arXiv subjects

Mithun Chakraborty

Publications and source records attributed to Mithun Chakraborty.

At least 19 recordsLinked to original sources

arXiv2026

Computing Perfect Bayesian Equilibria, with Application to Empirical Game-Theoretic Analysis

Perfect Bayesian Equilibrium (PBE) is a refinement of the Nash equilibrium for imperfect-information extensive-form games (EFGs) that enforces consistency between the two components of a solution: agents' strategy profile describing their decisions at information sets and the belief system quantifying their uncertainty over histories within an information set. We present a scalable approach for computing a PBE of an arbitrary two-player EFG. We adopt the definition of PBE enunciated by Bonanno in 2011 using a consistency concept based on the theory of belief revision due to Alchourrón, Gärdenfors, and Makinson. Our algorithm for finding a PBE is an adaptation of Counterfactual Regret Minimization (CFR) that minimizes the expected regret at each information set given a belief system, while maintaining the necessary consistency criteria. We prove that our algorithm is correct for two-player zero-sum games and has a reasonable slowdown in time-complexity relative to classical CFR given the additional computation needed for refinement. We also experimentally demonstrate the competent performance of PBE-CFR in terms of equilibrium quality and running time on medium-to-large non-zero-sum EFGs. Finally, we investigate the effectiveness of using PBE for strategy exploration in empirical game-theoretic analysis. Specifically, we compute PBE as a meta-strategy solver (MSS) in a tree-exploiting variant of Policy Space Response Oracles (TE-PSRO). Our experiments show that PBE as an MSS leads to higher-quality empirical EFG models with complex imperfect information structures compared to MSSs based on an unrefined Nash equilibrium.

cs.GT↗

arXiv2026

FAIRFORMER: A transformer architecture for discrete fair division

We propose a deep neural network-based solution to the problem of allocating indivisible goods under additive subjective valuations without monetary transfers, trading off economic efficiency with envy-based fairness. We introduce FairFormer, an amortized, permutation-equivariant two-tower transformer that encodes items and agents as unordered token sets, applies self-attention within each set, and uses item-to-agent cross-attention to produce per-item assignment distributions in a single forward pass. FairFormer is trained end-to-end to maximize expected log-Nash welfare on sampled instances, requiring no solver supervision, unrolled allocation procedures, or fairness labels. At test time, we discretize by row-wise $\arg\max$ and apply a lightweight post-processing routine that transfers items to eliminate violations of envy-freeness up to one item while prioritizing improvements in Nash welfare. Our approach generalizes beyond its training regime and achieves near-optimal welfare (e.g., for uniformly sampled valuations, $96$--$97\%$ for Nash welfare; $95$--$96\%$ for utilitarian welfare), outperforming strong baselines in solution quality and/or runtime.

cs.GT↗

arXiv2025

Policy Abstraction and Nash Refinement in Tree-Exploiting PSRO

Policy Space Response Oracles (PSRO) interleaves empirical game-theoretic analysis with deep reinforcement learning (DRL) to solve games too complex for traditional analytic methods. Tree-exploiting PSRO (TE-PSRO) is a variant of this approach that iteratively builds a coarsened empirical game model in extensive form using data obtained from querying a simulator that represents a detailed description of the game. We make two main methodological advances to TE-PSRO that enhance its applicability to complex games of imperfect information. First, we introduce a scalable representation for the empirical game tree where edges correspond to implicit policies learned through DRL. These policies cover conditions in the underlying game abstracted in the game model, supporting sustainable growth of the tree over epochs. Second, we leverage extensive form in the empirical model by employing refined Nash equilibria to direct strategy exploration. To enable this, we give a modular and scalable algorithm based on generalized backward induction for computing a subgame perfect equilibrium (SPE) in an imperfect-information game. We experimentally evaluate our approach on a suite of games including an alternating-offer bargaining game with outside offers; our results demonstrate that TE-PSRO converges toward equilibrium faster when new strategies are generated based on SPE rather than Nash equilibrium, and with reasonable time/memory requirements for the growing empirical model.

cs.GT↗

arXiv2024

Weighted Fairness Notions for Indivisible Items Revisited

We revisit the setting of fairly allocating indivisible items when agents have different weights representing their entitlements. First, we propose a parameterized family of relaxations for weighted envy-freeness and the same for weighted proportionality; the parameters indicate whether smaller-weight or larger-weight agents should be given a higher priority. We show that each notion in these families can always be satisfied, but any two cannot necessarily be fulfilled simultaneously. We then introduce an intuitive weighted generalization of maximin share fairness and establish the optimal approximation of it that can be guaranteed. Furthermore, we characterize the implication relations between the various weighted fairness notions introduced in this and prior work, and relate them to the lower and upper quota axioms from apportionment.

cs.GT↗

arXiv2023

Exploiting Extensive-Form Structure in Empirical Game-Theoretic Analysis

Empirical game-theoretic analysis (EGTA) is a general framework for reasoning about complex games using agent-based simulation. Data from simulating select strategy profiles is employed to estimate a cogent and tractable game model approximating the underlying game. To date, EGTA methodology has focused on game models in normal form; though the simulations play out in sequential observations and decisions over time, the game model abstracts away this temporal structure. Richer models of \textit{extensive-form games} (EFGs) provide a means to capture temporal patterns in action and information, using tree representations. We propose \textit{tree-exploiting EGTA} (TE-EGTA), an approach to incorporate EFG models into EGTA\@. TE-EGTA constructs game models that express observations and temporal organization of activity, albeit at a coarser grain than the underlying agent-based simulation model. The idea is to exploit key structure while maintaining tractability. We establish theoretically and experimentally that exploiting even a little temporal structure can vastly reduce estimation error in strategy-profile payoffs compared to the normal-form model. Further, we explore the implications of EFG models for iterative approaches to EGTA, where strategy spaces are extended incrementally. Our experiments on several game instances demonstrate that TE-EGTA can also improve performance in the iterative setting, as measured by the quality of equilibrium approximation as the strategy spaces are expanded.

cs.GT↗

arXiv2022

A Game-Theoretic Approach for Hierarchical Epidemic Control

We design and analyze a multi-level game-theoretic model of hierarchical policy interventions for epidemic control, such as those in response to the COVID-19 pandemic. Our model captures the potentially mismatched priorities among a hierarchy of policy-makers (e.g., federal, state, and local governments) with respect to two cost components that have opposite dependence on the policy strength -- post-intervention infection rates and the socio-economic cost of policy implementation. Additionally, our model includes a crucial third factor in decisions: a cost of non-compliance with the policy-maker immediately above in the hierarchy, such as non-compliance of counties with state-level policies. We propose two novel algorithms for approximating solutions to such games. The first is based on best response dynamics (BRD), and exploits the tree structure of the game. The second combines quadratic integer programming (QIP), which enables us to collapse the two lowest levels of the game, with best response dynamics. Through extensive experiments, we show that our QIP-based approach significantly outperforms the BRD algorithm both in running time and the quality of equilibrium solutions. Finally, we apply the QIP-based algorithm to experiments based on both synthetic and real-world data under various parameter configurations and analyze the resulting (approximate) equilibria to gain insight into the impact of decentralization on overall welfare (measured as the negative sum of costs) as well as emergent properties like free-riding and fairness in cost distribution among policy-makers.

cs.GT↗

arXiv2022

Solving Structured Hierarchical Games Using Differential Backward Induction

From large-scale organizations to decentralized political systems, hierarchical strategic decision making is commonplace. We introduce a novel class of structured hierarchical games (SHGs) that formally capture such hierarchical strategic interactions. In an SHG, each player is a node in a tree, and strategic choices of players are sequenced from root to leaves, with root moving first, followed by its children, then followed by their children, and so on until the leaves. A player's utility in an SHG depends on its own decision, and on the choices of its parent and all the tree leaves. SHGs thus generalize simultaneous-move games, as well as Stackelberg games with many followers. We leverage the structure of both the sequence of player moves as well as payoff dependence to develop a gradient-based back propagation-style algorithm, which we call Differential Backward Induction (DBI), for approximating equilibria of SHGs. We provide a sufficient condition for convergence of DBI and demonstrate its efficacy in finding approximate equilibrium solutions to several SHG models of hierarchical policy-making problems.

cs.GT↗

arXiv2021

Picking Sequences and Monotonicity in Weighted Fair Division

We study the problem of fairly allocating indivisible items to agents with different entitlements, which captures, for example, the distribution of ministries among political parties in a coalition government. Our focus is on picking sequences derived from common apportionment methods, including five traditional divisor methods and the quota method. We paint a complete picture of these methods in relation to known envy-freeness and proportionality relaxations for indivisible items as well as monotonicity properties with respect to the resource, population, and weights. In addition, we provide characterizations of picking sequences satisfying each of the fairness notions, and show that the well-studied maximum Nash welfare solution fails resource- and population-monotonicity even in the unweighted setting. Our results serve as an argument in favor of using picking sequences in weighted fair division problems.

cs.GT↗

arXiv2021

Finding Fair and Efficient Allocations When Valuations Don't Add Up

In this paper, we present new results on the fair and efficient allocation of indivisible goods to agents whose preferences correspond to {\em matroid rank functions}. This is a versatile valuation class with several desirable properties (such as monotonicity and submodularity), which naturally lends itself to a number of real-world domains. We use these properties to our advantage; first, we show that when agent valuations are matroid rank functions, a socially optimal (i.e. utilitarian social welfare-maximizing) allocation that achieves envy-freeness up to one item (EF1) exists and is computationally tractable. We also prove that the Nash welfare-maximizing and the leximin allocations both exhibit this fairness/efficiency combination, by showing that they can be achieved by minimizing any symmetric strictly convex function over utilitarian optimal outcomes. To the best of our knowledge, this is the first valuation function class not subsumed by additive valuations for which it has been established that an allocation maximizing Nash welfare is EF1. Moreover, for a subclass of these valuation functions based on maximum (unweighted) bipartite matching, we show that a leximin allocation can be computed in polynomial time. Additionally, we explore possible extensions of our results to fairness criteria other than EF1 as well as to generalizations of the above valuation classes.

cs.AI↗

arXiv2021

Weighted Envy-Freeness in Indivisible Item Allocation

We introduce and analyze new envy-based fairness concepts for agents with weights that quantify their entitlements in the allocation of indivisible items. We propose two variants of weighted envy-freeness up to one item (WEF1): strong, where envy can be eliminated by removing an item from the envied agent's bundle, and weak, where envy can be eliminated either by removing an item (as in the strong version) or by replicating an item from the envied agent's bundle in the envying agent's bundle. We show that for additive valuations, an allocation that is both Pareto optimal and strongly WEF1 always exists and can be computed in pseudo-polynomial time; moreover, an allocation that maximizes the weighted Nash social welfare may not be strongly WEF1, but always satisfies the weak version of the property. Moreover, we establish that a generalization of the round-robin picking sequence algorithm produces in polynomial time a strongly WEF1 allocation for an arbitrary number of agents; for two agents, we can efficiently achieve both strong WEF1 and Pareto optimality by adapting the adjusted winner procedure. Our work highlights several aspects in which weighted fair division is richer and more challenging than its unweighted counterpart.

cs.AI↗

arXiv2020

The Price of Quota-based Diversity in Assignment Problems

We introduce and analyze an extension to the matching problem on a weighted bipartite graph: Assignment with Type Constraints. The two parts of the graph are partitioned into subsets called types and blocks; we seek a matching with the largest sum of weights under the constraint that there is a pre-specified cap on the number of vertices matched in every type-block pair. Our primary motivation stems from the public housing program of Singapore, accounting for over 70% of its residential real estate. To promote ethnic diversity within its housing projects, Singapore imposes ethnicity quotas: each new housing development comprises blocks of flats and each ethnicity-based group in the population must not own more than a certain percentage of flats in a block. Other domains using similar hard capacity constraints include matching prospective students to schools or medical residents to hospitals. Limiting agents' choices for ensuring diversity in this manner naturally entails some welfare loss. One of our goals is to study the trade-off between diversity and social welfare in such settings. We first show that, while the classic assignment program is polynomial-time computable, adding diversity constraints makes it computationally intractable; however, we identify a $\tfrac{1}{2}$-approximation algorithm, as well as reasonable assumptions on the weights that permit poly-time algorithms. Next, we provide two upper bounds on the price of diversity -- a measure of the loss in welfare incurred by imposing diversity constraints -- as functions of natural problem parameters. We conclude the paper with simulations based on publicly available data from two diversity-constrained allocation problems -- Singapore Public Housing and Chicago School Choice -- which shed light on how the constrained maximization as well as lottery-based variants perform in practice.

cs.AI↗

arXiv2015

How to show a probabilistic model is better

We present a simple theoretical framework, and corresponding practical procedures, for comparing probabilistic models on real data in a traditional machine learning setting. This framework is based on the theory of proper scoring rules, but requires only basic algebra and probability theory to understand and verify. The theoretical concepts presented are well-studied, primarily in the statistics literature. The goal of this paper is to advocate their wider adoption for performance evaluation in empirical machine learning.

stat.ML↗

arXiv2014

On Manipulation in Prediction Markets When Participants Influence Outcomes Directly

Prediction markets are often used as mechanisms to aggregate information about a future event, for example, whether a candidate will win an election. The event is typically assumed to be exogenous. In reality, participants may influence the outcome, and therefore (1) running the prediction market could change the incentives of participants in the process that creates the outcome (for example, agents may want to change their vote in an election), and (2) simple results such as the myopic incentive compatibility of proper scoring rules no longer hold in the prediction market itself. We introduce a model of games of this kind, where agents first trade in a prediction market and then take an action that influences the market outcome. Our two-stage two-player model, despite its simplicity, captures two aspects of real-world prediction markets: (1) agents may directly influence the outcome, (2) some of the agents instrumental in deciding the outcome may not take part in the prediction market. We show that this game has two different types of perfect Bayesian equilibria, which we term LPP and HPP, depending on the values of the belief parameters: in the LPP domain, equilibrium prices reveal expected market outcomes conditional on the participants' private information, whereas HPP equilibria are collusive -- participants effectively coordinate in an uninformative and untruthful way.

cs.GT↗

arXiv2012

Near-Optimal Target Learning With Stochastic Binary Signals

We study learning in a noisy bisection model: specifically, Bayesian algorithms to learn a target value V given access only to noisy realizations of whether V is less than or greater than a threshold theta. At step t = 0, 1, 2, ..., the learner sets threshold theta t and observes a noisy realization of sign(V - theta t). After T steps, the goal is to output an estimate V^ which is within an eta-tolerance of V . This problem has been studied, predominantly in environments with a fixed error probability q < 1/2 for the noisy realization of sign(V - theta t). In practice, it is often the case that q can approach 1/2, especially as theta -> V, and there is little known when this happens. We give a pseudo-Bayesian algorithm which provably converges to V. When the true prior matches our algorithm's Gaussian prior, we show near-optimal expected performance. Our methods extend to the general multiple-threshold setting where the observation noisily indicates which of k >= 2 regions V belongs to.

cs.LG↗

arXiv2008

A Novel Approach for Complete Identification of Dynamic Fractional Order Systems Using Stochastic Optimization Algorithms and Fractional Calculus

This contribution deals with identification of fractional-order dynamical systems. System identification, which refers to estimation of process parameters, is a necessity in control theory. Real processes are usually of fractional order as opposed to the ideal integral order models. A simple and elegant scheme of estimating the parameters for such a fractional order process is proposed. This method employs fractional calculus theory to find equations relating the parameters that are to be estimated, and then estimates the process parameters after solving the simultaneous equations. The said simultaneous equations are generated and updated using particle swarm optimization (PSO) technique, the fitness function being the sum of squared deviations from the actual set of observations. The data used for the calculations are intentionally corrupted to simulate real-life conditions. Results show that the proposed scheme offers a very high degree of accuracy even for erroneous data.

cs.OH↗

arXiv2008

A Study of the Grunwald-Letnikov Definition for Minimizing the Effects of Random Noise on Fractional Order Differential Equations

Of the many definitions for fractional order differintegral, the Grunwald-Letnikov definition is arguably the most important one. The necessity of this definition for the description and analysis of fractional order systems cannot be overstated. Unfortunately, the Fractional Order Differential Equation (FODE) describing such a systems, in its original form, highly sensitive to the effects of random noise components inevitable in a natural environment. Thus direct application of the definition in a real-life problem can yield erroneous results. In this article, we perform an in-depth mathematical analysis the Grunwald-Letnikov definition in depth and, as far as we know, we are the first to do so. Based on our analysis, we present a transformation scheme which will allow us to accurately analyze generalized fractional order systems in presence of significant quantities of random errors. Finally, by a simple experiment, we demonstrate the high degree of robustness to noise offered by the said transformation and thus validate our scheme.

cs.OH↗

arXiv2008

The Application of Stochastic Optimization Algorithms to the Design of a Fractional-order PID Controller

The Proportional-Integral-Derivative Controller is widely used in industries for process control applications. Fractional-order PID controllers are known to outperform their integer-order counterparts. In this paper, we propose a new technique of fractional-order PID controller synthesis based on peak overshoot and rise-time specifications. Our approach is to construct an objective function, the optimization of which yields a possible solution to the design problem. This objective function is optimized using two popular bio-inspired stochastic search algorithms, namely Particle Swarm Optimization and Differential Evolution. With the help of a suitable example, the superiority of the designed fractional-order PID controller to an integer-order PID controller is affirmed and a comparative study of the efficacy of the two above algorithms in solving the optimization problem is also presented.

cs.OH↗

arXiv2008

Tuning PID and FOPID Controllers using the Integral Time Absolute Error Criterion

Particle swarm optimization (PSO) is extensively used for real parameter optimization in diverse fields of study. This paper describes an application of PSO to the problem of designing a fractional-order proportional-integral-derivative (FOPID) controller whose parameters comprise proportionality constant, integral constant, derivative constant, integral order (lambda) and derivative order (delta). The presence of five optimizable parameters makes the task of designing a FOPID controller more challenging than conventional PID controller design. Our design method focuses on minimizing the Integral Time Absolute Error (ITAE) criterion. The digital realization of the deigned system utilizes the Tustin operator-based continued fraction expansion scheme. We carry out a simulation that illustrates the effectiveness of the proposed approach especially for realizing fractional-order plants. This paper also attempts to study the behavior of fractional PID controller vis-a-vis that of its integer order counterpart and demonstrates the superiority of the former to the latter.

cs.OH↗