Theory of agent-based market models with controlled levels of greed and anxiety

We use generating functional analysis to study minority-game type market models with generalized strategy valuation updates that control the psychology of agents' actions. The agents' choice between trend following and contrarian trading, and their vigor in each, depends on the overall state of the market. Even in `fake history' models, the theory now involves an effective overall bid process (coupled to the effective agent process) which can exhibit profound remanence effects and new phase transitions. For some models the bid process can be solved directly, others require Maxwell-construction type approximations.Comment: 30 pages, 10 figure

Optimization frameworks and sensitivity analysis of Stackelberg mean-field games

This paper proposes and studies a class of discrete-time finite-time-horizon Stackelberg mean-field games, with one leader and an infinite number of identical and indistinguishable followers. In this game, the objective of the leader is to maximize her reward considering the worst-case cost over all possible $\epsilon$-Nash equilibria among followers. A new analytical paradigm is established by showing the equivalence between this Stackelberg mean-field game and a minimax optimization problem. This optimization framework facilitates studying both analytically and numerically the set of Nash equilibria for the game; and leads to the sensitivity and the robustness analysis of the game value. In particular, when there is model uncertainty, the game value for the leader suffers non-vanishing sub-optimality as the perturbed model converges to the true model. In order to obtain a near-optimal solution, the leader needs to be more pessimistic with anticipation of model errors and adopts a relaxed version of the original Stackelberg game

Social Optima in Leader-Follower Mean Field Linear Quadratic Control

This paper investigates a linear quadratic mean field leader-follower team problem, where the model involves one leader and a large number of weakly-coupled interactive followers. The leader and the followers cooperate to optimize the social cost. Specifically, for any strategy provided first by the leader, the followers would like to choose a strategy to minimize social cost functional. Using variational analysis and person-by-person optimality, we construct two auxiliary control problems. By solving sequentially the auxiliary control problems with consistent mean field approximations, we can obtain a set of decentralized social optimality strategy with help of a class of forward-backward consistency systems. The relevant Stackelberg equilibrium is further proved under some proper conditions

Herd behavior, the "Penguin effect", and the suppression of informational diffusion: An analysis of informational externalities and payoff interdependency

Technology;Information;Econometric Models;Group Behaviour

Control and estimation with limited information: a game-theoretic approach

Modern control systems can be viewed as interconnections of spatially distributed multiple subsystems, where the individual subsystems share their information with each other through an underlying network that inherently introduces limitations on information flow. Inherent limitations on the flow of information among individual subsystems may stem from structural constraints of the network and/or communication constraints of the network. Hence, in order to design optimal control and estimation mechanisms for modern control systems, we must answer the following two practical but important questions: (1) What are the fundamental communication limits to achieve a desired control performance and stability? (2) What are the approaches one has to adopt to design a decentralized controller for a complex system to deal with structural constraints? In this thesis, we consider four different problems within a game-theoretic framework to address the above questions. The first part of the thesis considers problems of control and estimation with limited communication, which correspond to question (1) above. We first consider the minimax estimation problem with intermittent observations. In this setting, the disturbance in the dynamical system as well as the sensor noise are controlled by adversaries, and the estimator receives the sensor measurements only sporadically, with availability governed by an independent and identically distributed (i.i.d.) Bernoulli process. This problem is cast in the thesis within the framework of stochastic zero-sum dynamic games. First, a corresponding stochastic minimax state estimator (SMSE) is obtained, along with an associated generalized stochastic Riccati equation (GSRE). Then, the asymptotic behavior of the estimation error in terms of the GSRE is analyzed. We obtain threshold-type conditions on the rate of intermittent observations and the disturbance attenuation parameter, above which 1) the expected value of the GSRE is bounded from below and above by deterministic quantities, and 2) the norm of the sequence generated by the GSRE converges weakly to a unique stationary distribution. We then study the minimax control problem over unreliable communication channels. The transmission of packets from the plant output sensors to the controller, and from the controller to the plant, are over sporadically failing channels governed by two independent i.i.d. Bernoulli processes. Two different scenarios for unreliable communication channels are considered. The first one is when the communication channel provides perfect acknowledgments of successful transmissions of control packets through a clean reverse channel, which is the TCP (Transmission Control Protocol), and the second one is when there is no acknowledgment, which is the UDP (User Datagram Protocol). Under both scenarios, the thesis obtains output feedback minimax controllers; it also identifies a set of explicit existence conditions in terms of the disturbance attenuation parameter and the communication channel loss rates, above which the corresponding minimax controller achieves the desired performance and stability. In the second part of the thesis, we consider two different large-scale optimization problems via mean field game theory, which address structural constraints in the complex system stated in question (2) above. We first consider two classes of mean field games. The first problem (P1) is one where each agent minimizes an exponentiated performance index, capturing risk-sensitive behavior, whereas in the second problem (P2) each agent minimizes a worst-case risk-neutral performance index, where a fictitious agent or an adversary enters each agent's state system. For both problems, a mean field system for the corresponding problem is constructed to arrive at a best estimate of the actual mean field behavior in various senses in the large population regime. In the finite population regime, we show that there exist epsilon-Nash equilibria for both P1 and P2, where the corresponding individual Nash strategies are decentralized as functions of the local state information. In both cases, the positive parameter epsilon can be taken to be arbitrarily small as the population size grows. Finally, we show that the Nash equilibria for P1 and P2 both feature robustness due to the risk-sensitive and worst-case behaviors of the agents. In the last main chapter of the thesis, we study mean field Stackelberg differential games. There is one leader and a large number, say N, of followers. The leader holds a dominating position in the game, where he first chooses and then announces his optimal strategy, to which the N followers respond by playing a Nash game. The followers are coupled with each other through the mean field term, and are strongly influenced by the leader's strategy. From the leader's perspective, he is coupled with the N followers through the mean field term. In this setting, we characterize an approximated stochastic mean field process of the followers governed by the leader's strategy, which leads to a decentralized epsilon-Nash-Stackelberg equilibrium. As a consequence of decentralization, we subsequently show that the positive parameter epsilon can be picked arbitrarily small when the number of followers is arbitrarily large. In the thesis, we also include several numerical computations and simulations, which illustrate the theoretical results

Long ties accelerate noisy threshold-based contagions

Network structure can affect when and how widely new ideas, products, and behaviors are adopted. In widely-used models of biological contagion, interventions that randomly rewire edges (generally making them "longer") accelerate spread. However, there are other models relevant to social contagion, such as those motivated by myopic best-response in games with strategic complements, in which an individual's behavior is described by a threshold number of adopting neighbors above which adoption occurs (i.e., complex contagions). Recent work has argued that highly clustered, rather than random, networks facilitate spread of these complex contagions. Here we show that minor modifications to this model, which make it more realistic, reverse this result: we allow very rare below-threshold adoption, i.e., rarely adoption occurs when there is only one adopting neighbor. To model the trade-off between long and short edges we consider networks that are the union of cycle-power-$k$ graphs and random graphs on $n$ nodes. Allowing adoptions below threshold to occur with order $1/\sqrt{n}$ probability along some "short" cycle edges is enough to ensure that random rewiring accelerates spread. Simulations illustrate the robustness of these results to other commonly-posited models for noisy best-response behavior. Hypothetical interventions that randomly rewire existing edges or add random edges (versus adding "short", triad-closing edges) in hundreds of empirical social networks reduce time to spread. This revised conclusion suggests that those wanting to increase spread should induce formation of long ties, rather than triad-closing ties. More generally, this highlights the importance of noise in game-theoretic analyses of behavior