Game-Theoretic Question Selection for Tests

Abstract

Conventionally, the questions on a test are assumed to be kept secret from test takers until the test. However, for tests that are taken on a large scale, particularly asynchronously, this is very hard to achieve. For example, TOEFL iBT and driver's license test questions are easily found online. This also appears likely to become an issue for Massive Open Online Courses (MOOCs, as offered for example by Coursera, Udacity, and edX). Specifically, the test result may not reflect the true ability of a test taker if questions are leaked beforehand.





In this paper, we take the loss of confidentiality as a fact. Even so, not all hope is lost as the test taker can memorize only a limited set of questions' answers, and the tester can randomize which questions to let appear on the test. We model this as a Stackelberg game, where the tester commits to a mixed strategy and the follower responds. Informally, the goal of the tester is to best reveal the true ability of a test taker, while the test taker tries to maximize the test result (pass probability or score). We provide an exponential-size linear program formulation that computes the optimal test strategy, prove several NP-hardness results on computing optimal test strategies in general, and give efficient algorithms for special cases (scored tests and single-question tests). Experiments are also provided for those proposed algorithms to show their scalability and the increase of the tester's utility relative to that of the uniform-at-random strategy. The increase is quite significant when questions have some correlation---for example, when a test taker who can solve a harder question can always solve easier questions.