Quantum signaling game
We present a quantum approach to a signaling game; a special kind of extensive games of incomplete information. Our model is based on quantum schemes for games in strategic form where players perform unitary operators on their own qubits of some fixed initial state and the payoff function is given by a measurement on the resulting final state. We show that the quantum game induced by our scheme coincides with a signaling game as a special case and outputs nonclassical results in general. As an example, we consider a quantum extension of the signaling game in which the chance move is a three-parameter unitary operator whereas the players’ actions are equivalent to classical ones. In this case, we study the game in terms of Nash equilibria and refine the pure Nash equilibria adapting to the quantum game the notion of a weak perfect Bayesian equilibrium.
The fifteen-year period of the development of quantum games has brought some of the ideas that tell us how special extensive form games might be played in the quantum domain, for example, the quantum model of Stackelberg duopoly  or games with multiple rounds . However, the previous results do not explain (even in a simple two-stage quantum game) how to identify behavioral strategies, information sets and other terms connected with extensive game theory. In our recent papers [3, 4] we have proposed a way of quantizing extensive games without chance moves through their normal representation which covers not only two-stage extensive games but also more complex games, including games with imperfect information. In this paper, with the use of a signaling game, we generalize our idea by allowing a chance mover to perform a quantum operation.
The key feature of our research is the study of the extensive structure of the quantum scheme so that we are able to introduce the notion of perfect Bayesian equilibrium–a Nash equilibrium refinement for games of incomplete information. For convenience, we consider the case where the chance mover and the players are equipped with unitary operations. Thus, we assume that the only interaction with the environment is by a quantum measurement. Certainly, a more general scheme could be constructed. According to [5, 6], the most natural generalization is to allow the players to use general quantum operations, i.e., trace-preserving, completely-positive maps. Our aim is not to construct the most general scheme but to show that quantum game theory can be developed by applying advanced terms from classical game theory.
To make the paper self-contained, we begin with recalling the notion of a signaling game and perfect Bayesian equilibrium.
2 Signaling game
The signaling game that we are going to study was introduced by In-Koo Cho and David M Kreps in . The game begins with a chance move that determines the type of player 1. After player 1 is informed about her type, she chooses her action. Then player 2 observes this action and moves next. The extensive form of such a game is illustrated in figure 1.
In this game each player has got two information sets–points of the game that describe the player’s knowledge about previous actions chosen in the game. Player 1’s information sets are represented by single nodes and since player 1 knows exactly her type. On the other hand, player 2’s information sets are determined by the actions of player 1. They are represented by the nodes connected by dashed lines that follow player 1’s actions. These information sets point out that player 2 learns about an action chosen by player 1. She does not know, however, the type of player 1. This lack of knowledge is a key feature of a signaling game. The only way for player 2 to find out player 1’ type is to analyse her chosen actions that might be a signal about this type.
Solution concepts for a signaling game
One of the most commonly used solution concepts for noncooperative games is a Nash equilibrium  (see also ). It is a strategy profile such that no player gains by unilateral deviation from the equilibrium strategy. The formal definition of a pure Nash equilibrium for a game in strategic form is as follows.
Let be a game in strategic form, where , is the set of players, is the set of strategies of player and is the payoff function of player that assigns for every strategy profile payoff .
A profile of strategies is a
pure Nash equilibrium in a strategic game
if for each player and for all
A Nash equilibrium is treated as a necessary condition for a strategy profile to be a reasonable solution of any noncooperative game and it may be very useful for determining a probable outcome of a strategic game if the number of Nash equilibria is quite low. However, the significance of a Nash equilibrium may decline if one considers a game in extensive form. An extensive game may have a lot of Nash equilibria and/or some of the Nash equilibria may include actions which are not optimal off the equilibrium path (see for example [9, 10]). As an example, let us consider an extensive game in figure 1 with . One way to find pure Nash equilibria in this game is first to determine its normal form. It is a strategic game defined by the number of players, all the possible strategies of the extensive game and the payoffs corresponding to the strategy profiles. We recall that a player’s strategy in an extensive game is a function assigning an action to each information set of the player. As a result, the normal form with is as follows:
where, for example, strategy means that player 1 plays action at her first information set and at the second one, and means that player 2 chooses action at both her information sets. Using the system of inequalities (1), pure Nash equilibria in (2) are then and . Although it might seem that these two equilibria are equally likely scenarios of the game, it is supposed that equilibrium strategy will not be chosen by a rational player. It follows from non-optimal action off the equilibrium path. Indeed, note that action is strictly dominated by action at the right information set of player 2, i.e., it always gives a worse outcome than action at this information set. Thus, in case player 2 specifies her action at the right information set, she ought to choose instead.
Equilibrium refinements can exclude Nash equilibria containing non-optimal actions. For the signaling game it is sufficient to consider a (weak) perfect Bayesian equilibrium, where, for our needs, we restrict ourselves to pure strategies. Following Kreps and Wilson , let us first define an assessment to be a pair of a pure strategy profile and a belief system , i.e., a map that assigns to each information set a probability distribution over the nodes of this information set. Thus, in our example in figure (1), player 2’s beliefs are probability distributions and . In turn, player 1’s beliefs assign to her decision nodes a probability equal to 1. Now, for any node from an information set let denote the probability that is reached given and chance moves, if any.
An assessment is Bayesian consistent if belief at node is equal to for all for which and all .
Denote by the expected payoff of player from playing action , conditional on being at node and strategy profile . Then, is the expected payoff of player from playing the action , conditional on being at information set .
An assessment in an extensive game is sequentially rational if for each player , each information set of this player and action from the set of available actions at , if is consistent with then
In other words, condition (3) requires for each player that action prescribed by strategy is optimal given and beliefs . The two definitions above allow one to formulate the following equilibrium refinement:
An assessment in an extensive game is a (weak) perfect Bayesian equilibrium if it is sequentially rational and Bayesian consistent.
Let us consider, for example, Nash equilibrium with a view to a perfect Bayesian equilibrium. If probability distribution determines the chance moves, the Bayesian consistency on player 2’s beliefs requires whereas beliefs are not forced by the consistency requirement. In this case action at the left information set is optimal. However, given arbitrary beliefs , action at the right information set gives player 2 a lower payoff than action . As a consequence, the profile is not a perfect Bayesian Nash equilibrium. A similar analysis would show that the profile together with player 2’ beliefs and satisfies Bayesian consistency and sequential rationality.
3 Quantum model for a signaling game
In paper  we introduced a model for describing extensive games, where we focused on the normal form and studied Nash equilibria in the resulting quantum game. Here, we are going to justify our scheme with respect to the dynamic nature of an extensive game.
Motivation for the model construction
Let us consider the generalized Eisert-Wilkens-Lewenstein (EWL) quantum approach to an -player strategic game with two-element strategy sets  (we encourage readers who are not familiar with the EWL scheme to first see ). According to an alternative notation for the EWL scheme introduced in  and generalized in , the quantum protocol is defined by 4-tuple
where the components specify the game in the following way:
is a Hilbert space with basis defined for all by the formula
where is the negation of .
is called the initial state, and .
defines the unitary operators available for each player. The matrix representation of the operators from (with respect to the computational basis) can be written as follows:
for each is an observable given by the formula
The measurement is performed on the final state . The possible outcomes of the measurement correspond to player ’s payoffs.
It turns out that we can adapt scheme (4) for any extensive game with two actions at each information set. Since operators represent classical moves in the EWL scheme for a bimatrix game, it is natural to assume that they correspond to classical moves in any quantum game defined by the generalized scheme. The argumentation is as follows. The final state after each player performs her unitary operator is as follows:
Let us denote by an action at the th information set and by
the projectors onto the respective subspaces of . Let us assign to each action projection of the state vector . Then, and . Taking , we obtain a probability distribution over the actions equivalent to one given by a classical behavioral strategy . Thus, in particular, and represent pure actions. In general, let us assign to a sequence of actions the product of projections . Then and this corresponds to the product of probabilities given by applying a sequence of classical behavioral strategies at the th information set, where .
As a result, we have obtained the procedure how to describe an extensive game in terms of the mathematical methods of quantum information. At the same time, we have obtained a scheme that places an extensive game in quantum domain whenever the set of unitary operators of at least one player is .
A detailed description of the quantum scheme for a game in figure 1 is as follows. This is a 6-tuple
components and are the special case of those from (4) for a Hilbert space ;
is a set of players and is a chance mover;
specifies the players’ and the chance mover’s actions. It is assumed that a unitary operation performed by the chance mover is known to the players;
is a map that relates qubits to players and the chance mover. It is a map given by formula
that assigns to each index of in a player or the chance mover;
is an observable that describes a measurement on the final state ,
Then the average value of measurement ,
determines a payoff for player .
Thus, the quantum model for the signaling game in figure 1 requires a five-qubit state. The chance mover’s action is represented by a unitary operation on the first qubit. In turn, a unitary operation on the second and third qubit, and a unitary operation on the fourth and fifth one are player 1’s and player 2’s strategies, respectively. The form of observable (12) is based on our motivation for the scheme construction. Following this line of thought, and then the link between projections and actions as it is given in figure 2, each term in corresponds to measurement that the state of the game is in the respective end node.
4 A signaling game with a quantum chance mover
In the literature of quantum games one can find many examples that show the advantages of quantum strategies over classical ones. The same could be done for the quantum signaling game if, for example, one player’s strategy set were extended to the full range of unitary operators. We are going to consider another case where the players’ actions are still classical ones, i.e., they are in the form of , and the full set is available only for the chance mover. In this case, we obtain an interesting example, where the normal form of the resulting quantum game has the same dimension as the classical game. As it is shown below, this feature makes the classical and quantum game easy to compare. Moreover, the fact that the players are equipped with operators enables us, easily, to refine Nash equilibria in the quantum game by using the notion of perfect Bayesian equilibrium.
More precisely, let us consider 6-tuple (10) in which the set is available only for the chance mover, and the players are equipped with the set . The possible measurement ’s outcomes , for of measurement correspond to payoffs from the game in figure (1), i.e.,
Let us assume that the chance mover specifies as her move. We recall that a chance mover’s action corresponds to probability distribution over her classical actions and non zero coordinates and in place the game into the quantum domain. The final state after the first and second player specify and , respectively, is in the form
Let us calculate the expected values and for each ; values corresponding to pure strategy profiles. For example, quadruple implies the following final state:
Then, the pair equals . By calculating the average values of measurements and for the other quadruples we obtain the following bimatrix:
As a result, quantum scheme (10) provides the players with a quite different bimatrix compared with (2). In particular, the classical game and the quantum counterpart have two pure Nash equilibria but differ in payoff outcomes. Indeed, in contrast to the classical case, profiles and are Nash equilibria with the same payoff outcome .
Perfect Bayesian-type equilibria
Let us study profile . According to definition 1, no player gains by unilaterally deviating from unitary operator . It turns out that it can be said more about the profile in terms of Bayesian consistency and sequential rationality. First, let us carry out perfect Bayesian equilibrium analysis for player 1. Following figure 2, the probability that the game reaches the upper node given chance move and player 2’s strategy equals where
and means the identity operator on . Since player 1’s information sets are singletons, reaching the upper node is equivalent to reaching the corresponding information set, so her belief of being at the upper node attaches probability 1 to this node. Therefore, after player 1 learns that the state of the chance mover’s qubit corresponds to , she believes that with probability 1 faces the following state:
Denote by the state obtained when player 1 performs unitary strategy on . Then
where . Then, is optimal given the belief (19) about the quantum state if
Thus, condition (21) is satisfied.
Similar computation for the case when player 1 learns that the state of the chance mover’s qubit corresponds to proves that is also optimal on state
As a result, player 1’s strategy is sequentially-type rational given her beliefs.
Let us consider now player 2’s strategy in the terms of perfect Bayesian equilibrium. The state after the chance mover and player 1 use operators and , respectively, is as follows
Then the probability that the left information set is reached is equal to . By Bayesian consistency, player 2’s beliefs of being at the upper and lower node at the left information set are
As a consequence, specifying her beliefs, player 2 faces post-measurement state with probability 3/4, and state with probability 1/4. In other words, player 2 is faced with the following mixed state
Thus, mixed state (26) after player 2 uses her unitary operator takes the form
and player 2’s expected payoff is given by . In order to prove that, given her beliefs, is optimal for player 2 let us determine ,
Result (4) shows that is also sequentially-type rational given player 2’s beliefs at the left information set. In a similar way, we can prove sequential-type rationality of at the right information set.
As a result, strategy profile consists of strategies that are optimal with respect to unilateral deviation in both cases: when only the payoff measuerment is performed (a Nash equilibrium) and when a player performs the additional measurement before her move (sequential rationality). It can be shown that the other pure equilibrium given by bimatrix (17) is also a perfect Bayesian-type equilibrium.
5 Conclusion and further research
The purpose of the research was to translate signaling games into the formalism of quantum information and to examine how playing the game would then change. We showed that there exists a quantum approach to a signaling game that constitutes a generalization of the classical game. In particular, we proved (with the use of Eisert et al quantum scheme for strategic games) that the special one-parameter unitary strategies are equivalent to classical moves in the game and a broader range of unitary operators affects the game. The key result of our work was to show that optimal strategy analysis in quantum games can go beyond the concept of Nash equilibrium. A player measuring the state after the other players operate but before her move gives rise to a new solution concept in quantum games that can be treated as a counterpart of a perfect Bayesian equilibrium in classical game theory. It is worth noting that there is more than one way to define a quantum counterpart of a perfect Bayesian equilibrium consistent with the classical term. For example, when solving optimization problems (21) and (4) it does not matter whether we maximize over and or over and . However, it may have great importance when the full set of unitary operators is also available for players and may constitute independent subject of research. Another interesting problem would be to specify how the players’ strategic positions change when one or both players are provided with the set . In particular, studying player 2’ position in the game seems significant. In the classical game, she is deprived of knowing the type of player 1 and we suppose that the access to quantum strategies may improve player 2’ strategic position. Finally, one may investigate the relation between Nash equilibrium and perfect Bayesian equilibrium. We suppose that, in contrast to the classical case, the perfect Bayesian conditions in the way we have presented in the paper may not imply Nash equilibrium. This would point out another distinction between classical and quantum game theory.
The author is very grateful to Prof. J. Pykacz from the Institute of Mathematics, University of Gdańsk, Poland for very useful discussions and great help in putting this paper into its final form. The project was supported by the Polish National Science Center under the project number DEC-2011/03/N/ST1/02940.
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