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

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 GameTheory.m2:1740:0.

konstanzMatrix -- construct the Konstanz matrix of a given game

Description

See also

Ways to use konstanzMatrix:

For the programmer