Quantum Game Theory
In a classical game, the rules and outcomes can be predicted with certainty. Take, for example, flipping a fair coin: The possible outcomes are {H,T} with probabilities 0.5. In Quantum game theory, it is possible to end up with a probabilistic outcome. An inevitable result of probabilistic outcomes is that theoretically you can have less information to determine outcomes. It should be no surprise that there is plenty of research in Quantum Finance, where the access of information is very limited, to predict outcomes related to financial behavior.
Quantum game theory is not intuitive. There is a basic example, covered in Quantum strategies and quantum gambling by Sumana Abeyratne [2], that may help in understanding the complexity of the theory. In this game, there is an Alien named Q and a human named P. How the game works is a coin is placed under a container with the head side up and Q has to choice to flip, F, or not flip, N, or to do a quantum flip. After Q’s turn, P has the option to flip or not flip. Finally Q has the final turn to flip, not flip, or to do a quantum flip. Neither player can see the coin at any point. Q tells P that he bets his spacecraft he can win every time. Why is Q so confident? Well in the classical game, ignoring Q quantum flip, we have the following table for P outcomes:
. Q
. | | NN | NF | FN | FF |
P | F | +1 | -1 | -1 | +1 |
. | N | -1 | +1 | +1 | -1 |
Where +1 is P wins and -1 is P lost. In this classical game, Q has no advantage so there is no way he can guarantee his victory. Now we add in Q possible choice of a quantum flip. If Q does a quantum flip the coin is theoretically in both states: heads and tails. So Q does a quantum flip his first turn, and it doesn’t matter what P does because it will still be in the mixed state. Now Q does another quantum flip which untangles the mixed state and causes the coin to end up back on heads. Therefore, Q wins the game every time without ever knowing any information about the coins state. This example provides insight into the complexity of the quantum field. Interesting enough, there is a lot of research into Quantum gambling, algorithms and Cryptography games.
Currently, quantum games are very theoretical and not quite applicable. But the games help build the foundation and insight into the potential future of quantum computing and other cool futuristic things. This material relates to the class because it shows the unique direction game theory has taken and where it’s currently being applied. Finally, I find it necessary to stress that Quantum games have advantages over Classical games because the quantum player can have less information to determine outcomes.
References:
[1]http://www.google.com/url?sa=t&rct=j&q=quantum%20game%20theory&source=web&cd=6&ved=0CGIQFjAF&url=http%3A%2F%2Fwww.cs.duke.edu%2F~reif%2Fcourses%2Fcomplectures%2FAltModelsComp%2FQuantumGames%2FTame%2FQuantumGamesTame.ppt&ei=93OtTp_UAqTz0gGJsOibDw&usg=AFQjCNFmLeHiQ3U0QOjRA7oaX9hUJQ2MZA&sig2=R_XMcEig2fUsiw6U4TK3Ow&cad=rja
[2]http://etd.ohiolink.edu/view.cgi?acc_num=bgsu1150470447