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Jan 26, 2018 Libratus versus humans. Pitting artificial intelligence (AI) against top human players demonstrates just how far AI has come. Brown and Sandholm built a poker-playing AI called Libratus that decisively beat four leading human professionals in the two-player variant of poker called heads-up no-limit Texas hold'em (HUNL). Libratus the Poker Bot is Crushing Professional Poker Players in no-limit Texas Hold'em tournament. By admin 2017-01-25T19:05:25+00:00 January 25th, 2017.
Libratus played 120,000 hands in a 20-day poker competition against four top-ranked Texas Hold 'em players. It won by a staggering amount and walked away with $1.8 million in chips. Dec 18, 2017 Poker players, by contrast, contend with hidden information — what cards their opponents hold and whether an opponent is bluffing,' according to an interesting account on the CMU website. The proof, of course, is in the winning; Libratus did this in spades at a 20-day, 120,000-hand competition last year at Rivers Casino, Pittsburgh.
Poker-playing AIs typically perform well against human opponents when the play is limited to just two players. Now Carnegie Mellon University and Facebook AI research scientists have raised the bar even further with an AI dubbed Pluribus, which took on 15 professional human players in six-player no-limit Texas Hold 'em and won. The researchers describe how they achieved this feat in a new paper in Science.
Playing more than 5,000 hands each time, five copies of the AI took on two top professional players: Chris 'Jesus' Ferguson, six-time winner of World Series of Poker events, and Darren Elias, who currently holds the record for most World Poker Tour titles. Pluribus defeated them both. It did the same in a second experiment, in which Pluribus played five pros at a time, from a pool of 13 human players, for 10,000 hands.
Co-author Tuomas Sandholm of Carnegie Mellon University has been grappling with the unique challenges poker poses for AI for the last 16 years. No-Limit Texas Hold 'em is a so-called 'imperfect information' game, since there are hidden cards (held by one's opponents in the hand) and no restrictions on the size of the bet one can make. By contrast, with chess and Go, the status of the playing board and all the pieces are known by all the players. Poker players can (and do) bluff on occasion, so it's also a game of misleading information.
Claudico begat Libratus
In 2015, Sandholm's early version of a poker-playing AI, called Claudico, took on four professional players in heads-up Texas Hold 'em—where there are only two players in the hand—at a Brains vs. Artificial Intelligence tournament at the Rivers Casino in Pittsburgh. After 80,000 hands played over two weeks, Claudico didn't quite meet the statistical threshold for declaring victory: the margin must be large enough that there is 99.98% certainty that the AI's victory is not due to chance.
Sandholm et al. followed up in 2017 with another AI, dubbed Libratus. This time, rather than focusing on exploiting its opponents' mistakes, the AI focused on improving its own play–apparently a more reliable approach. 'We looked at fixing holes in our own strategy because it makes our own play safer and safer,' Sandholm told IEEE Spectrum at the time. 'When you exploit opponents, you open yourself up to exploitation more and more.' The researchers also upped the number of games played to 120,000.
AdvertisementThe AI prevailed, even though the four human players tried to conspire against it, coordinating on making strange bet sizes to confuse Libratus. As Ars' Sam Machkovech wrote at the time, 'Libratus emerged victorious after 120,000 combined hands of poker played against four human online-poker pros. Libratus' $1.7 million margin of victory, combined with so many hands, clears the primary bar: victory with statistical significance.'
But Libratus was still playing against one other player in heads-up action. A far more challenging conundrum is playing poker with multiple players. So Pluribus builds on that earlier work with Libratus, with a few key innovations to allow it to come up with winning strategies in multiplayer games.
Sandholm and his former graduate student, Noam Brown—who is now working on his PhD with the Facebook Artificial Intelligence Research (FAIR) group—employed 'action abstraction' and 'information abstraction' approaches to reduce how many different actions the AI must consider when devising its strategy. Whenever Pluribus reaches a point in the game when it must act, it forms a subgame—a representation that provides a finer-grained abstraction of the real game, akin to a blueprint, according to Sandholm.
'It goes back a few actions and does a type of game theoretical reasoning,' he said. Each time, Pluribus must come up with four continuation strategies for each of the five human players via a new limited-lookahead search algorithm. This comes out to 'four to the power of six million different continuation strategies overall,' per Sandholm.
Like Libratus, Pluribus does not use poker-specific algorithms; it simply learns the rules of this imperfect information game and then plays against itself to devise its own winning strategy. So Pluribus figured out on its own it was best to devise a mixed strategy of play and being unpredictable—the conventional wisdom among today's top human players. 'We didn't even say, 'The strategy should be randomized,' said Sandholm. 'The algorithm automatically figured out that it should be randomized, and in what way, and with what probabilities in what situations.'
No limping
Pluribus actually confirmed one bit of conventional poker-playing wisdom: it's just not a good idea to 'limp' into a hand, that is, calling the big blind rather than folding or raising. The exception, of course, is if you're in the small blind, when mere calling costs you half as much as the other players. But while human players typically avoid so-called 'donk betting'—in which a player ends one round with a call but starts the next round with a bet—Pluribus placed donk bets far more often than its human opponents.
AdvertisementSo, 'In some ways, Pluribus plays the same way as the humans,' said Sandholm. 'In other ways, it plays completely Martian strategies.' Specifically, Pluribus makes unusual bet sizes and is better at randomization.
'Its major strength is its ability to use mixed strategies,' said Elias. 'That's the same thing that humans try to do. It's a matter of execution for humans—to do this in a perfectly random way and to do so consistently. Most people just can't.'
'It was incredibly fascinating getting to play against the poker bot and seeing some of the strategies it chose,' said Michael 'Gags' Gagliano, another participating poker player. 'There were several plays that humans simply are not making at all, especially relating to its bet sizing. Bots/AI are an important part in the evolution of poker, and it was amazing to have first-hand experience in this large step toward the future.'
This type of AI could be used to design drugs to take on antibiotic-resistant bacteria, for instance, or to improve cybersecurity or military robotic systems. Sandholm cites multi-party negotiation or pricing—such as Amazon, Walmart, and Target trying to come up with the most competitive pricing against each other—as a specific application. Optimal media spending for political campaigns is another example, as well as auction bidding strategies. Astuce roulette. Sandholm has already licensed much of the poker technology developed in his lab to two startups: Strategic Machine and Strategy Robot. The first startup is interested in gaming and other entertainment applications; Strategy Robot's focus is on defense and intelligence applications.
Potential for fraud
When Libratus beat human players in 2017, there were concerns about whether poker could still be considered a skill-based game and whether online games in particular would soon be dominated by disguised bots. Some took heart in the fact that Libratus needed major supercomputer hardware to analyze its game play and figure out how to improve its play: 15 million core hours and 1,400 CPU cores during live play. But Pluribus needs much less processing capability, completing its blueprint strategy in eight days using just 12,400 core hours and 28 cores during live play.
So is this the death knell for skill-based poker? Well, the algorithm was so successful that the researchers have decided not to release its code. 'It could be very dangerous for the poker community,' Brown told Technology Review.
Sandholm acknowledges the risk of sophisticated bots swarming online poker forums, but destroying poker was never his aim, and he still thinks it's a game of skill. 'I have come to love the game, because these AIs have really shown there's a whole additional depth to the game that humans haven't understood, even brilliant professional players who have played millions of hands,' he said. 'So I'm hoping this will contribute to the excitement of poker as a recreational game.'
DOI: Science, 2019. 10.1126/science.aay2400 (About DOIs).
Listing image by Steve Grayson/WireImage/Getty Images
In January 2017, four world-class poker players engaged in a three-week battle of heads-up no-limit Texas hold 'em.
They were not competing against each other. Instead, they were fighting against a common foe: an AI system called Libratus that was developed by Carnegie Mellon researchers Noam Brown and Tuomas Sandholm. The official competition between human and machine took place over three weeks, but it was clear that the computer was king after only a few days of play. Libratus eventually won[1] by a staggering 14.7 big blinds per 100 hands, trouncing the world's top poker professionals with 99.98% statistical significance.
This was the first AI agent to beat professional players in heads-up no-limit Texas hold 'em.
Libratus is not the only game-playing AI to make recent news headlines, but it is uniquely impressive.
In 2015, DeepMind's Deep Q-network mastered[2] a number of Atari games. A Deep Q-network learns how to play under the reinforcement learning framework, where a single agent interacts with a fixed environment, possibly with imperfect information.
Also in 2015, DeepMind's AlphaGo used similar deep reinforcement learning techniques to beat professionals at Go for the first time in history. Go is the opposite of Atari games to some extent: while the game has perfect information, the challenge comes from the strategic interaction of multiple agents.
Libratus, on the other hand, is designed to operate in a scenario where multiple decision makers compete under imperfect information. This makes it unique: poker is harder than games like chess and Go because of the imperfect information available. (On a chessboard, every piece is visible, but in poker no player can see another player's cards.)
At the same time, it's harder than other imperfect information games, like Atari games, because of the complex strategic interactions involved in multi-agent competition. (In Atari games, there may be a fixed strategy to 'beat' the game, but as we'll discuss later, there is no fixed strategy to 'beat' an opponent at poker.)
This combined uncertainty in poker has historically been challenging for AI algorithms to deal with. That is, until Libratus came along. Libratus used a game-theoretic approach to deal with the unique combination of multiple agents and imperfect information, and it explicitly considers the fact that a poker game involves both parties trying to maximize their own interests.
Libratus Poker Reddit Streams
The poker variant that Libratus can play, no-limit heads up Texas Hold'em poker, is an extensive-form imperfect-information zero-sum game. We will first briefly introduce these concepts from game theory.
A normal form game
For our purposes, we will start with the normal form definition of a game. In this two-player game, each player has a collection of available actions: $A_1$ for Player 1, and $A_2$ for Player 2. The two players simultaneously choose actions $a_1 in A_1$, $a_2 in A_2$ and receive rewards $r_1(a_1, a_2)$ and $r_2(a_1, a_2)$ respectively. The game concludes after a single turn. These games are called normal form because they only involve a single action. An extensive form game, like poker, consists of multiple turns. Before we delve into that, we need to first have a notion of a good strategy.
The Nash equilibrium
Multi-agent systems are far more complex than single-agent games. An optimal policy always exists in a single-agent decision process, but with multiple agents, optimal play depends on your opponent's strategy. To account for this, mathematicians use the concept of the Nash equilibrium.
A Nash equilibrium is a scenario where none of the game participants can improve their outcome by changing only their own strategy. Formally, a Nash equilibrium consists of strategies $sigma_1$ and $sigma_2$ such that if Player 1 chooses another strategy $sigma_1'$, we have
$$
r_1(sigma_1, sigma_2) geq r_1(sigma_1', sigma_2)
$$
This is because a rational player will change their actions to maximize their own game outcome. When the strategies of the players are at a Nash equilibrium, none of them can improve by changing his own. Thus this is an equilibrium. When allowing for mixed strategies (where players can choose different moves with different probabilities), Nash proved that all normal form games with a finite number of actions have Nash equilibria, though these equilibria are not guaranteed to be unique or easy to find.
Zero-sum games
While the Nash equilibrium is an immensely important notion in game theory, it is not unique. Thus, is hard to say which one is the optimal. Many two player games that people play (including poker) have an additional feature that one player's reward is the negative of their opponent's. Such games are called zero-sum.
Importantly, the Nash equilibria of zero-sum games are computationally tractable and are guaranteed to have the same unique value.
Consider a zero-sum game. For all $a_1 in A_1$ and $a_2 in A_2$, we have:
$$
r_1(a_1, a_2) + r_2(a_1, a_2) = 0
$$
Zero-sum games are interesting since any Nash equilibrium can be computed efficiently using the minmax theorem.
We define the maxmin value for Player 1 to be the maximum payoff that Player 1 can guarantee regardless of what action Player 2 chooses: Roulette trick automaten.
$$
maxmin_1 = max_{a_1}(min_{a_{2}} r_1(a_1, a_{2}))
$$
The corresponding minmax value of Player 1 is then the minimum payoff that Player 2 can force Player 1 into taking, if Player 2 chooses their action first:
$$
minmax_1 = min_{a_2}(max_{a_{1}} r_1(a_1, a_{2}))
$$
Libratus Poker Reddit Ufc
The minmax theorem states that minmax and maxmin are equal for a zero-sum game (allowing for mixed strategies) and that Nash equilibria consist of both players playing maxmin strategies. As an important corollary, the Nash equilibrium of a zero-sum game is the optimal strategy. Crucially, the minmax strategies can be obtained by solving a linear program in only polynomial time.
More Complex Games - Extensive Form Games
While many simple games are normal form games, more complex games like tic-tac-toe, poker, and chess are not. In normal form games, two players each take one action simultaneously. In contrast, games like poker are usually studied as extensive form games, a more general formalism where multiple actions take place one after another.
See Figure 1 for an example. Player 1 first decides between $L$ and $R$. The action $R$ ends the game with payoff $(5,2)$, while $L$ continues the game, offering Player 2 choice between $l$ and $r$ (and the corresponding rewards). All the possible games states are specified in the game tree.
The good news about extensive form games is that they reduce to normal form games mathematically. Since poker is a zero-sum extensive form game, it satisfies the minmax theorem and can be solved in polynomial time. However, as the tree illustrates, the state space grows quickly as the game goes on. Even worse, while zero-sum games can be solved efficiently, a naive approach to extensive games is polynomial in the number of pure strategies and this number grows exponentially with the size of game tree. Thus, finding an efficient representation of an extensive form game is a big challenge for game-playing agents. AlphaGo[3] famously used neural networks to represent the outcome of a subtree of Go. In a similar vein, many of Libratus's innovations boil down to novel ways to abstract the game to a reasonable size while preserving the essential information.
Knowing What You Do Not Know - Imperfect Information
While Go and poker are both extensive form games, the key difference between the two is that Go is a perfect information game, while poker is an imperfect information game. In a game of Go, the state of the game is determined entirely by the player's moves, which both players see. In poker however, the state of the game depends on how the cards are dealt, and only some of the relevant cards are observed by every player. To illustrate the difference, we look at Figure 2, a simplified game tree for poker.
In this simplified game tree, a random card is first dealt to Player 1 with some probability specified at chance node $R_1$. Then, another random card is dealt to Player 2 at chance node $R_2$. Here, randomness is depicted by the white chance nodes $R_1$ and $R_2$, where no player takes action but a random event decides the possible outcomes. After receiving the two cards, it is Player 1's turn to bet, who can choose to bet or to fold.
Note that players do not have perfect information and cannot see what cards have been dealt to the other player.
Let's suppose that Player 1 decides to bet. Player 2 sees the bet but does not know what cards player 1 has. In the game tree, this is denoted by the information set, or the dashed line between the two states. An information set is a collection of game states that a player cannot distinguish between when making decisions, so by definition a player must have the same strategy among states within each information set.
Thus, imperfect information makes a crucial difference in the decision-making process. A Go-playing agent asks itself the question: 'based on the current state of the game, what should I do?' But a poker agent has to make a decision based on an estimate: 'based on my current observations, what is the estimated distribution of all possible ground truth states of the game, and what is the best action based on that distribution?'
Libratus Poker Reddit Games
To decide their next action, player 2 needs to evaluate the possibility of all possible underlying states (which means all possible hands of player 1). Because the player 1 is making decisions as well, if player 2 changes strategy, player 1 may change as well, and player 2 needs to update their beliefs about what player 1 would do.
Ufc Reddit
Now we know what are some of the main challenges of poker:
- While theoretically solvable in polynomial time as a massive extensive form game, poker contains a tremendous amount of states that forbids a naive approach.
- Imperfect information complicates the decision-making process and makes solving poker even harder.
Libratus tackles poker's difficulty through three main modules:
- Creating and solving an abstraction of poker in advance
- Subgame solving during the contest
- Self improvement after each day of competition
Game abstraction
Libratus played a poker variant called heads up no-limit Texas Hold'em. Heads up means that there are only two players playing against each other, making the game a two-player zero sum game.
No-limit means that there are no restrictions on the bets you are allowed to make, meaning that the number of possible actions is enormous. In no-limit Texas Hold'em, bet sizes that differ by just one dollar cause different game states. In contrast, limit poker forces players to bet in fixed increments and was solved in 2015[4]. Nevertheless, it is quite costly and wasteful to construct a new betting strategy for a single-dollar difference in the bet. Libratus abstracts the game state by grouping the bets and other similar actions using an abstraction called a blueprint. In a blueprint, similar bets are be treated as the same and so are similar card combinations (e.g. Ace and 6 vs. Ace and 5).
Solving the blueprint
The blueprint is orders of magnitude smaller than the possible number of states in a game. Libratus solves the blueprint using counterfactual regret minimization (CFR), an iterative, linear time algorithm that solves for Nash equilibria in extensive form games. In CFR, two agents play against each other and try to minimize their own counterfactual regret with respect to the other agent's current strategy. Libratus uses a Monte Carlo-based variant that samples the game tree to get an approximate return for the subgame rather than enumerating every leaf node of the game tree.
Nested safe subgame solving
Libratus Poker Paper
While it's true that the blueprint simplifies the game state, Libratus doesn't let a bad approximation get in the way of winning. It expands the game tree in real time and solves that subgame, going off the blueprint if the search finds a better action.
Solving the subgame is more difficult than it may appear at first since different subtrees in the game state are not independent in an imperfect information game, preventing the subgame from being solved in isolation. 'Unsafe' subgame solving refers to the naive approach where one assumes that the opponent's strategy is fixed. This decouples the problem and allows one to compute a best strategy for the subgame independently. However, the assumption makes the algorithm exploitable as it does not adjust its play against changes to its opponent's strategy.
A 'safe' subgame solving method, on the other head, augments the subgame by giving the opponent some alternatives where they can modify their play to not enter the subtree being solved. In short, this ensures that for any possible situation, the opponent is no better-off reaching the subgame after the new strategy is computed. Thus, it is guaranteed that the new strategy is no worse than the current strategy. This approach, if implemented naively, while indeed 'safe', turns out to be too conservative and prevents the agent from finding better strategies. Libratus therefore uses 'reach' subgame solving which gives slightly weaker guarantees that the opponent is no better off for those cases where he is likely to reach this subgame instead of accounting for all possible strategy changes. The new method[5] is able to find better strategies and won the best paper award of NIPS 2017.
Self improvement
In addition, while its human opponents are resting, Libratus looks for the most frequent off-blueprint actions and computes full solutions. Thus, as the game goes on, it becomes harder to exploit Libratus for only solving an approximate version of the game.
Final words
While poker is still just a game, the accomplishments of Libratus cannot be understated. Bluffing, negotiation, and game theory used to be well out of reach for artificial agents, but we may soon find AI being used for many real-life scenarios like setting prices or negotiating wages. Soon it may no longer be just humans at the bargaining table.
Correction: A previous version of this article incorrectly stated that there is a unique Nash equilibrium for any zero sum game. The statement has been corrected to say that any Nash equilibria will have the same value. Thanks to Noam Brown for bringing this to our attention.
DOI: Science, 2019. 10.1126/science.aay2400 (About DOIs).
Listing image by Steve Grayson/WireImage/Getty Images
In January 2017, four world-class poker players engaged in a three-week battle of heads-up no-limit Texas hold 'em.
They were not competing against each other. Instead, they were fighting against a common foe: an AI system called Libratus that was developed by Carnegie Mellon researchers Noam Brown and Tuomas Sandholm. The official competition between human and machine took place over three weeks, but it was clear that the computer was king after only a few days of play. Libratus eventually won[1] by a staggering 14.7 big blinds per 100 hands, trouncing the world's top poker professionals with 99.98% statistical significance.
This was the first AI agent to beat professional players in heads-up no-limit Texas hold 'em.
Libratus is not the only game-playing AI to make recent news headlines, but it is uniquely impressive.
In 2015, DeepMind's Deep Q-network mastered[2] a number of Atari games. A Deep Q-network learns how to play under the reinforcement learning framework, where a single agent interacts with a fixed environment, possibly with imperfect information.
Also in 2015, DeepMind's AlphaGo used similar deep reinforcement learning techniques to beat professionals at Go for the first time in history. Go is the opposite of Atari games to some extent: while the game has perfect information, the challenge comes from the strategic interaction of multiple agents.
Libratus, on the other hand, is designed to operate in a scenario where multiple decision makers compete under imperfect information. This makes it unique: poker is harder than games like chess and Go because of the imperfect information available. (On a chessboard, every piece is visible, but in poker no player can see another player's cards.)
At the same time, it's harder than other imperfect information games, like Atari games, because of the complex strategic interactions involved in multi-agent competition. (In Atari games, there may be a fixed strategy to 'beat' the game, but as we'll discuss later, there is no fixed strategy to 'beat' an opponent at poker.)
This combined uncertainty in poker has historically been challenging for AI algorithms to deal with. That is, until Libratus came along. Libratus used a game-theoretic approach to deal with the unique combination of multiple agents and imperfect information, and it explicitly considers the fact that a poker game involves both parties trying to maximize their own interests.
Libratus Poker Reddit Streams
The poker variant that Libratus can play, no-limit heads up Texas Hold'em poker, is an extensive-form imperfect-information zero-sum game. We will first briefly introduce these concepts from game theory.
A normal form game
For our purposes, we will start with the normal form definition of a game. In this two-player game, each player has a collection of available actions: $A_1$ for Player 1, and $A_2$ for Player 2. The two players simultaneously choose actions $a_1 in A_1$, $a_2 in A_2$ and receive rewards $r_1(a_1, a_2)$ and $r_2(a_1, a_2)$ respectively. The game concludes after a single turn. These games are called normal form because they only involve a single action. An extensive form game, like poker, consists of multiple turns. Before we delve into that, we need to first have a notion of a good strategy.
The Nash equilibrium
Multi-agent systems are far more complex than single-agent games. An optimal policy always exists in a single-agent decision process, but with multiple agents, optimal play depends on your opponent's strategy. To account for this, mathematicians use the concept of the Nash equilibrium.
A Nash equilibrium is a scenario where none of the game participants can improve their outcome by changing only their own strategy. Formally, a Nash equilibrium consists of strategies $sigma_1$ and $sigma_2$ such that if Player 1 chooses another strategy $sigma_1'$, we have
$$
r_1(sigma_1, sigma_2) geq r_1(sigma_1', sigma_2)
$$
This is because a rational player will change their actions to maximize their own game outcome. When the strategies of the players are at a Nash equilibrium, none of them can improve by changing his own. Thus this is an equilibrium. When allowing for mixed strategies (where players can choose different moves with different probabilities), Nash proved that all normal form games with a finite number of actions have Nash equilibria, though these equilibria are not guaranteed to be unique or easy to find.
Zero-sum games
While the Nash equilibrium is an immensely important notion in game theory, it is not unique. Thus, is hard to say which one is the optimal. Many two player games that people play (including poker) have an additional feature that one player's reward is the negative of their opponent's. Such games are called zero-sum.
Importantly, the Nash equilibria of zero-sum games are computationally tractable and are guaranteed to have the same unique value.
Consider a zero-sum game. For all $a_1 in A_1$ and $a_2 in A_2$, we have:
$$
r_1(a_1, a_2) + r_2(a_1, a_2) = 0
$$
Zero-sum games are interesting since any Nash equilibrium can be computed efficiently using the minmax theorem.
We define the maxmin value for Player 1 to be the maximum payoff that Player 1 can guarantee regardless of what action Player 2 chooses: Roulette trick automaten.
$$
maxmin_1 = max_{a_1}(min_{a_{2}} r_1(a_1, a_{2}))
$$
The corresponding minmax value of Player 1 is then the minimum payoff that Player 2 can force Player 1 into taking, if Player 2 chooses their action first:
$$
minmax_1 = min_{a_2}(max_{a_{1}} r_1(a_1, a_{2}))
$$
Libratus Poker Reddit Ufc
The minmax theorem states that minmax and maxmin are equal for a zero-sum game (allowing for mixed strategies) and that Nash equilibria consist of both players playing maxmin strategies. As an important corollary, the Nash equilibrium of a zero-sum game is the optimal strategy. Crucially, the minmax strategies can be obtained by solving a linear program in only polynomial time.
More Complex Games - Extensive Form Games
While many simple games are normal form games, more complex games like tic-tac-toe, poker, and chess are not. In normal form games, two players each take one action simultaneously. In contrast, games like poker are usually studied as extensive form games, a more general formalism where multiple actions take place one after another.
See Figure 1 for an example. Player 1 first decides between $L$ and $R$. The action $R$ ends the game with payoff $(5,2)$, while $L$ continues the game, offering Player 2 choice between $l$ and $r$ (and the corresponding rewards). All the possible games states are specified in the game tree.
The good news about extensive form games is that they reduce to normal form games mathematically. Since poker is a zero-sum extensive form game, it satisfies the minmax theorem and can be solved in polynomial time. However, as the tree illustrates, the state space grows quickly as the game goes on. Even worse, while zero-sum games can be solved efficiently, a naive approach to extensive games is polynomial in the number of pure strategies and this number grows exponentially with the size of game tree. Thus, finding an efficient representation of an extensive form game is a big challenge for game-playing agents. AlphaGo[3] famously used neural networks to represent the outcome of a subtree of Go. In a similar vein, many of Libratus's innovations boil down to novel ways to abstract the game to a reasonable size while preserving the essential information.
Knowing What You Do Not Know - Imperfect Information
While Go and poker are both extensive form games, the key difference between the two is that Go is a perfect information game, while poker is an imperfect information game. In a game of Go, the state of the game is determined entirely by the player's moves, which both players see. In poker however, the state of the game depends on how the cards are dealt, and only some of the relevant cards are observed by every player. To illustrate the difference, we look at Figure 2, a simplified game tree for poker.
In this simplified game tree, a random card is first dealt to Player 1 with some probability specified at chance node $R_1$. Then, another random card is dealt to Player 2 at chance node $R_2$. Here, randomness is depicted by the white chance nodes $R_1$ and $R_2$, where no player takes action but a random event decides the possible outcomes. After receiving the two cards, it is Player 1's turn to bet, who can choose to bet or to fold.
Note that players do not have perfect information and cannot see what cards have been dealt to the other player.
Let's suppose that Player 1 decides to bet. Player 2 sees the bet but does not know what cards player 1 has. In the game tree, this is denoted by the information set, or the dashed line between the two states. An information set is a collection of game states that a player cannot distinguish between when making decisions, so by definition a player must have the same strategy among states within each information set.
Thus, imperfect information makes a crucial difference in the decision-making process. A Go-playing agent asks itself the question: 'based on the current state of the game, what should I do?' But a poker agent has to make a decision based on an estimate: 'based on my current observations, what is the estimated distribution of all possible ground truth states of the game, and what is the best action based on that distribution?'
Libratus Poker Reddit Games
To decide their next action, player 2 needs to evaluate the possibility of all possible underlying states (which means all possible hands of player 1). Because the player 1 is making decisions as well, if player 2 changes strategy, player 1 may change as well, and player 2 needs to update their beliefs about what player 1 would do.
Ufc Reddit
Now we know what are some of the main challenges of poker:
- While theoretically solvable in polynomial time as a massive extensive form game, poker contains a tremendous amount of states that forbids a naive approach.
- Imperfect information complicates the decision-making process and makes solving poker even harder.
Libratus tackles poker's difficulty through three main modules:
- Creating and solving an abstraction of poker in advance
- Subgame solving during the contest
- Self improvement after each day of competition
Game abstraction
Libratus played a poker variant called heads up no-limit Texas Hold'em. Heads up means that there are only two players playing against each other, making the game a two-player zero sum game.
No-limit means that there are no restrictions on the bets you are allowed to make, meaning that the number of possible actions is enormous. In no-limit Texas Hold'em, bet sizes that differ by just one dollar cause different game states. In contrast, limit poker forces players to bet in fixed increments and was solved in 2015[4]. Nevertheless, it is quite costly and wasteful to construct a new betting strategy for a single-dollar difference in the bet. Libratus abstracts the game state by grouping the bets and other similar actions using an abstraction called a blueprint. In a blueprint, similar bets are be treated as the same and so are similar card combinations (e.g. Ace and 6 vs. Ace and 5).
Solving the blueprint
The blueprint is orders of magnitude smaller than the possible number of states in a game. Libratus solves the blueprint using counterfactual regret minimization (CFR), an iterative, linear time algorithm that solves for Nash equilibria in extensive form games. In CFR, two agents play against each other and try to minimize their own counterfactual regret with respect to the other agent's current strategy. Libratus uses a Monte Carlo-based variant that samples the game tree to get an approximate return for the subgame rather than enumerating every leaf node of the game tree.
Nested safe subgame solving
Libratus Poker Paper
While it's true that the blueprint simplifies the game state, Libratus doesn't let a bad approximation get in the way of winning. It expands the game tree in real time and solves that subgame, going off the blueprint if the search finds a better action.
Solving the subgame is more difficult than it may appear at first since different subtrees in the game state are not independent in an imperfect information game, preventing the subgame from being solved in isolation. 'Unsafe' subgame solving refers to the naive approach where one assumes that the opponent's strategy is fixed. This decouples the problem and allows one to compute a best strategy for the subgame independently. However, the assumption makes the algorithm exploitable as it does not adjust its play against changes to its opponent's strategy.
A 'safe' subgame solving method, on the other head, augments the subgame by giving the opponent some alternatives where they can modify their play to not enter the subtree being solved. In short, this ensures that for any possible situation, the opponent is no better-off reaching the subgame after the new strategy is computed. Thus, it is guaranteed that the new strategy is no worse than the current strategy. This approach, if implemented naively, while indeed 'safe', turns out to be too conservative and prevents the agent from finding better strategies. Libratus therefore uses 'reach' subgame solving which gives slightly weaker guarantees that the opponent is no better off for those cases where he is likely to reach this subgame instead of accounting for all possible strategy changes. The new method[5] is able to find better strategies and won the best paper award of NIPS 2017.
Self improvement
In addition, while its human opponents are resting, Libratus looks for the most frequent off-blueprint actions and computes full solutions. Thus, as the game goes on, it becomes harder to exploit Libratus for only solving an approximate version of the game.
Final words
While poker is still just a game, the accomplishments of Libratus cannot be understated. Bluffing, negotiation, and game theory used to be well out of reach for artificial agents, but we may soon find AI being used for many real-life scenarios like setting prices or negotiating wages. Soon it may no longer be just humans at the bargaining table.
Correction: A previous version of this article incorrectly stated that there is a unique Nash equilibrium for any zero sum game. The statement has been corrected to say that any Nash equilibria will have the same value. Thanks to Noam Brown for bringing this to our attention.
Citation
For attribution in academic contexts or books, please cite this work as
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Jiren Zhu, 'Libratus: the world's best poker player', The Gradient, 2018.
BibTeX citation:
@article{zhu2018libratus,
author = {Zhu, Jiren}
title = {Libratus: the world's best poker player},
journal = {The Gradient},
year = {2018},
howpublished = {url{https://thegradient.pub/libratus-poker// } },
}
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Brown, Noam, and Tuomas Sandholm. 'Superhuman AI for heads-up no-limit poker: Libratus beats top professionals.' Science (2017): eaao1733. ↩︎
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Silver, David, et al. 'Mastering the game of go without human knowledge.' Nature 550.7676 (2017): 354. ↩︎
Bowling, Michael, et al. 'Heads-up limit hold'em poker is solved.' Science 347.6218 (2015): 145-149. ↩︎
Brown, Noam, and Tuomas Sandholm. 'Safe and nested subgame solving for imperfect-information games.' Advances in Neural Information Processing Systems. 2017. ↩︎
