Facebook Blackjack

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In the Lemonade Stand game, each player tries to be as far away as possible from the other participants. There are infinitely many ways to achieve this, however.

If each player independently chooses one of the infinitely many equilibria, the players are unlikely to all be spaced equally far apart.

The shortcomings of Nash equilibria outside of two-player zero-sum games have raised the question among researchers of what the right goal should even be in such games.

In the case of six-player poker, we take the viewpoint that our goal should not be a specific game-theoretic solution concept, but rather to create an AI that empirically defeats human opponents in the long run, including elite human professionals.

The algorithms we used to construct Pluribus are not guaranteed to converge to a Nash equilibrium outside of two-player zero-sum games.

Nevertheless, we observe that Pluribus plays a strategy that consistently defeats elite human poker professionals in six-player poker, and that the algorithms are therefore capable of producing superhuman strategies in a wider class of settings beyond two-player zero-sum games.

Hidden information in a more complex environment. No other game embodies the challenge of hidden information quite like poker, where each player has information his or her cards that the others lack.

A successful poker AI must reason about this hidden information and carefully balance its strategy to remain unpredictable while still picking good actions.

For example, bluffing occasionally can be effective, but always bluffing would be too predictable and would likely result in losing a lot of money.

It is therefore necessary to carefully balance the probability with which one bluffs with the probability that one bets with strong hands.

In other words, the value of an action in an imperfect-information game is dependent on the probability with which it is chosen and on the probability with which other actions are chosen.

In contrast, in perfect-information games, players need not worry about balancing the probabilities of actions; a good move in chess is good regardless of the probability with which it is chosen.

Adding additional players in poker, however, increases the complexity of the game exponentially. Those previous techniques could not scale to six-player poker even with 10,x as much compute.

Pluribus uses new techniques that can handle this challenge far better than anything that came before. The core of Pluribus's strategy was computed via self-play, in which the AI plays against copies of itself, without any human gameplay data used as input.

The AI starts from scratch by playing randomly and gradually improves as it determines which actions, and which probability distribution over those actions, lead to better outcomes against earlier versions of its strategy.

The version of self-play used in Pluribus is an improved variant of the iterative Monte Carlo CFR MCCFR algorithm.

At the start of the iteration, MCCFR simulates a hand of poker based on the current strategy of all players which is initially completely random.

Once the simulated hand is completed, the algorithm reviews each decision the traverser made and investigates how much better or worse it would have done by choosing the other available actions instead.

Next, the AI assesses the merits of each hypothetical decision that would have been made following those other available actions, and so on.

In Pluribus, this traversal is actually done in a depth-first manner for optimization purposes. Exploring other hypothetical outcomes is possible because the AI is playing against copies of itself.

If the AI wants to know what would have happened if some other action had been chosen, then it need only ask itself what it would have done in response to that action.

The difference between what the traverser would have received for choosing an action versus what the traverser actually achieved in expectation on the iteration is added to the counterfactual regret for the action.

At the end of the iteration, the traverser's strategy is updated so that actions with higher counterfactual regret are chosen with higher probability.

To reduce the complexity of the game, we ignore some actions and also bucket similar decision points together in a process called abstraction. After abstraction, the bucketed decision points are treated as identical.

Pluribus's self-play outputs what we refer to as the blueprint strategy for the entire game. During actual play, Pluribus improves upon this blueprint strategy using its search algorithm.

But Pluribus does not adapt its strategy to the observed tendencies of its opponents. Performance is measured against the final snapshot of training.

We do not use search in these comparisons. Typical human and top human performance are estimated based on discussions with human professionals.

We trained the blueprint strategy for Pluribus in eight days on a core server and required less than GB of RAM. No GPUs were used.

This is in sharp contrast to other recent AI breakthroughs, including those involving self-play in games, which commonly cost millions of dollars to train.

We are able to achieve superhuman performance at such a low computational cost because of algorithmic improvements, which are discussed below.

It looks ahead several moves as it does so, but does not require looking ahead all the way to the end of the game, which would be computationally prohibitive.

Limited-lookahead search is a standard approach in perfect-information games, but is extremely challenging in imperfect-information games.

A new limited-lookahead search algorithm is the main breakthrough that enabled Pluribus to achieve superhuman multiplayer poker.

Specifically, the search is an imperfect-information-game solve of a limited-lookahead subgame. At the leaves of that subgame, the AI considers five possible continuation strategies it and each opponent and itself might adopt for the rest of the game.

The number of possible continuation strategies is far larger, but the researchers found that their algorithm only needs to consider five continuation strategies per player at each leaf to compute a strong, balanced overall strategy.

Pluribus also seeks to be unpredictable. For instance, betting would make sense if the AI held the best possible hand, but if the AI bets only when it has the best hand, opponents will quickly catch on.

So Pluribus calculates how it would act with every possible hand it could hold and then computes a strategy balanced across all of those possibilities.

Though poker is an incredibly complicated game, Pluribus made efficient use of computation. Pluribus computed its blueprint strategy in eight days using only 12, core hours and used just 28 cores during live play.

Sandholm has founded two companies, Strategic Machine Inc.

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