Counterfactuals with Latent Information

We describe a methodology for making counterfactual predictions in settings where the information held by strategic agents and the distribution of payoff-relevant states of the world are unknown. The analyst observes behavior assumed to be rationalized by a Bayesian model, in which agents maximize expected utility, given partial and differential information about the state. A counterfactual prediction is desired about behavior in another strategic setting, under the hypothesis that the distribution of the state and agents’ information about the state are held fixed. When the data and the desired counterfactual prediction pertain to environments with finitely many states, players, and actions, the counterfactual prediction is described by finitely many linear inequalities, even though the latent parameter, the information structure, is infinite dimensional. (JEL D44, D82, D83)

Heuristics for Finding Sparse Solutions of Linear Inequalities

In this paper, we consider the problem of finding a sparse solution, with a minimal number of nonzero components, for a set of linear inequalities. This optimization problem is combinatorial and arises in various fields such as machine learning and compressed sensing. We present three new heuristics for the problem. The first two are greedy algorithms minimizing the sum of infeasibilities in the primal and dual spaces with different selection rules. The third heuristic is a combination of the greedy heuristic in the dual space and a local search algorithm. In numerical experiments, our proposed heuristics are compared with the weighted-[Formula: see text] algorithm and DCA programming with three different non-convex approximations of the zero norm. The computational results demonstrate the efficiency of our methods.

Real subset sums and posets with an involution

In this paper, we carry out in an abstract order context some real subset combinatorial problems. Specifically, let [Formula: see text] be a finite poset, where [Formula: see text] is an order-reversing and involutive map such that [Formula: see text] for each [Formula: see text]. Let [Formula: see text] be the Boolean lattice with two elements and [Formula: see text] the family of all the order-preserving 2-valued maps [Formula: see text] such that [Formula: see text] if [Formula: see text] for all [Formula: see text]. In this paper, we build a family [Formula: see text] of particular subsets of [Formula: see text], that we call [Formula: see text]-bases on [Formula: see text], and we determine a bijection between the family [Formula: see text] and the family [Formula: see text]. In such a bijection, a [Formula: see text]-basis [Formula: see text] on [Formula: see text] corresponds to a map [Formula: see text] whose restriction of [Formula: see text] to [Formula: see text] is the smallest 2-valued partial map on [Formula: see text] which has [Formula: see text] as its unique extension in [Formula: see text]. Next we show how each [Formula: see text]-basis on [Formula: see text] becomes, in a particular context, a sub-system of a larger system of linear inequalities, whose compatibility implies the compatibility of the whole system.

The Feasibility Analysis of Mathematics Textbook Material based on Bell's Criteria

The purpose of this study is to determine the feasibility of Mathematics textbook material for Grade VII 1st semester on 2017 Revised Edition based on Bell's criteria. So that if there is an error, it can be fixed. Then, this type of research is descriptive research. In the descriptive analysis, there is no need to find or explain its relationships and test hypotheses. The approach used is qualitative. Moreover, this study uses a qualitative approach because the data collected are described in sentences. This study will be obtained from two sources, namely the main source of the Mathematics Textbook Grade VII of 2013 curriculum and peer-discussion support sources. The method used in this research is the documentation method. The documentation data used in this research is the textbook itself. However, the data obtained by using the Analysis Instrument Result shows that the suitability of the mathematics textbook material of Grade VII with the material based on Bell's criteria for the numbers and algebra chapters includes a good category. Meanwhile, the sets and equations and linear inequalities of one variable chapter belong to be an excellent category. The average suitability of the mathematics material from each chapter in the textbook toward Bell's criteria is 83.75% and is supported by the validity data results through forms and peer discussions. As a result, the feasibility of the mathematics textbook material for grade VII of the 2017 revised edition based on Bell's criteria is very good.

The Feasibility Analysis of Mathematics Textbook Material based on Bell's Criteria

The purpose of this study is to determine the feasibility of Mathematics textbook material for Grade VII 1st semester on 2017 Revised Edition based on Bell's criteria. So that if there is an error, it can be fixed. Then, this type of research is descriptive research. In the descriptive analysis, there is no need to find or explain its relationships and test hypotheses. The approach used is qualitative. Moreover, this study uses a qualitative approach because the data collected are described in sentences. This study will be obtained from two sources, namely the main source of the Mathematics Textbook Grade VII of 2013 curriculum and peer-discussion support sources. The method used in this research is the documentation method. The documentation data used in this research is the textbook itself. However, the data obtained by using the Analysis Instrument Result shows that the suitability of the mathematics textbook material of Grade VII with the material based on Bell's criteria for the numbers and algebra chapters includes a good category. Meanwhile, the sets and equations and linear inequalities of one variable chapter belong to be an excellent category. The average suitability of the mathematics material from each chapter in the textbook toward Bell's criteria is 83.75% and is supported by the validity data results through forms and peer discussions. As a result, the feasibility of the mathematics textbook material for grade VII of the 2017 revised edition based on Bell's criteria is very good.