SlowGuess/NeurIPS_2023_OCR huggingface dataset:
We are going to consider a hidden variant of the above game among  $N = 30$  players. Each player  $i$  controls a 5-dimensional vector of control variables  $\theta_{i} \in \mathbb{R}^{5}$ , which feeds to an individual, differentiable, and preconfigured MLP. The MLP's output  $x_{i} = \chi_{i}(\theta_{i})$  is guaranteed to lie in [0,1] and is interpreted as the probability of player  $i$  visiting the El Farol bar. Regarding the MLP configuration, we follow a similar structure as in the previous examples. Specifically, the representation maps  $\chi_{i}: \mathbb{R}^{5} \to [0,1]$  are defined as
$$
\chi_ {i} \left(\theta_ {i}\right) = \operatorname {s i g m o i d} \left(\alpha_ {i} ^ {(2)} \cdot C e L U \left(A _ {i} ^ {(1)} \times \theta_ {i}\right)\right) \quad \text {f o r a l l} i = 1, 2, \tag {D.9}
$$
where  $A_{i}^{(1)} \in [-0.85, 0.85]^{4 \times 5}$ , and  $\alpha_{i}^{(2)} \in [-1, 1]^{4}$  are randomly chosen. The additional restriction to the domain of  $A^{(1)}$  reduces the chance for over-flow numerical errors of the sigmoid activation function. The activation function  $CeLU: \mathbb{R} \to (-1, \infty)$  is given as in (D.2b) and is applied pair-wise to the output of  $A_{i}^{(1)} \times \theta_{i}$ . The loss functions of this game follow the standard variant's definitions, namely, we have that
$$
f _ {i} (x) = S + x _ {i} \left(G - S + \mathbb {P} \left(\sum_ {i \neq j} x _ {i} \geq C\right) (B - G)\right) + h _ {0. 5} (x) \quad \text {f o r a l l} i \in \mathcal {N}, \tag {D.10}
$$
and the latent game's unique equilibrium is at  $x_{i}^{*} = \frac{C}{N}, i \in \mathcal{N}$ .
The large dimensionality of the above game prohibits the usage of a detailed visualization as opposed to the previous examples. However, sufficient information about the behavior PHGD, and GD can be gathered by the performance log-plot across 100 random trajectories in Fig. 10. Observe that, regardless of the increased number of players, the behavior of PHGD is unaffected, i.e., it converges to the game's equilibrium at an exponential rate. On the other hand, the GD fails to converge in the game's equilibrium. In fact it eventually maintains a constant distance from it, unable to proceed further; an indication of a cycling behavior.