Choose that by drawing the circle:
There are basically two ways to enter the data:
Here you can enter the values of the game. These should be non-negative integers. The value for the empty set is automatically set to zero and the initial pre-defined values for all sets are zeros.
The standard min-representation of the game consisting of the list of all vertices of the core will be automatically computed using R package rcdd. It will appear in the array entitled Core of the game.
Our extremity criterion requires the represented game to be in a standardized form, which means that the values for singletons must be zeros. If you click on the button Standardize then the represented game will be turned into its standardized version (by subtracting a certain additive game). Moreover, the (vertices of the corresponding) core will be modified accordingly. This standardization operation geometrically means a translation of the core and does not influence the extremity of the represented game.
Our extremity criterion also requires the core to be non-empty and its vertices to have integers as components. However even if the input game has a non-empty core it may happen that the vertices of the core are fractional. To ensure that the standard min-representation is composed of integers click on the button Multiply to Integers. If the input game has the empty core then the user will be notified about that. Otherwise, the represented game will be replaced by a game whose core is an integer multiple of the original core. Of course, the game may remain unchanged if it already has integer-composed standard min-representation. This adaptation operation does not influence the extremity of the represented game, but it can lead to a non-standardized game, in which case one has should click on the button Standardize.
You can enter the values of an input min-representation using the array Input/saving array on the right-hand side. The values should be non-negative integers and the sum in any row should be the same for different rows. Moreover, to ensure the resulting represented game to be standardized, every column of the input min-representation should contain at least once zero. Additional rows for the input can be generated using the button called Add row.
The initial pre-defined values in every newly generated row are zeros. After entering the input min-representation click on the button with the left-directed arrow. The represented game will be computed and will appear in the frame Represented game. Moreover, the vertices of the core of the represented game will be automatically computed and will appear in the array entitled Core of the game. Note that this standard min-representation can have a different number of rows than the input min-representation.
All tests available below are based on the standard min-representation in the array entitled Core of the game. Using the button with the right-directed arrow you can save it in the (originally input) array on the right-hand side.
Here you can perform various tests.
By clicking on the button bellow you can start a test whether the represented game is supermodular. If it is supermodular the answer will be TRUE, otherwise FALSE.
If the represented game is indeed supermodular then you can test whether it is extreme in the supermodular cone. If you click on the button bellow then the nullity of the respective linear equation system given by the standard min-representation will be computed. (The nullity of the equation system is the dimension of space of solutions.) The game is extreme iff this nullity is 1, otherwise it is not extreme in the supermodular cone.
The criterion for the extremity in the cone of supermodular games is based on the main result (Theorem 5 in Section 3) of the paper M.Studený, T.Kroupa: Core-based criterion for extreme supermodular functions. Discrete Applied Mathematics 206 (2016) 122 -151.
This web platform was created by Václav Kratochvíl.