A vector bundle approach to Nash equilibria
Supplementary Material
The scripts are ending with .m2 and are run in \(\verb|Macaulay2|\).
We provide the following code:
example_333_real_root_isolation.m2provides a numerical computation of the Nash equilibrium scheme of a \((3,3,3)\) game (Example 2.5). It also certifies that four of the isolated points of the Nash equilibrium scheme are real.singular_strata_222.m2computes the singular strata of the Nash discriminant variety for \((2,2,2)\) games (Section 3.2).variety_three_non_concurrent_lines.m2computes the Chern-Mather volumes of the singular projective toric variety parametrizing global sections whose Nash equilibrium scheme consists of space connected curve, union of three non concurrent lines (Remark C).variety_three_concurrent_lines.m2computes the Chern-Mather volumes of the singular projective toric variety parametrizing global sections whose Nash equilibrium scheme consists of space connected curve, union of three concurrent lines (Remark D).case_223.m2computes the two irreducible components of the Nash discriminant variety for \((2,2,3)\) games (Example 3.28).game223_one_double_tmNe.m2computes the Nash equilibrium scheme of a \((2,2,3)\) game (Example 3.29). It verifies that the Nash equilibrium scheme consists of a double point whose support is the unique totally mixed Nash equilibrium.
\(i\) |
\(\text{Sing}^{(i)}(\Delta(E))\) |
\(\dim(\text{Sing}^{(i)}(\Delta(E)))\) |
Number of irreducible components |
|---|---|---|---|
\(1\) |
\(\Delta^{\text{cub}}(E) \cup \Delta^{\text{con}}(E) \cup \Delta^{\text{lin}}(E)\) |
\(8\) |
\(7 = 1 + 3 + 3\) |
\(2\) |
\(\Delta^{\text{ll}}(E) \cup \Delta^{\text{cl}}(E)\) |
\(7\) |
\(6 = 3 + 3\) |
\(3\) |
\(\Delta^{\text{lll}}(E) \cup \Delta^{\text{scr}}(E)\) |
\(6\) |
\(6 = 3 + 3\) |
\(4\) |
\(\Delta^{\text{3l}}(E)\) |
\(5\) |
\(1\) |
\(5\) |
\(\Delta^{\text{qq}}(E)\) |
\(2\) |
\(3\) |
Singular strata of the Nash discriminant variety for \((2,2,2)\) games
The singular locus \(\text{Sing}^{(1)}(\Delta(E))\) of the Nash discriminant variety \(\Delta(E)\) of \((2,2,2)\) games is studied in Section 3.2 of the mentioned paper. We present the full computation of the singular strata of \(\Delta(E)\) here as appendix with accompanied code in \(\verb|Macaulay2|\).
Proposition 1.
The locus \(\Delta^{\mathrm{ll}}(E)\) consists of three irreducible components, each having dimension \(7\) and degree \(8\).
Proof. Given \(f\in H^0(\mathbb{P}^{\boldsymbol{d}},E)\), if \(\mathcal Z_f\) is a singular plane conic, then all three coordinate functions of \(f\) are reducible, and exactly two of them share a linear factor. Thus, three different components exist, each corresponding to which coordinate function does not share a linear factor with the other two. If this component is \(f^{(i)}\), then the corresponding component of \(\Delta^{\mathrm{ll}}(E)\) is denoted by \(\Delta_i^{\mathrm{ll}}(E)\). We only prove that \(\Delta_1^{\mathrm{ll}}(E)\) is irreducible of dimension \(7\).
Let \(\mathbb{P} := \mathbb{P} (V_2^*) \times \mathbb{P}(V_3^*)\), \(\mathbb{P}' := \mathbb{P}(V_1^*) \times \mathbb{P}(V_2^* \oplus V_3^*)\), and let
\[U := \{([M'],[N'],[L],[(M,N)]) \in \mathbb{P}\times \mathbb{P}' \mid \text{$\{M,M'\}$ and $\{N,N'\}$ are linearly independent}\}.\]Define the map \(\Psi\colon \mathbb{P}\times \mathbb{P}' \to \mathbb{P} H^0(\mathbb{P}^{\boldsymbol{d}}, E)\) by
\[\Psi([M'],[N'],[L],[(M,N)]) := [(M'N', LM, LN)]\,.\]The image of \(U\) under \(\Psi\) forms an open subset of the image of \(\Psi\). It also lies in \(\Delta_1^{\mathrm{ll}}(E)\), and hence so is its Zariski closure (which coincides with the image of \(\Psi)\).
On the one hand, the map from \(\mathbb P\) to \(\mathbb P(V_2^* \otimes V_3^*)\) defined by sending \(([M'],[N'])\) to \([M'N']\) is the Segre embedding of \(\mathbb P(V_2^*)\times \mathbb P(V_3^*)\). On the other hand, the map from \(\mathbb P'\) to \(\mathbb P((V_1^* \otimes V_2^*) \oplus (V_1^* \otimes V_3^*))\) defined by sending \(([L], [(M,N)])\) to \([(LM,LN)]\) is the Segre embedding of \(\mathbb P(V_1^*)\times \mathbb P(V_2^* \oplus V_3^*)\). Hence the image of \(\Psi\) is the join over these two irreducible varieties living in two complementary linear subspaces of \(\mathbb P H^0(\mathbb{P}^{\boldsymbol{d}}, E)\). Using Example 18.17 from [Har92], we conclude that the image of \(\Psi\) is irreducible of dimension \(1\) plus the dimensions of \(\mathbb P\) and \(\mathbb P'\), that equals \(1+2+4=7\). Furthermore, its degree is equal to the product of the degrees of \(\mathbb P\) and \(\mathbb P'\) in their Segre embeddings, that equals \(2\times 4=8\). ◻
Remark A.
This remark concerns a determinantal expression for the equations for \(\Delta_i^{\mathrm{ll}}(E)\). We only discuss the case when \(i=1\) as the remaining cases can be treated in a similar manner. As was shown in the proof of Proposition 1, the locus \(\Delta_1^{\mathrm{ll}}(E)\) is the join of two disjoint varieties, namely the images of the two Segre embeddings \(\mathbb P(V_2^*) \times \mathbb P(V_3^*)\to\mathbb{P}(V_2^* \otimes V_3^*)\) and \(\mathbb P(V_1^*)\times \mathbb P(V_2^* \oplus V_3^*)\to\mathbb P((V_1^* \otimes V_3^*) \oplus (V_1^* \otimes V_2^*))\). Thus, the equations for \(\Delta_1^{\mathrm{ll}}(E)\) are the union of the equations coming from the two Segre embeddings. The first Segre embedding gives the condition
\[\begin{split}\begin{vmatrix} a^{(1)}_{11} & a^{(1)}_{12} \\[2pt] a^{(1)}_{22} & a^{(1)}_{22} \end{vmatrix} = a^{(1)}_{11}a^{(1)}_{22}-a^{(1)}_{12}a^{(1)}_{21} = 0\,,\end{split}\]
which is the condition for \(f^{(1)}\) to be a product of linear forms. Instead, the equations for the second Segre embedding are discussed in [Remark 3.22, APS25].
Proposition 2.
The locus \(\Delta^{\mathrm{cl}}(E)\) consists of three irreducible components, each having dimension \(7\).
Proof. Given \(f\in H^0(\mathbb P^{\boldsymbol{d}},E)\), if \(\mathcal Z_f\) is a space connected cubic, union of a non-singular conic and a line, then one of the coordinate functions of \(f\) is irreducible, and the others are reducible and share a linear factor. Thus, three different components exist, each corresponding to which coordinate function is irreducible. If it is \(f^{(i)}\), then the corresponding component of \(\Delta^{\mathrm{cl}}(E)\) is denoted by \(\Delta_i^{\mathrm{cl}}(E)\). We only prove that \(\Delta_1^{\mathrm{cl}}(E)\) is irreducible of dimension \(7\).
Let \(U\) be the open subset of points of \(\Delta_1^{\mathrm{cl}}(E)\) of the form \([(MN'+M'N, LN, LM)]\) with \(L \in V_1^*\setminus \{{\boldsymbol{0}}\}, M, M' \in V_2^* \setminus \{{\boldsymbol{0}}\}\), and \(N, N' \in V_3^*\setminus \{{\boldsymbol{0}}\}\), where both \(\{M,M'\}\) and \(\{N,N'\}\) are linearly independent so that \(MN'+M'N\) is irreducible. Define the map from \(U\) to \(\mathbb P((V_1^* \otimes V_3^*) \oplus (V_1^* \otimes V_2^*))\) by \(\Psi([(MN'+M'N, LN, LM)]) := [(LN,LM)]\). The image of \(\Psi\) is the Segre embedding of \(\mathbb P(V_1^*) \times \mathbb{P}(V_2^*\oplus V_3^*)\) in \(\mathbb P((V_1^* \otimes V_3^*) \oplus (V_1^* \otimes V_2^*))\) given by sending \(([L],[(M,N)])\) to \([LN,LM]\). So, it is irreducible of dimension \(4\). The fiber of \(\Psi\) over \([(LN,LM)]\) is
\[\{[(MN'+M'N,LN,LM)] \mid M' \in V_2^* \setminus \{{\boldsymbol{0}}\}, N' \in V_3^* \setminus \{{\boldsymbol{0}}\}\},\]and thus, it can be identified with an open set of the \(3\)-plane \(\{[(M',N')] \mid M' \in V_2^*, N' \in V_3^*\}\), which implies that \(U\), and hence its closure, is irreducible and has dimension \(7\). ◻
Remark B.
Following a similar argument as in [Remark 3.22, APS25] and [Remark 3.24, APS25], one verifies that the six \(2\times 2\) minors of the \(2\times 4\) matrix in [Equation (3.28), APS25] and the four polynomials in [Remark 3.24, APS25] cut out the variety \(\Delta_1^{\mathrm{cl}}(E)\) set-theoretically. We verified in \(\verb|Macaulay2|\) that \(\Delta_1^{\mathrm{cl}}(E)\) has degree \(10\), and similarly for the other two components of \(\Delta^{\mathrm{cl}}(E)\).
Proposition 3.
The locus \(\Delta^{\mathrm{lll}}(E)\) consists of three irreducible components, each having dimension \(6\).
Proof. Suppose that the zero scheme of \(f:=(f^{(1)},f^{(2)},f^{(3)}) \in H^0(\mathbb P^{\boldsymbol{d}},E)\) is a space connected cubic, union of three non concurrent lines. Then there exists one line meeting the other two. Furthermore, all coordinate functions of \(f\) are reducible, and for exactly one \(i\in[3]\), the linear factors in \(f^{(i)}\) divide one of the other two components of \(f\). Thus, three different components exist, each corresponding to which index \(i\in[3]\) is chosen, and the corresponding component of \(\Delta^{\mathrm{lll}}(E)\) is denoted by \(\Delta_i^{\mathrm{lll}}(E)\). We only prove that \(\Delta_1^{\mathrm{lll}}(E)\) is irreducible of dimension \(6\).
Define the map \(\Psi\colon\mathbb P((V_1^*)^{\oplus 2}\oplus V_2^*\oplus V_3^*)\cong \mathbb P^7 \to \mathbb P H^0(\mathbb P^{\boldsymbol{d}}, E)\) by
\[\Psi([(L,L',M,N)]):= [(MN, LN, L'M)]\,.\]The Zariski closure of the image of \(\Psi\) is \(\Delta_1^{\mathrm{lll}}(E)\), hence \(\Delta_1^{\mathrm{lll}}(E)\) is an irreducible toric variety. The fiber of \(\Psi\) over \([(MN, LN, L'M)]\) is \(\{[(\lambda L,\mu L',\lambda M,\mu N)] \mid [\lambda,\mu]\in\mathbb{P}^1\}\), and thus, it has dimension \(1\). Therefore \(\Delta_1^{\mathrm{lll}}(E)\) has dimension \(6\). ◻
Remark C.
Similarly as in [Remark 3.22, APS25], and choosing suitable coordinates for the vector spaces involved, one verifies that \(\Delta_1^{\mathrm{lll}}(E)\) is cut out set-theoretically by the \(2\times 2\) minors of the two matrices
\[\begin{split}\label{eq: matrices defining conditions three lines} \begin{pmatrix} a^{(3)}_{11} & a^{(3)}_{21} & a^{(1)}_{11} & a^{(1)}_{12} \\[2pt] a^{(3)}_{12} & a^{(3)}_{22} & a^{(1)}_{21} & a^{(1)}_{22} \end{pmatrix}\,,\quad \begin{pmatrix} a^{(2)}_{11} & a^{(2)}_{12} & a^{(1)}_{11} & a^{(1)}_{12} \\[2pt] a^{(2)}_{21} & a^{(2)}_{22} & a^{(1)}_{21} & a^{(1)}_{22} \end{pmatrix}\,.\end{split}\]
We verified in \(\verb|Macaulay2|\) with the command cmModules of the package ToricInvariants that the list of Chern-Mather volumes of the singular projective toric variety \(\Delta_1^{\mathrm{lll}}(E)\), ordered from \(0\) to \(6=\dim(\Delta_1^{\mathrm{lll}}(E))\), is \(\{20, 52, 90, 108, 88, 44, 10\}\). In particular \(\Delta_1^{\mathrm{lll}}(E)\) has degree \(10\), and similarly for the other two components of \(\Delta^{\mathrm{lll}}(E)\).
Proposition 4.
The locus \(\Delta^{\mathrm{3l}}(E)\) is irreducible of dimension \(5\).
Proof. Suppose that the zero scheme of \(f:=(f^{(1)},f^{(2)},f^{(3)}) \in H^0(\mathbb P^{\boldsymbol{d}},E)\) is a degenerate twisted cubic, union of three concurrent lines. Then all coordinate functions of \(f\) are reducible, and for each \(i\in[3]\), the linear factors in \(f^{(i)}\) divide one of the other two components of \(f\). Thus, defining the map \(\Psi\colon\mathbb P(V_1^*\oplus V_2^*\oplus V_3^*)\cong \mathbb P^5 \to \mathbb P H^0(\mathbb P^{\boldsymbol{d}}, E)\) by
\[\Psi([(L,M,N)]):= [(MN, LN, LM)]\,,\]then the Zariski closure of the image of \(\Psi\) is \(\Delta^{\mathrm{3l}}(E)\), hence \(\Delta^{\mathrm{3l}}(E)\) is an irreducible toric variety. Since \(\Psi\) is birational over its image, then \(\Delta^{\mathrm{3l}}(E)\) has dimension \(5\). ◻
Remark D.
Following up on Remark C, one verifies that \(\Delta^{\mathrm{3l}}(E)\) is cut out set-theoretically by the \(2\times 2\) minors of the three matrices
\[\begin{split}\label{eq: matrices defining conditions three concurrent lines} \begin{pmatrix} a^{(3)}_{11} & a^{(3)}_{12} & a^{(2)}_{11} & a^{(2)}_{12} \\[2pt] a^{(3)}_{21} & a^{(3)}_{22} & a^{(2)}_{21} & a^{(2)}_{22} \end{pmatrix}\,,\quad \begin{pmatrix} a^{(3)}_{11} & a^{(3)}_{21} & a^{(1)}_{11} & a^{(1)}_{12} \\[2pt] a^{(3)}_{12} & a^{(3)}_{22} & a^{(1)}_{21} & a^{(1)}_{22} \end{pmatrix}\,,\quad \begin{pmatrix} a^{(2)}_{11} & a^{(2)}_{12} & a^{(1)}_{11} & a^{(1)}_{12} \\[2pt] a^{(2)}_{21} & a^{(2)}_{22} & a^{(1)}_{21} & a^{(1)}_{22} \end{pmatrix}\,.\end{split}\]
Similarly as in Remark Remark C, we verified in \(\verb|Macaulay2|\) that the list of Chern-Mather volumes of the singular projective toric variety \(\Delta^{\mathrm{3l}}(E)\), ordered from \(0\) to \(5=\dim(\Delta^{\mathrm{3l}}(E))\), is \(\{24, 48, 68, 66, 42, 14\}\). In particular \(\Delta^{\mathrm{3l}}(E)\) has degree \(14\).
Proposition 5.
The locus \(\Delta^{\mathrm{ql}}(E)\) consists of three irreducible components, each having dimension \(4\) and degree \(4\).
Proof. Suppose that the zero scheme of \(f:=(f^{(1)},f^{(2)},f^{(3)}) \in H^0(\mathbb P^{\boldsymbol{d}},E)\) is the union of a quadric and a line. Then exactly one coordinate function of \(f\) is zero, and the other two are reducible and share a linear factor. Thus, three different components exist, each corresponding to which coordinate function \(f^{(i)}\) is zero, and the corresponding component of \(\Delta^{\mathrm{ql}}(E)\) is denoted by \(\Delta_i^{\mathrm{ql}}(E)\). The proof of the statement follows by a similar argument used in [Proposition 3.21, APS25]. ◻
Remark E.
Similarly as in [Remark 3.22, APS25], the locus \(\Delta_1^{\mathrm{ql}}(E)\) is cut out by the \(2\times 2\) minors of the matrix [Equation (3.28), APS25], and by the linear conditions \(a_{11}^{(1)}=a_{12}^{(1)}=a_{21}^{(1)}=a_{22}^{(1)}=0\). The cases \(i=2\) and \(i=3\) are similar.
Proposition 6.
The locus \(\Delta^{\mathrm{qq}}(E)\) consists of three pairwise disjoint non-singular irreducible components, each having dimension \(2\) and degree \(2\).
Proof. Suppose that the zero scheme of \(f:=(f^{(1)},f^{(2)},f^{(3)}) \in H^0(\mathbb P^{\boldsymbol{d}},E)\) is the union of two quadric surfaces. Then only one coordinate functions of \(f\), say \(f^{(i)}\), is nonzero, and \(f^{(i)}\) is reducible. Thus, three different components exist, each corresponding to which index \(i\in[3]\) is chosen, and the corresponding component of \(\Delta^{\mathrm{qq}}(E)\) is denoted by \(\Delta_i^{\mathrm{qq}}(E)\). Choose \(i=1\) for simplicity. Then \(\Delta_1^{\mathrm{qq}}(E)\) is the transversal intersection between the \(3\)-plane \(\Delta_1^{\mathrm{scr}}(E)\) and the cone over the Segre embedding of \(\mathbb P(V_2^*)\times\mathbb P(V_3^*)\) in \(\mathbb P(V_2^*\otimes V_3^*)\). In particular it is an irreducible quadric surface in \(\mathbb{P} H^0(\mathbb{P}^{\boldsymbol{d}},E)\). If \(i\neq j\), then \(\Delta_i^{\mathrm{qq}}(E)\) and \(\Delta_j^{\mathrm{qq}}(E)\) are disjoint because the corresponding \(3\)-planes \(\Delta_i^{\mathrm{scr}}(E)\) and \(\Delta_j^{\mathrm{scr}}(E)\) are. ◻