SlowGuess/NeurIPS_2023_OCR huggingface dataset:
In game theory and artificial intelligence, Goofspiel is commonly used as an example of a multi-stage simultaneous move game. In two-player Goofspiel, each player has one suit of cards, which is ranked  $A$  (low), 2, ..., 10,  $J$ ,  $Q$ ,  $K$  (high). Another suit of cards serves as the prize (competition cards). Play proceeds in a series of rounds. The players make sealed bids for the top (face up) prize by selecting a card from their hand (keeping their choice secret from their opponent). Once these cards are selected, they are simultaneously revealed, and the player making the highest bid takes the competition card. If tied, the competition card is discarded. The cards used for bidding are discarded, and play continues
with a new upturned prize card. After 13 rounds, there are no remaining cards and the game ends. Typically, players earn points equal to the sum of the ranks of cards won (i.e.  $A$  is worth one point, 2 is worth two points, etc.,  $J$  11,  $Q$  12, and  $K$  13 points). Goofspiel demonstrates high non-transitivity. Any pure strategy in this game has a simple counter-strategy where the opponent bids one rank higher, or as low as possible against the King bid. As an example, consider the strategy of matching the upturned card value mentioned in the previous section. The final score will be 78 - 13 with the  $K$  being the only lost prize. We focus on the simple versions of this game: goofspiel with 5 point cards and 8 point cards settings.
Similar to the setting in Leduc poker, we compare PSD-PSRO with P-PSRO,  $\mathrm{PSRO}_{rN}$ , and PSRO in this benchmark. We implement the PSRO framework with Nash solver, using DQN as the Oracle agent. We use Linear Programming to solve NE from the payoff table. As the game size of Goofspiel is larger than Leduc poker, we improve the capacity of the model by setting the hidden dimension in MLP to be 512 and the episodes for each BR training to be  $3 \times 10^4$ . Other settings are similar to Leduc poker in Table 6.