Game theory is very important in economics and it’s interesting to ask how games change when they’re played using quantum logic instead of classical logic.

Quantum game theory seems to have started back in 1999 when quantum computing was still in its infancy. There are two games that we’re going to talk about, one is the prisoner’s dilemma which will be the subject of the next segment, and the first one is the penny flip game. This is an extremely simple game where we have two players A and B. Player A starts by positioning a coin in the up state and player B can choose to flip the coin or not without seeing. Then player A can choose to flip the coin or not without B seeing. Player B then chooses to flip the coin or not and if the coin ends heads up then B wins, otherwise A wins.

We’ll denote the up/down states of the coin as usual , so up will be or (1,0)^{T} and down will be or (0,1)^{T}. The choices to flip or not flip a coin can then be represented by the NOT gate which is flip and the identity I which keeps things the same. A quantum circuit for this game would look like this where we’ve got the various different moves, and measure the final outcome to see whether it’s heads up or tails up.

When you play the game with random moves you find that half the time the result is going to be up, so B wins, and half the time the result is going to be down, so A wins. Each player should therefore win 50 percent of the time. Suppose though that after playing a number of games, player B wins every time.

The situation is a bit like a trick performed by magician Darren Brown, where he takes somebody from the audience onto the stage, and that person holds a a coin behind their back and then holds both hands out in front, and Darren Brown has to guess which hand is holding the coin, and he does this several times in a row. What is going on?

In his case of course the answer is magic, but for our case it’s that player B is cheating by applying the Hadamard transformation (see video screenshot below). This puts the coin in a superposed state of up and down. Whether the coin is then flipped or not flipped by A has no effect on the superposed date. Player B then gets to apply the Hadamard transformation again at the end because they get the last move, and that has the effect of always putting the coin back in the up state. The answer is therefore always up and B wins.

A classical analogy of this would be that player B turns a normal coin by 90 degrees so it’s on its edge. If player A flips the coin or not it is still going to remain on its edge, and then player B turns it by 90 degrees again so it is face up and and B wins the game. Of course in the classical version a coin on its edge has a 50-50 chance of falling either way but the quantum coin can exist in a superposed state. As David Meyer, who invented this game back in 1999, pointed out quantum strategies can be more successful than classical ones and the reason quantum computers promise to have vastly stronger computational power than classical computers is because they can they can play these quantum tricks, and do moves which are simply not possible using classical computers. It is also exactly these moves that seem to play such a key role in human cognition.

**Further reading:**

Meyer DA (1999) Quantum Strategies. Physical Review Letters 82, 1052.

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