Shapley weights of test items

Abstract

One of the problems of the theory of testing is the problem of determining the weights of the test item. For most tests, the weights of assignments are considered to be equal, but as the complexity of test design and increasing the number of test items, the need for determining the complexity of the test increases. This is especially important when comparing students to solve different tests or in cases when the time for the solution to all the test items is not enough. In addition, the resolution of the test increases significantly if different test items are attributed to different weights. In the work for a course that has a hierarchical structure and is known for the duration of the development of its sections, we built a “sum of attainments” cooperative game in which players are the parts of the course, and the value of the characteristic function on a coalition is the time necessary for a student to study the coalition of the parts. We have found simple formulas for the calculation of the Shapley value as constructed in the game and so expanded the class of cooperative games for which the Shapley value is calculated analytically. The basis of the proof lies in the decomposition of the characteristic function in a sum of the dual unanimity games and using the properties of a cone in the set of partially ordered elements (parts of course). The component of the Shapley value is the average time of learning the part of the course and it may be used as the weight of the item for a final test. The theory of duality (complementarity), developed for cooperative games permits us to link “the sum of attainments game”, “the airport game” and “the peer group game”. Examples of the calculation of weights for test items has been prepared by the use of a particular model term course. Refs 16. Figs 2.