SlowGuess/NeurIPS_2023_OCR huggingface dataset:
We begin with the proof of Theorem D.1, the statement of which is recalled below. In the following proof, we will denote by  $\mathcal{L}:\Xi \times X\ni (\pmb {\mu},(\pmb {x}_i)_{i = 1}^n)\mapsto \pmb {c}^\top \pmb {\mu} - \lambda \sum_{i = 1}^n\pmb {\mu}^\top \mathbf{A}_i\pmb {x}_i$  the induced Lagrangian, for a fixed  $\lambda >0$
1. For any strategy  $\pmb{\mu}^{*}\in \Xi$  such that  $\max_{i\in [[n]]}\max_{\pmb{x}_i^*\in X_i}(\pmb{\mu}^*)^\top \mathbf{A}_i\pmb{x}_i^*\leq 0,$
Proof. Let  $\bar{\mu} \in \Xi$  be the average strategy of the mediator and  $\bar{x}_i \in X_i$  be the average strategy of each player  $i \in [[n]]$  over the  $T$  iterations. We first argue about the approximate optimality of  $\bar{\mu}$ . In particular, we have that
$$
\begin{array}{l} \boldsymbol {c} ^ {\top} \bar {\boldsymbol {\mu}} \geq \max  _ {\boldsymbol {\mu} \in \Xi} \left\{\boldsymbol {c} ^ {\top} \boldsymbol {\mu} - \lambda \sum_ {i = 1} ^ {n} \boldsymbol {\mu} ^ {\top} \mathbf {A} _ {i} \bar {\boldsymbol {x}} _ {i} \right\} - \frac {1}{T} \left(\sum_ {i = 1} ^ {n} \operatorname {R e g} _ {X _ {i}} ^ {T} + \operatorname {R e g} _ {\Xi} ^ {T}\right) (3) \\ \geq \boldsymbol {c} ^ {\top} \boldsymbol {\mu} ^ {*} - \lambda \sum_ {i = 1} ^ {n} \left(\boldsymbol {\mu} ^ {*}\right) ^ {\top} \mathbf {A} _ {i} \bar {\boldsymbol {x}} _ {i} - \frac {1}{T} \left(\sum_ {i = 1} ^ {n} \operatorname {R e g} _ {X _ {i}} ^ {T} + \operatorname {R e g} _ {\Xi} ^ {T}\right) (4) \\ \geq \boldsymbol {c} ^ {\top} \boldsymbol {\mu} ^ {*} - \lambda \sum_ {i = 1} ^ {n} \max  _ {\boldsymbol {x} _ {i} ^ {*} \in X _ {i}} \left(\boldsymbol {\mu} ^ {*}\right) ^ {\top} \mathbf {A} _ {i} \boldsymbol {x} _ {i} ^ {*} - \frac {1}{T} \left(\sum_ {i = 1} ^ {n} \operatorname {R e g} _ {X _ {i}} ^ {T} + \operatorname {R e g} _ {\Xi} ^ {T}\right) \\ \geq \boldsymbol {c} ^ {\top} \boldsymbol {\mu} ^ {*} - \frac {1}{T} \left(\sum_ {i = 1} ^ {n} \operatorname {R e g} _ {X _ {i}} ^ {T} + \operatorname {R e g} _ {\Xi} ^ {T}\right), (5) \\ \end{array}
$$
where (3) follows from the fact that
$$
\max  _ {\boldsymbol {\mu} ^ {*} \in \Xi} \mathcal {L} \left(\boldsymbol {\mu} ^ {*}, \left(\bar {\boldsymbol {x}} _ {i}\right) _ {i = 1} ^ {n}\right) - \min  _ {\left(\boldsymbol {x} _ {i} ^ {*}\right) _ {i = 1} ^ {n} \in X} \mathcal {L} \left(\bar {\boldsymbol {\mu}}, \left(\boldsymbol {x} _ {i} ^ {*}\right) _ {i = 1} ^ {n}\right) \leq \frac {1}{T} \left(\sum_ {i = 1} ^ {n} \operatorname {R e g} _ {X _ {i}} ^ {T} + \operatorname {R e g} _ {\Xi} ^ {T}\right), \tag {6}
$$
in turn implying (3) since  $\sum_{i=1}^{n} \max_{\boldsymbol{x}_i^* \in X_i} \bar{\boldsymbol{\mu}}^\top \mathbf{A}_i \boldsymbol{x}_i^* \geq \sum_{i=1}^{n} \bar{\boldsymbol{\mu}}^\top \mathbf{A}_i \boldsymbol{d}_i = 0$ ; (4) uses the notation  $\boldsymbol{\mu}^*$  to represent any equilibrium strategy optimizing the objective  $c^\top \boldsymbol{\mu}$ ; and (5) follows from the fact that, by assumption,  $\boldsymbol{\mu}^*$  satisfies the equilibrium constraint:  $\max_{\boldsymbol{x}_i^* \in X_i} (\boldsymbol{\mu}^*)^\top \mathbf{A}_i \boldsymbol{x}_i^* \leq 0$  for any player  $i \in [[n]]$ , as well as the nonnegativity of the Lagrange multiplier. This establishes Item 1 of the statement.
Next, we analyze the equilibrium gap of  $\bar{\mu}$ . Consider any mediator strategy  $\pmb{\mu} \in \Xi$  such that  $\pmb{\mu}^{\top} \mathbf{A}_i \pmb{x}_i \leq 0$  for any  $\pmb{x}_i \in X_i$  and player  $i \in [[n]]$ . By (6),
$$
\boldsymbol {c} ^ {\top} \boldsymbol {\mu} - \lambda \sum_ {i = 1} ^ {n} \boldsymbol {\mu} ^ {\top} \mathbf {A} _ {i} \bar {\boldsymbol {x}} _ {i} - \boldsymbol {c} ^ {\top} \bar {\boldsymbol {\mu}} + \lambda \sum_ {i = 1} ^ {n} \max  _ {\boldsymbol {x} _ {i} ^ {*} \in X _ {i}} \bar {\boldsymbol {\mu}} ^ {\top} \mathbf {A} _ {i} \boldsymbol {x} _ {i} ^ {*} \leq \frac {1}{T} \left(\sum_ {i = 1} ^ {n} \operatorname {R e g} _ {X _ {i}} ^ {T} + \operatorname {R e g} _ {\Xi} ^ {T}\right). \tag {7}
$$
But, by the equilibrium constraint for  $\pmb{\mu}$ , it follows that  $\pmb{\mu}^{\top}\mathbf{A}_{i}\pmb{x}_{i}\leq 0$  for any  $\pmb {x}_i\in X_i$  and player  $i\in [[n]]$ , in turn implying that  $\sum_{i = 1}^{n}\pmb{\mu}^{\top}\mathbf{A}_{i}\pmb{x}_{i}\leq 0$ . So, combining with (7),
$$
\lambda \sum_ {i = 1} ^ {n} \max  _ {\boldsymbol {x} _ {i} ^ {*} \in X _ {i}} \bar {\boldsymbol {\mu}} ^ {\top} \mathbf {A} _ {i} \boldsymbol {x} _ {i} ^ {*} \leq \boldsymbol {c} ^ {\top} \bar {\boldsymbol {\mu}} - \boldsymbol {c} ^ {\top} \boldsymbol {\mu} + \frac {1}{T} \left(\sum_ {i = 1} ^ {n} \operatorname {R e g} _ {X _ {i}} ^ {T} + \operatorname {R e g} _ {\Xi} ^ {T}\right). \tag {8}
$$
Finally, given that  $\max_{\boldsymbol{x}_{i^{\prime}}^{*}\in X_{i^{\prime}}}\bar{\boldsymbol{\mu}}^{\top}\mathbf{A}_{i^{\prime}}\boldsymbol{x}_{i^{\prime}}^{*}\geq \bar{\boldsymbol{\mu}}^{\top}\mathbf{A}_{i^{\prime}}\boldsymbol{d}_{i^{\prime}} = 0$  for any player  $i^{\prime}$ , it follows that
$$
\sum_ {i = 1} ^ {n} \max  _ {\boldsymbol {x} _ {i} ^ {*} \in X _ {i}} \bar {\boldsymbol {\mu}} ^ {\top} \mathbf {A} _ {i} \boldsymbol {x} _ {i} ^ {*} \geq \max  _ {i \in [   [ n ]   ]} \max  _ {\boldsymbol {x} _ {i} ^ {*} \in X _ {i}} \bar {\boldsymbol {\mu}} ^ {\top} \mathbf {A} _ {i} \boldsymbol {x} _ {i} ^ {*},
$$
and (8) implies Item 2 of the statement.
Bounding the regret of the players To instantiate Theorem D.1 for our problem, we first bound the regret of Player min in (L1) in terms of the magnitude of the Lagrange multiplier. As we explained in Appendix D, the regret minimization problem faced by Player min can be decomposed into subproblems over the sequence-form polytope, one for each player. To keep the exposition self-contained, let us first recall the standard regret guarantee of CFR under the sequence-form polytope.
Proposition E.1 ([106]). Let  $\operatorname{Reg}_{X_i}^T$  be the regret cumulated by CFR [106] over the sequence-form polytope  $X_{i}$ . Then, for any  $T\in \mathbb{N}$
$$
\operatorname {R e g} _ {X _ {i}} ^ {T} \leq C D _ {i} | \Sigma_ {i} | \sqrt {T},
$$
where  $D_{i} > 0$  is the range of utilities observed by player  $i\in \llbracket n\rrbracket$  and  $C > 0$  is an absolute constant.
An analogous regret guarantee holds for online mirror descent [91]. As a result, given that the range of observed utilities for each player is  $O(\lambda)$ , for a fixed Lagrange multiplier  $\lambda$ , we arrive at the following result.
Corollary E.2. If all players employ CFR, the regret of Player min in (L1) can be bounded as
$$
\operatorname {R e g} _ {X} ^ {T} = \sum_ {i = 1} ^ {n} \operatorname {R e g} _ {X _ {i}} ^ {T} = O (\lambda \sqrt {T}).
$$
Here, we used the simple fact that regret over a Cartesian product can be expressed as the sum of the regrets over each individual set [29].
Bounding the regret of the mediator We next turn our attention to the regret minimization problem faced by the mediator. The complexity of this problem depends on the underlying notion of equilibrium at hand. In particular, for the correlated equilibrium concepts studied in this paper—namely, NFCCE, EFCCE and EFCE, we employ the framework of DAG-form sequential decision problem (DFSDP) [103]. In particular, DFSDP is a sequential decision process over a DAG. We will denote by  $E$  the set of edges of the DAG, and by  $\mathcal{S}$  the set of its nodes; we refer to Zhang et al. [103] for the precise definitions. A crucial structural observation is that a DFSDP can be derived from the probability simplex after repeated Cartesian products and scaled-extensions operations; suitable DGFs arising from such operations have been documented [32]. As such, we can use the following guarantee shown by Zhang et al. [103].
Proposition E.3 ([103]). Let  $\mathrm{Reg}_{\Xi}^{T}$  be the regret cumulated by the regret minimization algorithm  $\mathcal{R}_{\Xi}$  used by the mediator (Player max in (L1)) up to time  $T \in \mathbb{N}$ . Then, if  $\mathcal{R}_{\Xi}$  is instantiated using CFR, or suitable variants thereof,  $\mathrm{Reg}_{\Xi}^{T} = O(|S|\sqrt{T}D)$ , where  $D$  is the range of the utilities observed by the mediator. Further, the iteration complexity is  $O(|E|)$ .
As a result, combining Theorem D.1 with Proposition E.3 and Corollary E.2, and setting the Lagrange multiplier  $\lambda \coloneqq T^{1 / 4}$ , we establish the statement of Corollary 3.3.
Faster rates through optimism Next, to obtain Corollary 3.4, let us parameterize the regret of optimistic gradient descent in terms of the maximum utility, which can be directly extracted from the work of Rakhlin and Sridharan [87].
Proposition E.4. If both agents in the saddle-point problem (L1) employ optimistic gradient descent with a sufficiently small learning rate  $\eta > 0$ , then the sum of their regrets is bounded by  $O(\lambda)$ , for any fixed  $\lambda > 0$ .
Proof Sketch. By the RVU bound [94, 87], the sum of the agents' regrets can be bounded as  $(\mathrm{diam}_{\Xi}^{2} + \mathrm{diam}_{X}^{2}) / \eta$ , for a sufficiently small  $\eta = O(1 / \lambda)$ , where  $\mathrm{diam}_{\Xi}$  and  $\mathrm{diam}_X$  denote the  $\ell_2$ -diameter of  $\Xi$  and  $X$ , respectively. Thus, taking  $\eta = \Theta (1 / \lambda)$  to be sufficiently small implies the statement.
As a result, taking  $\lambda \coloneqq T^{1 / 2}$  and applying Theorem D.1 leads to the bound claimed in Corollary 3.4.
Last-iterate convergence Finally, let us explain how known guarantees can be applied to establish Theorem 3.5. By applying [4], it follows that for a sufficiently small learning rate  $\eta = O(1 / \lambda)$  there is an iterate of optimistic gradient descent with  $O\left(\frac{1}{\eta\sqrt{T}}\right)$  duality gap. Thus, setting  $\eta = \Theta (1 / \lambda)$  to be sufficiently small we get that the duality gap is bounded by  $O\left(\frac{\lambda}{\sqrt{T}}\right)$ . As a result, for  $\lambda \coloneqq T^{1 / 4}$  Theorem D.1 implies a rate of  $T^{-1 / 4}$ , as claimed in Theorem 3.5. We remark that while the guarantee of Theorem D.1 has been expressed in terms of the sum of the agents' regrets, the conclusion readily applies for any pair of strategies  $(\bar{\mu},\bar{x})\in \Xi \times X$  by replacing the term  $\mathrm{Reg}_{\Xi}^{T} + \sum_{i = 1}^{n}\mathrm{Reg}_{X_{i}}$  with the duality gap of  $(\bar{\mu},\bar{x})$  with respect to (L1) (for the fixed value of  $\lambda$ ). We further note that once the desirable duality gap  $O\left(\frac{1}{\eta\sqrt{T}}\right)$  has been reached, one can fix the players' strategies to obtain a last-iterate guarantee as well.
E.1 An alternative approach
In this subsection, we highlight an alternative approach for solving the saddle-point (L1) using regret minimization. In particular, we first observe that it can be expressed as the saddle-point problem
$$
\max  _ {\boldsymbol {\mu} \in \Xi} \min  _ {\bar {x} _ {i} \in X _ {i}: i \in [   [ n   ]   ]} \quad \boldsymbol {c} ^ {\top} \boldsymbol {\mu} - \sum_ {i = 1} ^ {n} \boldsymbol {\mu} ^ {\top} \mathbf {A} _ {i} \bar {\boldsymbol {x}} _ {i}, \tag {9}
$$
where  $\bar{X}_i\coloneqq \{\lambda_i\pmb {x}_i:\lambda_i\in [0,K],\pmb {x}_i\in X_i\}$  is the conic hull of  $X_{i}$  truncated to a sufficiently large parameter  $K > 0$  . Analogously to our approach in Appendix D, suitably tuning the value of  $K$  will allow us trade off between the optimality gap and the equilibrium gap. In this context, we point out below that how to construct a regret minimizer over a conic hull.
Regret minimization over conic hulls Suppose that  $\mathcal{R}_{X_i}$  is a regret minimizer over  $X_{i}$  and  $\mathcal{R}_+$  is a regret minimizer over the interval  $[0,K]$ . Based on those two regret minimizers, Algorithm 2 shows how to construct a regret minimizer over the conic hull  $\bar{X}_i$ . More precisely, Algorithm 2 follows the convention that a generic regret minimizer  $\mathcal{R}$  interacts with its environment via the following two subroutines:
-  $\mathcal{R}$ .NEXTSTRATEGY:  $\mathcal{R}$  returns the next strategy based on its internal state; and
-  $\mathcal{R}$ . OBSERVEUTILITY  $(\pmb{u}^{(t)})$ :  $\mathcal{R}$  receives as input from the environment a (compatible) utility vector  $\pmb{u}^{(t)}$  at time  $t \in \mathbb{N}$ .
ALGORITHM 2: Regret minimization over a conic hull
1 function NEXTSTRATEGY()
2  $\lambda_{i} \gets \mathcal{R}_{+}.\mathrm{NEXTSTRATEGY}$ 
3  $\pmb{x}_i \gets \mathcal{R}_{X_i}.\mathrm{NEXTSTRATEGY}$ 
4 return  $\tilde{\pmb{x}}_i \coloneqq \lambda_i\pmb{x}_i$ 
5 function OBSERVEUTILITY( $\pmb{u}_i$ )
6  $\mathcal{R}_{X_i}.\mathrm{OBSERVEUTILITY}(\pmb{u}_i)$ 
7  $\mathcal{R}_{+}.\mathrm{OBSERVEUTILITY}(\pmb{u}_i^\top \pmb{x}_i)$
The formal statement regarding the cumulated regret of Algorithm 2 below is cast in the framework of regret circuits [29].
Proposition E.5 (Regret circuit for the conic hull). Suppose that  $\mathrm{Reg}_{X_i}^T$  and  $\mathrm{Reg}_+^T$  is the cumulative regret incurred by  $\mathcal{R}_{X_i}$  and  $\mathcal{R}_+$ , respectively, up to a time horizon  $T \in \mathbb{N}$ . Then, the regret  $\mathrm{Reg}_{\bar{X}_i}^T$  of  $\mathcal{R}_{\bar{X}_i}$  constructed based on Algorithm 2 can be bounded as
$$
\operatorname {R e g} _ {\bar {X} _ {i}} ^ {T} \leq K \max  \left\{0, \operatorname {R e g} _ {X _ {i}} ^ {T} \right\} + \operatorname {R e g} _ {+} ^ {T}.
$$
Proof. By construction, we have that  $\operatorname{Reg}_{\bar{X}_i}^T$  is equal to
$$
\begin{array}{l} \max  _ {\bar {\boldsymbol {x}} _ {i} ^ {*} \in \bar {X} _ {i}} \left\{\sum_ {t = 1} ^ {T} \langle \bar {\boldsymbol {x}} _ {i} ^ {*} - \bar {\boldsymbol {x}} _ {i} ^ {(t)}, \boldsymbol {u} ^ {(t)} \rangle \right\} = \max  _ {\lambda_ {i} ^ {*} \boldsymbol {x} _ {i} ^ {*} \in \bar {X} _ {i}} \left\{\sum_ {t = 1} ^ {T} \langle \lambda_ {i} ^ {*} \boldsymbol {x} _ {i} ^ {*} - \lambda_ {i} ^ {(t)} \boldsymbol {x} _ {i} ^ {(t)}, \boldsymbol {u} _ {i} ^ {(t)} \rangle \right\} \\ = \max  _ {\lambda_ {i} ^ {*} \boldsymbol {x} _ {i} ^ {*} \in \bar {X} _ {i}} \left\{\lambda_ {i} ^ {*} \sum_ {t = 1} ^ {T} \langle \boldsymbol {x} _ {i} ^ {*} - \boldsymbol {x} _ {i} ^ {(t)}, \boldsymbol {u} _ {i} ^ {(t)} \rangle + (\lambda_ {i} ^ {*} - \lambda_ {i} ^ {(t)}) (\boldsymbol {u} _ {i} ^ {(t)}) ^ {\top} \boldsymbol {x} _ {i} ^ {(t)} \right\} \\ \leq K \max  \{0, \operatorname {R e g} _ {X _ {i}} ^ {T} \} + \operatorname {R e g} _ {+} ^ {T}, \\ \end{array}
$$
where the last derivation uses that  $\lambda_i^* \in [0, K]$ .
As a result, by suitable instantiating  $\mathcal{R}_{X_i}$  and  $\mathcal{R}_+$  (e.g., using Proposition E.1), the regret circuit of Proposition E.5 enables us to construct a regret minimizer over  $\bar{X}_i$  with regret bounded as  $O(K\sqrt{T})$ . In turn, this directly leads to a regret minimizer for Player min in (9) with regret bounded by  $O(K\sqrt{T})$ . We further remark that Theorem D.1 can be readily cast in terms of the saddle-point problem (9) as well, parameterized now by  $K$  instead of  $\lambda$ . As a result, convergence bounds such as Corollary 3.3 also apply to regret minimizers constructed via conic hulls.
F Description of game instances
In this section, we provide a detailed description of the game instances used in our experiments in Section 4.1.
F.1 Liar's dice (D), Goofspiel (GL), Kuhn poker (K), and Leduc poker (L)
Liar's dice At the start of the game, each of the three players rolls a fair  $k$ -sided die privately. Then, the players take turns making claims about the outcome of their roll. The first player starts by stating any number from 1 to  $k$  and the minimum number of dice they believe are showing that value among all players. On their turn, each player has the option to make a higher claim or challenge the previous claim by calling the previous player a "liar." A claim is higher if the number rolled is higher or the number of dice showing that number is higher. If a player challenges the previous claim and the claim is found to be false, the challenger is rewarded  $+1$  and the last bidder receives a penalty of -1. If the claim is true, the last bidder is rewarded  $+1$ , and the challenger receives -1. All other players receive 0 reward. We consider two instances of the game, one with  $k = 2$  (D32) and one with  $k = 3$  (D33).
Goofspiel This is a variant of Goofspiel with limited information. In this variation, in each turn the players do not reveal the cards that they have played. Instead, players show their cards to a neutralUMPire, who then decides the winner of the round by determining which card is the highest. In the event of a tie, theUMPire directs the players to divide the prize equally among the tied players, similar to the Goofspiel game. The instance GL3 which we employ has 3 players, 3 ranks, and imperfect information.
Kuhn poker Three-player Kuhn Poker, an extension of the original two-player version proposed by Kuhn [61], is played with three players and  $r$  cards. Each player begins by paying one chip to the pot and receiving a single private card. The first player can check or bet (i.e., putting an additional chip in the pot). Then, the second player can check or bet after a first player's check, or fold/call the first player's bet. The third player can either check or bet if no previous bet was made, otherwise they must fold or call. At the showdown, the player with the highest card who has not folded wins all the chips in the pot. We use the instance K35 which has rank  $r = 5$ .
Leduc poker In our instances of the three-player Leduc poker the deck consists of  $s$  suits with  $r$  cards each. Our instances are parametric in the maximum number of bets  $b$ , which in limit hold'em is not necessarily tied to the number of players. The maximum number of raise per betting round can be either 1, 2 or 3. At the beginning of the game, players each contribute one chip to the pot. The game proceeds with two rounds of betting. In the first round, each player is dealt a private card, and in the second round, a shared board card is revealed. The minimum raise is set at 2 chips in the first round and 4 chips in the second round. We denote by L3brs an instance with three players with  $b$  bets per round,  $r$  ranks, and  $s$  suits. We employ the following five instances: L3132, L3133, L3151, L3223, L3523.
F.2 Battleship game (B) and Sheriff game (S)
Battleship The game is a general-sum version of the classic game Battleship, where two players take turns placing ships of varying sizes and values on two separate grids of size  $h \times w$ , and then take turns firing at their opponent. Ships which have been hit at all their tiles are considered destroyed. The game ends when one player loses all their ships, or after each player has fired  $r$  shots. Each player's payoff is determined by the sum of the value of the opponent's destroyed ships minus  $\gamma \geq 1$  times the number of their own lost ships. We denote by Bphwr an instance with  $p$  players on a grid of size  $h \times w$ , one unit-size ship for each player, and  $r$  rounds. We consider the following four instances: B2222, B2322, B2323, B2324.
Sheriff This game is a simplified version of the Sheriff of Nottingham board game, which models the interaction between a Smuggler—who is trying to smuggle illegal items in their cargo—and the Sheriff—who's goal is stopping the Smuggler. First, the Smuggler has to decide the number  $n \in \{0, \dots, N\}$  of illegal items to load on the cargo. Then, the Sheriff decides whether to inspect the cargo. If they choose to inspect, and find illegal goods, the Smuggler has to pay  $p \cdot n$  to the Sheriff. Otherwise, the Sheriff has to compensate the Smuggler with a reward of  $s$ . If the Sheriff decides not to inspect the cargo, the Sheriff's utility is 0, and the Smuggler's utility is  $v \cdot n$ . After the Smuggler has loaded the cargo, and before the Sheriff decides whether to inspect, the Smuggler can try to bribe the Sheriff to avoid the inspection. In particular, they engage in  $r$  rounds of bargaining and, for each round  $i$ , the Smuggler proposes a bribe  $b_i \in \{0, \dots, B\}$ , and the Sheriff accepts or declines it. Only the proposal and response from the final round  $r$  are executed. If the Sheriff accepts a bribe  $b_r$  then they get  $b_r$ , while the Smuggler's utility is  $vn - b_r$ . Further details on the game can be found in Farina et al. [30]. An instance SpNBr has  $p$  players,  $N$  illegal items, a maximum bribe of  $B$ , and  $r$  rounds of bargaining. The other parameters are  $v = 5$ ,  $p = 1$ ,  $s = 1$  and they are fixed across all instances. We employ the following five instances: S2122, S2123, S2133, S2254, S2264.
F.3 The double-dummy bridge endgame (TP)
The double-dummy bridge endgame is a benchmark introduced by Zhang et al. [101] which simulates a bridge endgame scenario. The game uses a fixed deck of playing cards that includes three ranks (2, 3, 4) of each of four suits (spades, hearts, diamonds, clubs). Spades are designated as the trump suit. There are four players involved: two defenders sitting across from each other, the dummy, and the declarer. The dummy's actions will be controlled by the declarer, so there are only
three players actively participating. However, for clarity, we will refer to all four players throughout this section.
The entire deck of cards is randomly dealt to the four players. We study the version of the game that has perfect information, meaning that all players' cards are revealed to everyone, creating a game in which all information is public (i.e., a double-dummy game). The game is played in rounds called tricks. The player to the left of the declarer starts the first trick by playing a card. The suit of this card is known as the lead suit. Going in clockwise order, the other three players play a card from their hand. Players must play a card of the lead suit if they have one, otherwise, they can play any card. If a spade is played, the player with the highest spade wins the trick. Otherwise, the highest card of the lead suit wins the trick. The winner of each trick then leads the next one. At the end of the game, each player earns as many points as the number of tricks they won. In this adversarial team game, the two defenders are teammates and play against the declarer, who controls the dummy.
The specific instance that we use (i.e., TP3) has 3 ranks and perfect information. The dummy's hand is fixed as  $2 \spadesuit 2 \heartsuit 3 \heartsuit$ .
F.4 Ridesharing game (RS)
This benchmark was first introduced by Zhang et al. [101], and it models the interaction between two drivers competing to serve requests on a road network. The network is defined as an undirected graph  $G^{\mathrm{U}} = (V^{\mathrm{U}}, E^{\mathrm{U}})$ , where each vertex  $v \in V^{\mathrm{U}}$  corresponds to a ride request to be served. Each request has a reward in  $\mathbb{R}_{\geq 0}$ , and each edge in the network has some cost. The first driver who arrives on node  $v \in V^{\mathrm{U}}$  serves the corresponding ride, and receives the corresponding reward. Once a node has been served, it stays clean until the end of the game. The game terminates when all nodes have been cleared, or when a timeout is met (i.e., there's a fixed time horizon  $T$ ). If the two drivers arrive simultaneously on the same vertex they both get reward 0. The final utility of each driver is computed as the sum of the rewards obtained from the beginning until the end of the game. The initial position of the two drivers is randomly selected at the beginning of the game. Finally, the two drivers can observe each other's position only when they are simultaneously on the same node, or they are in adjacent nodes.
Ridesharing games are particularly well-suited to study the computation of optimal equilibria because they are not triangle-free [28].
Setup We denote by  $\mathsf{RS}p i T$  a ridesharing instance with  $p$  drivers, network configuration  $i$ , and horizon  $T$ . Parameter  $i \in \{1,2\}$  specifies the graph configuration. We consider the two network configurations of Zhang et al. [101], their structure is reported in Figure 2. All edges are given unitary cost. We consider a total of four instances: RS212, RS222, RS213, RS223.