I am getting too many requests to referee quantum games’ papers, and while I am very enthusiastic about the interface of game theory and quantum physics, there is a certain strand of this literature which in my view misses the point of game theory. Since I find myself copy-pasting from previous reports I wrote, I thought I should make my stance public. This post is a generic referee report. If you think my criticism shows that I am too narrow minded to understand your paper then I recommend that you ask the editor not to send it to me. If you have already read this post in a rejection letter then I hope we can still be friends. I am mostly going to rely on EWL’s paper which is a seminal paper in this literature (350 citations in google scholar) and the most mathematically coherent that I know.

I start with what I like about the approach of this literature. (Here (pdf) is good paper to look at which explains this issue better than I will) Consider an ${n}$-person normal form game which is given by sets of actions ${A_i}$ for ${i\in\{1,\dots,n\}}$ and payoff functions ${p_i:\prod_iA_i\rightarrow \mathbb{R}^n}$. This is just a mathematical concept, but it is supposed to model some real life situation in which ${n}$ players have to choose actions and the action profile determines payoffs to the players. The various solutions that game theory proposes are motivated by arguments about what will happen in real life when the game is actually played. For example, if we think about a real life situation in which a player has to communicate her action to a  coordinator then because we cannot prevent her from tossing a coin before she speaks with the coordinator, we might as well allow it also in the model. Thus mixed strategies.

Consider now how the interaction between the player and the coordinator happens the physical world. In EWL’s quantum implementation’ of this process, each player receives some particle, performs some physical operation on it, and then a joint measurement on the pair of particles is performed by the coordinator. The outcome of this measurement dictates the entry in the payoff matrix that determines the outcome of the game. So in the new game the player has to choose what physical operation to perform on his particle, and the two operations together dictates the outcome of the game.

Two explanatory notes before I start bitching about the model: First, a quantum extension of the game is a special case of commitment device (pdf). Second, there are many possible quantum extensions. Every such extension is given by the initial state of the particles delivered to the players, the physical operations that the players are allowed to perform on their particles, and the measurement that the coordinator performs.

Now for the juicy part: my first objection is that the quantum analogue of the game cannot be considered as an extension of the classical game because the players lost the ability to choose their actions. (Defect,Defect) is no longer an equilibrium in the EWL `quantum version’ of the prisoner’s dilemma simply because when a player plays the quantum operation that is called Defect in this game, the referee does not necessarily end up looking at an entry in the payoff matrix in which that players play Defect. In the terminology of commitments devices that was introduced in the paper of Kalai-Kalai-Lehrer-Samet (link above) the quantum extension corresponds to a device that is not voluntary.

My second objection is that even if we consider the quantum game as a separate entity, abandoning pretense of relationship to the classical game, the model studied by EWL is unsatisfactory as a physical description of a game, because players are restricted to a specific set of quantum operations — mixtures of unitary transformations. since the underlying assumption is that we cannot detect what the players are doing with their particles, how can they be forced not to perform an arbitrary physical operation ? (that is, trace preserving completely positive map ?) this is as if we would consider a classical game in which the players are only allowed to use a specific set of mixtures. Possible, but not very natural, at least without an additional justification.