## A Game with New Rules

First an anecdote:

A businessman was walking home when he noticed a small, rumpled-looking man standing behind a small wooden table with a top hat in one hand and a coin in the other. Curious, he sauntered over to the fellow, who cordially greeted and asked him, “My kind sir, would you like to play a small game?” More curious, the businessman responded, “What kind of game?” to which the man replied “Just a simple coin game, nothing to it. I’ll flip a coin and then you’ll flip it. I’ll predict which side the coin will land on, and then I’ll flip it a second time and that’ll be the end of the round. If my prediction is correct, we’ll start the round over again and repeat, but if it’s wrong, I’ll pay you a hundred dollars and that’s the end of the game.” The businessman gawked at this strange individual, wondering whether the man was crazy. “There’s just one more thing,” the man added. “If I manage to make ten correct prediction in a row, then you have to give me your business suit.” “What?” exclaimed the businessman. “Yes, your suit, my kind sir,” the man replied. “These are my rules.” The businessman hesitated and then agreed. After all, the odds were definitely in his favor.

The stranger flipped the coin into the top hat and then handed it to the businessman.  Repeating what he saw, he flipped the coin into the top hat, and then gave it back to man. “Tails,” the man declared, and he flipped the coin—and sure enough, it landed tails up right into the hat. “A lucky guess,” he laughed. The businessman chuckled apprehensively, realizing that he might have a cold walk home somewhere in the near future. Nine more rounds were repeated, and nine more times the man guessed correctly. The dazed businessman took off clothes and handed them to the grinning stranger.

So how did the stranger manage such a remarkable feat? The answer should be obvious. Cheating, of course! But done in quite a remarkable way: the stranger had used a quantum coin. A normal flip and the coin would land heads or tails, or a “quantum flip,” and it would land heads and tails. The stranger had to simply remembered which side the coin had started on, quantum flipped the coin, let the businessman flip the coin, and had simply quantum flipped it back to its original state. Meanwhile, the businessman, flipping normally, would never affect the orientation of the coin.

These duality states, among other strange properties of quantum particles, are the basis of a new branch of game theory, appropriately named “quantum game theory.” Just as the coin can end up in a quantum state, a player can choose a quantum strategy. Take the Prisoner’s Dilemma, for example. In this classic illustration of a two-player game, each prisoner has a choice to make: he can either confess or keep silent. However, in a modified quantum version, a prisoner can confess, stay silent, or both confess and stay silent simultaneously! In addition, the two choices can become “entangled.” Whereas two quantum particles can directly affect each other’s behavior, two players—prisoners in this case—can directly affect each other’s choice of strategy. If two players’ strategies become entangled, they can be considered to be a single unit making one choice.

It sounds strange, but is it practical? Currently, the answer to this question is unknown. Quantum game theory is still in the theoretical stage and is far from becoming widely used. However, if proven to work, it has great potential: it could eliminate suboptimal Nash equilibria of classical game theory, and it might even be able to model previously unsolved physical and biological processes.

The future is uncertain, but one thing’s for sure: if a stranger in a top hat asks you to play a mysterious game, say no every time. Quantum game theory or not, that’s probably your dominant strategy.

References:

http://www.sciencenews.org/pages/sn_arc99/11_20_99/bob2.htm

http://arxiv.org/pdf/quant-ph/0211191v1.pdf

-A Quantum Enthusiast