ollege

an ovom qinnindeci

lassica

Morgee in decision supportrecent papers in this]). Thory f comteresantumis stafferr equ

more. This survey seeks to introduce and provide a sum-

those already familiar with these ideas.

1.1. Quantum computing basics

like an ion, quantum mechanics dictates the state and the

Decision Support Systems 46 (2008) 318332

Contents lists available at ScienceDirect

Decision Supp

j ourna l homepage: www.emary of quantum games to the decision sciences community.We note the interesting point that not only did John vonNeumann provide seminal work in game theory but was alsoa key researcher in quantum mechanics, so there was animplicit historic relationship between the two areas.

In order to understand these ideas, the basics of quantum

item is called a qubit (for quantum bit multiple qubits areoften called qudits for d qubits). The qubit state can be ineither pure state 0 or 1 or in a superposition where both existsimultaneously. This counterintuitive superposition of both 0and 1 is one of the hallmarks of quantum phenomena and hasbeen experimentally veried in countless ways and times.of related papers. Quantum games ooperate, to remove dilemmas, to altecomputing are required. Towards this endbrief summary of quantum computing conceare many other good sources (e.g., see Nie

Corresponding author.E-mail addresses: guohong@u.edu (H. Guo), juhen

koehler@u.edu (G.J. Koehler).

0167-9236/$ see front matter. Published by Elsevierdoi:10.1016/j.dss.2008.07.001new ways to co-ilibria and much

bit represents a 0 or a 1. When very small items are used torepresent a zero or one, say, for example, a charged particlejournal include [4,11,67,75,129,162,167a survey of a new form of game thetheory. In recent years a new type ocomputing, has gained considerable in[127] merged game theory with quproposed the rst quantized game. This paper providesquantum game

puting, quantumts. In 1999, Meyercomputing and

rted an avalanche

1.1.1. Quantum phenomenaGeneral purpose quantum computers do not exist yet nor

are they likely to exist for 2030 years, although small-scalelaboratory models and small specialized commercial modelshave been developed (http://www.dwavesys.com/). On stan-dard digital computers, the basic unit of information is a bit. AGame theory (von Neumann andNash [131]) plays an important rolsystems and decision sciences (e.g.,nstern [134] and1. Introduction [135] for a good overview). Section 1.1 may be skipped byA survey of quantum games

Hong Guo, Juheng Zhang, Gary J. KoehlerDepartment of Information Systems and Operations Management, Warrington C

a r t i c l e i n f o a b s t r a c t

Article history:Received 9 April 2008Received in revised form 27 June 2008Accepted 3 July 2008Available online 10 July 2008

This paper presentsideas emanating frcomputing at the begand possibilities intobroad spectrum of c

Keywords:Quantum computingGame theoryNash equilibriawe start with apts, though therelsen and Chuang

g@u.edu (J. Zhang),

B.V.of Business, University of Florida, Gainesville, FL 32611, United States

erview and survey of a new type of game-theoretic setting based onuantum computing. (We provide a brief overview of quantumg of the paper.) Initial results suggest this view bringsmore exibilitysions involving game-theoretic considerations. Applications cover al games as well as games in economics, nance and other areas.

Published by Elsevier B.V.

ort Systems

l sev ie r.com/ locate /dssWhen performing calculations on a superposition, we end upperforming the calculation on all pure states in the super-position. So if there are n qubits involved in a superposition,then the calculation takes place over all of the 2n pure states.This parallelism has no counterpart in classical computing.

When one examines ormeasures a qubit in a superpositionstate, he will observe only a pure state. This interaction of aquantum system with the environment is called decoherence

The much celebrated result by Shor [158] provides a quantum

319H. Guo et al. / Decision Support Systems 46 (2008) 318332factorization algorithm that operates in polynomial-time.

1.1.2. Mathematical representationMathematically, a qubit is represented by a 2-vector in a

complex vector space as 10

01

. Here, and

are complex numbers called complex amplitudes. Themodulussquared of each gives the probability that nature will select

pure state 0 given by 10

and pure state 1 given by 01

respectively. Physicists often employ Dirac notation to repre-sent quantum states. |0 represents pure state 0 and |1 purestate 1. So, an equivalent representation of a qubit is =|0+|1 where 1=||2+ ||2. An n-qubit register is given by

ji xa 0;1f gn

axjxi; where 1 xa 0;1f gn

jaxj2.

This takes 2n complex numbers to completely describe itsstate.

The state of a composite system is the tensor product of thestate of the components. For example, a composite system oftwo qubits, 1 and 2, written as |12, is

j12i j1i j2i 11

22

12121212

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1.1.3. Quantum computing and informationClosed quantummechanical systems evolve under unitary

transformations. A unitary matrix, U, is a matrix whose ad-joint (i.e., the conjugate transpose), U, is its inverse. That is,UU=UU= I. Quantum computer computation is carried-outby subjecting qubits to unitary transforms. Developingquantum algorithms requires specifying starting states andunitary transformations to achieve the desired result. Sincethe nal measured state is probabilistically chosen based onamplitudes, part of the process of crafting an algorithm eitherproduces a state with an amplitude whose modulus squaredis 1.0 or very close to 1.0. Several small sets of unitary trans-forms are universal, meaning any quantum computation canbe reduced to using just the transforms in the set [135]. Also,simple algorithms (see [165]) can be used to do normal arith-metic operations (like addition and multiplication).and can be thought of as nature randomly picking one of thepure states in the superposition. That is, the act of observingthe superposition appears to have forced nature to randomlychoose one state.

Qubits can be coordinated so that they share theirquantum states, even if they are not physically close to eachother. This is called entanglement. For example, two qubits canbe entangled so that if one is observed in state 0 the other willbe in state 1, and vice versa. Until an observation is made, bothqubits are in a superposition, but once one state is known, theother state is also known, even if it is extremely far away. Thisis the non-locality feature Einstein found objectionable [45].

Quantum phenomena of superposition, entanglement andnon-locality open the door to all new ways of computing andprocessing information. Many interesting algorithms havebeen discovered that take advantage of quantum phenomena.Unitary transforms function as gates do in classical com-

puters. For example, the transform X 0 11 0

performs a

NOT operation, switching pure state 0 to a 1 and vice versa.The identity transform, I, preserves the current state. The

Z 1 00 1

gate just switches the sign on state 1. The X, Z

and Y 0 ii 0

gates are known as Pauli gates. Here i is the

usual imaginary number i 1

p.

Some transforms are of great use. The Hadamard transform,

H 12

p 1 11 1

changes |0 to a superposition with equal

probabilities for each state. That is

Hj0i 12

p 1 11 1

10

1

2p 1

1

1

2p j0i 1

2p j1i:

Note that states |0 and |1 both have probability j 12

p j2 12of being observed. For n qubits, we just use a tensor productto get an equal superposition.

In summary, a quantum computer algorithm takes thegeneral form:

Prepare the initial state (usually taken to be |000).Apply a sequence of unitary transforms.Perform a measurement to read out a state.

1.1.4. Ensemble systems and density matricesAnother formalism for representing and working with

quantum computing uses positive, semidenite, Hermitianmatrices with trace 1 to represent states. These are calleddensity matrices. Any density matrix, , can be written as n

i1ivivi where i0, 1

n

i1i and 1

vTi vi

q. States

represented by vectors earlier, such as , would be repre-sented by the density matrix . Density matrices representmixed states that are viewed as a probabilistic mixture (thei's) of pure states, vivi. This pays dividends when ensemblesof states arise naturally in a problem, as is typical in mixed-strategy, game-theoretic models.

1.1.5. Miscellaneous issuesWootters and Zurek [172] showed that it is impossible to

copy a qubit in a superposition state (only pure states can becopied). This is called the No Clone theorem. Interestingly,Bennett et al. [8] showed how to teleport a qubit. In theprocess, the original qubit's state is destroyed but the recipientends with a particle indistinguishable from the original.

Any 2 by 2 unitary matrix U can be represented byU=eiaeibZeicYeidZ for appropriate choices of real-valued a, b, cand d. This is known as the ZY decomposition or Cartandecomposition. The matrix exponential notation means thefollowing. If x is a real number and A is a matrix such thatAA=AA= I. Then eixA=cos(x)I+ i sin (x)A. Y and Z satisfy therequirement (i.e., YY=YY= I and ZZ=ZZ= I), so for example,eidZ=cos(d)I+ i sin(d)Z.

1.2. Quantum games

It is not hard to imagine settingswherequantumgamesmaybe the natural game-theoretic choice because the phenomenaare taking place at microscopic scales where the laws of

320 H. Guo et al. / Decision Support Systems 46 (2008) 318332quantummechanics reign. For example, evolutionary biologistsstudy the interaction of competition at the gene level. Dawkin'sSelsh Gene [29] comes to mind. Similar arguments have beenmade about protein interactions and protein folding [68]. Inanother setting, information communication with possibleeavesdroppers can be viewed as a game and if the commu-nication medium employs quantum components, as many do,quantum games are a natural framework. Even fairness can beguaranteed in remote gambling [69].

Along these lines, if classic games are played on a quantumcomputer or played by a quantum computer, the games canbecome quantum games. In [36] the Prisoner's Dilemma isactually played on an NMR quantum computer and Zhuanget al. [177] discuss how a quantum gambling machine,supporting a class of games, could be built using opticalelements. Chen et al. actually do experimental studies withhuman players playing quantum games (see [17,18]). In theprisoner's dilemma game, humans did better in the quantumgame than they do in a classic game.

This view quickly leads one to ask what happens whenwereplace other classical components in games, such asprobabilities, with quantum counterparts or if there areparallels between classical systems and quantum systems?For example, Demski et al. [30] nd interesting parallelsbetween accounting information systems and quantumsystems in areas such as measurement (a key task inAccounting) and interactions with the environment. Iqbal[90] argues situations akin to this areworthwhile since there isa basic relationship between quantum algorithms andquantum games and that the study of quantum games maylead to new quantum algorithms.

Perhaps less tangible is Witte's [170] view that imaginesquantum games as classical games played by quantum players.He argues that a human player plays virtual games in his mindbefore choosing a strategy. Since these are never actually played,they are reminiscent of the many worlds view of quantummechanics [79] and thus lead to considering the game as aquantumgame. Alternatively, if our brain functions are describedby quantum mechanisms as viewed by Penrose [139], thenquantum games may be a natural choice for modeling games.

More concretely, Grib et al. [72,73] look at macroscopicgames where the rules of the game lead to a non-distributivityproperty that not only prevents the usual use of probabilitycalculation in mixed strategies but yields a system whereHilbert Space amplitudes capture the appropriate character-istics. That is, macroscopic games can be described by quantumformalisms.

Conversely, Levine [116] argues that quantum games offernothing new. He claims quantum entanglement gives featuressimilar to correlated equilibrium and phenomena like cheap-talk equilibrium (where players can get advice). This view iscontested. Dahl and Landsburg [28] show that quantumequilibrium with entanglement is not the same as correlatedequilibrium in classical games. Eisert et al. [48] also givecounterexamples. Lee and Johnson [109] show that whatevera quantum game can do, so can some more complicatedclassic game but the quantum game is more efcient and maybe the appropriate model for microscopic games.

In Section 2 we introduce and discuss the details of twosimple quantum games. In Section 3 we give a survey of over100 recent papers in this area, including ones in Accounting,Finance and Economics. Section 4 provides some futuredirections for research.

2. Quantum game framework

In this section,we present the basic ideas of quantumgamesusing two specic examples discussed in the literature. Thesedemonstrate how unique features of quantummechanics, suchas superposition and entanglement, can bring new aspects toclassical games.

2.1. Quantum game basics

We start by recalling some notation from classic games(e.g., [136]). Assume there are n players with strategy spacesSi, i=1,,n and payoff functions ui(s1,,sn), i=1,,n wheresiSi. Then a game can be denoted by G(n, S, u) whereS=S1Sn and u=u1un. This is the normal-formrepresentation of a game commonly used in static games.For dynamic games, extensive-form representations are usedto include the timing of moves by each player, the possiblestrategies for each player at each move; what each playerknows at each of his/her opportunity to move, etc. Classicgames implicitly assume an information exchange mechan-ism to facilitate game implementation. For example, in theclassic prisoners' dilemma game, the judge explains the gamerules to the two prisoners, collects each prisoner's decisionand then calculates the corresponding payoffs.

Quantumgames explicitly depict the processes of informa-tion exchanges among players and payoff realization. Insteadof a triple, a more general form is used to describe a quantumgame (based on [47,113]). We denote a quantum game byG n; H ;; S;uf g where n is the number of players, H is thetwo dimensional Hilbert space, H is the game's state space,a H is the starting state, S=S1Sn is the strategy spaceand u=u1un is the utility function with ui : H YR forplayer i. In quantum games, the state of the game can berepresented by a qubit or tensor products of multiple qubits. Areferee (also called a judge or arbiter) serves as the coordinatorof the game. As shown in the following gures, at the game'sbeginning, the referee may alter the starting state using anoptional transform U to change the default starting spacea H (which is typically the zero pure state) to a desiredstate such as an entanglement of pure states. Then individualplayers move strategically. In other words, player i choosessiSi and all the si's are quantumoperationswhich usually canbe represented by unitary transformations Ui's. For example,in a simple two-person game, unitary transformations U1 andU2 are applied either simultaneously (e.g., in a static gameas inFig. 1) or sequentially (e.g., in a dynamic game as in Fig. 2) tothe game state. After the players move, the referee applies theinverse transform U to undo the earlier application of U andtakes the nal measurement to reveal the nal state of thegame. Based on the observed nal state the referee calculatesthe payoffs to all players according to each player's ui's.

2.2. Example one: PQ penny ip

To illustrate these notions, we start withMeyer's PQ pennyip game [127] which is recognized as the rst quantizedgame. This two-player game pitting player Q against player P

e. At t if the s head , thenpayo Q los payoff rwise,

and P The cl nny i can be

quant

ic qua

321H. Guo et al. / Decision Support Systems 46 (2008) 318332marized in the following payoff matrix.

Q NN NF FN FF

P:N (1, +1) (+1, 1) (+1, 1) (1, +1)P:F (+1, 1) (1, +1) (1, +1) (+1, 1)

This is a dynamic, two-person zero-sum game and has no pure-strategy Nashequilibrium.

Now we turn to a simple quantum version of this game inwhichweallowplayerQ to adopt quantumstrategies and restrictplayerP to classic strategies.Weuse |0 to representheads and |1+1to represenby a qubitHere theUthe refereeversionof tbe represeshown in Fany unitarythe game iwhere S1=the payoffs

Considetwice. Afte12

p j0i j1(i.e., guramatter whremains thof |0. In ostrategy allbe alteredgets 1.t tails. Then the=|0+|1. Tas shown in Fig.leaves the initiahis game. The twnted by unitaryig. 3, P can onlytransform (wes specied by G{I, X}, S2 is the sdiscussed abovr the case wher Q's rst movi. By playingHtively, the coinat P chooses, we same. Then Qther words, Q aows him to preby P and thus gassic pestate of the gamhe initial state1 is just the idenl state as is. Figo classicmovetransforms Xplay either X orshow theHadamn 2; H fet of all unitarye.re Q plays H (te, the state o,Q puts the sysstands on itshether X or I,chooses H agalways wins! Tsent a superimuarantees himp gamee can be repreof the game istity transform.. 3 shows the qusip andnotand I, respectivIwhile Qmayard transformH; j0i; S1 matrices, and

he Hadamard mf the game istem in a superpside). As a resthe state of thein giving a nahe quantizationposed state thaself a win.sum-

with ff +1 and es with 1. Othe Q gets

mov he end, coin end s-down P winsoffers interesting nuances over its classic counterpart. Thereare four steps in the classic version: a penny is placed heads-up in a box by the referee; Q either ips the penny over (F) ornot (N); then P either ips the penny over (F) or not (N); andthen Q takes the nal move either ipping the penny over (F)or not (N). Note that both P and Q don't know each other's

Fig. 1. Static

Fig. 2. Dynamsented=|0.That is,antumip canely. Aschoose). Thus,S2;ug

u gives

atrix)Hj0i ositionult, nogame

l resultof Q'st can'tNow let us examine the classic mixed strategies in thisgame and the quantum counterpart. Standard game theoryresults of two-person zero-sum games tell us that the classicPQ penny ip game has a mixed-strategy Nash equilibrium. Itcan be easily shown that themixed-strategy Nash equilibriumhas player P choose F or N with equal probability 0.5 andplayer Q chooses among his four possible strategies (NN, NF,FN, or FF) with equal probability, 0.25. As a result, bothplayers get zero expected payoffs. For the quantum version,we still restrict P to classic strategies but now consider mixedstrategies as shown in Fig. 4. Let P's mixed-strategy be ipwith probability p and not ip with probability 1p. Onceagain we allow Q to employ a quantum move. FollowingMeyers [127], let his transform be given by unitary matrices

with the form UQ1 a bT

b a

where aa+bb=1. Since now

the game involves mixed states, we use a density matrix (asdiscussed in Section 1.1) to represent the state of the game.

In the form of a density matrix, the initial state is 0 j0ih0j 1 00 0

. The state after Q's move is 1 UQ10U

Q1

aaT abT

. After P's move we haventum games.um games.baT bbT

2 pX1X 1p I1I

pbbT 1p aaT pbaT 1p abT

pabT 1p baT paaT 1p bbT

:

Consider what happens if the game ends here. Recall thatthe diagonal elements of a density matrix correspond to theprobabilities of pure states. The probabilities of |0 and |1 arepbb+(1p)aa and paa+(1p)bb respectively. On the onehand, given Q's strategy, P's best response is to choose p=1 ifaaNbb; p=0 if aabbb; any p [0,1] if aa=bb. On theother hand, player Q would choose bb=1 if pN0.5; bb=0 ifpb0.5; and any a,b if p=0.5. Therefore the (classic mixed-strategy, quantum strategy) equilibrium is p=0.5 and aa=bb=0.5 with both players getting zero payoff. This resultis the same as the classic mixed-strategy for both players.This two-step game suggests that a player with an optimalquantum strategy has an expected payoff at least as great ashis expected payoff with an optimal mixed-strategy.

In thisThe state o

. s

ip (mixed-strateg

ip (pure-strategy).

322 H. Guo et al. / Decision Support Systems 46 (2008) 318332matrix again) and

1 UQ10UQ1 12

1 11 1

; 2 pX1X 1p I1I

12

1 11 1

; 3 UQ22UQ2

1 00 0

By doing so, Q wins with probability 1. Thus Q can alwayswin by playing a Hadamard matrix in both of his turns nomatter whether P uses a pure or mixed-strategy. Meyer alsoproved that a two-person zero-sum game may not have a(pure-strategy quantum, pure-strategy quantum) equilibriumbut always has a (mixed-strategy quantum, mixed-strategyNowwe continue the third step in the PQ penny ip gameQ has a second chance to ip. Suppose Q chooses a strategy

witha b 12

p . ThenUQ1 UQ2 12p 1 11 1

(theHadamard

Fig. 4. PQ pennyquantum) equilibrium.

2.3. Example two: prisoner

The penny ip game dequantum phenomena slooked at a quantized prisanother unique quantumthe outcome of the gamedilemma, each player hasdefect (D). The correspon

Player 1: CPlayer 1: D

The resulting Nash eqsuboptimal because both(C, C).

Now we consider a qEisert [48] as shown belos' dilemma

monstrated one unique ouuperposition. Eisert et al.oner's dilemma game showfeature entanglement . In the classic game of ptwo choices, either cooperding payoff is:

Player 2: C

(3,3)(5,0)

uilibrium is (D, D) whichplayers would be better off

uantization of this gamew in Fig. 5.. 3. PQ pennyFigtcome of[48] rsting how

can affectrisoners'ate (C) or

Player 2: D

(0,5)(1,1)

is Paretochoosing

given bying the case where both players cooperate. The initial Utransform produces an entangled state, U|00=(|00+ i|11) /2

p. Then the players may choose one of the classic moves I

(maintains cooperate) or X (switch to defect) or a quantummove restricted to unitary matrix Z. Thus, the game isG n 2; H H2; j00i; S1 S2;u

where S1=S2={I, X,Z} and u gives the normal payoffs. The nine possible results are:

Player 2: I Player 2: X Player 2: Z

Player 1: I I I Uj00i

12

p j00i ij11i

I X Uj00i

12

p j01i ij10i

I Z Uj00i

12

p j00iij11i of the twoPlayer 1: X

Player 1: Z

After a

Player 1: I

Player 1: X

Player 1: Zplayers' choicesX I Uj00i

12

p j10i ij01i

Z I Uj00i

12

p j00iij11i

pplying U we ge

Player 2: I

U I I Uj00 j00i

U X I Uj00 j10i

U Z I Uj00 ij11iThe initial state iX X Uj00i

12

p j11i ij00i

Z X Uj00i

12

p j01iij10i

t:

Player 2: X

i U I X Uj00i j01i

i U X X Uj00i j11i

i U Z X Uj00i ij10i|00 represent-

so the statgame, let |0 denof the game is detee of the game cante cooperate and |rminedby the plabe represented as1denote defect.yers' choices andthe combinationy).X Z Uj00i

12

p j10iij01i

Z Z Uj00i

12

p j00i ij11i

Player 2: Z

U I Z Uj00i ij11i

U X Z Uj00i ij01i

Z Z Uj00i j00i

ibZeicYe ei bd co ei bd siei bd i bd co

oners'

323H. Guo et al. / Decision Support Systems 46 (2008) 318332UQ2 eieiZeiYeiZ ei ei cos ei sin

ei sin ei cos UQ1 e ewhere a, b, c, d, the game in theresponding densi

0 1 00 0

;

3 pUQ2XUQ10U UQ2UQ10UQ1 e, , and are athree steps can bty matrix as follo

1 UQ10UQ1;Q1X

UQ2 1p UQUQ2 p UQ2XUQ1

sin c ell real-valued. The represented byws:

2 pX1X 1

2UQ10UQ1U

Q2

0UQ1X

UQ2UQ2UQ1s c

ia idZ ia

s c n c

tion discu Section 1.1The nal measurement gives us the following payoffs:

Player 2: I Player 2: X Player 2: Z

Player 1: I (3,3) (0,5) (1,1)Player 1: X (5,0) (1,1) (0,5)Player 1: Z (1,1) (5,0) (3,3)

In this quantized game, (Z, Z) is not only the unique Nashequilibrium but is also Pareto optimal. The quantum general-ization allows players to escape the usual prisoner's dilemma.

Amore general class of strategies is considered in [48]where

Uj eicos =2 sin =2

sin =2 e icos =2

and

U cos =2 0 0 isin =2

0 cos =2 isin =2 00 isin =2 cos =2 0

isin =2 0 0 cos =2

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3775

with [0, /2]. Parameter measures the degree of entangle-ment with =0 representing zero entanglement making thegame separable (i.e., U is I) and =/2 represents maximumentanglement. Eisert [48] showed that separable games do notdisplay any feature that goes beyond the classic game. The gamewith maximum entanglement has a unique equilibrium.

2.4. Generalizations

We introduce one of many extensions proposed byresearchers generalizing the strategy spaces.

2.4.1. A general form of PQ penny ipWe rst reexamine the PQ penny ip gamewherewe allow

Q to take general quantum moves using the ZY decomposi-ssed in . Let

Fig. 5. Prise state ofthe cor-

pI1I

0UQ1U

Q2

Since the probability of heads is (3)1,1 and the probabilityof tails is (3)2,2, the expected payoff for P is EV(P)=(3)2,2(3)1,1 and the expected payoff for Q is EV(Q)=(3)1,1 (3)2,2.Now we derive the conditions for Q to win and look forconditions giving EV(P)=1 and EV(Q)=1. For this result to holdrequires both (UQ2XUQ10UQ1XUQ2 UQ2UQ10UQ1UQ2)1,1=0and (UQ2XUQ10UQ1XUQ2 UQ2UQ10UQ1UQ2)2,2=0 for all p.Substituting UQ1 and UQ2 gives cos(2b)sin(2c)sin(2)cos(2)=1. Notice the winning strategy for Q discussed above ofUQ1=UQ2=H is a special case of this condition (e.g., a=b===/2, c==/4, and d==). However, other uni-tary matrices will also work. For example,

UQ1 12

p e2i=3 e2i=3

e2i=3 e2i=3

1

2p 1

1=3 1 2=31 1=3 1 2=3

" #;

UQ2 i2

p 1 11 1

with a=, b=0, c==/4, d=/3, ===/2.Now we constrain the available quantum strategies

for Q to derive conditions for the classic results of this game,i.e., EV(P)=EV(Q)=0. Substituting the previous results givesEV P 12p cos 2c cos 2 sin 2b sin 2c sin 2 sin 2 cos 2b sin 2c sin 2 cos 2 :

The maximal payoff P can achieve by selecting p as jcos 2c cos 2 sin 2b sin 2c sin 2 sin 2 j cos 2b sin 2c sin 2 cos 2 :

Therefore the game will have classic results when jcos 2c cos 2 sin 2b sin 2c sin 2 sin 2 j cos 2b sin 2csin 2 cos 2 0:

Can P actually win? The following conditions provide anafrmative result for this case: jcos 2c cos 2 sin 2b sin 2c sin 2 sin 2 j cos 2b sin 2c sin 2 cos 2 1:

Thus, the strategy space can be partitioned into four areascorresponding to the three special cases P wins, Q wins, atie which is the classic result - and all other cases. We can seethat restrictions on Q's strategy space determine the outcomeof the generalized PQ penny ip game: Depending on Q'sstrategy space, he can win, tie or lose.

2.4.2. A general form of prisoner's dilemma

dilemma.For a general form of prisoner's dilemma strategies letstrategies be given by

U1 eiaeibZeicYeidZ eia ei bd cos c ei bd sin c

ei bd sin c ei bd cos c

U2 eieiZeiYeiZ ei ei cos ei sin

ei sin ei cos

:

:324 H. Guo et al. / Decision Support Systems 46 (2008) 318332ei ei bd cos c ei 3=2bd sin c ei 3=2bd sin c ei bd cos c . So the best response forplayer 2 is to play XUundo, i.e., =c/2, =b5/4, and =d/4, which results in the nal state of |01with probability 1and the highest possible payoff 5 for player 2. Similarly, player 1will adopt the same strategy and results in the nal state of |10with probability 1 and the highest possible payoff 5 for player 1.Therefore there is no pure-strategy equilibrium.

In thenext sectionwepresenta literature reviewof quantumgames. Other summaries, expositions and reviews are availableu01 cos c sin cos b d sin c cos sin bd 2u10 sin c cos cos bd cos c sin sin b d 2u11 sin c sin cos bd cos c cos sin b d 2

Given player 1's choices a, b, c, and d, player 2 can choose= c, =3 / 4 b, and = d3 / 4 to undo player 1'smove which restores the initial state of the game |00.The corresponding undo matrix for player 2 is Uundo 2

pei b ei d sin c cos iei d cos c sin ei b iei d cos c cos ei d sin c sin

64 75

After applying U we get

U U1 U2 Uj00i

cos a isin a cos c cos cos b d sin c sin sin bd cos a isin a cos c sin cos b d sin c cos sin bd cos a isin a sin c cos cos bd cos c sin sin b d cos a isin a sin c sin cos bd cos c cos sin b d

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There are four possible outcomes |00, |01, |10, and |11with probabilities u00, u01, u10, and u11 respectively. Theprobabilities u00, u01, u10, and u11 can be specied as

u00 cos c cos cos b d sin c sin sin bd 2EV2 3cos2 c cos2 5cos2 c sin2 0sin2 c cos2 1sin2 c sin2 :

Given , player 1's best response is to set sin2(c)=1. Givenc, player 2's best response is to set sin2()=1. So the equi-librium would be c=k /2 and = l /2 where k and l areintegers. This result corresponds to the Pareto suboptimalequilibrium of (D, D) in the classic game. The prisoners cannotescape the dilemma without entanglement.

Nowwe consider themaximally entangledU 12

p I2 iX2 .After the players' moves:

U1 U2 Uj00i

ei a ei b ei d cos c cos iei d sin c sin ei b iei d sin c cos ei d cos c sin

266

377The resulting expected payoff for players 1 and 2 are

EV1 3cos2 c cos2 0cos2 c sin2 5sin2 c cos2 1sin2 c sin2 If there is no preprocessing transform U, then the nalstate of the game is

U1 U2j00i ei abd cos c cos ei abd cos c sin ei abd sin c cos ei abd sin c sin

2664

3775:targeting different areas of interest and covering differentpublication time-frames (e.g., see [53,54,56,57,71,90,109,130,145,147]).

3. Literature review of quantum game theory

Although Meyer [127] is usually cited as originatingquantum game theory, Piotrowski and Sladowski [147]claim Vaidman [164] was the rst to use the term game inthe context of quantum phenomena and that Wiesner [169]deserves credit for considering quantum money in 1983giving the rst glimmer of market-based quantum gamenotions.

We start our literature review by collecting in Table 1 thevariousquantumgame counterparts to classic games studied byresearchers. With each game type, we give citations, adescription of the game and discuss what aspect is uncoveredin thequantumgameversions. (WeuseNE forNashEquilibriumin the singular or plural.)

As one can see, many types of games have been studied,including most of the classic cases typically encountered ingame theory courses. Most of the initial results showed thatquantum games offered ways around classic dilemmas.However, as researcher's focused more carefully, some ofthese results vanished. For example, in the prisoner'sdilemma, as shown in Section 2, only restricted strategyspaces resulted in the usual dilemma being removed. Oncestrategy spaces were allowed to be more general, this aspectdisappeared.

Several avenues of research emerged. Many researchersfocused on the representation of quantum games and varioustypes of generalization therein. Others looked at additionalaspects of quantummechanics, like decoherence, and the rolethey play in quantum games. Still others looked at theoreticalproperties of quantum games, like Nash Equilibria, or atextensions such as into evolutionary games. Finally, a numberof researchers opened new areas of quantum game applica-tions especially in communication and econophysics applica-tions. We briey look at each of these areas.

3.1. Generalizations

Many avenues of generalization have been explored. Table 1shows generalizations from simple two-player static, non-cooperative games to dynamic games, multiplayer games,cooperative games, etc.

As discussed in Section 2, some generalizations focused onexpanding the strategy space from restricted sets of unitarymatrices to more general collections of unitary transforms.Papers that fall in this category are [5,6,123,124]. Someresearchers use vectors to represent player strategies insteadof unitary transformations (e.g., [24,25,85,86]). Among theseresearchers, Ichikawa [86] uses Schmidt decompositionapproaches and nds that eight phase structures exist forgeneral symmetric games when varying correlation betweenplayers. A similar phase transition phenomenon is found inthe minority game by Challet [13] and in the prisoner'sdilemma game by Du [43]. Iqbal [100] looks at whetherquantum games are the same as classic games. He comparesthem under two constraints, one where the classic andquantum game have the same set of strategies and the other

Table 1Quantum games

Game types Game description Contribution

Bar game [101] A bar has 2-seats but all n players want to sit and they alsowant more people present.

In this multiplayer quantum game, a noisy demon thatcontrols the input qubit's corruption reduces the valueof the game.

Bargaining game [142] The game is a model of interactive bargaining with a xedresource amount. Two players get what they want if thesum of their demands is less than or equal to the xed amount.Otherwise, they get nothing.

In a complex quantum bargaining game, the prot is in asuperposition and market transaction is polarized.

Battle of the sexes[2,5,37,38,123,124,132]

Two players (husband and wife) choose to go to the theateror to a football game and have different preference, but theywould benet more if they pick the same option.

Innite NE exist with the same payoff as does the classicalgame. In the game with a general strategy space, there is noequilibrium. The classical game can be reproduced from thequantum game without entanglement.

Byzantine agreement [55] There are three players and one of them is the enemy.They use pair-wise channels to communicate and player Asends a bit to player B and C separately. Player C shouldnd that his bit is the same as the bit player B tells him.However, this is false information since one of player Aor B is cheating.

A quantum solution is found by using pair-wise quantumchannels and entangled qudits. The game can be appliedto cryptography.

Card game [34,71] Player A puts three different cards in a blackbox. The rst cardhas dots on both sides, the second card has circles, and the thirdcard has a dot on one side and a circle on the other side. Player Bpicks one card from the box. If the card has the same pattern onboth sides, player A wins. Otherwise, player B wins.

Without the help of entanglement, this unfair gamebecomes fair when the classical card game is quantized.

Chicken game[47,64,85,86]

Also known as the HawkDove game, this is similar to theBattle of the Sexes except the players' payoffs are higher whenthey pick different strategies.

The usual dilemma can be removed if players use quantumstrategies within a restricted strategy space.

Chinos game [77] In each turn, every player guesses the total number of coinshidden in hands of a group of n players and the player whoseguess is right wins. After m plays, the player whose totalwinning time is worst looses the game.

The partial quantum strategy is not stable while the classicalstrategy is. A full quantum analogy gives a winning strategy.

Cooperative game[26,28,88,96,100,102,105]

Groups of players make an agreement to coordinatetheir strategies.

In the three player case, conditions on the pure initial statesare found that prevent coalition formation.

Guess the numbergame [71]

Player A picks any integer and lets player B guess the number.Player B wins if he can guess right within k tries.

Player B wins more often in the quantum game than in theclassical game since she can put player A's oracle insuperposition states. She uses Grover's database searchalgorithm [74] to search for the number.

Gun duel [60] Two or more gunghters shoot each other and the winneris the player who is still standing after the ght. With respectto the quantum analogy, players lay out their strategies whichare not contingent on prior outcomes since a measurementisn't taken till the nal round is over.

One round quantum duels are equivalent to the classicalversion, but not so with longer turns. Interference effectsplay a big role.

Minority game [3,7,1315,18,23,49,50,62,63,146]

An odd number of players have two strategies. Players winif their strategy is in the minority group.

New NE arises when the number, n, of players is even.When n is odd, the quantum game is like the classicalone with non-maximal entanglement.

Monty Hall problem[27,58,118]

Based on a TV game show, a contestant chooses one of threedoors behind which there is a car or a goat. The host, MontyHall, opens one of two doors that have not been chosen andasks whether the player wants to switch her choice. Theproblem in this game is whether the player should switch ornot. She should switch since the probability of winning thegame is 2/3 then.

The quantum game becomes a fair game between thecontestant and host. If both play quantum strategies thereis no NE in pure strategies but there is in mixed strategieswhich is the same as the classic game. With entanglement,one quantum player has an advantage over the other classicplayer. The classical game is the same as the quantumone with no entanglement of initial states.

Newcombs game [144] In this game, there is an open box with $1000 and a closed boxthat has one million dollars or nothing. A human player can onlypick the closed box or pick both boxes. The player always wins$1,000 more if he chooses to open two boxes. However, there isa predictor (Omega) who decides how much money was put inthe closed box before the game starts and can predict the player'sbehavior. If he predicts the player will pick both boxes, he willput in the closed box. Otherwise one million dollars is put in.

This problem is quantized with alien Omega against thehuman player and the paradox is removed. In this gamehumans retain their free will but cannot prot from theirdecisions.

Parrondo's ([117]) game[59,78,112,115,128,138]

When played in an alternating order, two losing games becomea winning game.

The quantum game can have a spatial state-dependentor historical-dependent version. The quantum game canremove the dependence on a modulo rule based on thecurrent capital in classical game.

Penny ip [71,89,127] Two players each have a coin. Simultaneously, they ip thecoins to heads or tails secretly and then reveal it to each other.If the coins match, player A wins, otherwise player B wins.A special case of this game is Spin-ip game, in which twoplayers take turns in ipping an electron up or down twiceand then the electron's nal state is measured. Player B winsif the measured state is up.

If player B uses quantum strategies while player A uses mixedclassical ones, player B can win the game with certainty.

(continued on next page)

325H. Guo et al. / Decision Support Systems 46 (2008) 318332

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326 H. Guo et al. / Decision Support Systems 46 (2008) 318332Table 1 (continued)

Game types Game description

Prisoner's dilemma[17,19,24,33,39,42,43,8587,94,98,137,160]

Two players (prisoners) choose either to confesscorporate with each other and deny a crime. Whtheir payoff is higher than the payoff when they

Rockscissorspaper[53,90,97,99]

Two players use their hands to represent their thRock, Scissors or Paper. Rock beats Scissors, Scissand Paper beats Rock. If their gestures are same

RSA game [71] One player sends an encrypted message Me andand e to the other player. If the message can bethe player who encrypts the message losses.

Stag hunt game (Samaritangame) [85,86,147,163]

Two players choose to cooperate or not. Their pahigher if they cooperate with each other than ifTwo Nash equilibrium exists, which is the differethe stag hunt game and prisoner's dilemma gam

Trucker's game [41] A number of truckers choose one of n roads to dThe worst case is that all truckers chose the sambest is that every road is chosen by a single truc

Ultimatum game [166] Two players divide a xed amount of money. Plaproposes a plan. If player B agrees with the propthe players get what was agreed upon. Howeverrejects the plan, then both get nothing.

Vaidman's game[12,90,164]

Three players reach agreement before they are bseparate places. They are asked questions what

Table 1 (continued)with the same explicit payoffs, and claims that the quantumgames are not reproducible with classic games.

Others worried about the preparation of the initial state.Some early papers viewed the initial state prepared by thereferee [94,114]. Witte [170] formulates quantum gameswhere initial states are prepared by players. Others look atthe entanglement between the initial state and the measure-ment basis [133], at special types of entangled initial states[44,63,157,163] or at a general initial states [132]. An interest-ing issue ariseswhen the refereemight lie to the players aboutthe initial state [155].

In classic games, players can be viewed as randomvariables. Boukas [9] generalizes this to viewing players asquantum entities. De Sousa et al. [160] treats the payoffs asquantum information, allowing entanglement between pay-offs and strategies. This can make a game where somepossibilities of the payoff table are forbidden or for the gameto remain only in equilibrium states.

Some generalizations occur because of quantum mechan-ical considerations. One avenue looks at the changes to thebasic game theory model under decoherence. In closedsystems, unitary matrices dictate state changes. In opensystems where decoherence is likely to occur, other classes oftransforms are needed [1,47,82,174,175].

or what is X and their answers can be 0 or 1 onlIf they are all asked the question about Z and the numanswers is odd, the team wins. The team also can winwhere two of them are asked Z question and one is asquestion and the number of 0 answers is even.

Wise and foolishAlice [147]

Player B puts a ball on one of four corners and player Aguesses the location of the ball. Player A can ask questabout where B put the ball, and B gives yes or noanswers assuming that he is honest. If the answer is is satised, otherwise player B losses money to playerContribution

e or toy cooperatet.

The usual dilemma disappears if the players userestricted quantum strategies, but it cannot be removedwithin a general strategy space. The payoffs for the playersare dependent on the degree of entanglement. The classicalanalogy is embedded.

rategies,ats Paper,me is tied.

Although the mixed-strategy classic solution is not stable,the modied game (where draws get a slight premium)yields stable mixed-strategy NE for some initial state forthe quantum game.

key Nted, then

The message is decrypted and the player who guesses thenumber M wins. Shor's quantum algorithm [158] andFourier transformation techniques are used in thedecryption process.

areo not.etween

The usual dilemma is ameliorated if we use Schmidtdecomposition in describing two-player joint strategies.The classical game can be reproduced from entangledstates when GreenbergerHorneZeilinger (GHZ) statesare used. A GHZ state is 12p j000i j111i

a city., and the

The worst case is removed in the quantum game.

enyer B

The ultimatum game can be quantized, and entanglementeffects cause interdependency between players.

t to It is possible to win the Vaidman's game. Vaidman's gameimplicitly assumed that 3-players shared a common3.2. Additional quantum mechanical aspects

The most common quantum mechanical feature exploredin quantum games is entanglement. Even within this topic,there are additional issues. For example, there are differenttypes of entanglement. Dur et al. [44] look at two differentrepresentations of entanglement (GreenbergerHorneZei-linger state and W state). The W state is more stable than theGHZ state since if one qubit is removed, the W state remainsentangled. This is used in the Minority quantum game byFlitney and Hollenberg [63] where they show the GHZ statesbecome very fragile as the number of players increases.Interestingly, Shimamura et al. [157] show that classicalgames can be reproduced fromquantum games if initial statesare N-product states, or entangled GHZ states.

A number of papers have looked at effects other thanentanglement that quantum mechanics brings to gametheory. As shown in Section 2 with the Penny Flip game,superposition can play a role. This is seen in Table 1 forexample in the Bargaining Game and the Number Game.Another quantum mechanical feature that plays a prominenttopic in quantum games is decoherence which can be viewedas introducing noise into a game. Some papers look at theimpact of decoherence on specic games (e.g., [19,21,56,61,

y.ber of 0in the caseked X

eference frame.

ions

yes then AA.

The Wise Alice game has the same payoff matrix as theFoolish Alice game showing that the payoff matrix doesn'ttell us about the rules of the game. However, the averagepayoff to both types of Alice is the same for a single game.Repeated games lead to differences. A non-distributiveproperty arises which is easily and naturally handled in aquantum version of the game.

327H. Guo et al. / Decision Support Systems 46 (2008) 31833263,133]) with many nding that noise reduces the quantumeffects of entanglement. Most such papers view decoherenceas a statistically independent stochastic process but evidencesuggest it is likely a correlated noise that can be suppressedusing Parrondo type games [115].

zdemir et al. [137] look at how corruption of the sourceinput qubits strongly and negatively inuences payoffs andthe number of NE if players are unaware of corruption removing advantages seen in quantum games. If they knowabout the corruption they can handle it.

Somemore esoteric aspects have been explored too. Hruby[82] uses ideas from supersymmetric quantum mechanics inexploring quantum games. This leads to another type ofgeneralization to unitary matrices discussed earlier in Section3.1. Grover's quantumdatabase search [74] is used to search forthe number in the Guess the Number game [71].

3.3. Theoretical properties of quantum games and extensions

Boukas [9] provides a general Minimax result assumingplayers are quantum entities. Lee [110,113] shows all classicalgames are subset of quantum games and that the Minimaxtheorem for zero-sum games and NE for general static gamesholds by assuming that the quantum feature in games is not theplayers but the strategies that players have. This assumption iswidely used.

Landsburg [108] considers computing NE when the playerscan communicate with the referee, thus effectively expandingtheir strategy sets. In [107], he argues that a refereewould neverbe able to distinguish a player's mixed-strategy from a purestrategy in a single game. Others also nd that the arbiter iscritical in computing NE. Cheon et al. [25] see that an arbiter canprovideentanglementbetween the two strategies chosenby theplayers, and Ichikawa et al. [85] show that the arbiter furnishes acorrelation. However, Iqbal et al. [94] note that the referee canremove any motivation for coalitions in cooperative games.

Mixed strategies are studied in quantumgames. Landsburg[107] formalizes mixed strategies in general games. Stohlerand Fischbach [161] show two ways to study mixed quantumstrategies, either by using classical probabilities or a singlematrix that combines classical probabilities and operators.Even though in some cases nopure-strategyequilibriumexistsin games with general strategy spaces, mixed-strategyequilibria may exist, as shown by Eisert et al. [47].

Some studies study how to quantizemultiplayer games andtogeneralize two-playerquantumgames intonplayerquantumgames. For example, Du et al. [40,42] extend the Prisoner'sDilemma game to an n player prisoner's dilemma game. Wu[173,175] presents a different view on how to representmultiplayer games. More specically, a base vector set of theHilbert space that is a direct product of each player's strategyvector base set is used to represent multiplayer games. Theproblem that the number of parameters and equationsincreases as the number of players expands is resolved. This isvery helpful in solvingnplayers game. In entangledmultiplayergames, Flitney [63] gives an interesting result that as thenumber of players increases, the entanglement of the gamedecreases. This diminishes the added contribution quantizationbrings over their classic counterpart.

When quantum games are allowed to repeat, resultsdifferent from the single game may appear. In the Wise andFoolish Alice game, Grib and Paronov [73] nd that theexpected payoff of Alice is higher than that of her opponent inrepeated games even though they have the same expectedpayoffs in one round. Iqbal [98] show that different resultsemerge between the repeated quantum Prisoner's Dilemmagame and the single round version. For example, if the game isrepeated twice, players in the game cooperate instead ofdefect.

Cooperation in games has also been explored. Some papers[33,39,40] show that entanglement improves the coordinationbetweenplayers and thus payoffs (if thepayoff functiondependsonplayers' types.) The quantumcoordinationproblem is studiedin a number of papers [83,88,96,100,102,105]. Iqbal et al. [96]focus on the correlations of outcomes and nd that NE changebased on correlation and quantum games are not reproduciblewith classic games. Witte [171] shows correlations betweenpayoffs yields neutral or winning/losing strategies. Flitney andGreentree [62] compare coalitions using entanglement versusclassic communication, and nd that as the coalition sizeincreases, classic communication is better when measurementis done in an arbitrary basis.

Evolutionary game theory originated from the combina-tion of evolutionary biology and game theory with a focus onplayers' strategy dynamics. The associated equilibrium con-cept is evolutionary stable strategy (ESS) dened as a strategythat cannot be invaded by mutant strategies once xed in thepopulation. Evolutionary quantum games were introducedthrough papers like [65,66,125]. Iqbal et al. [9193,95,97,99]show that, under a symmetric information condition, classicalgames can evolve to quantum games while NE remainunchanged, and that a non-ESS attractor of replicatordynamics can be changed into an ESS or, conversely, when aclassic game is changed to a quantum game. However,entanglement in asymmetric or symmetric games can disturbESS equilibria. Kay et al. [103] concludes that a quantumevolutionary game dominates a classic game if time steps arelarge, but the quantum evolutionary game is not better withsmall time intervals. Application of ESS in nancial markets isstudied by Gonalves and Gonalves [70].

3.4. Information and communication theory

Quantum information theory [135,104] deals with infor-mation theory that relies on quantum effects. These problemsoften arise in communication applications including suchtopics as error correction, coding, teleportation and others.Iqbal [90] states that the theory of quantum communicationcan be viewed as quantum games. Although most researchersin quantum information don't use game-theoretic paradigms,it is clear that this view offers many unique perspectives. Weonly touch on quantum games in this area.

Fitzi and Gisin [55] show that a quantum solution to theByzantine Agreement problem exists with the help ofentangled qudits. Eisert et al. [46] discuss an entanglement-assisted local operator in classical communication (ELOCC)and shows that ELOCC is greater than LOCC (local operator inclassical communication.)

Lee et al. [111] tighten the link between quantum gametheory and quantum information. They address two technicalissues that arise in quantum information areas by studyingquantum games that capture the phenomena. Werner [168]

Table 2Econophysics games

Researchstream

Type Description Contribution

Competitivemodels

Cournot [22,35,117] There is a quantity competition between duopolyrms who produce a homogeneous product. Firmscompete in quantities and choose quantitiessimultaneously.

Entanglement can increase payoffs by inducingcooperation. Information asymmetry underentanglement impacts payoffs for both players.The effect of entanglement becomes complicatedwhen information is uncertain. here quantumgames do not perform better than classical ones.

Bertrand [121,156] There is a price competition between duopolyrms who produce a homogeneous product.Firms compete in prices and choose pricessimultaneously.

Prot is a monotonic function of entanglementdegree.

Stackelberg[94,120,122]

In a Stackelberg game, a leader moves rstand followers second. The leader becomes betteroff at the expense of the followers.

In a Stackelberg duopoly game with completeinformation, the rst move advantage is a monotonicfunction of entanglement degree. Positive entanglementmeasurement enhances the advantage, while negativedegree decreases the benet. In a Stackelberg duopolygame with incomplete information, entanglement helpsthe rst mover advantage. Negative entanglement hurts.The uncertainty of information for the rst player hurtshis rst move advantage.

Oligopolies [33,119] Oligopoly refers to a market form within whicha small number of sellers dominate the market.

Entanglement can be used to study oligopolies in aquantum setting. A measure of informationincorrectness can be varied to regulate bad aspects.

Financialmodels

Financial markets[140,141,143,148,150153,159]

There are changing and unlimited number ofplayers whose payoffs are non-constant. Eachtrader possesses asset B and money $. ArbiterA considers traders' strategies, assets and moneyto decide the output.

Quantum market models can explain sudden largeprice changes (quantum zeno effect), undividity ofattention of traders, etc.

Financial markets asa minority game[14,15,50]

Financial markets can be modeled as minoritygames with minor modication.

If players in a market act as price takers, the minoritygame reproduces stylized facts of nancial marketssuch as fat tailed distribution of returns and volatilityclustering.

Financial markets asan evolutionaryquantum game [70]

Financial markets can be modeled as evolutionaryquantum games.

Financial markets can be modeled as evolutionaryquantum games to explain multifractal behavior.

Treasury bonds [3] A treasury bond is a nancial instrument forwhich there is no risk of default in receiving thepayments. The forward rate of a treasury bondrepresents the spot interest rate at a future timefor a contract entered at a previous time which isa stochastic variable.

Path integral approaches are used to model forwardrates of treasury bonds. For forward rates, the entireyield curve impacts a bond's price.

Principalagent

Production [31,51,52] The principal is risk neutral and the risk averseagent can be either a worker or a shirker. If theagent is a worker, then the outcome would bea success with probability p and a failure withprobability 1p; If the agent is a shirker, thenthe outcome would be a failure for sure.

The agency model is analyzed under no synergy, puresynergy and partial synergy. Nature's productionmodels synergy as an entangled state. With maxsynergy, an aggregate (group) measure is moreeffective. With partial synergy, an aggregate measuremay be more or less effective depending on otherfactors.

Public goods N-player free ridingproblem [20]

An n-player public goods game involves anorganization that decides whether to provide acommon good which can be consumed by n players.Some players may free ride on the efforts of otherplayers.

By using entangled two particles, players achievebetter (near optimal) expected payoff than in theclassical problem when they play mixed strategies.

Auctions General [76,81] An auction is a mechanism of bidding anddeciding the winning bidder(s) and payment rule.

Quantum auctions work better than classical oneswith respect to payoffs, privacy, and other aspects.Privacy is not necessarily improved in a quantumauction. A dishonest auctioneer can violate theprivacy guarantees but bidders can counter this.(See also [146]).

English auction[18,149]

An English auction is an open-bid auction.The auctioneer announces a reserve price(the lowest acceptable price) and then thecompeting buyers increase their bids. The highestbidder wins and pays the highest bid.

Experiments with humans [18] show that peopletend to overbid compared to NE even in classicsettings. Lower revenue and allocative efcienciesresulted probably because of the many no win cases.

Asset valuationmodels

Binomial (CRR)market [16]

There is a stock with a current price S0 and an optionon the stock with a current price C. The stock pricecan either go up or down. Assumes no arbitrage.

The classic binomial market model results in aparadox the option price does not depend on theprobabilities of the stock price moving up and down.Quantization of the binomial market model resolvesthe paradox.

328 H. Guo et al. / Decision Support Systems 46 (2008) 318332

329H. Guo et al. / Decision Support Systems 46 (2008) 318332also argues that quantum games are a good way to viewquantum information.

Goldenberg et al. [69] offer a secure communicationscheme using a quantum game. This was extended byHwang et al. [84]. Bennett et al [8] show that while encodingtwo bits and then transmitting them as a single particle of twostates, players can non-locally communicate with each otherand a player can know the other player's operation. La Mura[105] concludes that quantum signals are better than classicalones in classic games.

3.5. Quantum econophysics

The application of ideas in physics to economics [50] is oftencalled econophysics. Some have argued that mathematicaleconomics and quantum mechanics are isomorphic [80,106].With the advent of quantum game theory, a particularlyinteresting stream of research explores quantum game theoryapplications in economic areas. Table 2 gives a summary ofarticles organized along several economic themes. We brieydiscuss each theme and some quantum results.

Three of the most common competitive models (Cournot,Bertrand, and Stackelberg) are quantized in the context ofboth duopoly and oligopoly. Quantum nancial models(minority game and evolutionary game) are proposed todepict nancial markets and nancial instruments. Thesequantum nancial models can explain special features such asquantum zeno effect, undividity of attention of traders, fattailed distribution of returns, volatility clustering, multifractalbehavior, etc. Quantum public goods games are more efcientin terms of players' expected payoff compared to classicresults. Quantum auctions sometimes outperform classicauctions with respect to payoffs, privacy, and other aspects.Concepts and techniques from quantum game theory are alsoapplied to principalagent theory. The effectiveness ofdifferent measures depends on the synergy of agents.Quantization of a simple asset valuationmodel the binomialmodel resolves the classic paradox.

4. Concluding remarks and future research

Although much has been accomplished in this nascenteld, much remains to be explored. None of the themesdiscussed in Section 3 are mature leaving plenty of room forfuture work. Some streams are likely to be extended byspecialists in quantum phenomena, like the generalizationsinvolving decoherence. However, formulating and studyingapplications of quantum games within the decision sciencesis an important area for future research.

Many papers in quantum games involve quantization ofclassic games as presented in Table 1. However, there is anoticeable concentration on static games and dynamic gameswith complete information. Numerous categories of classicgames still remain unexplored. The only quantized gamesstudied with incomplete information are continuous gamessuch as Cournot duopoly, Stackelberg duopoly and Bertrandduopoly. There is no discussion of quantum games with perfectinformation versus imperfect information. Quantization ofclassic games and comparisons between quantum and classiccounterparts are of special importance because they represent anatural start to explore the power of quantum games.Many decision science problems can be viewed as principalagent problems, including learning involving strategic agents[10]. Only one principleagent model has been discussed in aquantum form [31,51,52] leaving much to be explored in futureresearch. Trust in eCommerce (e.g., [126]) has received con-siderable attention in recent years. Ar [2] used the Stag Huntand Prisoner's Dilemma games to illustrate how trust can beconstructed by a quantum approach. This avenue may proveuseful in eCommerce especially since communication systemsare inherently quantum mechanical. Related ideas of cheating[154] and reputation [176]might alsobenet fromsuch research.

Although several papers address the gap between quantummechanical level phenomena and the macro level mostactivities take place in, more work is needed here. Aninteresting contribution by Deutsch [32] shows how quantumformulas relating amplitudes to probabilities can be derivedfrom other axioms of quantum mechanics. He does thisassuming a rational economic agent, who believes everythingabout quantum mechanics but does not assume the probabil-istic postulates, makes decisions. The result is the usual zero-sum game-theoretic setting.

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Hong Guo is a Ph.D. student in the Department of Information Systems andOperations Management in the Warrington School of Business at theUniversity of Florida. Her interests are in economics of information systems,computer-mediated social networks, quantum computing and quantumgames.

Juheng Zhang is a Ph.D. student in the Department of Information Systemsand Operations Management in the Warrington School of Business at theUniversity of Florida. Her interests are in quantum computing and quantumgames, real options, and economics of information systems.

Gary J. Koehler is the John B. Higdon Eminent Scholar of ManagementInformation Systems at the University of Florida. He received his Ph. D. fromPurdue University in 1974. He has held academic positions at NorthwesternUniversity and Purdue University and between 19791987 was a cofounderand CEO of a high-tech company which grew to over 260 employees duringthat period. His research interests are in areas formed by the intersection ofthe Operations Research, Articial Intelligence and Information Systemsareas and include such areas as genetic algorithm theory, machine learning,e-commerce, quantum computing and decision support systems. He haspublished in journals including Management Science, Operations Research,Informs Journal on Computing, Evolutionary Computation, Decision Sciences,Decision Support Systems, the European Journal on Operational Research, theJournal of Management Information Systems, Information Systems and e-Business Management, SIAM Journal on Control and Optimization, DiscreteApplied Mathematics, Journal of Finance, and others. He is an area editor forDecision Support System and is on several other editorial boards. He hasserved as an expert witness for many large rms (including AT&T), has beenan External Examiner for several Universities and has worked under grantsfrom IBM and the National Science Fo