Macaulay2 » Documentation
Packages » GameTheory :: konstanzMatrix
next | previous | forward | backward | up | index | toc

# konstanzMatrix -- construct the Konstanz matrix of a given game

- Usage:
    `konstanzMatrix(PR, X)`
- Inputs:
  - PR, a ring, a probability ring obtained via probabilityRing(Di), where $Di = \{d_1 , \cdots , d_n\}$ is the format of the game
  - X, a list, a list of n tensors of format $D_i$ describing the payoffs of the game
- Optional inputs:
  - KonstanzVariableName (missing documentation) => ..., default value "k",
- Outputs:
  - a matrix, the $(d_1 + \ldots + d_n) \times (d-1 \cdots d_n)$-dimensional Konstanz matrix

## Description

It is crucial that the formats in PR and X match up. The Konstanz matrix $K_X(k)$ is the unique matrix with monic polynomials as entries such that the Spohn variety is the union $\bigcup_{k \in (\mathbb P^1)^n} \ker K_X(k)$.

    i1 : Di = {2,2,3};

    i2 : PR = probabilityRing(Di);

    i3 : X = randomGame(Di);

    i4 : K = konstanzMatrix(PR,X)

    o4 = | k_0-9/2 k_0-9/4 k_0-3/4  k_0-7/4 k_0-7/9 k_0-7/10 0        0
         | 0       0       0        0       0       0        k_0-7/10 k_0-7/3
         | k_1-5/4 k_1-2/9 k_1-3/10 0       0       0        k_1-10   k_1-3/2
         | 0       0       0        k_1-3/7 k_1-5   k_1-10/9 0        0
         | k_2-5/3 0       0        k_2-6/5 0       0        k_2-5/3  0
         | 0       k_2-7/2 0        0       k_2-5/7 0        0        k_2-1/10
         | 0       0       k_2-2/5  0       0       k_2-5/9  0        0
         ------------------------------------------------------------------------
         0       0       0        0       |
         k_0-7   k_0-3/7 k_0-6/7  k_0-6   |
         k_1-7/8 0       0        0       |
         0       k_1-5/6 k_1-5    k_1-2/5 |
         0       k_2-3/7 0        0       |
         0       0       k_2-9/10 0       |
         k_2-4/3 0       0        k_2-4/7 |

                            7                 12
    o4 : Matrix (PR[k ..k ])  <-- (PR[k ..k ])
                     0   2             0   2

The optional argument KonstanzVariableName allows to change the name of the variables. If no variable name choice is specified, the variables will be named with k.

    i5 : Di = {2,2};

    i6 : PR = probabilityRing(Di);

    i7 : X = randomGame(Di);

    i8 : K = konstanzMatrix(PR,X, KonstanzVariableName => "z")

    o8 = | z_0-5/9 z_0-5/9 0       0       |
         | 0       0       z_0-6/7 z_0-6   |
         | z_1-5/4 0       z_1-8/5 0       |
         | 0       z_1-2/9 0       z_1-9/4 |

                            4                 4
    o8 : Matrix (PR[z ..z ])  <-- (PR[z ..z ])
                     0   1             0   1

## See also

- probabilityRing -- ring of probability distributions of a game indexed by ordered multi-indices
- randomGame -- construct a game of a given format with arbitrary payoffs
- spohnMatrices -- compute the list of Spohn matrices of a given game
- spohnIdeal -- compute the ideal of the Spohn variety of a given game

## Ways to use konstanzMatrix:

- konstanzMatrix(Ring,List) (missing documentation)

## For the programmer

The object konstanzMatrix is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/GameTheory.m2:1740:0.