perspective we derive new variational formulations for first-order MFG systems with congestion. Finally, we use these findings to prove the existence of time-periodic solutions for viscous MFG systems with a coupling that is not a non-decreasing function of density.Comment: 31 page