YOU slump in your chair and smile as your on-screen bankroll notches up another $1000 first prize. Knocking back the last of your coffee, you check your watch. It's 3 am. You've been playing these tournaments since lunchtime. One more, you tell yourself, and then bed. After all, it's worth it: this is the game that has made you rich. Well, that's not quite true. You're rich all right - but your talents at online poker are only part of the story. You glance down at the quantum processor buzzing softly under your desk as the virtual cards hit the felt once more.
According to one US academic, this fictional scenario will not remain fantasy forever. Steve Bleiler, a mathematician at Portland State University in Oregon, is figuring out what will happen to the game of online poker when today's computers are eventually superseded by the quantum computing technology of tomorrow.
"The strategies you find in classical poker textbooks work just fine for online games right now," he says. "But when your 'Quantum iMac' arrives you can just take all those poker textbooks and throw them in the dumpster."
Bleiler's calculations suggest that when the game of poker is played by the rules of the quantum world, it undergoes a radical transformation. New strategies open up that simply didn't exist before and any gamers ready to exploit them stand to make a tidy profit - at the expense of those who are not. Players with quantum computers but with no access to "quantised strategies" will be at a terminal disadvantage, he says. "I can't wait."
These startling predictions come from recasting poker within the framework of a branch of mathematics called quantum game theory. The most basic version of game theory arose in the 1920s as a way to evaluate the different strategies available to the players in any kind of competitive game, from poker to military campaigns.
When the playing pieces in your game are everyday-scale objects that obey the laws of classical physics, this is relatively straightforward. But at the scale of subatomic particles, physics is governed by the counter-intuitive rules of quantum theory.
Quantum particles do bizarre things, like being in different positions at the same time, a phenomenon called quantum superposition. They can also stay connected to other particles even when they are separated by vast distances, a property known as entanglement.
This means that when you play games in the quantum realm, actions like betting or not betting are suddenly replaced by complex superpositions of the two. This opens up new strategies by allowing players to both bet and not bet, and to do so in varying proportions simultaneously. Not only that, but thanks to entanglement, the decisions you take instantly affect your competitors' options.
The foundations of quantum game theory were laid in 1999 by physicist David Meyer at the University of California in San Diego. He was motivated by what he saw as the coming revolution in quantum information technology (Physical Review Letters, vol 82, p 1052). Researchers were starting to realise that storing and processing information in quantum form was both faster and more efficient than doing it classically.
Take computing as an example. Computers today perform calculations using classical bits of information that can take the value 1 or 0. Quantum computers, on the other hand, work with quantum bits, or qubits, formed by encoding information in the quantum states of subatomic particles.
Owing to quantum weirdness, a qubit can be in a superposition of both 1 and 0 at the same time. That means that while a single byte, made up of a string of eight bits, can represent any single number up to 256, a quantum byte, made of eight qubits, can store and process 256 different numbers all at once. This leads to an enormous hike in processing speed and computational power.
Meyer realised that when everyone has a quantum computer on their desk, hooked up to one another via quantum communication channels - a sort of quantum internet - then any kind of competitive pursuit conducted over this network is going to obey quantum rules. And that's where quantum game theory comes in. You need more than just quantum computers, though. "Quantum games should be thought of as games with quantum communication between the players and the referee rather than classical communication," Meyer stresses.
Several well-known games from classical game theory, such as the prisoners' dilemma, have already had the quantum treatment. So how will poker fare when the referee and the players adopt quantum strategies?
Ace in the hole
Enter Bleiler, a mathematician and professional poker player who has more than 30 years' experience and has competed twice in the World Series of Poker in Las Vegas.
Bleiler considered the impact on online poker, where the referee is a computer program that deals the cards, mediates the players' actions and spits back the results.
The most popular variation of the game today is Texas hold 'em. In hold 'em, players are dealt two "pocket cards" each, which they keep hidden. During the hand and between betting rounds, a total of five "community cards" are dealt face-up.
Each player makes the strongest five-card hand they can by combining their pocket cards and the community cards. There is a maximum of four rounds of betting during each hand. The first player to act in each betting round can either bet or decide not to open the betting by declaring "check". If he checks then the action passes to the next player, who has the same two options. However, if the first player bets then every subsequent player must at least match that bet or fold their cards, which would exclude them from further participation in that hand. A third option is to raise the stakes and increase the amount bet, in which case the other players must match the new bet in order to stay in the hand. If all but one player folds, that player wins the money in the pot. If, after all four rounds of betting, more than one player remains, the game goes to a showdown - they compare hands, and the strongest wins the pot.
There are at least a billion billion possible ways that each hand of poker can unfold. So to keep the mathematics tractable, Bleiler confined his attention to the last round of betting after all the community cards had been dealt. He assumed that the community cards form a "straight" comprising a nine, ten, jack, queen and king from different suits. According to the rules of hold 'em, any player who holds an ace in this situation wins. To simplify this showdown, Bleiler assumed that the cards can take one of two denominations - high (an ace) or low (any other card) - with high beating low.
One of the first things Bleiler noticed when he set this up was that quantum rules completely reverse the tactical benefit of what poker players call "late position". In classical hold 'em, the last player to act in a round of betting has a big advantage: they get to see how everyone else has bet before deciding on their own tactics. For example, if you're in late position with a middling hand and there have been several big raises ahead of you, it's a fair assumption that someone else has stronger cards and you're better off folding. On the other hand, if everyone else has checked or made small bets then it might be worthwhile to raise the betting, to scare the other players into folding.
In quantum poker, it's the players acting first, in "early position", who have the advantage. Bleiler has found this is due to quantum entanglement.
Communicating during the game with quantum particles such as photons means that your decision to bet or fold can be represented by a photon's "up" and "down" spin states. Being quantum, your photons exist in a superposition of states pointing up and down at the same time. Only when the referee measures the photon you sent is it forced into either the up or down state.
If you play with entangled photons the situation is complicated further. With entanglement, all the players' actions are linked, so as each player adjusts the state of their photon according to their tactics it instantly affects everyone else's photons. This introduces an unavoidable connection between the players, so that seizing the initiative comes to outweigh the benefits of acting later.
Winning at poker is generally about formulating a solid strategy and sticking to it. Playing on a quantum computer can help here too - enabling players to communicate not just their individual game play actions but entire winning strategies all in a single qubit.
Call my bluff
For example, how often should you bet or raise with weak cards in order to scare everyone else into folding? Clearly, if you bluff like this all the time, other players will soon get wise and catch you out. But if you never bluff then you will miss out on lucrative opportunities that a bluffy reputation can provide when you actually have a very strong hand. For best results, and to keep your opponents off-balance, poker players should bluff a certain optimal percentage of the time - call it p - and play straightforwardly the rest of the time (100 - p).
Game theory predicts what p should be and how it varies in different scenarios but players rarely stick to their strategy. The key advantage when you play on a quantum computer is that p becomes automatically built into your actions. The skill is in using quantum game theory to work out p, and in preparing the qubits you send to the referee so that they are in a superposition of p parts bluff and 100 - p parts not-bluff. This guarantees that when each qubit is measured it will have you bluff, or not, in exactly the right proportions.
This alone, Bleiler believes, forces players to stick to their strategy when they might otherwise have felt tempted to alter it.
He thinks this effect might even make quantum poker a more exciting game - reasoning that once everyone is playing by quantum rules, the optimal strategies will lead to greater swings between profit and loss.
Quantum poker could become reality sooner than you might think (see "Quantum computing's first killer app?"). In October 2008, a team in Austria switched on the world's largest quantum communication network. Spanning 200 kilometres, it connected six locations in Vienna and the nearby town of St Pölten. Meanwhile, Swiss firm ID Quantique is already selling quantum communication systems that boast absolutely secure business transactions.
"There's no doubt in my mind that engineers are going to get quantum computation to work," says Bleiler.
Primitive quantum games have been played already. A team led by Prem Kumar at Northwestern University in Evanston, Illinois, built an experimental fibre-optic apparatus on which three people can play a quantum version of the "public goods" game of economics. In it, players decide how much of their personal wealth to invest in a communal pot, which is then used to buy shared rewards for the whole group. "Quantum poker could be played in a quantum communication system such as the one they built," says Meyer.
Intrigued by potential parallels between quantum games and business, Bleiler and his colleagues are now exploring other instances where quantum strategies might hold advantages over classical ones. "There's a nascent community of modellers that are just starting to realise the power of using a quantised game as a model over a classical one," he says.
This could elucidate not just poker and economics, but also the hazy link between the quantum world of subatomic particles and the classical reality that we inhabit. As games go, that's a pretty good payout.
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