SlowGuess/NeurIPS_2023_OCR huggingface dataset:
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Bernhard von Stengel. Efficient computation of behavior strategies. Games and Economic Behavior, 14(2):220-246, 1996.
FrancescoOrabona.A Modern Introduction to Online Learning,2022.
Geoffrey Gordon. Regret bounds for prediction problems. In Proceedings of 12th Annual Conference on Computational Learning Theory (COLT '99), pages 29 - 40, July 1999.
Martin Zinkevich. Online convex programming and generalized infinitesimal gradient ascent. In Proceedings of the Twentieth International Conference on International Conference on Machine Learning, ICML'03, page 928-935. AAAI Press, 2003.
Brian Hu Zhang, Gabriele Farina, Andrea Celli, and Tuomas Sandholm. Optimal correlated equilibria in general-sum extensive-form games: Fixed-parameter algorithms, hardness, and two-sided column-generation. In David M. Pennock, Ilya Segal, and Sven Seuken, editors, EC '22: The 23rd ACM Conference on Economics and Computation, Boulder, CO, USA, July 11 - 15, 2022, pages 1119-1120. ACM, 2022.
Nisheeth K. Vishnoi. Algorithms for Convex Optimization. Cambridge University Press, 2021.
Bernd Gartner and Jiri Matousek. Approximation Algorithms and Semidefinite Programming. Springer Publishing Company, Incorporated, 2014.
Gurobi Optimization, LLC. Gurobi Optimizer Reference Manual, 2023. URL https://www.gurobi.com.
H. W. Kuhn. A Simplified Two-Person Poker. In H. W. Kuhn and A. W. Tucker, editors, Contributions to the Theory of Games, volume 1 of Annals of Mathematics Studies, 24, pages 97-103. Princeton University Press, Princeton, New Jersey, 1950.
A Additional Extensive-Form Game Notation
In the proofs, we will make use of the following symbols and notation.
Symbol	Description
Ji	Set of all Player i's infosets.
Aj	Set of actions available at any node in the information set j.
Σi	Set of sequences for Player i, defined as Σ(i) := {(j,a): j ∈ J, a ∈ Aj} ∪ {∅}, where the special element ∅ is called the empty sequence.
∅	where the special element ∅ is called the empty sequence.
Σi⊥	Set of terminal sequences for Player i.
pj	Parent sequence of j, defined as the last pair (j,a) ∈ Σi encountered on the path from the root to any information set j.
Cσ	Set of all "children" of sequence σ, defined as the information sets j ∈ J having as parent pj = σ.
j' < j	Information set j ∈ J is an ancestor of j' ∈ J, that is, there exists a path in the game tree connecting a node h ∈ j to some node h' ∈ j'.
σ < σ'	Sequence σ precedes sequence σ', where σ, σ' belong to the same player.
σ ≥ j	Sequence σ = (j', a') is such that j' ≥ j.
Σ≥j	Sequences at j ∈ J and all of its descendants, Σ≥j := {σ ∈ Σ: σ ≥ j}.
Qi	Sequence-form strategies of Player i (Definition 2.2).
Q≥j	Sequence-form strategies for the subtree rooted at j ∈ J (Definition A.1).
Πi	Reduced-normal-form plans (a.k.a. deterministic sequence-form strategies) of Player i.
Π≥j	Reduced-normal-form plans (a.k.a. deterministic sequence-form strategies) for the subtree rooted at j ∈ J.
Table 1: Summary of game-theoretic notation used in this paper. Note that we might skip player-specific subscripts when they can be inferred.
Furthermore, when the subscript referring to players can be inferred or is irrelevant (that is, the quantities refer to a generic player), then we might skip it.
As hinted by some of the rows in the above table, we will sometimes find it important to consider partial strategies that only specify behavior at a decision node  $j$  and all of its descendants  $j' \succ j$ . We make this formal through the following definition.
Definition A.1. The set of sequence-form strategies for the subtree rooted at  $j$ , denoted  $\mathcal{Q}_{\geq j}$ , is the set of all vectors  $\pmb{x} \in \mathbb{R}_{\geq 0}^{\Sigma_{\succ j}}$  such that probability-mass-conservation constraints hold at decision node  $j$  and all of its descendants  $j' \succ j$ , specifically
$$
\mathcal {Q} _ {\succeq j} := \left\{\boldsymbol {x} \in \mathbb {R} _ {\geq 0} ^ {\Sigma_ {\succ j}}: \begin{array}{l l} (9) & \sum_ {a \in \mathcal {A} _ {j}} \boldsymbol {x} [ j a ] = 1 \\ (1 0) & \sum_ {a \in \mathcal {A} _ {j ^ {\prime}}} \boldsymbol {x} [ j ^ {\prime} a ] = \boldsymbol {x} [ p _ {j ^ {\prime}} ] \end{array} \forall j ^ {\prime} \succ j \right\}.
$$
Finally, we define the symbol  $\Sigma_i^\perp$  to be the set of all terminal sequences for Player  $i$ . Thus, the set  $\Sigma_i \setminus \Sigma_i^\perp$  would give us all the non-terminal sequences of that player.
Access to coordinates By definition, sequence-form strategies are vectors indexed by sequences. To access the coordinate corresponding to sequence  $\sigma$ , we will use the notation  $\pmb{x}[\sigma]$ . Occasionally, we will need to extract a subvector corresponding to all sequences that are successor of an information set  $j$ , that is, all sequences  $\sigma \succeq j$ . For that, we use the notation  $\pmb{x}[\succeq j]$ .
Remark on the structure of sequence-form strategies We further remark the following known fact about the structure of sequence-form strategies. Intuitively, it crystallizes the idea that sequence-form strategies encode product of probabilities of actions on the path from the root to any decision point. The proof follows directly from the definitions.
Lemma A.2. Let  $j \in \mathcal{J}_i$  be an information set for a generic player. Then, given any sequence-form strategy  $x \in \mathcal{Q}_{\succeq j}$ , action  $a \in \mathcal{A}_j$ , and child information set  $j' \in \mathcal{C}_{ja}$ , there exists a sequence-form strategy  $x_{\succeq j'} \in \mathcal{Q}_{\succeq j'}$  such that
$$
\boldsymbol {x} [ \succeq j ^ {\prime} ] = \boldsymbol {x} [ j a ] \boldsymbol {x} _ {\succeq j ^ {\prime}}.
$$
B Proof of the Characterization Theorem (Theorem 3.1)
In this section we prove the central result of this paper, the characterization given in Theorem 3.1 of linear functions from the sequence-form strategy polytope  $\mathcal{Q}$  to the generic polytope  $\mathcal{P} \coloneqq \{\pmb{x} \in \mathbb{R}^d : \pmb{P}\pmb{x} = \pmb{p}, \pmb{x} \geq \pmb{0}\}$ , where  $\mathbf{P} \in \mathbb{R}^{k \times d}$  and  $\pmb{p} \in \mathbb{R}^k$ .
We will prove the characterization theorem by induction on the structure of the extensive-form strategy polytope. To do so, it will be useful to introduce a few additional objects and notations. We do so in the next subsection.
B.1 Additional Objects and Notation Used in the Proof
First, we introduce a parametric version of the polytope  $\mathcal{P}$ , where the right-hand side vector is made variable.
Definition B.1. Given any  $\pmb{b} \in \mathbb{R}^k$ , we will denote with  $\mathcal{P}(\pmb{b})$  the polytope
$$
\mathcal {P} (\boldsymbol {b}) := \left\{\boldsymbol {x} \in \mathbb {R} ^ {d}: \mathbf {P} \boldsymbol {x} = \boldsymbol {b}, \boldsymbol {x} \geq \mathbf {0} \right\}.
$$
In particular,  $\mathcal{P} = \mathcal{P}(\pmb {p})$
Furthermore, we introduce the equivalence relation  $\cong_{\mathcal{D}}$  to indicate that two matrices induce the same linear function when restricted to domain  $\mathcal{D}$ .
Definition B.2. Given two matrices  $\mathbf{A},\mathbf{B}$  of the same dimension, we write  $\mathbf{A}\cong_{\mathcal{D}}\mathbf{B}$  if  $\mathbf{Ax} = \mathbf{Bx}$  for all  $\pmb {x}\in \mathcal{D}$ . Similarly, given two sets  $\mathcal{U},\mathcal{V}$  of matrices we write  $\mathcal{U}\cong_{\mathcal{D}}\mathcal{V}$  to mean that for any  $\mathbf{A}\in \mathcal{U}$  there exists  $\mathbf{B}\in \mathcal{V}$  with  $\mathbf{A}\cong_{\mathcal{D}}\mathbf{B}$ , and vice versa.
Additionally, we introduce a symbol to denote the set of all matrices that induce linear functions from a set  $\mathcal{U}$  to a set  $\mathcal{V}$ .
Definition B.3. Given any sets  $\mathcal{U},\mathcal{V}$  we denote with  $\mathcal{L}_{\mathcal{U}\to \mathcal{V}}$  the set of all matrices that induce linear transformations from  $\mathcal{U}$  to  $\mathcal{V}$ , that is,
$$
\mathcal {L} _ {\mathcal {U} \rightarrow \mathcal {V}} := \left\{\mathbf {A}: \mathbf {A} \mathbf {x} \in \mathcal {V} f o r a l l \mathbf {x} \in \mathcal {U} \right\}.
$$
Finally, we remark that for a matrix  $\mathbf{A}$  whose columns are indexed using sequences  $\sigma \in \Sigma$ , we represent its columns as  $\mathbf{A}_{(\sigma)}$ . Furthermore, for sequence-form strategies  $\pmb{x} \in \mathcal{Q}$ , we use  $\pmb{x}[\sigma]$  to represent their entries, and  $\pmb{x}[\Sigma_{\geq j}]$  to represent a vector consisting only of the entries corresponding to sequences  $\sigma \in \Sigma_{\geq j}$ .
B.2 A Key Tool: Linear Transformations of Cartesian Products
We are now ready to introduce the following Proposition, which will play an important role in the proof of Theorem 3.1.
Proposition B.4. Let  $\mathcal{U}_1, \ldots, \mathcal{U}_m$  be sets, with  $\mathbf{0} \notin \operatorname{aff} \mathcal{U}_i$ ^2 for all  $i = 1, \ldots, m$ . Furthermore, for any  $i = 1, \ldots, m$  and any  $\pmb{b}_i \in \mathbb{R}^k$ , let  $\mathcal{M}_{\mathcal{U}_i \to \mathcal{P}(\pmb{b}_i)}$  be such that  $\mathcal{M}_{\mathcal{U}_i \to \mathcal{P}(\pmb{b}_i)} \cong_{\mathcal{U}_i} \mathcal{L}_{\mathcal{U}_i \to \mathcal{P}(\pmb{b}_i)}$ . Then, for all  $\pmb{b} \in \mathbb{R}^k$ ,
$$
\mathcal {L} _ {\left(\mathcal {U} _ {1} \times \dots \times \mathcal {U} _ {m}\right)\rightarrow \mathcal {P} (\boldsymbol {b})} \cong_ {\left(\mathcal {U} _ {1} \times \dots \times \mathcal {U} _ {m}\right)} \left\{\begin{array}{l l}(1 1) \quad \mathbf {A} _ {i} \in \mathcal {M} _ {\mathcal {U} _ {i} \rightarrow \mathcal {P} (\boldsymbol {b} _ {i})}&\forall i \in \{1, \dots , m \}\\\left[ \mathbf {A} _ {1} \mid \dots \mid \mathbf {A} _ {m} \right]:&(1 2) \quad \boldsymbol {b} _ {1} + \dots + \boldsymbol {b} _ {m} = \boldsymbol {b}\\(1 3) \quad \boldsymbol {b} _ {i} \in \mathbb {R} ^ {k}&\forall i \in \{1, \dots , m \}\end{array}\right\}. \tag {14}
$$
Instead of the condition  $\mathbf{0} \notin \operatorname{aff} \mathcal{U}_i$ , we could equivalently state that there exists  $\pmb{\tau}_i$  such that  $\pmb{\tau}_i^\top \pmb{x}_i = 1$  for all  $\pmb{x}_i \in \mathcal{U}_i$  using the properties of affine sets.
Proof. We prove the result by showing the two directions of the inclusion separately.
$(\supseteq)$  First, we show that for any  $\pmb{b} \in \mathbb{R}^k$ , any matrix  $\mathbf{A} = [\mathbf{A}_1 \mid \dots \mid \mathbf{A}_m]$  that belongs to the set on the right-hand side of (14) induces a linear transformation from  $\mathcal{U}_1 \times \dots \times \mathcal{U}_m$  to  $\mathcal{P}(\pmb{b})$  and thus belongs to  $\mathcal{L}_{(\mathcal{U}_1 \times \dots \times \mathcal{U}_m) \to \mathcal{P}(\pmb{b})}$ . To that end, we note that for any  $\pmb{x} = (\pmb{x}_1, \dots, \pmb{x}_m) \in \mathcal{U}_1 \times \dots \times \mathcal{U}_m$ ,
$$
\mathbf {A} \boldsymbol {x} = \sum_ {i = 1} ^ {m} \mathbf {A} _ {i} \boldsymbol {x} _ {i} \stackrel {(1 1)} {\geq} \mathbf {0}, \quad \text {a n d} \quad \mathbf {P} (\mathbf {A} \boldsymbol {x}) = \sum_ {i = 1} ^ {m} \mathbf {P} \mathbf {A} _ {i} \boldsymbol {x} _ {i} \stackrel {(1 1)} {=} \sum_ {i = 1} ^ {m} \boldsymbol {b} _ {i} \stackrel {(1 2)} {=} \boldsymbol {b},
$$
where in both cases we used the fact that  $\mathbf{A}_i$  maps any point in  $\mathcal{U}_i$  to a point in  $\mathcal{P}(\pmb{b}_i) = \{\pmb{y} : \pmb{P}\pmb{y} = \pmb{b}_i, \pmb{y} \geq 0\}$  by (11). Hence  $\mathbf{A}\pmb{x} \in \mathcal{P}(\pmb{b})$  for all  $\pmb{x} \in \mathcal{U}_1 \times \dots \times \mathcal{U}_m$ , as we wanted to show.
$(\subseteq)$  We now look at the converse, showing that for any  $\pmb{b} \in \mathbb{R}^k$  and matrix  $\mathbf{B} = [\mathbf{B}_1 \mid \dots \mid \mathbf{B}_m] \in \mathcal{L}_{(\mathcal{U}_1 \times \dots \times \mathcal{U}_m) \to \mathcal{P}(\pmb{b})}$ , there exists a matrix  $\mathbf{A} = [\mathbf{A}_1 \mid \dots \mid \mathbf{A}_m]$  that satisfies constraints (11)-(13) and such that  $\mathbf{B}\mathbf{x} = \mathbf{A}\mathbf{x}$  for all  $\mathbf{x} \in \mathcal{U}_1 \times \dots \times \mathcal{U}_m$ . As a first step, in the next lemma we show that  $\mathbf{B}$  is always equivalent to another matrix  $\mathbf{B}' = [\mathbf{B}_1' \mid \dots \mid \mathbf{B}_m'] \cong_{(\mathcal{U}_1 \times \dots \times \mathcal{U}_m)} \mathbf{B}$  that satisfies  $\mathbf{B}_i' \pmb{x}_i \geq 0$  for all  $i = 1, \ldots, m$  and  $\pmb{x}_i \in \mathcal{U}_i$ .
Lemma B.5. There exist  $\mathbf{B}_1^{\prime},\ldots ,\mathbf{B}_n^{\prime}$  such that  $\mathbf{B}\cong_{(\mathcal{U}_1\times \dots \times \mathcal{U}_m)}\mathbf{B}^{\prime}:= \left[\mathbf{B}_{1}^{\prime}\mid \ldots \mid \mathbf{B}_{n}^{\prime}\right]$ , and furthermore, for all  $i = 1,\ldots ,m$ ,  $\pmb {B}_i^\prime \pmb {x}_i\geq \mathbf{0}$  for all  $\pmb {x}_i\in \mathcal{U}_i$
Proof of Lemma B.5. Since  $\mathbf{0} \notin \operatorname{aff}\mathcal{U}_i$  for all  $i = 1, \ldots, m$  by hypothesis, then there exist vectors  $\pmb{\tau}_i$  such that  $\pmb{\tau}_i^\top \pmb{x}_i = 1$  for all  $\pmb{x}_i \in \mathcal{U}_i$ . For any  $k \in \{1, \ldots, d\}$  and  $i \in \{1, \ldots, m-1\}$ , let
$$
\boldsymbol {\beta} _ {i} [ k ] := \min  _ {\boldsymbol {x} \in \mathcal {U} _ {i}} (\mathbf {B} _ {i} \boldsymbol {x}) [ k ] \quad \forall   k \in \{1, \ldots , d \}, \qquad \mathbf {B} _ {i} ^ {\prime} := \mathbf {B} _ {i} - \boldsymbol {\beta} _ {i} \boldsymbol {\tau} _ {i} ^ {\top}
$$
Furthermore, let  $\beta_{m} \coloneqq \sum_{i=1}^{m-1} \beta_{i}$  and  $\mathbf{B}_{m}' \coloneqq \mathbf{B}_{i} + \beta_{m} \pmb{\tau}_{m}^{\top}$ . It is immediate to check that the matrix  $\mathbf{B}' \coloneqq [\mathbf{B}_{1}' | \dots | \mathbf{B}_{m}']$  is such that  $\mathbf{B}' \pmb{x} = \mathbf{B} \pmb{x}$  for all  $\pmb{x} \in \mathcal{U}_{1} \times \dots \times \mathcal{U}_{m}$ , that is,  $\mathbf{B}' \cong_{(\mathcal{U}_{1} \times \dots \times \mathcal{U}_{m})} \mathbf{B}$ . We now show that  $\mathbf{B}_{i}' \pmb{x}_{i} \geq 0$  for all  $i = 1, \dots, m$  and  $\pmb{x}_{i} \in \mathcal{U}_{i}$ . Expanding the definition of  $\mathbf{B}_{i}'$  and  $\beta_{i}$ , for all  $i \in \{1, \dots, m-1\}$ ,  $\pmb{x}_{i} \in \mathcal{U}_{i}$  and  $k \in \{1, \dots, d\}$ ,
$$
(\mathbf {B} _ {i} ^ {\prime} \pmb {x} _ {i}) [ k ] = (\mathbf {B} _ {i} \pmb {x} _ {i}) [ k ] - (\pmb {\beta} _ {i}) [ k ] \cdot (\pmb {\tau} _ {i} ^ {\top} \pmb {x} _ {i}) = (\mathbf {B} _ {i} \pmb {x} _ {i}) [ k ] - \min _ {\hat {\pmb {x}} _ {i} \in \mathcal {U} _ {i}} (\mathbf {B} _ {i} \hat {\pmb {x}} _ {i}) [ k ] \geq 0.
$$
Hence, it only remains to prove that the same holds for  $i = m$ . To that end, fix any  $k \in \{1, \ldots, d\}$  and  $\mathbf{x}_m \in \mathcal{U}_m$ , and let  $\mathbf{x}_i^* \in \arg \min_{\mathbf{x}_i \in \mathcal{U}_i} (\mathbf{B}_i \mathbf{x}_i)[k]$  for all  $i \in \{1, \ldots, m-1\}$ . Using the fact that all vectors in  $\mathcal{P}(\mathbf{b})$  are nonnegative,  $\mathbf{x}^* := (\mathbf{x}_1^*, \ldots, \mathbf{x}_{m-1}^*, \mathbf{x}_m) \in \mathcal{U}_1 \times \cdots \times \mathcal{U}_m$  must satisfy  $\mathbf{B} \mathbf{x}^* \geq 0$ . Hence,
$$
0 \leq (\mathbf {B} \boldsymbol {x} ^ {*}) [ k ] = (\mathbf {B} _ {m} \boldsymbol {x} _ {m}) [ k ] + \sum_ {i = 1} ^ {m - 1} (\mathbf {B} _ {i} \boldsymbol {x} _ {i} ^ {*}) [ k ] = (\mathbf {B} _ {m} \boldsymbol {x} _ {m}) [ k ] + \sum_ {i = 1} ^ {m - 1} \beta_ {i} [ k ] = (\mathbf {B} _ {m} ^ {\prime} \boldsymbol {x} _ {m}) [ k ],
$$
thus concluding the proof of the lemma.
Since  $\mathbf{B}' \cong_{(\mathcal{U}_1 \times \dots \times \mathcal{U}_m)} \mathbf{B}$ , and  $\mathbf{B}$  maps to  $\mathcal{P}(\boldsymbol{b})$ , for all  $\boldsymbol{x} = (\boldsymbol{x}_1, \ldots, \boldsymbol{x}_m) \in \mathcal{U}_1 \times \dots \times \mathcal{U}_m$  we must have
$$
\boldsymbol {b} = \mathbf {P} (\mathbf {B} ^ {\prime} \boldsymbol {x}) = \sum_ {i = 1} ^ {m} \mathbf {P B} _ {i} ^ {\prime} \boldsymbol {x} _ {i}.
$$
Since we can pick the  $\pmb{x}_i$  for different indices  $i$  independently, it follows that  $\mathbf{P}\mathbf{B}_i\pmb{x}_i$  must be a constant function of  $\pmb{x}_i \in \mathcal{U}_i$ , that is, there must exist vectors  $\pmb{b}_1, \dots, \pmb{b}_m \in \mathbb{R}^k$  such that
$$
\boldsymbol {b} _ {1} + \dots + \boldsymbol {b} _ {m} = \boldsymbol {b}, \quad \text {a n d} \quad \mathbf {P B} _ {i} ^ {\prime} \boldsymbol {x} _ {i} = \boldsymbol {b} _ {i} \quad \forall \boldsymbol {x} _ {i} \in \mathcal {U} _ {i}.
$$
Since in addition  $\mathbf{B}_i' \mathbf{x}_i \geq 0$  (by construction of the  $\mathbf{B}_i'$ ), this means that  $\mathbf{B}_i' \in \mathcal{L}_{\mathcal{U}_i \to \mathcal{P}(\mathbf{b}_i)}$ . Finally, using the hypothesis that  $\mathcal{L}_{\mathcal{U}_i \to \mathcal{P}(\mathbf{b}_i)} \cong_{\mathcal{U}_i} \mathcal{M}_{\mathcal{U}_i \to \mathcal{P}(\mathbf{b}_i)}$ , there must exist  $\mathbf{A}_i \in \mathcal{M}_{\mathcal{U}_i \to \mathcal{P}(\mathbf{b}_i)}$ , with  $\mathbf{A}_i \cong_{\mathcal{U}_i} \mathbf{B}_i'$ , for all  $i = 1, \ldots, m$ . This concludes the proof.
B.3 Characterization of Linear Functions of Subtrees
The following result can be understood as a version of Theorem 3.1 stated for each subtree, rooted at some decision node, of the decision space.
Theorem B.6. For any decision node  $j \in \mathcal{J}$  and vector  $\pmb{b}_j \in \mathbb{R}^k$ , let
$$
\tilde {\mathcal {M}} _ {\mathcal {Q} _ {\succeq j} \rightarrow \mathcal {P} (\boldsymbol {b} _ {j})} := \left\{\underbrace {\left[ \cdots \mid \mathbf {A} _ {(j a)} \mid \cdots \right]} _ {\in \mathbb {R} ^ {d \times \Sigma_ {\succeq j}}} \colon\begin{array}{l l l}(1 5)&\mathbf {P A} _ {(j ^ {\prime} a ^ {\prime})} = \boldsymbol {b} _ {j ^ {\prime}}&\forall j ^ {\prime} a ^ {\prime} \in \Sigma_ {\succeq j} \cap \Sigma^ {\perp}\\(1 6)&\mathbf {A} _ {(j ^ {\prime} a ^ {\prime})} = \mathbf {0}&\forall j ^ {\prime} a ^ {\prime} \in \Sigma_ {\succeq j} \backslash \Sigma^ {\perp}\\(1 7)&\sum_ {j ^ {\prime \prime} \in \mathcal {C} _ {j ^ {\prime} a ^ {\prime}}} \boldsymbol {b} _ {j ^ {\prime \prime}} = \boldsymbol {b} _ {j ^ {\prime}}&\forall j ^ {\prime} a ^ {\prime} \in \Sigma_ {\succeq j} \backslash \Sigma^ {\perp}\\(1 8)&\mathbf {A} _ {(j ^ {\prime} a ^ {\prime})} \geq \mathbf {0}&\forall j ^ {\prime} a ^ {\prime} \in \Sigma_ {\succeq j}\\(1 9)&\boldsymbol {b} _ {j ^ {\prime}} \in \mathbb {R} ^ {k}&\forall j ^ {\prime} \succ j\end{array}\right\}. \tag {20}
$$
Then,  $\tilde{\mathcal{M}}_{\mathcal{Q}_{\succeq j}\to \mathcal{P}(\pmb {b}_j)}\cong_{\mathcal{Q}_{\succeq j}}\mathcal{L}_{\mathcal{Q}_{\succeq j}\to \mathcal{P}(\pmb {b}_j)}$
Before continuing with the proof, we remark a subtle point: unlike (7), which constraints each column to have entries in  $[0,\gamma ]$ , (18) only specifies the lower bound at zero, but no upper bound. Hence the tilde above the symbol of this Theorem. Consequently, the matrices in the set  $\tilde{\mathcal{M}}_{\mathcal{Q}\succeq j\to \mathcal{P}(\boldsymbol {b}_j)}$  need not have bounded entries. In that sense, Theorem B.6 is slightly different from Theorem 3.1. We will strengthen (18) to enforce a bound on each column when completing the proof of Theorem 3.1 in the next subsection.
Proof. To aid us with the proof, we first express the definition of  $\tilde{\mathcal{M}}_{\mathcal{Q}_{\succeq j}\to \mathcal{P}(\pmb {b}_j)}$  in a way that better captures the inductive structure we need. By direct inspection of the constraints, the set  $\tilde{\mathcal{M}}_{\mathcal{Q}_{\succeq j}\to \mathcal{P}(\pmb {b}_j)}$  satisfies the inductive definition
$$
\tilde {\mathcal {M}} _ {\mathcal {Q} _ {\geq j} \rightarrow \mathcal {P} (\boldsymbol {b} _ {j})} = \left\{\begin{array}{l l}\mathbf {A} = \left[ \dots \mid \mathbf {A} _ {(j a)} \mid \dots \right] \in \mathbb {R} ^ {d \times \Sigma_ {\geq j}},&\text {s u c h t h a t :}\\(2 1)&\mathbf {P A} _ {(j a)} = \boldsymbol {b} _ {j}\\(2 2)&\mathbf {A} _ {(j a)} = \mathbf {0}\\(2 3)&\sum_ {j ^ {\prime} \in \mathcal {C} _ {j a}} \boldsymbol {b} _ {j ^ {\prime}} = \boldsymbol {b} _ {j}\\(2 4)&[ \mathbf {A} _ {(\sigma)} ] _ {\sigma \geq j ^ {\prime}} \in \tilde {\mathcal {M}} _ {\mathcal {Q} _ {\geq j ^ {\prime}} \rightarrow \mathcal {P} (\boldsymbol {b} _ {j ^ {\prime}})}\\(2 5)&\mathbf {A} _ {(j a)} \geq \mathbf {0}\\(2 6)&\boldsymbol {b} _ {j ^ {\prime}} \in \mathbb {R} ^ {k}\end{array}\right\}. \tag {27}
$$
We prove the result by structural induction on the tree-form decision process.
- Base case. We start by establishing the result for any terminal decision node  $j \in \mathcal{I}$ , that is, one for which all sequences  $\{ja : a \in \mathcal{A}_j\}$  are terminal. In this case, the set  $\mathcal{Q}_{\succeq j}$  is the probability simplex  $\Delta(\{ja : a \in \mathcal{A}_j\})$ . Thus, for a matrix  $\mathbf{A}$  to map all  $x \in \mathcal{Q}_{\succeq j}$  to elements in the convex polytope  $\mathcal{P}(b_j)$  it is both necessary and sufficient that all columns of  $\mathbf{A}$  be elements of  $\mathcal{P}(b_j)$ . It is necessary because if  $\mathbf{A}x \in \mathcal{P}(b_j)$  for all  $x \in \mathcal{Q}_{\succeq j}$ , then for the indicator vector  $x$  with  $x[ja] = 1$  we get  $\mathbf{A}x = \mathbf{A}_{(ja)} \in \mathcal{P}(b_j)$ . And, it is sufficient because any  $x \in \mathcal{Q}_{\succeq j}$  represents a convex combination of the columns  $\mathbf{A}_{(ja)}$ .
The set defined by these constraints matches exactly the set  $\tilde{\mathcal{M}}_{\mathcal{Q}_{\geq j} \to \mathcal{P}(\pmb{b}_j)}$  defined in the statement: since all sequences  $ja$  are terminal, in this case it reduces to
$$
\tilde {\mathcal {M}} _ {\mathcal {Q} _ {\succeq j} \to \mathcal {P} (\boldsymbol {b} _ {j})} = \left\{\left[ \begin{array}{c c c} \dots & \mathbf {A} _ {(j a)} & \dots \end{array} \right] \in \mathbb {R} ^ {d \times \Sigma_ {\succeq j}}: \begin{array}{l l l} (2 8) & \mathbf {P A} _ {(j a)} = \boldsymbol {b} _ {j} & \forall a \in \mathcal {A} _ {j} \\ (2 9) & \mathbf {A} _ {(j a)} \geq \mathbf {0} & \forall a \in \mathcal {A} _ {j} \end{array} \right\},
$$
that is, the set of matrices whose columns are elements of  $\mathcal{P}(\pmb{b}_j)$ . So, we have  $\tilde{\mathcal{M}}_{\mathcal{Q}_{\succeq j}\to \mathcal{P}(\pmb{b}_j)} = \mathcal{L}_{\mathcal{Q}_{\succeq j}\to \mathcal{P}(\pmb{b}_j)}$  with equality, which immediately implies the claim  $\tilde{\mathcal{M}}_{\mathcal{Q}_{\succeq j}\to \mathcal{P}(\pmb{b}_j)}\cong_{\Sigma_{\succeq j}}\mathcal{L}_{\mathcal{Q}_{\succeq j}\to \mathcal{P}(\pmb{b}_j)}$ .
- Inductive step. We now look at a general decision node  $j \in \mathcal{J}$ , assuming as inductive hypothesis that the claim holds for any  $j' \succ j$ . Below we prove that  $\tilde{\mathcal{M}}_{\mathcal{Q}_{\geq j} \to \mathcal{P}(\pmb{b}_j)} \cong_{\mathcal{Q}_{\geq j}} \mathcal{L}_{\mathcal{Q}_{\geq j} \to \mathcal{P}(\pmb{b}_j)}$  as well.
$(\subseteq)$  We start by showing that for any  $\pmb{b}_j \in \mathbb{R}^k$ ,  $\pmb{x} \in \Pi_{\succeq j}$  and  $\mathbf{A} \in \tilde{\mathcal{M}}_{\mathcal{Q}_{\succeq j} \to \mathcal{P}(\pmb{b}_j)}$ , we have  $\mathbf{A}\pmb{x} \in \mathcal{P}(\pmb{b}_j)$ . From (18) it is immediate that  $\mathbf{A}$  has nonnegative entries, and since any vector  $\pmb{x} \in \Pi_{\succeq j}$  also has nonnegative entries, it follows that  $\mathbf{A}\pmb{x} \geq \mathbf{0}$ . Hence, it only remains to show that  $\mathbf{P}(\mathbf{A}\pmb{x}) = \pmb{b}_j$ . Using Lemma A.2, for any  $j' \in \sqcup_{a \in \mathcal{A}_j} \mathcal{C}_{ja}$  there exists  $\pmb{x}_{\succeq j'} \in \mathcal{Q}_{\succeq j'}$  such that  $\pmb{x}[\Sigma_{\succeq j'}] = \pmb{x}[ja] \cdot \pmb{x}_{\succeq j'}$ . Hence, we have