SlowGuess/NeurIPS_2023_OCR huggingface dataset:
where we reordered the variables according to  $(w_{1},w_{3},w_{2},\xi_{1},v_{1},\xi_{3},v_{3},\xi_{2},v_{2})$  for convenience. Following the same arguments in proving Lemma 5.1, if  $CA^{m}B = 0$  for all  $m$ , then eigenvalues of  $A - BC$  are the same as the eigenvalues of  $A$ . Writing  $A$  in block matrix form yields
$$
A = \left( \begin{array}{c c} A _ {1 1} & A _ {1 2} \\ A _ {2 1} & A _ {2 2} \end{array} \right),
$$
where  $A_{11}$  is  $3 \times 3$ . Notice that  $A_{11}$  is strictly lower triangular, and  $A_{21}$  is strictly block lower triangular. Now examine
$$
\det  \left[ s I - A \right] = \det  \left[ s I - A _ {1 1} \right] \det  \left[ \left(s I - A _ {2 2}\right) - A _ {2 1} \left(s I - A _ {1 1}\right) ^ {- 1} A _ {1 2} \right].
$$
One can show that  $A_{21}(sI - A_{11})^{-1}A_{12}$  is strictly block lower triangular. Therefore, we have  $\det [sI - A] = \det [sI - A_{11}] \det [sI - A_{22}]$ . Thus,  $A$  has eigenvalues at 0 with multiplicity 3 or more because of  $A_{11}$ . By exponential stability, there exists  $m \geq 2$  such that  $CA^{m}B \neq 0$ . Since  $CB = 0$ , and  $CAB = 0$ , we can now apply Lemma 5.1.
B.4 Proof of Proposition 6.1
Reference (37) presents a necessary and sufficient condition for strong stabilizability. First, we compute
$$
T (s) = \mathcal {C} (s I - \mathcal {A}) ^ {- 1} \mathcal {B} = \frac {s}{s + 1} \mathcal {M} (s I - \mathcal {M}) ^ {- 1}.
$$
There is blocking zero (i.e.,  $T(s) = 0$ ) at  $s = 0$  and  $|s| \to \infty$ . According to reference (37), a necessary and sufficient condition for strong stabilizability is that there should be an even number of eigenvalues in between such pairs of real zeros. This property is known as the "parity interlacing principle". The eigenvalues of  $\mathcal{A}$  are  $-1, -1, \pm \sqrt{m_{12}m_{21}}$ , and so there is a single (and hence, an odd number) of real eigenvalue in between two real zeros of  $T(s)$ .
C Destabilization with a sufficiently small  $\mu$
The proof in Section 5 can be modified to consider non-convergence over bounded games, e.g.,
$$
\| M _ {i j} \| _ {\alpha} <   1 \quad \forall i, j.
$$
Consider first the following lemma.
Lemma C.1. Let  $A \in \mathbb{R}^{n \times n}$ ,  $B \in \mathbb{R}^{n \times 1}$  and  $C \in \mathbb{R}^{1 \times n}$ . Assume that  $A$  has eigenvalues at 0 with multiplicity 3 or more. Then for sufficiently small  $\mu > 0$ ,  $A - \mu BC$  is not a stability matrix.
Proof. As in the proof of Lemma 5.1, we have
$$
\det  \left[ s I - (A - \mu B C) \right] = \det  \left[ s I - A \right] \left(q (s) + \mu p (s)\right) \frac {1}{q (s)}.
$$
Recall that the roots of  $q(s)$  are a subset of the roots of  $\det[sI - A]$ . If  $q(s)$  does not have at least 3 roots at zero, then  $A - \mu BC$  is not a stability matrix. Otherwise, root-locus arguments in references (30), and (31) (angle of departure rule) imply there exist roots of  $q(s) + \mu p(s)$  with positive real parts for small  $\mu$ .
Using the structure of the local dynamics and the fact that  $A$  in (17a) has eigenvalues at 0 with multiplicity 3 or more, one can directly use Lemma C.1 to show the existence of a sufficiently small offending  $\mu$ .