Many tests have only two possible outcomes, “positive” or “negative”. Such binary tests are widely used in various scientific fields, including machine learning, statistical testing, medicine, engineering, social science, and pattern recognition. The probability of a positive response is the main characteristic of a binary test and is called the Item Response Function (IRF). This work looks at binary tests in antagonistic settings, in which the IRF depends on the relationship between the difficulty of the test item – d and ability of the object under test – a.

Antagonistic situations arise from confrontation or competition between two parties with mutually opposite, conflicting interests. In some cases, the “success” (i.e. positive outcome) of one side may formally be considered as a “loss” (i.e. negative outcome) to the opposite side. For example: arm–wrestling, hypothesis of a researcher vs. conservative hypothesis in statistical testing, pursuit – evasion, strength – stress relationship, signal vs. noise, struggle between competitors for the same resource. In some sense, the same goes for survivability in a situation of mortal danger. A conflict can be defined in a purely formal manner, even if it is not as such in fact. For example, the failure of the examinee does not really mean the success of the examiner, but formally it can be described in this way. For such situations, ability and difficulty are two interchangeable sides of the same coin; for example, speed / maneuverability of a policeman (his ability) can be considered as a difficulty for the thief running away from him, and speed / maneuverability of the thief poses the challenge to the policeman trying to catch him.

The above considerations mean that, for antagonistic situations, the permutation of difficulty and ability in the IRF leads to the probability complementary to the original probability one. Far-reaching implications regarding a feasible mathematical form of the IRF stem from this simple fact, which allows scientists to significantly narrow down the search for adequate IRF models to describe an antagonistic situation. In the article, particular attention, accompanied by a number of practical examples, is directed to two main models: differential (for which IRF is determined by the difference between ability and difficulty) and relative (for which IRF is determined by the ratio between ability and difficulty).