Search Results (5)

We introduce Causal Incentive Design (CID), a framework that applies causal inference to canonical single-stage principal–agent problems (PAPs) characterized by bilateral private information. Within CID, the operating rules of PAPs are formalized using an additive-noise causal graphical model (CGM). Incentives are modeled as interventions on a function space variable, $\Gamma$, which correspond to policy interventions in the principal–follower causal relation. The causal inference target estimand $V(\Gamma )$ is defined as the expected value of the principal’s utility variable under a specified policy intervention in the post-intervention distribution. In the context of additive-Gaussian independent noise, the estimand $V(\Gamma )$ decomposes into a two-layer expectation: (i) an inner Gaussian smoothing of the principal’s utility regression; and (ii) an outer averaging over the conditional probability of the follower’s action given the incentive policy. A Gauss–Hermite quadrature method is employed to efficiently estimate the first layer, while a policy-local kernel reweighting approach is used for the second. For offline selection of a single incentive policy, a Functional Causal Bayesian Optimization (FCBO) algorithm is introduced. This algorithm models the objective functional $\gamma \mapsto V\left(\gamma \right)$ using a functional Gaussian process surrogate defined on a Reproducing Kernel Hilbert Space (RKHS) domain and utilizes an Upper Confidence Bound (UCB) acquisition functional. Consequently, the policy value $V\left(\gamma \right)$ becomes an interventional query that can be answered using offline observational data under standard identifiability assumptions. High-probability cumulative-regret bounds are established in terms of differential information gain for the proposed FBO algorithm. Collectively, these elements constitute the central contributions of the CID framework, which integrates causal inference through identification and estimation with policy search in principal–agent problems under private information. This approach establishes a causal decision-making pipeline that enables commitment to a high-performing incentive in a single-shot game, supported by regret guarantees. Provided that the data used for estimation is sufficient, the resulting offline pipeline is appropriate for scenarios where adaptive deployment is impractical or costly. Beyond the methodological contribution, this work introduces a novel application of causal graphical models and causal reasoning to incentive design and principal–agent problems, which are central to economics and multi-agent systems. Full article

Consider the problem of approximating an integral of a real-valued integrand on a real interval by a Gauss quadrature rule. The classical approach to estimate the quadrature error of a Gauss rule is to evaluate an associated Gauss–Kronrod rule and compute the difference between the value of the Gauss–Kronrod rule and that of the Gauss rule. However, for a variety of measures and a number of nodes of interest, Gauss–Kronrod rules do not have real nodes or positive weights. This makes these rules impossible to apply when the integrand is defined on a real interval only. This has spurred the development of several averaged Gauss quadrature rules for estimating the quadrature error of Gauss rules. A significant advantage of the averaged Gauss rules is that they have real nodes and positive weights also in situations when Gauss–Kronrod rules do not. The most popular averaged rules include Laurie’s averaged Gauss quadrature rules, optimal averaged Gauss quadrature rules, weighted averaged Gauss quadrature rules, and two-measure-based generalized Gauss quadrature rules. This paper reviews the accuracy, numerical construction, and internality of averaged Gauss rules. Full article

Gaussian quadrature rules are commonly used to approximate integrals with respect to a non-negative measure $d\widehat{\sigma }$. It is important to be able to estimate the quadrature error in the Gaussian rule used. A common approach to estimating this error is to evaluate another quadrature rule that has more nodes and higher algebraic degree of precision than the Gaussian rule, and use the difference between this rule and the Gaussian rule as an estimate for the error in the latter. This paper considers the situation when $d\widehat{\sigma }$ is a Chebyshev measure that is modified by a linear factor and a linear divisor, and investigates whether the rules in a recently proposed new class of quadrature rules for estimating the error in Gaussian rules are internal, i.e., if all nodes of the new quadrature rules are in the interval $(-1,1)$. These new rules are defined by two measures, one of which is a modified Chebyshev measure $d\widehat{\sigma }$. The other measure is auxiliary. Full article

We propose higher-order adaptive energy-preserving methods for a charged particle system and a guiding center system. The higher-order energy-preserving methods are symmetric and are constructed by composing the second-order energy-preserving methods based on the averaged vector field. In order to overcome the energy drift problem that occurs in the energy-preserving methods based on the average vector field, we develop two adaptive algorithms for the higher-order energy-preserving methods. The two adaptive algorithms are developed based on using variable points of Gauss–Legendre’s quadrature rule and using two different stepsizes. The numerical results show that the two adaptive algorithms behave better in phase portrait and energy conservation than the Runge–Kutta methods. Moreover, it is shown that the energy errors obtained by the two adaptive algorithms can be bounded by the machine precision over long time and do not show energy drift. Full article

In this research, some new and efficient quadrature rules are proposed involving the combination of function and its first derivative evaluations at equally spaced data points with the main focus on their computational efficiency in terms of cost and time usage. The methods are theoretically derived, and theorems on the order of accuracy, degree of precision and error terms are proved. The proposed methods are semi-open-type rules with derivatives. The order of accuracy and degree of precision of the proposed methods are higher than the classical rules for which a systematic and symmetrical ascendancy has been proved. Various numerical tests are performed to compare the performance of the proposed methods with the existing methods in terms of accuracy, precision, leading local and global truncation errors, numerical convergence rates and computational cost with average CPU usage. In addition to the classical semi-open rules, the proposed methods have also been compared with some Gauss–Legendre methods for performance evaluation on various integrals involving some oscillatory, periodic and integrals with derivative singularities. The analysis of the results proves that the devised techniques are more efficient than the classical semi-open Newton–Cotes rules from theoretical and numerical perspectives because of promisingly reduced functional cost and lesser execution times. The proposed methods compete well with the spectral Gauss–Legendre rules, and in some cases outperform. Symmetric error distributions have been observed in regular cases of integrands, whereas asymmetrical behavior is evidenced in oscillatory and highly nonlinear cases. Full article