Macaulay2 » Documentation
Packages » GameTheory :: nashEquilibriumIdeal(Ring,List)
next | previous | forward | backward | up | index | toc

nashEquilibriumIdeal(Ring,List) -- make the Nash Equilibrium ideal

Description

For an $n$-player game, the totally mixed Nash equilibria are the zero loci of a system of polynomials in the interior of product of probability simplices. The coefficients of these polynomials are certain differences of the entries of the payoff tensors. These polynomials, together with the linear constraints $\sum^{d_i -1}_{j=0} p_{i,j}=1$ for each player $i$, generate an ideal in the polynomial ring computed by nashEquilibriumRing.

i1 : tensors = randomGame {2,2,2};
i2 : R = nashEquilibriumRing tensors;
i3 : I = nashEquilibriumIdeal(R, tensors)

              67                  1                 31                  7    
o3 = ideal (- --p      p       - --p      p       - --p      p       + --p   
              18 {1, 0} {2, 0}   20 {1, 1} {2, 0}   20 {1, 0} {2, 1}   12 {1,
     ------------------------------------------------------------------------
                  43                 19                 39                
       p      , - --p      p       - --p      p       + --p      p       +
     1} {2, 1}     7 {0, 0} {2, 0}   20 {0, 1} {2, 0}    7 {0, 0} {2, 1}  
     ------------------------------------------------------------------------
     13                  35                  1                 17           
     --p      p      , - --p      p       - --p      p       - --p      p   
     63 {0, 1} {2, 1}     9 {0, 0} {1, 0}   24 {0, 1} {1, 0}    2 {0, 0} {1,
     ------------------------------------------------------------------------
          23
        - --p      p      , p       + p       - 1, p       + p       - 1,
     1}    5 {0, 1} {1, 1}   {0, 0}    {0, 1}       {1, 0}    {1, 1}     
     ------------------------------------------------------------------------
     p       + p       - 1)
      {2, 0}    {2, 1}

o3 : Ideal of R

An introduction together with the relevant definitions is given in Chapter 6, Sturmfels, Bernd, Solving Systems of Polynomial Equations. American Mathematical Society, 2002 and in Abo, Hirotachi, Portakal, Irem, and Sodomaco, Luca, A vector bundle approach to Nash equilibria, arXiv:2504.03456.

See also

Ways to use this method:


The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/GameTheory.m2:1371:0.