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Libratus – Poker-Pros lassen $1,77 Millionen liegenLibratus, an artificial intelligence developed by Carnegie Mellon University, made history by defeating four of the world's best professional poker players in a. Libratus adjusted on the fly. The computations were carried out on the new 'Bridges' supercomputer at the Pittsburgh Supercomputing Center. It used another 4. Poker-Software Libratus "Hätte die Maschine ein Persönlichkeitsprofil, dann Gangster". Eine künstliche Intelligenz hat erfolgreicher gepokert.
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For our purposes, we will start with the normal form definition of a game. 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.
Multi-agent systems are far more complex than single-agent games. 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.
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.
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.
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.
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:.
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.
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. 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  famously used neural networks to represent the outcome of a subtree of Go. 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 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. 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. 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.
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 contrast, limit poker forces players to bet in fixed increments and was solved in . 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.
Ace and 6 vs. Ace and 5. 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.
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.
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.
This decouples the problem and allows one to compute a best strategy for the subgame independently. It used another 4 million core hours on the Bridges supercomputer for the competition's purposes.
Libratus had been leading against the human players from day one of the tournament. I felt like I was playing against someone who was cheating, like it could see my cards.
It was just that good. This is considered an exceptionally high winrate in poker and is highly statistically significant.
While Libratus' first application was to play poker, its designers have a much broader mission in mind for the AI. Because of this Sandholm and his colleagues are proposing to apply the system to other, real-world problems as well, including cybersecurity, business negotiations, or medical planning.
From Wikipedia, the free encyclopedia. Artificial intelligence poker playing computer program. IEEE Spectrum. Retrieved Artificial Intelligence".
Carnegie Mellon University. MIT Technology Review. Interesting Engineering. Categories : Computer poker players Carnegie Mellon University.
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