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numberTMNE(List) -- compute the maximum number of totally mixed Nash equilibria for a generic game

Description

For an $n$-player game where the $i$-th player has $d_i$ pure strategies, the maximum number of isolated totally mixed Nash equilibria is given by the degree $c(\mathbf{d})$ of the top Chern class of the following vector bundle:

\[ E \coloneqq \bigoplus_{i=0}^{n-1} \ko_{\mathbb{P}^{\mathbf{d}}}({\mathbf{1}}_i)^{\oplus(d_i-1)},\]

where $\mathbb{P}^{\mathbf{d}}=\prod_{i=0}^{n-1}\mathbb{P}^{d_i-1}$ and $\mathbf{1}_i=(1,\ldots,1,0,1,\ldots,1)$, where the entry $0$ is in the $i$-th component. In particular, this function computes the integer $c(\mathbf{d})$ as the coefficient of the monomial $\prod_{i=0}^{n-1} h_i^{d_i-1}$ in $\prod_{i=0}^{n-1} \hat{h}_i^{d_i-1}$ with $\hat{h}_i\coloneqq \sum_{j\neq i}h_j$, where $h_i$ denotes the pullback of the hyperplane class on the $i$-th factor $\mathbb{P}^{d_i-1}$ of $\PP^\bd$ via the projection map.

i1 : d = {2,2,2};
i2 : nTMNE = numberTMNE d

o2 = 2

Alternatively, if you have a tensor $T$, you can compute its maximum number as follows:

i3 : T = randomTensor {2,2,2}

o3 = Tensor{...11...}

o3 : Tensor
i4 : nTMNE2 = numberTMNE format T

o4 = 2

See also

Ways to use this method:

  • numberTMNE(List) -- compute the maximum number of totally mixed Nash equilibria for a generic game

The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/GameTheory.m2:1455:0.