Elicitation procedures for the comparison of decisional maps

In the field of land management, it is not uncommon to have a geographic map, representing a region under study, and divided into geographic units. Each unit is assessed on an ordinal scale describing its degree of suitability for some usage, for instance housing, or its state of degradation with respect to sustainable development criteria. We call such a map a decisional map. After a while, and for example, after the application of some policies aiming to improve the situation, the state of the units evolves. For some units, the state has improved, and for some others, it has deteriorated. What we want to know is whether the global state of the map has improved or not. The purpose of this work is to provide models to help a decision maker to compare the state of a region at different stages of its evolution. We formulate two mathematical models and we provide an interactive elicitation procedure to determine all the parameters of the models developed. For this purpose, we transpose a widely used method in the theory of decision under risk, namely the comparison of lotteries. We interpret lotteries as maps and we formulate questions to the decision maker in terms of comparisons of well-chosen maps.