Quantum Penny Flip game with unawareness
 219 Downloads
Abstract
Games with unawareness model strategic situations in which players’ perceptions about the game are limited. They take into account the fact that the players may be unaware of some of the strategies available to them or their opponents as well as the players may have a restricted view about the number of players participating in the game. The aim of the research is to introduce this notion into the theory of quantum games. We shall focus on PQ Penny Flip game introduced by D. Meyer. We shall formalize the previous results and consider other cases of unawareness in the game.
Keywords
Quantum game Game with unawareness Zerosum gameMathematics Subject Classification
91A05 81P681 Introduction
Game theory, launched in 1928 by von Neumann in [1] and developed in 1944 by von Neumann and Morgenstern in [2], is one of the youngest branches of mathematics. The aim of this theory is mathematical modeling of behavior of rational participants in conflict situations. Each participant is supposed to maximize their own gain and take into account all possible ways of behaving of remaining participants. Within this young theory, new ideas that improve already used models of conflict situations are still proposed. One of the latest trends is to study games with unawareness, i.e., games that describe situations in which a player behaves according to his own view of the game and considers how all the remaining players view the game. This way of describing a conflict situation goes beyond the most frequently used paradigm, according to which it is assumed that all participants in a game have full knowledge of the situation.
The other young field developed on the border of game theory and quantum information theory is quantum game theory. This is an interdisciplinary area of research within which considered games are supposed to be played with the aid of objects that behave according to the laws of quantum mechanics, and in which nonclassical features of these objects are relevant to the way of playing the game and its rational results.
Game with unawareness is a relatively new notion. The first attempts at formalizing that concept can be found in papers [3, 4] published already in twentyfirst century. In paper [5], there is a summary of results obtained in this area till 2012. Quantum counterparts of games with unawareness have not been studied yet. Papers on quantum games with incomplete information concerned only Bayesian games [6, 7, 8, 9] and games with imperfect recall [10, 11]. Our project is the first attempt to use the notion of game with unawareness in theory of quantum games. The main motivation for our interest in developing this branch of quantum game theory was our observation that already in the first paper on quantum games by Meyer [12] its author unconsciously considered a game with unawareness. We may conclude from the famous PQ Penny Flip game described in [12] that Captain Picard (player 2) agrees to join the game because his chance of winning is 1/2. In other words, the game he perceives is the classical one. Q (player 1) views the game in a different way. He is aware of unitary strategies. In addition, player 1 knows that player 2 is only aware of the classical strategies. This knowledge is crucial in the way he chooses his strategy. Choosing, for example, the Hadamard matrix always leads player 1 to getting the best possible outcome. It is optimal to player 1 to play that strategy since he is aware that player 2 has no strategy to counteract the Hadamard matrix. Once we learned the quantum PQ Penny Flip game is a game with unawareness, the description of the game, say by using normal form, requires a family of games rather than a single normalform game. This has numerous important consequences in the form of solution concepts supposed to predict rational results of the game. In particular, Nash equilibrium concept is not sufficient to fully describe the players’ rational choices. In the case of PQ Penny Flip game in which quantum strategies are available only for player 1, each player 2’s mixed strategy is an equilibrium strategy. However, taking into account player 2’s view about the game (he finds the game to be the classical one), we should predict that he chooses his pure strategies with equal probability.
2 PQ Penny Flip game
Inflation–Deflation The extensive games \(\varGamma \) and \(\varGamma '\) share the same reduced strategicform game if \(\varGamma '\) differs from \(\varGamma \) only in an information set of some player i in \(\varGamma \) that is a union of information sets of player i in \(\varGamma '\) (\(\{1.2a, 1.2b\}\) and \(\{1.3a, 1.3b\}\) in Fig. 1) with the following property: any two sequences of actions h and \(h'\) leading from the root of the game tree to different members of the union (for example, sequences \((I_{1}, X_{2})\) and \((X_{1}, X_{2})\)) have the subsequences that lead to the same information set of player i (empty sequence \(\emptyset \) in our case) and player i’s action at this information set is different in h and \(h'\).

\(\{1,2\}\) is a set of players,

\((U_{1}, U_{3})\) and \(U_{2}\) are strategies of player 1 and 2, respectively, and \(U_{j}\) is a \(2\times 2\) unitary matrix for each j,
 \(\rho _{\mathrm {f}}\) is a density matrix defined as follows$$\begin{aligned} \rho _{\mathrm {f}} = U_{3}U_{2}U_{1}0\rangle \langle 0 U^{\dagger }_{1}U^{\dagger }_{2}U^{\dagger }_{3}, \end{aligned}$$(6)
 P is a Hermitian operator in the form$$\begin{aligned} P = 0\rangle \langle 0  1\rangle \langle 1. \end{aligned}$$(7)
One of the main ideas behind the PQ Penny Flip game was to show that Alice can win the game every time she plays against Bob. It is possible if Alice has access to unitary strategies that Bob is not aware of. That is, Alice is fully aware of unitary operators available in the quantum PQ Penny Flip game, Bob is only aware of unitary operations identified with his strategies in the classical PQ Penny Flip game (for example, \(\mathbb {1}\) and \(\sigma _{x}\)).
2.1 Technical difficulties in describing PQ Penny Flip problem
We know from (10) that Alice can win the PQ Penny Flip game if she has access to the Hadamard matrix, and Bob is not aware of unitary matrices except \(\mathbb {1}\) and \(\sigma _{x}\). A natural question arises as to how this problem can be described from a game theory point of view.
The solution is to consider a family of games—the core of the definition of games with unawareness. The formal definition can take into account a player’s view about his strategy set or strategies of the other players, a player’s view about other players’ views, and even a player’s view about the number of players taking part in the game.
3 Preliminaries on games with unawareness
For the convenience of the reader, we review the relevant material from [5]. Before we begin the formal presentation, we will look at an example that illustrates that concept and the ideas behind it. The reader who is not familiar with this topic is encouraged to see a similar introductory example in [5].
Example 1
Let us consider the case that Alice is fully aware of Bob’s reasoning. Not only does she perceive her whole strategy set \(\{a_{1}, a_{2}, a_{3}\}\), Alice also finds that Bob does not realize that she is considering \(\{a_{1}, a_{2}, a_{3}\}\) but \(\{a_{1}, a_{2}\}\). Moreover, Alice finds that Bob views the game as in (12).
Although both players are aware of playing (12), it is not evident that the game ends with (2, 2). According to (14), Bob finds that Alice perceives game (13). Hence, he may deduce that Alice plays according to strategy profile \((a_{2}, b_{1})\) that is the most profitable Nash equilibrium in (13). Bob’s choice would be then \(b_{1}\). Alice, however, is aware of Bob’s thinking. She finds that Bob is considering (12) and also finds that Bob finds that she is considering (13). Alice can therefore deduce that Bob chooses strategy \(b_{1}\) that weakly dominates \(b_{2}\) in (13), i.e., it gives Bob a payoff at least as high as \(b_{2}\), and at the same time, is an element of the most beneficial Nash equilibrium in (13). Since Alice is aware of playing \(\varGamma _{1}\), it is not optimal for her to play according to \((a_{2}, b_{1})\) but to choose \(a_{3}\). As a result, the game described by (14) ends with outcome (4, 0) corresponding to the strategy profile \((a_{3}, b_{1})\).
3.1 The role of the notion of games with unawareness in quantum game theory
The notion of games with unawareness is designed to model game theory problems in which players’ perceptions of the game are restricted. It was shown in [5] that the novel structure extends the existing forms of games. Although it is possible to represent games with unawareness with the use of games with incomplete information (by associating probability 0 to games that a player is not aware of), the extended Nash equilibrium does not map to any known solution concept of incomplete information games. In particular, the set of extended Nash equilibria forms a strict subset of the Bayesian Nash equilibria.
Once we know that the notion of games with unawareness presents a new game form, it is natural to study that type of games in the quantum domain. Having given a quantum game scheme that maps a classical game G to the quantum one Q(G), and having given a family of games \(\{G_v\}\), a family of quantum games \({Q(G_v)}\) can be constructed in a natural way. Then we can study if, and to what extent, quantum strategies compensate restricted perception of players.
In what follows we provide a formal definition of game with unawareness and extended Nash equilibrium adapted from [5].
3.2 Strategicform games with unawareness
Let \(G = \left( N, \prod _{i\in N}S_{i}, (u_{i})_{i\in N}\right) \) be a strategicform game. This is the game considered by the modeler. Each player may not be aware of the full description of G. Hence \(G_{\mathrm {v}} = \left( N_{\mathrm {v}}, \prod _{i\in N_{\mathrm {v}}}(S_{i})_{\mathrm {v}}, ((u_{i})_{\mathrm {v}})_{i\in N_{\mathrm {v}}}\right) \) denotes player \(\mathrm {v}\)’s view of the game for \(\mathrm {v} \in N\). In general, each player also considers how each of the other players views the game. Formally, with a finite sequence of players \(v = (i_{1}, \dots , i_{n})\) there is associated a game \(G_{v} = \left( N_{v}, \prod _{i\in N_{v}}(S_{i})_{v}, ((u_{i})_{v})_{i\in N_{v}}\right) \). This is the game that player \(i_{1}\) considers that player \(i_{2}\) considers that ...player \(i_{n}\) is considering. A sequence v is called a view. The empty sequence \(v = \emptyset \) is assumed to be the modeler’s view (\(G_{\emptyset } = G\)). We denote a strategy profile in \(G_{v}\) by \((s)_{v}\). The concatenation of two views \(\bar{v} = (i_{1}, \dots , i_{n})\) followed by \(\tilde{v} = (j_{1}, \dots , j_{m})\) is defined to be \(v = \bar{v}\hat{}{\tilde{v}} = (i_{1}, \dots , i_{n}, j_{1}, \dots , j_{m})\). The set of all potential views is \(V = \bigcup ^{\infty }_{n=0}N^{(n)}\) where \(N^{(n)} = \prod ^{n}_{j=1}N\) and \(N^{(0)} = \emptyset \).
Definition 1
 1.For every \(v \in \mathcal {V}\),$$\begin{aligned} v\hat{}{\mathrm {v}} \in \mathcal {V} ~\hbox {if and only if}~\mathrm {v} \in N_{v}. \end{aligned}$$(18)
 2.For every \(v\hat{}{\tilde{v}} \in \mathcal {V}\),$$\begin{aligned} v \in \mathcal {V}, \quad \emptyset \ne N_{v\hat{}{\tilde{v}}} \subset N_{v}, \quad \emptyset \ne (A_{i})_{v\hat{}{\tilde{v}}} \subset (A_{i})_{v} ~\hbox {for all}~i \in N_{v\hat{}{\tilde{v}}} \end{aligned}$$(19)
 3.If \(v\hat{}{\mathrm {v}}\hat{}{\bar{v}} \in \mathcal {V}\), then$$\begin{aligned} {v\hat{}{\mathrm {v}}\hat{}{\mathrm {v}}}\hat{}{\bar{v}} \in \mathcal {V} ~\hbox {and}~ G_{v\hat{}{\mathrm {v}}\hat{}{\bar{v}}} = G_{v\hat{}{\mathrm {v}}\hat{}{\mathrm {v}}\hat{}{\bar{v}}}. \end{aligned}$$(20)
 4.For every strategy profile \((s)_{v\hat{}{\tilde{v}}} = \{s_{j}\}_{j\in N_{v\hat{}{\tilde{v}}}}\), there exists a completion to an strategy profile \((s)_{v} = \{s_{j}, s_{k}\}_{j\in N_{v\hat{}{\tilde{v}}}, k\in N_{v}\setminus N_{v\hat{}{\tilde{v}}}}\) such that$$\begin{aligned} (u_{i})_{{v\hat{}{\tilde{v}}}}((s)_{v\hat{}{\tilde{v}}}) = (u_{i})_{v}((s)_{v}). \end{aligned}$$(21)
3.3 Extended Nash equilibrium in strategicform games with unawareness
In order to define extended Nash equilibrium, it is needed to redefine the notion of strategy profile.
Definition 2
To illustrate (22) let us take the game \(G_{12}\)—the game that player 1 thinks that player 2 is considering. If player 1 assumes that player 2 plays strategy \((\sigma _{2})_{12}\) in the game \(G_{12}\), she must assume the same strategy in the game \(G_{1}\) that she considers, i.e., \((\sigma _{2})_{1} = (\sigma _{2})_{12}\). In other words, player 1 finds that player 2 is considering strategy \((\sigma _{2})_{12}\). Thus, player 1 considers that strategy in her game \(G_{1}\).
Next step is to extend rationalizability from strategicform games to the games with unawareness.
Definition 3
An ESP \(\{(\sigma )_{v}\}_{v\in \mathcal {V}}\) in a game with unawareness is called extended rationalizable if for every \(v\hat{}{\mathrm {v}} \in \mathcal {V}\) strategy \((\sigma _{\mathrm {v}})_{v}\) is a best reply to \((\sigma _{\mathrm {v}})_{v\hat{}{\mathrm {v}}}\) in the game \(G_{v\hat{}{\mathrm {v}}}\).
Consider a strategicform game with unawareness \(\{G_{v}\}_{v\in \mathcal {V}}\). For every relevant view \(v \in \mathcal {V}\) the relevant views as seen from v are defined to be \(\mathcal {V}^{v} = \{\tilde{v} \in \mathcal {V}:v\hat{}{\tilde{v}} \in \mathcal {V}\}\). For \(\tilde{v} \in \mathcal {V}^{v}\) define the game \(G^{v}_{\tilde{v}} = G_{v\hat{}{\tilde{v}}}\). Then the game with unawareness as seen from v is defined as \(\{G^{v}_{\tilde{v}}\}_{\tilde{v} \in \mathcal {V}^{v}}\).
We are now in a position to define the counterpart of Nash equilibrium in games with unawareness.
Definition 4
An ESP \(\{(\sigma )_{v}\}_{v\in \mathcal {V}}\) in a game with unawareness is called an extended Nash equilibrium (ENE) if it is rationalizable and for all \(v, \bar{v} \in \mathcal {V}\) such that \(\{G^{v}_{\tilde{v}}\}_{\tilde{v} \in \mathcal {V}^{v}} = \{G^{\bar{v}}_{\tilde{v}}\}_{\tilde{v} \in \mathcal {V}^{\bar{v}}}\) we have that \((\sigma )_{v} = (\sigma )_{\bar{v}}\).
The first part of the definition (rationalizability) is similar to the standard Nash equilibrium where it is required that each strategy in the equilibrium is a best reply to the other strategies of that profile. According to Definition 3, player 2’s strategy \((\sigma _{2})_{1}\) in the game of player 1 has to be a best reply to player 1’s strategy \((\sigma _{1})_{12}\) in the game \(G_{12}\). On the other hand, in contrast to the concept of Nash equilibrium, \((\sigma _{1})_{12}\) does not have to a best reply to \((\sigma _{2})_{1}\) but to strategy \((\sigma _{2})_{121}\).
We saw in (14) of Example 1 that for \(v \in \{21, 121, 212, 1212, \dots \}\) we have \(G_{v} = \varGamma _{2}\). It follows that \(\{G_{21\hat{}v}\}_{v\in \mathcal {V}_{0}} = \{G_{121\hat{}v}\}_{v\in \mathcal {V}_{0}} = \{\varGamma _{2}\}\). According to the second part of ENE, \((\sigma )_{21} = (\sigma )_{121}\).
The following proposition is useful to determine the extended Nash equilibria.
Proposition 1
 1.
\(\sigma \) is rationalizable for G if and only if \((\sigma )_{v} = \sigma \) is part of an extended rationalizable profile in \(\{G_{v}\}_{v\in \mathcal {V}}\).
 2.
\(\sigma \) is a Nash equilibrium for G if and only if \((\sigma )_{v} = \sigma \) is part of on an ENE for \(\{G_{v}\}_{v\in \mathcal {V}}\) and this ENE also satisfies \((\sigma )_{v} = (\sigma )_{v\hat{}{\bar{v}}}\).
Remark 1
4 Quantum PQ Penny Flip game with unawareness
We noted in Sect. 2.1 that a single bimatrix game does not properly reflect the variant of the PQ Penny Flip game where Alice can win every time. We now show that the problem may be regarded as a game with unawareness.
Example 2
It is worth emphasizing that an ordinary Nash equilibrium applied to (11) leads to an incorrect prediction about Bob’s optimal strategy. Since Alice has a winning strategy in \(F_{1}\), every probability distribution over \(\{\mathbb {1}, \sigma _{x}\}\) rather than a single strategy \(\sigma ^c_{2}\) is an optimal strategy of Bob in \(F_{1}\).
Our next example shows how a limited perception of a player affects the result of PQ Penny Flip game.
Example 3
The above examples shows that incomplete awareness may dramatically affect the result of the game. In what follows, we shall show that this is also true in a general setting, where the set of available actions for the players is the set of \(2\times 2\) unitary matrices U(2).
4.1 Relevant best replies in PQ Penny Fliptype games
Proposition 2
We are now in a position to prove the lemmas that determine players’ best replies to specific strategies. Let \(\pm \rangle = (0\rangle \pm 1\rangle )/\sqrt{2}\). The following lemma exhibits all winning strategies of player 1 in Meyer’s problem. It is a reformulation of the results appeared in [17, 18].
Lemma 1
Proof
As it was mentioned in [12], player 2 being aware of his unitary strategies has a counterstrategy \(V_{2}\) to player 1’s optimal strategy played in \(\varGamma _{QC}\). The following lemma provides us with the general form of \(V_{2}\).
Lemma 2
Proof
The next lemma characterizes player 2’s optimal unitary strategy in the game against player 1 equipped with the classical strategies.
Lemma 3
Proof
4.2 Extended Nash equilibria in generalized PQ Penny Fliptype games
Having determined the relevant best replies in \(\varGamma _{QQ}\), \(\varGamma _{QC}\) and \(\varGamma _{CQ}\), we are now in a position to study the quantum PQ Penny Flip game with unawareness in a more general setting.
Proposition 3
Proof
The fact that \((\sigma )_{v} = (\sigma ^c_{1}, \sigma ^c_{2})\) for \(v\in \mathcal {V}_{0}\setminus \{\emptyset , 1\}\) is obtained by the same line of reasoning as in Example 2.
Let us examine \((\sigma )_{1} = (\sigma _{1}, \sigma _{2})_{1}\). From (22) we deduce that \((\sigma _{2})_{1} = (\sigma _{2})_{12} = \sigma ^c_{2}\). By Definition 3, \((\sigma _{1})_{1}\) is a best reply to \((\sigma _{2})_{1} = (\sigma _{2})_{12} = \sigma ^c_{2}\). Lemma 1 leads to \((\sigma _{1})_{1} = (V_{1}, V_{3})\), which establishes formula (70). \(\square \)
Proposition 4
Proof
As in Example 3, the game \(G = \varGamma _{CC}\) meets condition (34). This fact justifies the first piece of (72). Let us justify \((\sigma )_{21}\). From (22) we obtain \((\sigma _{2})_{21} = (\sigma _{2})_{212} = \sigma ^c_{2}\). Turning to \((\sigma _{1})_{21}\), by Definition 3, we need to determine player 1’s best reply to \((\sigma _{2})_{21} = \sigma ^c_{2}\) in \(G_{21} = \varGamma _{QQ}\). Lemma 1 shows that player 1’s optimal strategy to any probability mixture over \(\mathbb {1}\) and \(\sigma _{x}\) is \((V_{1}, V_{3})\). Hence \((\sigma _{1})_{21} = (V_{1}, V_{3})\). Let us examine the strategy profile \((\sigma )_{2}\). Again, we see from (22) that \((\sigma _{1})_{2} = (\sigma _{1})_{21} = (V_{1}, V_{3})\). On the other hand, it follows from Lemma 2 that \(V_{2}\) defined by (54) is player 2’s best reply to \((V_{1}, V_{3})\) in \(G_{2} = \varGamma _{QQ}\). This gives \((\sigma _{2})_{2} = V_{2}\). Similar reasoning applies to the other profiles of (72). \(\square \)
Proposition 5
Proof
Analysis similar to that in the proof of Proposition 4 shows that \((\sigma )_{v} = (\sigma ^c_{1}, \sigma ^c_{2})\) for \(v\in \{12, 21, 121, 212, \dots \}\) and \((\sigma _{1})_{2} = \sigma ^c_{1}\). By Lemma 3, player 2’s best reply to \(\sigma ^c_{1}\) is given by (64). We thus obtain \((\sigma )_{2} = (\sigma ^c_{1}, W)\). We leave it to the reader to verify the other profiles of (75). \(\square \)
5 Conclusions
We have shown that the notion of game with unawareness is a useful tool in studying the quantum PQ Penny Flip game. Different players’ perceptions of strategies available in the game require using more sophisticated methods for describing the game and its possible rational results than an ordinary matrix game together with the concept of Nash equilibrium. The examples used in the paper indicate that not only the possibility of using quantum strategies but also incomplete awareness of the players may lead to unpredictable outcomes. This fact undoubtedly sheds new light on quantum game theory.
Our work provides new tools that might be utilized in allied sciences. The obtained results can be generalized to more complex games and then applied to study numerous economical problems formulated in terms of games with unawareness with the use of mathematical methods of quantum information. At the same time, these problems will enrich theory of quantum information through new examples that will show superiority of using quantum methods over methods of classical information theory.
References
 1.von Neumann, J.: Zur Theorie der Gesellschaftsspiele. Math. Ann. 100, 295 (1928)MathSciNetCrossRefGoogle Scholar
 2.von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944)zbMATHGoogle Scholar
 3.Halpern, J.: Alternative semantics for unawareness. Games Econ. Behav. 37, 321 (2001)MathSciNetCrossRefGoogle Scholar
 4.Feinberg Y.: Subjective reasoning—games with unawareness. Research Paper No. 1875, Stanford Graduate School of Business (2004)Google Scholar
 5.Feinberg, Y.: Games with Unawareness. Working Paper No. 2122, Stanford Graduate School of Business (2012)Google Scholar
 6.Han, Y.J., Zhang, Y.S., Guo, G.C.: Quantum game with incomplete information. Fluct. Noise Lett. 02, L263 (2002)MathSciNetCrossRefGoogle Scholar
 7.Iqbal, A., Chappell, J.M., Li, Q., Pearce, C.E.M., Abbott, D.: A probabilistic approach to quantum Bayesian games of incomplete information. Quantum Inf. Process. 13, 2783 (2014)ADSMathSciNetCrossRefGoogle Scholar
 8.Situ, H.: Quantum Bayesian game with symmetric and asymmetric information. Quantum Inf. Process. 14, 1827 (2015)ADSMathSciNetCrossRefGoogle Scholar
 9.Situ, H.: Twoplayer conflicting interest Bayesian games and Bell nonlocality. Quantum Inf. Process. 15, 137 (2016)ADSMathSciNetCrossRefGoogle Scholar
 10.Cabello, A., Calsamiglia, J.: Quantum entanglement, indistinguishability, and the absentminded driver’s problem. Phys. Lett. A 336, 441 (2005)ADSMathSciNetCrossRefGoogle Scholar
 11.Fra̧ckiewicz, P.: Application of the Eisert–Wilkens–Lewenstein quantum game scheme to decision problems with imperfect recall. J. Phys. A Math. Theor. 44, 325304 (2011)MathSciNetCrossRefGoogle Scholar
 12.Meyer, D.A.: Quantum strategies. Phys. Rev. Lett. 82, 1052 (1999)ADSMathSciNetCrossRefGoogle Scholar
 13.Maschler, M., Solan, E., Zamir, S.: Game Theory. Cambridge University Press, Cambridge (2013)CrossRefGoogle Scholar
 14.Thompson, F.B.: Equivalence of Games in Extensive Form, Research Memorandum RM759, U.S. Air Force Project Rand, Rand Corporation, Santa Monica, California, (1952), (Reprinted on pp. 36–45 of Classics in Game Theory (Kuhn, H.W. (ed.)) Princeton University Press, Princeton (1997)Google Scholar
 15.Osborne, M.J., Rubinstein, A.: A Course in Game Theory. The MIT Press, Cambridge (1994)zbMATHGoogle Scholar
 16.Shende, V.V., Markov, I.L., Bullock, S.S.: Minimal universal twoqubit controlledNOTbased circuits. Phys. Rev. A 69, 062321 (2004)ADSCrossRefGoogle Scholar
 17.Chappell, J.M., Iqbal, A., Lohe, M.A., von Smekal, L.: An analysis of the quantum penny flip game using geometric algebra. J. Phys. Soc. Jpn. 78, 054801 (2009)ADSCrossRefGoogle Scholar
 18.Balakrishnan, S., Sankaranarayanan, R.: Classical rules and quantum strategies in penny flip game. Quantum Inf. Process. 12, 1261 (2013)ADSMathSciNetCrossRefGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.