SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \mathbf{P}(\mathbf{A}\boldsymbol {x}) = \sum_{a\in \mathcal{A}_{j}}\mathbf{PA}_{(ja)}\boldsymbol {x}[ja] + \sum_{\substack{a\in \mathcal{A}_{j}\\ ja\notin \Sigma^{\perp}}}\sum_{j^{\prime}\in \mathcal{C}_{ja}}\mathbf{PA}_{\succeq j^{\prime}}(\boldsymbol {x}[ja]\cdot \boldsymbol{x}_{\succeq j^{\prime}}) \\ = \sum_ {\substack {a \in \mathcal {A} _ {j} \\ j a \in \Sigma^ {\perp}}} \boldsymbol {x} [ j a ] \cdot \mathbf {P A} _ {(j a)} + \sum_ {\substack {a \in \mathcal {A} _ {j} \\ j a \notin \Sigma^ {\perp}}} \left(\boldsymbol {x} [ j a ] \sum_ {j ^ {\prime} \in \mathcal {C} _ {j a}} \mathbf {P A} _ {\succeq j ^ {\prime}} \boldsymbol {x} _ {\succeq j ^ {\prime}}\right) (from(22)) \\ = \sum_ {\substack {a \in \mathcal {A} _ {j} \\ j a \in \Sigma^ {\perp}}} \boldsymbol {x} [ j a ] \cdot \boldsymbol {b} _ {j} + \sum_ {\substack {a \in \mathcal {A} _ {j} \\ j a \notin \Sigma^ {\perp}}} \left(\boldsymbol {x} [ j a ] \sum_ {j ^ {\prime} \in \mathcal {C} _ {j a}} \boldsymbol {b} _ {j ^ {\prime}}\right) \quad \text{(from (21) and (24))} \\ = \sum_ {\substack {a \in \mathcal {A} _ {j} \\ j a \in \Sigma^ {\perp}}} \boldsymbol {x} [ j a ] \cdot \boldsymbol {b} _ {j} + \sum_ {\substack {a \in \mathcal {A} _ {j} \\ j a \notin \Sigma^ {\perp}}} \boldsymbol {x} [ j a ] \cdot \boldsymbol {b} _ {j} (from(23)) \\ = \boldsymbol {b} _ {j} \cdot \sum_ {a \in \mathcal {A} _ {j}} \boldsymbol {x} [ j a ] = \boldsymbol {b} _ {j}. \quad (\text {f r o m D e f i n i t i o n 2 . 2}) \\ \end{array}
$$
$(\supseteq)$  Conversely, consider any  $\mathbf{B} \in \mathcal{L}_{\mathcal{Q}_{\succeq j} \to \mathcal{P}(\pmb{b}_j)}$ . We will show that there exists a matrix  $\mathbf{A} \in \tilde{\mathcal{M}}_{\mathcal{Q}_{\succeq j} \to \mathcal{P}(\pmb{b}_j)}$  such that  $\mathbf{B} \cong_{\mathcal{Q}_{\succeq j}} \mathbf{A}$ . First, we argue that there exists a matrix  $\mathbf{B}' \cong_{\mathcal{Q}_{\succeq j}} \mathbf{B}$  with the property that the column  $\mathbf{B}_{(ja)}'$  corresponding to any nonterminal sequence is identically zero.
Lemma B.7. There exists  $\mathbf{B}' \cong_{\mathcal{Q}_{\geq j}} \mathbf{B}$  such that  $\mathbf{B}_{(ja)}' = \mathbf{0}$  for all  $a \in \mathcal{A}_j$  such that  $ja \in \Sigma_{\geq j} \setminus \Sigma^\perp$ .
Proof. Fix any  $a \in \mathcal{A}_j$  such that  $ja$  is nonterminal. Then, by definition there exists at least one decision node  $j'$  whose parent sequence is  $ja$ . Consider now the matrix  $\mathbf{B}''$  obtained from "spreading" column  $\mathbf{B}_{(ja)}$  onto  $\mathbf{B}_{(j'a')}$  ( $a' \in \mathcal{A}_{j'}$ ), that is, the matrix whose columns are defined according to the following rules: (i)  $\mathbf{B}_{(ja)}'' = \mathbf{0}$ , (ii)  $\mathbf{B}_{(j'a')}' = \mathbf{B}_{(j'a')} + \mathbf{B}_{(ja)}$  for all  $a' \in \mathcal{A}_{j'}$ , (iii)  $\mathbf{B}_{(\sigma)}'' = \mathbf{B}_{(\sigma)}$  everywhere else. The column  $\mathbf{B}_{(ja)}''$  is identically zero by construction, and all other columns  $\mathbf{B}_{(ja')}'$ ,  $a' \in \mathcal{A}_j \setminus \{a\}$ , are the same as  $\mathbf{B}$ . Most importantly, since from the sequence-form constraints Definition 2.2 any sequence-form strategy  $x \in \mathcal{Q}_{\succeq j}$  satisfies the equality  $x[ja] = \sum_{a' \in \mathcal{A}_{j'}} x[j'a']$ , the matrix  $\mathbf{B}''$  satisfies  $\mathbf{B}''x = \mathbf{B}x$  for all  $x \in \mathcal{Q}_{\succeq j}$ , i.e.,  $\mathbf{B}'' \cong_{\mathcal{Q}_{\succeq j}} \mathbf{B}$ . Iterating the argument for all actions  $a \in \mathcal{A}_j$  yields the statement.
Consider now any  $a \in \mathcal{A}_j$  that leads to a terminal sequence  $ja \in \Sigma_{\succeq j} \cap \Sigma^{\perp}$ . The vector  $\mathbf{1}_{ja}$  defined as having a 1 in the position corresponding to  $ja$  and 0 everywhere else is a valid sequence-form strategy vector, that is,  $\mathbf{1}_{ja} \in \mathcal{Q}_{\succeq j}$ . Hence, since  $\mathbf{B}'$  maps  $\mathcal{Q}_{\succeq j}$  to  $\mathcal{P}(\boldsymbol{b}_j)$ , it is necessary that  $\mathbf{B}_{(ja)}' \in \mathcal{P}(\boldsymbol{b}_j)$ , that is,  $\mathbf{B}_{(ja)}' \geq \mathbf{0}$  and  $\mathbf{PB}_{(ja)}' = \boldsymbol{b}_j$ . In other words, we have just proved the following.
Lemma B.8. For any  $a \in \mathcal{A}_j$  such that  $ja \in \Sigma_{\succeq j} \cap \Sigma^{\perp}$ ,  $\mathbf{B}_{(ja)}' \geq 0$  and  $\mathbf{PB}_{(ja)}' = b_j$ .
Combined, Lemmas B.7 and B.8 show that  $\mathbf{B}'$  satisfies constraints (21), (22), and (25). Consider now any action  $a \in \mathcal{A}_j$  that defines a nonterminal sequence  $ja \in \Sigma_{\succeq j} \setminus \Sigma^{\perp}$ . For each child decision point  $j' \in \mathcal{C}_{ja}$ , let  $\boldsymbol{x}_{\succeq j'} \in \mathcal{Q}_{\succeq j'}$  be a choice of strategy for that decision point, and denote  $\mathbf{B}_{\succeq j'}'$  the submatrix of  $\mathbf{B}'$  obtained by only considering the columns  $\mathbf{B}_{(\sigma)}'$  corresponding to sequences  $\sigma \succeq j'$ . The vector  $\boldsymbol{x}$  defined according to  $\boldsymbol{x}[ja] = 1$ ,  $\boldsymbol{x}[\Sigma_{\succeq j'}] = \boldsymbol{x}_{\succeq j'}$  for all  $j' \in \mathcal{C}_{ja}$ , and 0 everywhere else is a valid sequence-form strategy  $\boldsymbol{x} \in \mathcal{Q}_{\succeq j}$ , and therefore  $\mathbf{B}'\boldsymbol{x} \in \mathcal{P}(\boldsymbol{b}_j)$  since  $\mathbf{B}' \cong_{\mathcal{Q}_{\succeq j}} \mathbf{B}$  and  $\mathbf{B} \in \mathcal{L}_{\mathcal{Q}_{\succeq j} \to \mathcal{P}(\boldsymbol{b}_j)}$  by hypothesis. Therefore, using the fact that  $\mathbf{B}_{(ja)}' = \mathbf{0}$  by Lemma B.7, we conclude that
$$
\mathcal {P} (\boldsymbol {b} _ {j}) \ni \mathbf {B} ^ {\prime} \boldsymbol {x} = \sum_ {j ^ {\prime} \in \mathcal {C} _ {j a}} \mathbf {B} _ {\succeq j ^ {\prime}} ^ {\prime} \boldsymbol {x} _ {\succeq j ^ {\prime}}.
$$
Because the above holds for any choice of  $\pmb{x}_{\succeq j'} \in \mathcal{Q}_{\succeq j'}$ , it follows that the matrix  $[\dots | \mathbf{B}_{\succeq j'}' | \dots ] \in \mathcal{L}_{\times_{j'} \in \mathcal{C}_{j_a}} \mathcal{Q}_{\succeq j'} \to \mathcal{P}(\pmb{b}_j)$ . Hence, applying Proposition B.4 (note that 0 is off  $\mathcal{Q}_{\succeq j'}$  since  $\sum_{a \in \mathcal{A}_{j'}} \pmb{x}[j'a] = 1$  for all  $\pmb{x} \in \mathcal{Q}_{\succeq j'}$  by Definition A.1) together with the inductive hypothesis, we conclude that for each  $j' \in \mathcal{C}_{j_a}$  there exist a vector  $\pmb{b}_{j'} \in \mathbb{R}^k$  and a matrix  $\mathbf{A}_{\succeq j'} \in \mathcal{M}_{\mathcal{Q}_{\succeq j'} \to \mathcal{P}(\pmb{b}_{j'})}$ , such that  $\sum_{j' \in \mathcal{C}_{j_a}} \pmb{b}_{j'} = \pmb{b}_j$  and  $[\dots | \mathbf{B}_{\succeq j'}' | \dots ] \cong_{x_{j'} \in \mathcal{C}_{j_a}} \mathcal{Q}_{\succeq j'} [ \dots | \mathbf{A}_{\succeq j'} | \dots ]$ . We can therefore replace all columns corresponding to  $\mathbf{B}_{\succeq j'}'$  with those of  $\mathbf{A}_{\succeq j'}$ , obtaining a new matrix  $\cong_{\mathcal{Q}_{\succeq j}} \mathbf{B}'$ . Repeating the argument for each  $ja \in \Sigma_{\succeq j} \setminus \Sigma^\perp$  finally yields a new matrix that is  $\cong_{\Sigma_{\succeq j}} \mathbf{B}$  and satisfies all constraints given in (27), as we wanted to show.
B.4 Putting all the Pieces Together
Finally, we are ready to prove the main result of the paper.
Theorem 3.1. Let  $\mathcal{Q}$  be a sequence-form strategy space and let  $\mathcal{P}$  be any bounded polytope of the form  $\mathcal{P} := \{\pmb{x} \in \mathbb{R}^d : \mathbf{P}\pmb{x} = \pmb{p}, \pmb{x} \geq \mathbf{0}\} \subseteq [0, \gamma]^d$ , where  $\mathbf{P} \in \mathbb{R}^{k \times d}$ . Then, for any linear function  $f : \mathcal{Q} \rightarrow \mathcal{P}$ , there exists a matrix  $\mathbf{A}$  in the polytope
Proof of Theorem 3.1. We prove the result in two steps. First, we show that
$$
\mathcal {L} _ {\mathcal {Q} \to \mathcal {P}} \cong_ {\mathcal {Q}} \tilde {\mathcal {M}} _ {\mathcal {Q} \to \mathcal {P}} := \left\{ \right.\mathbf {A} = \left[\begin{array}{c c c}&(3 0)&\mathbf {P A} _ {(j a)} = \boldsymbol {b} _ {j}\\&(3 1)&\mathbf {A} _ {(\sigma)} = \mathbf {0}\\&(3 2)&\sum_ {j ^ {\prime} \in \mathcal {C} _ {\varnothing}} \boldsymbol {b} _ {j ^ {\prime}} = \boldsymbol {p}\\&(3 3)&\sum_ {j ^ {\prime} \in \mathcal {C} _ {j a}} \boldsymbol {b} _ {j ^ {\prime}} = \boldsymbol {b} _ {j}\\&(3 4)&\mathbf {A} _ {(\sigma)} \geq \mathbf {0}\\&(3 5)&\boldsymbol {b} _ {j} \in \mathbb {R} ^ {k}\end{array}\right.   :\begin{array}{l l l}&(\text {a n d})\\&(\text {a n d})\\&(\text {a n d})\\&(\text {a n d})\\&(\text {a n d})\\&(\text {a n d})\\&(\text {a n d})\\&(\text {a n d})\\&(\text {a n d})\\&(\text {a n d})\\&(\text {a n d})\\&(\text {a n d})\\\end{array}\begin{array}{l l l}&(\text {a n d})\\&(\text {a n d})\\&(\text {a n d})\\&(\text {a n d})\\&(\text {a n d})\\&(\text {a n d})\\&(\text {a n d})\\&(\text {a n d})\\&(\text {a n d})\\&(\text {a n d})\\\end{array}
$$
where the difference between  $\tilde{\mathcal{M}}_{\mathcal{Q}\to \mathcal{P}}$  and  $\mathcal{M}_{\mathcal{Q}\rightarrow \mathcal{P}}$  lies in constraint (34), which only sets a lower bound (at zero) for each entry of the matrix, as opposed to a bound  $[0,\gamma ]$  as in (7). Using the definition of  $\tilde{\mathcal{M}}_{\mathcal{Q}_{\geq j}\to \mathcal{P}(\pmb {b}_j)}$  given in (20) (Theorem B.6), the set  $\tilde{\mathcal{M}}_{\mathcal{Q}\rightarrow \mathcal{P}}$  can be equivalently written as
$$
\tilde {\mathcal {M}} _ {\mathcal {Q} \to \mathcal {P}} = \left\{\mathbf {A} = \left[ \begin{array}{c c c} \dots & \mathbf {A} _ {(\sigma)} & \dots \end{array} \right] \in \mathbb {R} ^ {d \times \Sigma}: \begin{array}{l l l} (3 6) & \mathbf {A} _ {(\varnothing)} = \mathbf {0} \\ (3 7) & \sum_ {j \in \mathcal {C} _ {\varnothing}} \boldsymbol {b} _ {j} = \boldsymbol {p} \\ (3 8) & [ \mathbf {A} _ {(\sigma)} ] _ {\sigma \succeq j} \in \tilde {\mathcal {M}} _ {\mathcal {Q} _ {\succeq j} \to \mathcal {P} (\boldsymbol {b} _ {j})} & \forall j \in \mathcal {C} _ {\varnothing} \\ (3 9) & \boldsymbol {b} _ {j} \in \mathbb {R} ^ {k} & \forall j \in \mathcal {C} _ {\varnothing} \end{array} \right\}.
$$
To show that  $\tilde{\mathcal{M}}_{\mathcal{Q}\to \mathcal{P}}\cong_{\mathcal{Q}}\mathcal{L}_{\mathcal{Q}\to \mathcal{P}}$  we proceed exactly like in the inductive step of the proof of Theorem B.6. Specifically, let  $\mathbf{A}\in \tilde{\mathcal{M}}_{\mathcal{Q}\to \mathcal{P}}$  and  $\pmb {x}\in \mathcal{Q}$  be arbitrary. From Definition 2.2 it follows that  $\pmb {x}[\Sigma_{\geq j}]\in \mathcal{Q}_{\geq j}$  for any  $j\in \mathcal{C}_{\emptyset}$  and therefore, denoting  $\mathbf{A}_{\geq j}\coloneqq [\mathbf{A}_{(\sigma)}]_{\sigma \geq j}$ ,
$$
\mathbf {P} (\mathbf {A} \boldsymbol {x}) \stackrel {(3 6)} {=} \sum_ {j \in \mathcal {C} _ {\emptyset}} \mathbf {P A} _ {(\sigma)} \boldsymbol {x} [ \Sigma_ {\succeq j} ] \stackrel {(3 8)} {=} \sum_ {j \in \mathcal {C} _ {\emptyset}} \boldsymbol {b} _ {j} \stackrel {(3 8)} {=} \boldsymbol {p}, \quad \mathbf {A} \boldsymbol {x} \stackrel {(3 6)} {=} \sum_ {j \in \mathcal {C} _ {\emptyset}} \mathbf {A} _ {(\sigma)} \boldsymbol {x} [ \Sigma_ {\succeq j} ] \stackrel {(3 8)} {\geq} \mathbf {0}.
$$
which shows that  $\mathbf{A}\pmb{x} \in \mathcal{P}$ . Since  $\mathbf{A}$  and  $\pmb{x}$  were arbitrary, it follows that  $\tilde{\mathcal{M}}_{\mathcal{Q} \to \mathcal{P}} \subseteq \mathcal{L}_{\mathcal{Q} \to \mathcal{P}}$ . Conversely, let  $\mathbf{B} \in \mathcal{L}_{\mathcal{Q} \to \mathcal{P}}$  be arbitrary, and fix a root decision node  $j \in \mathcal{C}_{\emptyset}$ . Then, we can "spread out" the column  $\mathbf{B}_{(\emptyset)}$  by adding it to each  $\mathbf{B}_{(ja)}: a \in \mathcal{A}_j$  by constructing the matrix  $\mathcal{L}_{\mathcal{Q} \to \mathcal{P}} \ni \mathbf{B}' \cong_{\mathcal{Q}} \mathbf{B}$  defined by (i)  $\mathbf{B}_{(\emptyset)}' = \mathbf{0}$ , (ii)  $\mathbf{B}_{(ja)}' = \mathbf{B}_{(ja)} + \mathbf{B}_{(\emptyset)}$  for any  $a \in \mathcal{A}_j$ , and (iii)  $\mathbf{B}_{(\sigma)}' = \mathbf{B}_{(\sigma)}$  everywhere else. Pick now any vectors  $\{\pmb{x}_{\succeq j} \in \mathcal{Q}_{\succeq j}: j \in \mathcal{C}_{\emptyset}\}$ , and consider the vector  $\pmb{x}$  defined as  $\pmb{x}[\emptyset] = 1$ , and  $\pmb{x}[\Sigma_{\succeq j}] = \pmb{x}_{\succeq j}$  for all  $j \in \mathcal{C}_{\emptyset}$ . The vector  $\pmb{x}$  is a valid sequence-form strategy, that is,  $\pmb{x} \in \mathcal{Q}$ . Let now  $\mathbf{B}_{\succeq j}' := [\mathbf{B}_{(\sigma)}']_{\sigma \succeq j}$ . From the fact that  $\mathbf{B}'\pmb{x} \in \mathcal{P}$ , together with the fact that by construction  $\mathbf{B}_{(\emptyset)}' = \mathbf{0}$ , we conclude that
$$
\mathbf {B} ^ {\prime} \boldsymbol {x} = \sum_ {j \in \mathcal {C} _ {\emptyset}} \mathbf {B} _ {\succeq j} ^ {\prime} \boldsymbol {x} _ {\succeq j} \in \mathcal {P}.
$$
Since the inclusion above holds for any choice of  $\{\pmb{x}_{\succeq j} \in \mathcal{Q}_{\succeq j} : j \in \mathcal{C}_{\emptyset}\}$ , and since for all  $j \in \mathcal{C}_{\emptyset}$  the vector  $\mathbf{0} \notin \mathrm{aff} \mathcal{Q}_{\succeq j}$  (indeed,  $\sum_{a \in \mathcal{A}_j} \pmb{x}[ja] = 1$  for all  $\pmb{x} \in \Sigma_{\succeq j}$  by Definition A.1), from Proposition B.4 together with Theorem B.6 we conclude that for each  $j \in \mathcal{C}_{\emptyset}$  there exists a vector  $\pmb{b}_j \in \mathbb{R}^k$  and a matrix  $\mathbf{A}_{\succeq j} \in \tilde{\mathcal{M}}_{\mathcal{Q}_{\succeq j} \to \mathcal{P}(\pmb{b}_j)}$ , such that  $\sum_{j \in \mathcal{C}_{\emptyset}} \pmb{b}_j = \pmb{p}$  and  $\mathbf{A}_{\succeq j} \cong_{\mathcal{Q}_{\succeq j}} \mathbf{B}_{\succeq j}'$ . By replacing the submatrices  $\mathbf{B}_{\succeq j}'$  with  $\mathbf{A}_{\succeq j}$  in  $\mathbf{B}'$  we then obtain an equivalent matrix that satisfies all constraints that define  $\tilde{\mathcal{M}}_{\mathcal{Q} \to \mathcal{P}}$ . In summary, we have  $\tilde{\mathcal{M}}_{\mathcal{Q} \to \mathcal{P}} \cong \mathcal{L}_{\mathcal{Q} \to \mathcal{P}}$ .
To conclude the proof, we now show that  $\tilde{\mathcal{M}}_{\mathcal{Q}\to \mathcal{P}} = \mathcal{M}_{\mathcal{Q}\to \mathcal{P}}$ . First, we make the straightforward observation that any  $\mathbf{A} \in \mathcal{M}_{\mathcal{Q}\to \mathcal{P}}$  also belongs to  $\tilde{\mathcal{M}}_{\mathcal{Q}\to \mathcal{P}}$ , as the constraint that defines the latter set are only looser. Hence, we only need to show that any  $\mathbf{B} \in \tilde{\mathcal{M}}_{\mathcal{Q}\to \mathcal{P}}$  also satisfies constraint (7). Since  $\mathbf{B} \in \tilde{\mathcal{M}}_{\mathcal{Q}\to \mathcal{P}}$ , all columns of  $\mathbf{B}$  are nonnegative (constraint (34)). Furthermore, since  $\tilde{\mathcal{M}}_{\mathcal{Q}\to \mathcal{P}} \cong_{\mathcal{Q}} \mathcal{L}_{\mathcal{Q}\to \mathcal{P}}$ , clearly  $\mathbf{B}\pmb{x} \in \mathcal{P}$  for all  $\pmb{x} \in \mathcal{Q}$ . Fix now any sequence  $\sigma \in \Sigma$ , and consider any strategy  $\pmb{x} \in \mathcal{Q}$  that puts probability mass 1 on all the actions on the path from the root to  $\sigma$  included, that is, any  $\pmb{x} \in \mathcal{Q}$  with  $\pmb{x}[\sigma] = 1$ . Then, from the nonnegativity of the columns of  $\mathbf{B}$ , it follows that
$$
\mathcal {P} \ni \mathbf {B} x \geq \mathbf {B} _ {(\sigma)}.
$$
Since by definition of  $\gamma$  any point in  $\mathcal{P}$  belongs to  $[0,\gamma ]^d$ , we then conclude that  $\mathbf{B}_{(\sigma)}\in [0,\gamma ]^d$ , implying that  $\mathbf{B}\in \mathcal{M}_{\mathcal{Q}\to \mathcal{P}}$  as we wanted to show.
C Corollaries of the Characterization Theorem
We mention two direct corollaries of Theorem 3.1 that slightly extend the scope of the characterization. The first corollary asserts that the polytope  $\mathcal{M}_{\mathcal{Q}\rightarrow \mathcal{P}}$  characterizes not only all linear functions from  $\mathcal{Q}$  to  $\mathcal{P}$ , but also all affine functions.
Corollary C.1 (From linear to affine functions). Let  $\mathcal{Q}$  be a sequence-form strategy space and let  $\mathcal{P}$  be any polytope. Then, for any affine function  $g: \mathcal{Q} \rightarrow \mathcal{P}$ , there exists a matrix  $\mathbf{A}$  in the polytope  $\mathcal{M}_{\mathcal{Q} \rightarrow \mathcal{P}}$  defined in Theorem 3.1 such that  $g(\mathbf{x}) = \mathbf{A}\mathbf{x}$  for all  $\mathbf{x} \in \mathcal{Q}$ . Conversely, any  $\mathbf{A} \in \mathcal{M}_{\mathcal{Q} \rightarrow \mathcal{P}}$  induces an affine function from  $\mathcal{Q}$  to  $\mathcal{P}$ .
Proof. The second part of the statement is trivial since any linear function is also affine, and any  $\mathbf{A} \in \mathcal{M}_{\mathcal{Q} \to \mathcal{P}}$  induces a linear function from  $\mathcal{Q}$  to  $\mathcal{P}$ . Let  $g(\boldsymbol{x}) = f(\boldsymbol{x}) + \boldsymbol{b}$  be any affine function, where  $f$  is an appropriate linear function from  $\mathcal{Q}$  to  $\mathcal{P}$  and  $\boldsymbol{b} \in \mathbb{R}^n$ . Since  $\boldsymbol{q}[\varnothing] = 1$  for all  $\boldsymbol{q} \in \mathcal{Q}$ , the function  $g$  coincides on  $\mathcal{Q}$  with the function  $\tilde{g}: \mathcal{Q} \ni \boldsymbol{x} \mapsto f(\boldsymbol{x}) + \boldsymbol{b} \cdot \boldsymbol{x}[\varnothing]$ , which is a linear function of  $\boldsymbol{x}$ . Hence, from the first part of Theorem 3.1 there exists  $\mathbf{A} \in \mathcal{M}_{\mathcal{Q} \to \mathcal{P}}$  such that  $\mathbf{A}\boldsymbol{x} = \tilde{g}(\boldsymbol{x}) = g(\boldsymbol{x})$  for all  $\boldsymbol{x} \in \mathcal{Q}$ .
The second corollary of Theorem 3.1 extends the characterization to the alternative definition of polytope as a bounded set of the form  $\mathcal{C} \coloneqq \{\pmb{y} \in \mathbb{R}^n : \mathbf{C}\pmb{y} \leq \pmb{c}\}$ , by first introducing slack variables and rewriting the polytope in the form handled by Theorem 3.1.
Corollary C.2 (Alternative polytope representations). Let  $Q$  be a sequence-form strategy polytope, and  $\mathcal{C} \coloneqq \{\pmb{y} \in \mathbb{R}^n : \mathbf{C}\pmb{y} \leq \pmb{c}\} \subseteq [-\gamma, \gamma]^n$  be a bounded polytope, where  $\mathbf{C} \in \mathbb{R}^{m \times n}$ . Let  $k \coloneqq \max \{\| \mathbf{C} \|_{\infty}, \| \pmb{c} \|_{\infty}\}$  and introduce the polytope
$$
\tilde {\mathcal {P}} := \left\{\left(\tilde {\boldsymbol {y}}, \boldsymbol {s}\right) \in \mathbb {R} ^ {n} \times \mathbb {R} ^ {m}: \left[ \begin{array}{c c} \mathbf {C} & k n \mathbf {I} \\ & \boldsymbol {s} \end{array} \right] \left[ \begin{array}{c} \tilde {\boldsymbol {y}} \\ \boldsymbol {s} \end{array} \right] = \boldsymbol {c} + \gamma \mathbf {C} \mathbf {1}, \quad \left[ \begin{array}{c} \tilde {\boldsymbol {y}} \\ \boldsymbol {s} \end{array} \right] \geq \mathbf {0} \right\} \subseteq [ 0, 2 \gamma ] ^ {n + m},
$$
which is of the form handled by Theorem 3.1. For any affine function  $g: \mathcal{Q} \to \mathcal{C}$ , there exists a matrix  $\mathbf{A}$  in the polytope
$$
\tilde {\mathcal {M}} _ {\mathcal {Q} \rightarrow \mathcal {C}} := \left\{\left[ \tilde {\mathbf {M}} _ {(\varnothing)} - \gamma \mathbf {1} \mid \dots \mid \tilde {\mathbf {M}} _ {(\sigma)} \mid \dots \right] \in \mathbb {R} ^ {n \times \Sigma}: \left[\begin{array}{c}\tilde {\mathbf {M}}\\\tilde {\mathbf {Z}}\end{array}\right] \in \mathcal {M} _ {\mathcal {Q} \rightarrow \tilde {\mathcal {P}}}, \tilde {\mathbf {M}} \in \mathbb {R} ^ {n \times \Sigma}, \tilde {\mathbf {Z}} \in \mathbb {R} ^ {m \times \Sigma} \right\}
$$
such that  $g(\pmb{x}) = \mathbf{A}\pmb{x}$  for all  $\pmb{x} \in \mathcal{Q}$ . Conversely, any  $\mathbf{A} \in \tilde{\mathcal{M}}_{\mathcal{Q} \to \mathcal{C}}$  induces an affine function from  $\mathcal{Q}$  to  $\mathcal{C}$ .
Proof. We begin by proving that  $\mathcal{C} \subseteq [-\gamma, \gamma]^n$  implies  $\tilde{\mathcal{P}} \subseteq [0, 2\gamma]^{n+m}$ . Consider any  $(\tilde{\pmb{y}}, \pmb{s}) \in \tilde{\mathcal{P}}$  and set  $\pmb{y} = \tilde{\pmb{y}} - \gamma \mathbf{1}$ . Then, this is a valid  $\pmb{y} \in \mathcal{C} \subseteq [-\gamma, \gamma]^n$  and, consequently,  $\tilde{\pmb{y}} \in [0, 2\gamma]^n$ . For the slack variables  $\pmb{s}$  it holds that  $kns = c - \mathbf{Cy}$ , where  $\pmb{y} = \tilde{\pmb{y}} - \gamma \mathbf{1}$  from before. By definition of  $k$  and by  $\pmb{y} \geq -\gamma \mathbf{1}$ , we conclude that  $c - \mathbf{Cy} \leq (k + n\gamma k)\mathbf{1} \Rightarrow s \in [0, 2\gamma]^m$ .
Now let  $g: \mathcal{Q} \to \mathcal{C}$  be any affine function from  $\mathcal{Q}$  to  $\mathcal{C}$ . Then we can define an affine function  $f: \mathcal{Q} \to \tilde{\mathcal{P}}$  such that
$$
f (\boldsymbol {x}) = \left[ \begin{array}{c} g (\boldsymbol {x}) + \gamma \mathbf {1} \\ \boldsymbol {s} \end{array} \right]
$$
for all  $\pmb{x} \in \mathcal{Q}$ . By Corollary C.1 we know that  $\mathcal{M}_{\mathcal{Q} \to \tilde{\mathcal{P}}}$  characterizes all affine functions from  $\mathcal{Q}$  to  $\tilde{\mathcal{P}}$ , including the previous function  $f$ . Thus, there exists an  $\tilde{\mathbf{M}} \in \mathbb{R}^{n \times \Sigma}$  such that  $g(\pmb{x}) + \gamma \mathbf{1} = \tilde{\mathbf{M}}\pmb{x}$  for all  $\pmb{x} \in \mathcal{Q}$ . Since  $\pmb{x}[\varnothing] = 1$  for all  $\pmb{x} \in \mathcal{Q}$ , we conclude that there exists  $\mathbf{A} \in \tilde{\mathcal{M}}_{\mathcal{Q} \to \mathcal{C}}$  such that  $g(\pmb{x}) = \mathbf{A}\pmb{x} = \tilde{\mathbf{M}}\pmb{x} - \gamma \mathbf{1}$  for all  $\pmb{x} \in \mathcal{Q}$ .
Conversely, consider any  $\mathbf{A} \in \tilde{\mathcal{M}}_{\mathcal{Q} \to \mathcal{C}}$  and define  $g(\boldsymbol{x}) = \mathbf{A}\boldsymbol{x}$ . That is, there exist suitable  $\tilde{\mathbf{M}} \in \mathbb{R}^{n \times \Sigma}$ ,  $\tilde{\mathbf{Z}} \in \mathbb{R}^{m \times \Sigma}$  that satisfy the constraints of polytope  $\tilde{\mathcal{M}}_{\mathcal{Q} \to \mathcal{C}}$ . Then for all  $\boldsymbol{x} \in \mathcal{Q}$  it holds  $g(\boldsymbol{x}) = \tilde{\mathbf{M}}\boldsymbol{x} - \gamma \mathbf{1}$ , and  $(\tilde{\boldsymbol{y}}, s) \in \tilde{\mathcal{P}}$  where  $\tilde{\boldsymbol{y}} = \tilde{\mathbf{M}}\boldsymbol{x}$  and  $s = \tilde{\mathbf{Z}}\boldsymbol{x}$ . Thus, by construction of  $\tilde{\mathcal{P}}$ , as we also argued in the beginning of the proof, we conclude that  $g(\boldsymbol{x}) = \tilde{\boldsymbol{y}} - \gamma \mathbf{1} = \boldsymbol{y} \in \mathcal{C}$ .
D Additional proofs
Theorem D.1. Let  $\Sigma$  denote the set of sequences of the learning player in the extensive-form game, and let  $\eta^{(t)} = 1 / \sqrt{t}$  for all  $t$ . Then, for any sequence of loss vectors  $\ell^{(t)} \in [0,1]^{\Sigma}$ , Algorithm 1 guarantees linear-swap regret  $O(|\Sigma|^2\sqrt{T})$  after any number  $T$  of iterations, and runs in  $O(|\Sigma|^{10}\log (|\Sigma|)\log^2 t)$  time for each iteration  $t$ .
Proof of Theorem 3.2. First we focus on the linear-swap regret bound. Based on Gordon et al. [2008] the  $\Phi$ -regret equals external regret over the set  $\Phi$  of transformations. In our case  $\Phi$  is the set  $\mathcal{M}_{\mathcal{Q}\to \mathcal{Q}}$  of all valid linear transformations and the losses for the external regret minimizer are functions  $\mathbf{A} \mapsto \langle \ell^{(1)},\mathbf{A}\pmb{x}^{(1)}\rangle, \mathbf{A} \mapsto \langle \ell^{(2)},\mathbf{A}\pmb{x}^{(2)}\rangle, \ldots$ . Equivalently, we can write these as  $\mathbf{A} \mapsto \langle \mathbf{L}^{(t)},\mathbf{A}\rangle_F$ , where  $\langle \cdot ,\cdot \rangle_{F}$  is the component-wise inner product for matrices and  $\mathbf{L}^{(t)} = \boldsymbol{\ell}^{(t)}(\boldsymbol{x}^{(t)})^\top$ , which is a rank-one matrix. Let  $D$  be an upper bound on the diameter of  $\mathcal{M}_{\mathcal{Q}\to \mathcal{Q}}$ , and  $L$  be such that  $\| \mathbf{L}^{(t)}\|_F \leq L$  for all  $t$ . Then, based on Orabona [2022] we can bound the external regret for this instance of Online Linear Optimization by picking  $\eta^{(t)} = \frac{D}{L\sqrt{t}}$  which gives a regret of  $O(DL\sqrt{T})$ . Since  $\mathbf{A} \in [0,1]^{\Sigma \times \Sigma}$  we get  $D = |\Sigma|$ , and since  $\ell^{(t)},\pmb{x}^{(t)} \in [0,1]^{\Sigma}$  we get  $L = |\Sigma|$ . This results in the desired linear-swap regret of  $O(|\Sigma|^2\sqrt{T})$ .