Partial Information Setting Research Articles

An Inexact-Penalty Method for GNE Seeking in Games With Dynamic Agents

We consider a network of autonomous agents whose outputs are actions in a game with coupled constraints. In such network scenarios, agents seek to minimize coupled cost functions using distributed information while satisfying the coupled constraints. Current methods consider the small class of multi-integrator agents using primal-dual methods. These methods can only ensure constraint satisfaction in steady-state. In contrast, we propose an inexact penalty method using a barrier function for nonlinear agents with equilibrium-independent passive dynamics. We show that these dynamics converge to an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula> -GNE while satisfying the constraints for all time, not only in steady-state. We develop these dynamics in both the full-information and partial-information settings. In the partial-information setting, dynamic estimates of the others' actions are used to make decisions and are updated through local communication. Applications to optical networks, velocity synchronization of flexible robots and wind farm optimization are provided.

On the information content of implied liquidity measure: Evidence from the S&P 500 index options

This paper aims to unfold the information content of the implied liquidity measure, which is introduced through the Conic Finance theory and considered a proxy for the market liquidity level. We propose a partial information setting in which the dynamics of the implied liquidity, representing the noisy information on the unobserved true market liquidity, follow a continuous-time Markov-chain modulated exponential Ornstein–Uhlenbeck process. Model inference requires the filtering of the unobserved states of the true market liquidity, as well as the estimation of the unknown model parameters. We address the inference problem using the EM algorithm methodology, in which we provide novel results on robust filters leading to maximum likelihood estimates. We fit the proposed model to the implied liquidity series obtained from the prices of (closest to) 1-year ATM call options on the S&P 500 covering the period from January 2002 to August 2022. The data application shows that the unobserved true market liquidity follows three regimes. The implied liquidity series contains relevant information as the filtered trajectory of the underlying Markov chain moves according to the economic environment changes due to the Federal Reserve’s actions, the global financial crisis of 2007-08, and the COVID-19 pandemic.

Bidding Graph Games with Partially-Observable Budgets

Two-player zero-sum "graph games" are central in logic, verification, and multi-agent systems. The game proceeds by placing a token on a vertex of a graph, and allowing the players to move it to produce an infinite path, which determines the winner or payoff of the game. Traditionally, the players alternate turns in moving the token. In "bidding games", however, the players have budgets and in each turn, an auction (bidding) determines which player moves the token. So far, bidding games have only been studied as full-information games. In this work we initiate the study of partial-information bidding games: we study bidding games in which a player's initial budget is drawn from a known probability distribution. We show that while for some bidding mechanisms and objectives, it is straightforward to adapt the results from the full-information setting to the partial-information setting, for others, the analysis is significantly more challenging, requires new techniques, and gives rise to interesting results. Specifically, we study games with "mean-payoff" objectives in combination with "poorman" bidding. We construct optimal strategies for a partially-informed player who plays against a fully-informed adversary. We show that, somewhat surprisingly, the "value" under pure strategies does not necessarily exist in such games.

Optimal Retirement Under Partial Information

The optimal retirement decision is an optimal stopping problem when retirement is irreversible. We investigate the optimal consumption, investment, and retirement decisions when the mean return of a risky asset is unobservable and is estimated by filtering from historical prices. To ensure nonnegativity of the consumption rate and the borrowing constraints on the wealth process of the representative agent, we conduct our analysis using a duality approach. We link the dual problem to American option pricing with stochastic volatility and prove that the duality gap is closed. We then apply our theory to a hidden Markov model for regime-switching mean return with Bayesian learning. We fully characterize the existence and uniqueness of variational inequality in the dual optimal stopping problem, as well as the free boundary of the problem. An asymptotic closed-form solution is derived for optimal retirement timing by small-scale perturbation. We discuss the potential applications of the results to other partial-information settings.

On the quartet distance given partial information

AbstractLet be an arbitrary phylogenetic tree with leaves. It is well known that the average quartet distance between two assignments of taxa to the leaves of is . However, a longstanding conjecture of Bandelt and Dress asserts that is also themaximumquartet distance between two assignments. While Alon, Naves, and Sudakov have shown this indeed holds for caterpillar trees, the general case of the conjecture is still unresolved. A natural extension is when partial information is given: the two assignments are known to coincide on a given subset of taxa. The partial information setting is biologically relevant as the location of some taxa (species) in the phylogenetic tree may be known, and for other taxa it might not be known. What can we then say about the average and maximum quartet distance in this more general setting? Surprisingly, even determining theaveragequartet distance becomes a nontrivial task in the partial information setting and determining the maximum quartet distance is even more challenging, as these turn out to be dependent on the structure of . In this paper we prove nontrivial asymptotic bounds that are sometimes tight for the average quartet distance in the partial information setting. We also show that the Bandelt and Dress conjecture does not generally hold under the partial information setting. Specifically, we prove that there are cases where the average and maximum quartet distance substantially differ.

Information Acquisition in Matching Markets: The Role of Price Discovery

We explore the acquisition and flow of information in matching markets through a model of college admissions with endogenous costly information acquisition. We extend the notion of stability to this partial information setting, and introduce regret-free stability as a refinement that additionally requires optimal student information acquisition. We show regret-free stable outcomes exist, and finding them is equivalent to finding appropriately-defined market-clearing cutoffs. To understand information flows, we recast matching mechanisms as price-discovery processes. No mechanism guarantees a regret-free stable outcome, because information deadlocks imply some students must acquire information sub optimally. Our analysis suggests approaches for facilitating efficient price discovery, leveraging historical information or market sub-samples to estimate cutoffs. We show that mechanisms that use such methods to advise applicants on their admission chances yield approximately regret-free stable outcomes. A survey of university admission systems highlights the practical importance of providing applicants with information about their admission chances.

Optimal Retirement Problem Under Partial Information

The optimal retirement decision is an optimal stopping problem when retirement is irreversible. We investigate the optimal consumption, investment, and retirement decisions when the mean return of a risky asset is unobservable and is estimated by filtering from historical prices. To ensure non-negativity of the consumption rate and the borrowing constraints on the wealth process of the representative agent, we conduct our analysis using a duality approach. We link the dual problem to American option pricing with stochastic volatility and prove that the duality gap is closed. We then apply our theory to a hidden Markov model for regime-switching mean return with Bayesian learning or the Wonham filter. We fully characterize the existence and uniqueness of variational inequality in the dual optimal stopping problem, as well as the free boundary of the problem. An asymptotic closed-form solution is derived for optimal retirement timing by small-scale perturbation. We discuss the potential applications of the results to other partial information settings.

Optimal convergence trading with unobservable pricing errors

We study a dynamic portfolio optimization problem related to convergence trading, which is an investment strategy that exploits temporary mispricing by simultaneously buying relatively underpriced assets and selling short relatively overpriced ones with the expectation that their prices converge in the future. We build on the model of Liu and Timmermann (Rev Financ Stud 26(4):1048–1086, 2013) and extend it by incorporating unobservable Markov-modulated pricing errors into the price dynamics of two co-integrated assets. We characterize the optimal portfolio strategies in full and partial information settings under the assumption of unrestricted and beta-neutral strategies. By using the innovations approach, we provide the filtering equation which is essential for solving the optimization problem under partial information. Finally, in order to illustrate the model capabilities, we provide an example with a two-state Markov chain.

Recoverability from direct quantum correlations

We address the problem of compressing density operators defined on a finite dimensional Hilbert space which assumes a tensor product decomposition. In particular, we look for an efficient procedure for learning the most likely density operator, according to ‘Jaynes’ principle, given a chosen set of partial information obtained from the unknown quantum system we wish to describe. For complexity reasons, we restrict our analysis to tree-structured sets of bipartite marginals. We focus on the tripartite scenario, where we solve the problem for the couples of measured marginals which are compatible with a quantum Markov chain, providing then an algebraic necessary and sufficient condition for the compatibility to be verified. We introduce the generalization of the procedure to the n-partite scenario, giving some preliminary results. In particular, we prove that if the pairwise Markov condition holds between the subparts then the choice of the best set of tree-structured bipartite marginals can be performed efficiently. Moreover, we provide a new characterization of quantum Markov chains in terms of quantum Bayesian updating processes.

Dynamic NE Seeking for Multi-Integrator Networked Agents With Disturbance Rejection

In this paper, we consider game problems played by (multi)-integrator agents, subject to external disturbances. We propose Nash equilibrium seeking dynamics based on gradient-play, augmented with a dynamic internal-model based component, which is a reduced-order observer of the disturbance. We consider single-, double- and extensions to multi-integrator agents, in a partial-information setting, where agents have only partial knowledge on the others' decisions over a network. The lack of global information is offset by each agent maintaining an estimate of the others' states, based on local communication with its neighbours. Each agent has an additional dynamic component that drives its estimates to the consensus subspace. In all cases, we show convergence to the Nash equilibrium irrespective of disturbances. Our proofs leverage input-to-state stability under strong monotonicity of the pseudo-gradient and Lipschitz continuity of the extended pseudo-gradient.

Fair Division with a Secretive Agent

We study classic fair-division problems in a partial information setting. This paper respectively addresses fair division of rent, cake, and indivisible goods among agents with cardinal preferences. We will show that, for all of these settings and under appropriate valuations, a fair (or an approximately fair) division among n agents can be efficiently computed using only the valuations of n − 1 agents. The nth (secretive) agent can make an arbitrary selection after the division has been proposed and, irrespective of her choice, the computed division will admit an overall fair allocation.For the rent-division setting we prove that well-behaved utilities of n − 1 agents suffice to find a rent division among n rooms such that, for every possible room selection of the secretive agent, there exists an allocation (of the remaining n − 1 rooms among the n − 1 agents) which ensures overall envy freeness (fairness). We complement this existential result by developing a polynomial-time algorithm for the case of quasilinear utilities. In this partial information setting, we also develop efficient algorithms to compute allocations that are envy-free up to one good (EF1) and ε-approximate envy free. These two notions of fairness are applicable in the context of indivisible goods and divisible goods (cake cutting), respectively.One of the main technical contributions of this paper is the development of novel connections between different fairdivision paradigms, e.g., we use our existential results for envy-free rent-division to develop an efficient EF1 algorithm.

Optimal Converge Trading with Unobservable Pricing Errors

We study a dynamic portfolio optimization problem related to convergence trading, which is an investment strategy that exploits temporary mispricing by simultaneously buying relatively underpriced assets and selling short relatively overpriced ones with the expectation that their prices converge in the future. We build on the model of Liu and Timmermann (2013) and extend it by incorporating unobservable Markov-modulated pricing errors into the price dynamics of two co-integrated assets. We characterize the optimal portfolio strategies in full and partial information settings both under the assumption of unrestricted and beta-neutral strategies. By using the innovations approach, we provide the filtering equation that is essential for solving the optimization problem under partial information. Finally, in order to illustrate the model capabilities, we provide an example with a two-state Markov chain.

Worst-Case Range Value-at-Risk with Partial Information

In this paper, we study the worst-case scenarios of a general class of risk measures, the range value-at-risk (RVaR), in single and aggregate risk models with given mean and variance, as well as symmetry and/or unimodality of each risk. For different types of partial information settings, sharp bounds for RVaR are obtained for single and aggregate risk models, together with the corresponding worst-case scenarios of marginal risks and the corresponding copula functions (dependence structure) among them. Different from the existing literature, the sharp bounds under different partial information settings in this paper are obtained via a unified method combining convex order and the recently developed notion of joint mixability. As particular cases, bounds for value-at-risk and tail value-at-risk are derived directly. Numerical examples are also provided to illustrate our results.

A multi-criteria approach to sort and rank policies based on Delphi qualitative assessments and ELECTRE TRI: The case of smart grids in Brazil

This work presents a methodology to perform a multi-criteria decision analysis based on qualitative assessments from multiple experts or stakeholders. The application that motivated these methodological developments is also described, consisting of an assessment of policies to foster technological innovations in the electricity sector, namely in a context of evolution to smart grids. Qualitative assessments were elicited from 28 experts using a Delphi process. The assessments concern the impact of policies on different objectives, but also opinions concerning the importance of these objectives. These opinions are translated into constraints on the weights of the corresponding criteria, in a partial information setting. Then, policies are sorted into categories based on a stochastic multi-criteria analysis for the ELECTRE TRI method. A new ranking approach is also developed to be compatible with the sorting results, based on a robustness analysis of ELECTRE TRI credibility indices. Results are discussed considering the entire set of experts and considering subsets of experts representing different perspectives.

Portfolio optimization for a large investor under partial information and price impact

This paper studies portfolio optimization problems in a market with partial information and price impact. We consider a large investor with an objective of expected utility maximization from terminal wealth. The drift of the underlying price process is modeled as a diffusion affected by a continuous-time Markov chain and the actions of the large investor. Using the stochastic filtering theory, we reduce the optimal control problem under partial information to the one with complete observation. For logarithmic and power utility cases we solve the utility maximization problem explicitly and we obtain optimal investment strategies in the feedback form. We compare the value functions to those for the case without price impact in Bauerle and Rieder (IEEE Trans Autom Control 49(3):442–447, 2004) and Bauerle and Rieder (J Appl Prob 362–378, 2005). It turns out that the investor would be better off due to the presence of a price impact both in complete-information and partial-information settings. Moreover, the presence of the price impact results in a shift, which depends on the distance to final time and on the state of the filter, on the optimal control strategy.

Worst-Case Range Value-at-Risk with Partial Information

In this paper, we study the worst-case scenarios of a general class of risk measures, the Range Value-at-Risk (RVaR), in single and aggregate risk models with given mean and variance, as well as symmetry and/or unimodality of each risk. For different types of partial information settings, sharp bounds for RVaR are obtained for single and aggregate risk models, together with the corresponding worst-case scenarios of marginal risks and the corresponding copula functions (dependence structure) among them. Different from the existing literature, the sharp bounds under different partial information settings in this paper are obtained via a unified method combining convex order and the recently developed notion of joint mixability. As particular cases, bounds for Value-at-Risk (VaR) and Tail Value-at-Risk (TVaR) are derived directly. Numerical examples are also provided to illustrate our results.

Optimal investment in markets with over and under-reaction to information

In this paper we introduce a jump-diffusion model of shot-noise type for stock prices, taking into account over and under-reaction of the market to incoming news. We work in a partial information setting, by supposing that standard investors do not have access to the market direction, the drift, (modeled via a random variable) after a jump. We focus on the expected (logarithmic) utility maximization problem by providing the optimal investment strategy in explicit form, both under full (i.e., from the insider point of view, aware of the right kind of market reaction at any time) and under partial information (i.e., from the standard investor viewpoint, who needs to infer the kind of market reaction from data). We test our results on market data relative to Enron and Ahold. The three main contributions of this paper are: the introduction of a new market model dealing with over and under-reaction to news, the explicit computation of the optimal filter dynamics using an original approach combining enlargement of filtrations with Innovation Theory and the application of the optimal portfolio allocation rule to market data.

Load Balancing in Heterogeneous Networks Based on Distributed Learning in Near-Potential Games

We present a novel approach for distributed load balancing in heterogeneous networks that use cell range expansion (CRE) for user association and almost blank subframe (ABS) for interference management. First, we formulate the problem as a minimization of an $\alpha$ -fairness objective function with load and outage constraints. Depending on $\alpha$ , different objectives in terms of network performance or fairness can be achieved. Next, we model the interactions among the base stations for load balancing as a near-potential game, in which the potential function is the $\alpha$ -fairness function. The optimal pure Nash equilibrium (PNE) of the game is found by using distributed learning algorithms. We propose log-linear and binary log-linear learning algorithms for complete and partial information settings, respectively. We give a detailed proof of convergence of learning algorithms for a near-potential game. We provide sufficient conditions under which the learning algorithms converge to the optimal PNE. By running extensive simulations, we show that the proposed algorithms converge within few hundreds of iterations. The convergence speed in the case of partial information setting is comparable to that of the complete information setting. Finally, we show that outage can be controlled and a better load balancing can be achieved by introducing ABS.

Optimal Investment in Markets with Over and Under-Reaction to Information

In this paper we introduce a new jump-diffusion model for stock prices, which takes into account over and under-reaction of the market to incoming news. The jumps' impact on the assets dynamics is twofold: on one hand we use a Poisson process as a driver to obtain discontinuous trajectories and on the other hand the presence of jumps in the drift, via a shot noise process, allows to incorporate fade-away'' effects, meaning that the effects of these abrupt changes fade away as time goes by. Our model is a partial information one: the drift direction after a jump is not accessible to standard investors immediately after the jump.We focus on a maximization of expected utility from terminal wealth problem, providing, in a logarithmic utility setting, the optimal investment strategy in explicit form, both under full (i.e., from the insider point of view) and under partial information (i.e., from the standard investor viewpoint). We test our results on real market data relative to Enron and Ahold.The three main contributions of this paper are: the introduction of a new market model dealing with over and under-reaction to news, the explicit computation of the optimal filter dynamics using an approach based on enlargement of filtrations and the application of the optimal portfolio allocation rule to real market data in both full and partial information setting.

Mathematical Modeling and Analysis of Product Rating with Partial Information

Many Web services like Amazon, Epinions, and TripAdvisor provide historical product ratings so that users can evaluate the quality of products. Product ratings are important because they affect how well a product will be adopted by the market. The challenge is that we only have partial information on these ratings: each user assigns ratings to only a small subset of products. Under this partial information setting, we explore a number of fundamental questions. What is the minimum number of ratings a product needs so that one can make a reliable evaluation of its quality? How may users’ misbehavior, such as cheating in product rating, affect the evaluation result? To answer these questions, we present a probabilistic model to capture various important factors (e.g., rating aggregation rules, rating behavior) that may influence the product quality assessment under the partial information setting. We derive the minimum number of ratings needed to produce a reliable indicator on the quality of a product. We extend our model to accommodate users’ misbehavior in product rating. We derive the maximum fraction of misbehaving users that a rating aggregation rule can tolerate and the minimum number of ratings needed to compensate. We carry out experiments using both synthetic and real-world data (from Amazon and TripAdvisor). We not only validate our model but also show that the “average rating rule” produces more reliable and robust product quality assessments than the “majority rating rule” and the “median rating rule” in aggregating product ratings. Last, we perform experiments on two movie rating datasets (from Flixster and Netflix) to demonstrate how to apply our framework to improve the applications of recommender systems.