Theoretical evaluation of partial credit scoring of the multiple-choice test item

In multiple-choice tests, guessing is a source of test error which can be suppressed if its expected score is made negative by either penalizing wrong answers or rewarding expressions of partial knowledge. Starting from the most general formulation of the necessary and sufficient scoring conditions for guessing to lead to an expected loss beyond the test-taker’s knowledge, we formulate a class of optimal scoring functions of which that due to Zapechelnyuk (Economics Letters, 132, 2015) appears as a special case. We then consider an arbitrary multiple-choice test taken by a rational test-taker that knows an arbitrary fraction of its keys and distractors as a model of a test of factual knowledge. For this model, we study the statistical properties of the obtained score for both standard marking (where guessing is not penalized), and marking where guessing is suppressed either by expensive score penalties for incorrect answers or by different marking schemes that reward partial knowledge.