

Quantum Game Theory

Chiu Fan Lee  and  Neil F. Johnson

Center for Quantum Computation and Physics Department
Clarendon Laboratory, Oxford University
Parks Road, Oxford OX1 3PU, U.K.

We pursue a general theory of quantum games. We show that quantum games are more efficient than classical games, and provide a saturated upper bound for this efficiency. We demonstrate that the set of finite classical games is a strict subset of the set of finite quantum games. We also deduce the quantum version of the Minimax Theorem and the Nash Equilibrium Theorem.

The field of quantum games is currently attracting much attention within the physics community [1, 2, 3, 4]. In addition to their own intrinsic interest, quantum games offer a new vehicle for exploring the fascinating world of quantum information [3, 4, 5]. So far, research on quantum games has tended to concentrate on finding interesting phenomena when a particular classical game is quantized. As a result, studies of quantum games have centered on particular special cases rather than on the development of a general theoretical framework.

This paper aims to pursue the general theory of quantum games. We are able to identify a definite sense in which quantum games are ‘better’ than classical games, in terms of their efficiency. We show explicitly that, in terms of the number of (qu)bits required, there can be a factor of 2 increase in efficiency if we play games quantum mechanically. Hence we are able to quantify a distinct advantage of quantum games as compared to their classical counterparts. Furthermore, through the formalism developed, we are able to demonstrate that the set of finite classical games is a strict subset of finite quantum games. Namely, all finite classical games can be played by quantum rules but not vice versa. We also deduce the quantum version of two of the most important theorems in classical game theory: the minimax theorem for zero-sum games and the Nash theorem for general static games.

We start by defining what is meant by a game. In game theory, a game consists of a set of players, a set of rules which dictate what actions a player can take, and a payoff function specifying the reward for a given set of played strategies. In other words, it is a triple where is the number of players, with such that each set is the set of strategies for the -th player, and such that each with is the payoff function for the -th player. Without loss of generality, we can imagine the existence of a referee who computes the corresponding payoff function after he receives the strategies being played by each of the players. This formal structure includes all classical games and all quantum games. In other words any game, whether classical or quantum, is fully described by the corresponding triple . On the other hand, given any triple , it is not hard to imagine a purely classical game that might be associated to it.

So in what sense are quantum games any ‘better’ than classical games? We believe that the answer lies in the issue of efficiency. Although any game could be played classically, the physically feasible ones form a very restricted subset. As we have seen, playing a game is ultimately about information exchange between the players and the referee - hence if a particular game requires you to submit an infinite amount of information before the payoff functions can be computed, it will not be a playable game. In other words, we are interested in those games which require only a finite amount of resources and time to play. Hence a connection can be made between this consideration and the study of algorithms. In the study of computation, we learn that there are computable functions which may however not be computed efficiently. Shor’s great contribution to information theory was to advance the boundary of the set of efficiently computable functions [6]. This naturally begs the question as to whether it is more efficient to play games quantum mechanically than classically. We will show shortly that, in terms of efficiency, some quantum games can indeed outperform classical games.

The quantum game protocol that we study is a generalization of that described in [2]. We use the term static quantum games to reflect the similarity of the resulting games to static classical games. To play a static quantum game, we start with an initial state which is represented by qubits. The referee then divides the state into sets of qubit parts, sending the -th set to player . The players separately operate on the qubits that they receive, and then send them back to the referee. The referee then determines the payoff for the players with regard to the measurement outcome of a collection of POVM operators . Anticipating the focus on efficiency, we fix the dimension of to be where is the number of qubits, and we set . We also assume that the players share the initial qubits equally, i.e. each one of them will receive qubits. This assumption is inessential and is for ease of exposition only. Following the game’s protocol, the players operate independently on the states, and hence it is natural to allow them access to all possible physical maps. Specifically, we allow each player to have access to the set of trace-preserving completely positive maps, i.e. for each , we set to be the set of trace-preserving completely positive maps. Indeed, the only way to restrict the players’ strategy sets in this protocol is to perform some measurement at the referee’s end - this is incorporated into our formalism by allowing the referee the set of all POVM operators.

First, we restrict ourselves to two-player games. If a payoff of for player is associated with the measurement outcome , the payoff for player is then tr where and is the resulting state. For example, if player I decides to use operation and player II , then . Hence We now fix a set of operators which will form a basis for the set of operators in the state space. If and , then the payoff is

where . Letting and , then and are positive hermitian matrices with independent real parameters. Note that the number comes from the fact that real parameters are needed to specify a positive hermitian matrix, while comes from the fact that with the assumptions that is positive and hermitian. This procedure is the same as the so-called chi matrix representation [7]. For a general matrix , we now observe that tr. Hence we have . Therefore, the payoff is actually , which is always real as expected. To recap, the strategy sets for the players are : these are subsets of the set of positive hermitian matrices such that


The payoff is given by . As shown above, we may now identify to be the set of positive semi-definite hermitian matrices satisfying condition (1). It then follows that is a convex, compact Euclidean space.

The above analysis can easily be generalized to -player games. For a particular -player static game, where (index summation omitted for clarity).

We can now see a striking similarity between static quantum games and static classical finite games. The payoff for a classical finite two-player game has the form where belong to some multi-dimensional simplexes and is a general matrix—the payoff for a static quantum game is where belong to some multi-dimensional compact and convex sets . Indeed the multi-linear structure of the payoff function and the convexity and compactness of the strategy sets, are the essential features underlying both classical and quantum games. And one may exploit these similarities to extend some classical results into the quantum domain. Two immediate examples are the Nash equilibrium Theorem and the Minmax Theorem. To show this, we first need to recall some relevant definitions. For a vector , we set to be and we denote by . We also define the set of best replies for player to be . We note that for each , is convex and closed in and hence compact. Using the notion of best reply, we can easily define what a Nash equilibrium is: an operator profile is a Nash equilibrium if for all .

Theorem 1

(Quantum Nash Equilibrium Theorem)
For all static quantum games, at least one Nash equilibrium exists.

Proof: We know that is a convex compact subset of a Euclidean space. Since is an upper semi-continuous point-to-set map which takes each to a convex set , the theorem follows from Kakutani’s fixed point theorem [8]. Q.E.D.

We now restrict ourselves to two-player zero-sum game, i.e., for all . Trivially, given any , player I’s payoff is bounded above by

Similarly, given any , player II’s payoff is bounded below by

We therefore define

Theorem 2

(Quantum Minimax Theorem)

Proof: We have seen that for each , is compact and convex as a Euclidean space. Also the payoff is linear and continuous in and . Therefore, the theorem follows from the Minimax theorem in Ref. [9]. Q.E.D.

We note that the proofs of theorem 1 and theorem 2 are completely analogous to the corresponding classical proofs. This is because the underlying theorems involved - Katutani’s theorem and the Minmax theorem - are general enough to allow for compact and convex strategy sets without restricting them to only be simplexes. We also note that although quantum games can profit from some nice classical results, problematic issues in classical game theory also carry over to quantum games. For example, one would expect multiple Nash equilibira in general quantum games. Classical game theorists have invented evolutionary game theory [10] to deal with this problem and its quantum analogy awaits full development [11].

If quantum games were merely some replicas of classical games, or vice versa, the subject of quantum games would not be very interesting. Here we show that quantum games are more than that, by showing that in comparison with classical games, not only the set of finite quantum games is strictly larger, quantum games can also be played more efficiently. Before we do this, however, we need to discuss how to quantify efficiency in both classical and quantum cases, and how we then compare the resulting efficiencies. We have seen that the quantum strategy set is a conpact and convex Euclidean space. Since any compact and convex Euclidean space lies inside some -dimensional simplex, yet at the same time contains a smaller -dimensional simplex as a subset where is unique and equals for . The dimensionality will be the quantity we use to gauge efficiency, because it is well-defined and reflects the number of (qu)bits needed. For example, in order to play a two-player quantum game we need to exchange qubits in total. This is because the referee needs to send qubits to the two players and they then need to send them back. The strategy set for each player has dimension . If the same number of bit-transfers is allowed in a classical game, then the strategy set for each player will be a simplex of dimension . Therefore, does it mean that we have a factor of 2 increase in efficiency by playing quantum games? It is true in general, but is not immediately obvious: although the quantum strategy set has a higher dimension, we do not yet know whether many of the strategies are redundant or not.

In order to show that general quantum games are indeed more efficient, we first perform some concrete calculations. Since the choice of is arbitrary, we take where denotes a square matrix such that -entry and all other entries are equal to . Denoting by and using condition (1), we have the following restrictive conditions on : and , where the first two sub-indices represent while the latter two represent . A further calculation shows that . For arbitrary and , we find the following:

We are now ready to prove the following theorem:

Theorem 3

is diagonal with non-zero diagonal entries for some and .

Proof: We first note that the diagonal elements of are

Therefore for all , we set

We then set all the other entries of and to be 0. Q.E.D.

[We note that the above construction still holds true in multi-player games. will be a tensor with vanishing entries, except for entries with identical indices.] Any two operations by player I, and , are redundant if for all . However in the above game, in general. Therefore, the payoff depends on all of the independent parameters and there are of them. Hence, the upper bound on efficiency is indeed saturated. One could also envisage varying and infinitesimally to provide a continuum of quantum games with superior efficiency.

We now provide an example. We consider a two-qubit two-player zero-sum game, and take

This means that where and . For example, the referee may do a von Neumann measurement on the state with respect to the orthonormal basis ; and then award a payoff of 1 for each player if the outcome is , and a payoff of 0 for any other outcomes. The strategy set in this game is of dimension 12, which corresponds to13 independent strategies classically—therefore at least 8 bits have to be transferred. This is in contrast with the fact that only 4 qubits needed to be transferred in the quantum version. The above game, although reasonably simple, does therefore highlight the potential of quantum games.

Besides efficiency, we will now show that quantum games are indeed strictly more general as claimed. Firstly of all, we note that the classical strategy set and the quantum strategy set cannot be made identical if the linearity of the payoff function is to be preserved. This is because there is no linear homeomorphism that maps to a simplex of any dimension. In essense the positivity of , i.e, the conditions for all , spoils this possibility. Therefore, if we identify as some multi-dimensional simplex, we must lose linearity of the payoff function. On the other hand, we have seen that every strategy in is non-redundant in the above games, it is therefore impossible to play the above game classically no matter how you enlarge the strategy set (still finite, of course).

Conversely, we show by an example how classical games can be played within the formalism of quantum games. We consider a general two-player two-move game, the game being classical naturally suggests that the initial qubits are not entangled and the payoffs are determined by measuring the resulting qubits with respect to the computational basis. Hence, without loss of generality,

As a result,

So the only relevant dimensions for player I are and . We now recall the conditions imposed on , which are and . We therefore see that . Similarly, . Therefore, playing the above quantum game is the same as playing the classical game with the game matrix below:

In summary we have shown that playing games quantum mechanically can be more efficient, and have given a saturated upper bound on the efficiency. In particular, there is a factor of 2 increase in efficiency. We have also deduced the quantum version of the minimax theorem for zero-sum games and the Nash theorem for general static games. In addition, we have shown that finite classical games consist of a strict subset of finite quantum games. We have also pointed out the essential characteristics shared by static quantum and static classical games—these are the linearity of the payoff function and the convexity and compactness of the strategy sets. Indeed, the success of using linear programming to search for Nash equilibira in classical two-player zero-sum games relies on these characteristics—one would suspect the same method could be applicable to the quantum version as well [5]. Our final words would be a cautious speculation on the possibility that the physical and natural world might already be exploiting this efficiency advantage on the microscopic scale [12].

CFL thanks NSERC (Canada), ORS (U.K.) and Clarendon Fund (Oxford) for financial support.


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