Theory of Quantum Games and Quantum Economic Behavior

Shoto Aoki and Kazuki Ikeda Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan Research Institute for Advanced Materials and Devices, Kyocera Corporation, Soraku, Kyoto 619-0237, Japan

E-mail: [email protected], [email protected]

Abstract: The quest of this work is to present a discussion of some fundamental questions of in the era of , which require a treatment different from economics studied thus far in the literature. A study of quantum economic behavior will become the center of attention of in the coming decades. We analyze a quantum economy in which players produce and consume quantum . They meet randomly and barter with neighbors bilaterally for quantum goods they produced. We clarify the conditions where certain quantum goods emerge endogenously as media of exchange, called quantum commodity . As quantum strategies are entangled, we find distinctive aspects of quantum games that cannot be explained by conventional classical games. In some situations a quantum player can acquire a quantum good from people regardless of their strategies, while on the other hand people can find quantum strategies that improve their welfare based on an agreement. Those novel properties imply that quantum games also shed new light on theories of , auction and contract in the quantum era. arXiv:2010.14098v1 [quant-ph] 27 Oct 2020 Contents

1 Introduction 1

2 Quantum Money as a Medium of Quantum Exchange2 2.1 Quantum Commodity Money2 2.2 for Consensus Building3 2.3 Entangled Quantum Goods, Strategies and Quantum Money as a Medium of Exchange5 2.4 Emergence of Commodity Money from Quantum Exchange 11 2.5 Emergence of Coalition and Quantum Effects on Deals 13

3 Conclusion and Discussion 20

1 Introduction

Recent progress in quantum technologies has been stood us in good stead to acquire abilities to control quantum systems for various usages. It runs the gamut from [1,2] to communication with [3–7]. Those quantum technologies are bound to have a significant impact on economy in the literature. In particular the appearance of programmable quantum computers [8,9] heralds potential growth in demand for goods or services available via the quantum internet [10], by interconnecting quantum computers with the between nodes [11–14]. With quantum communication devises and networks, economic activities will be buoyed in conformity with the laws of quantum physics. Consequently they promise the rise of Quantum Economics, which is a branch of economics in the era of noisy intermediatescale quantum (NISQ) technologies [15] and full-fledged quantum technologies in future [16, 17]. Quantum is a theoretical study on efficient and rational ways of programming quantum devises. It will accrue to multifarious activities of humanbeings. In this article we aim at providing a theoretical study on the quantum economical behavior in a newly emerging and communication technology (QICT) . To our best knowledge this is the first paper that expounds on economic activities from a viewpoint of quantum games. Indeed, mathematical or physical aspects of quantum games, such as the role of entanglement or quantum strategies for mathematical toy models, have been investigated in many works [18–20]. However, less is known from an economic perspective that leads to a meaningful analysis of a quantum market doctrine and of economical behavior in the market. Therefore a critical milestone in quantum games that shows a significant difference from the traditional games has not been reached. This unfortunate lack of impact on economics casts doubt on the power of quantum computational protocols for games. The major criticisms toward the present situation of quantum games are summarized in [21, 22]. In this article we tackle this challenge and succeed in showing novel key differentiating features of a quantum game model. The classical game theory today [23] plays a crucial role in modern economics [24] and based on its history we can foresee prosperous days of quantum games as the foundation of Quantum Economics. We study economies where quantum commodities that serve as media of exchange are determined endoge- nously. In the modern economy, almost all products are exchanged for money, which is defined as an object with functions of a , a and a store of . In the long period of history of money, it was only as late as the second half of the 20 century that the modern fiat money system set by command of

– 1 – the was established. Precious metals had been widely used as a commodity money till the U.S. gov- ernment abandoned the convertibility of the dollar into gold. A physical form of money also keeps changing. The recent advances in technologies popularize the use of digital money as an official alternative of legal tender. More recently cryptoassets and cryptocurrencies [25] exude presence in various and sundry fields [26–28]. Therefore it is meaningful to investigate the possibility of a system and a form of money in the era of quantum technologies. The concepts of quantum money are proposed multifariously [29–32], however, whether quantum objects can be media of exchange is yet another question. Quantum money is a naive quantum extension of classical money, but there are no formal modelings that show the medium of exchange function as an equilibrium of a quantum game. The main purpose of this article is to analyze a simple quantum model, where quantum objects which become media of exchange are pursed as a theory of non-cooperative repeated quantum game [33, 34]. The medium of exchange function is a crucial property of money. The historical evolution of money has long been studied by many authors. [35] listed the functions of money for hoarding and [36] described that the idea to overcome the double [37] leads to the use of money as a medium of exchange. The formal modeling of equilibria with classical commodity money and classical fiat money is known as the Kiyotaki-Wright model [38], which explores the economic foundation of the emergence of money as a perishable medium of exchange. The model is tested numerically [39–42] and a mixed strategy case is also proposed [43]. Kiyotaki and Wright initially addressed infinitely durable goods and later their model was extended to goods with a different durability [44, 45]. In this article we purse the conditions and welfare properties of quantum objects in order to clarify how using quantum media of exchange can emerge in equilibrium. The main achievement in Sec.2.4 is Theorem 2.2, which clarifies a mechanism of the emergence of money as a medium of exchange. We instantiate some strategies and equilibria in Sec.2.3. In Sec.2.5 we investigate more details of entangled quantum strategies. A type of quantum goods which can be a medium depends on strategies. Some concrete examples are summarized in Table2. It is notable that among strategies allow Bob to obtain his consuming good from Alice regardless of her inclination. This is a distinctive aspect of our quantum economy and not found in classical economy. In other words, those characteristic properties of quantum games incentivize us to investigate theories of mechanism design, auction and contract from a perspective of quantum information theory. We assume all quantum goods are infinitely durable by means of various quantum information processing technologies, such as [46–49], fault-tolerant quantum computation [50–52] and robust quantum memories [53, 54]. As a generic has finite lifetime, it will be also interesting to consider quantum goods with a different durability.

2 Quantum Money as a Medium of Quantum Exchange

2.1 Quantum Commodity Money A quantum object is called a quantum commodity money if it is accepted in trade not to be consumed or used in production, but to be used to facilitate further trade. If a quantum object with no intrinsic value becomes a medium of exchange, it is called quantum fiat money. In this article we discuss how a quantum commodity can be a medium of exchange. As a simple model, we consider a quantum information system in which there are three commodities called goods 1,2 and 3. Quantum agents in the community are classified into one of three types: type i quantum agent can produce good j(= i) and receive positive only from consumption of good i. In model A, type 6 i = 1, 2, 3 agents produce goods j = 2, 3, 1, respectively, while in model B type i = 1, 2, 3 produce good j = 3, 1, 2, respectively. At every round, agents who are allowed to store one unit of any good need to decide whether they carry over their goods to the next round or exchange. In each period, agents are matched randomly (with the probability 1/3) in pairs and must decide whether or not to trade bilaterally. When type i gets good i, he or she immediately consumes it, produces a new unit of good, and stores it until the next date. Storing a good

– 2 – is costly, and an agent incurs storage costs cj for holding good j at the end of every round. When an agent of type i successfully for good i, then he immediately consumes that good and produces a new unit of his production good j without cost. All agents receive the same profit u(> ck k) from consuming goods. Therefore ∀ the total classical profit πi of an agent of type i is given by

t X τ πi(t) = (fi(τ)u pij(τ)cij)δ , (2.1) τ=0 − where δ (0, 1) is a discount factor, pij(t) is i’s probability to have a good j , fi(t) is i’s probability to get and ∈ | i consume i at date t, and cij is the cost to type i of storing good j. Note that fi(t) depends on strategies and pij(t 1). − Agents try to maximize their total profit. In both models, there are two equilibria: fundamental equilibrium and speculative equilibrium. One equilibrium is referred to as a fundamental equilibrium, which is dominant in model A, since agents always prefer a lower-storage-cost commodity to a higher-storage-cost commodity unless the latter is their own consumption good. On the other hand, in speculative equilibrium, which is dominant in model B, agents trade a lower for a higher-storage-cost commodity not because they wish to consume it, but because they rationally expect that this is the best way to ultimately trade for another good that they do want to consume, that is, because it is more marketable. In this article, we address the type A model and assume there are only three persons Alice, Bob and Charlie who represent type 1,2 and 3 agents respectively. So, Alice creates a type 2 quantum good 2 and consumes a | i type 1 quantum good 1 . Bob creates a type 3 quantum good 3 and consumes a type 2 quantum good 2 . | i | i | i Charlie creates a type 1 quantum good 1 and consumes a type 3 quantum good 3 . For example, Alice gives | i | i Bob a type 2 or 3 quantum good to get a type 1 quantum good (Fig.1).

Alice

2,3 1,3 2,3 1,2

1,2 Bob Charlie 1,3

Figure 1: The diagram of trades.

2.2 Quantum Circuit for Consensus Building As the most fundamental case of consensus building, we first describe a trade between two persons. To clarify our idea, we exhibit a quantum circuit (Fig.3) in which Alice and Bob exchange her 0 and his 1 . They operate | i | i their unitary matrices to their ancilla 0 and 0 in such a way that | iA | iB

UA 0 A = a0 0 A + a1 1 A =: τA(0, 1) | i | i | i | i (2.2) UB 0 = b0 0 + b1 1 =: τB(1, 0) , | iB | iB | iB | i 2 2 where we interpret a1 ( b1 ) as Alice’s (Bob’s) probability of willing to trade. Their deal can be made if and | | | | only if 1 1 is observed in the ancilla state. | iA | iB To draw a complete quantum circuit explicitly, we express i i U + i 1 i 1 I by the diagram | i h | ⊗ | ⊕ i h ⊕ | ⊗ (Fig.2). Using this gate, we present a quantum circuit Fig.3 by which Alice and Bob exchange her 0 and his 1 . | i | i

– 3 – i

U

Figure 2: Quantum gate for the operation i i U + i 1 i 1 I. The symbol means 0 0 = 1 1 = | i h | ⊗ | ⊕ i h ⊕ | ⊗ ⊕ ⊕ ⊕ 0, 0 1 = 1 0 = 0. ⊕ ⊕ 0 ψ | A⟩ K 1 ψ | B⟩ 1 0 U | ⟩A A † J J 1 0 U | ⟩B B

Figure 3: Quantum circuit in which Alice and Bob exchange her 0 and his 1 , when they have. creates | i | i K entanglement between states ψA and ψB , and yields entanglement between quantum strategies UA and UB. † | i | i J is operated so the game recovers classical results when UA and UB are classical strategies. The swap operator J acts on the first two qubits if two agents agree to trade.

We first consider the case where quantum strategies are not entangled ( = † = I). Generic states which J J Alice and Bob have can be written as ψA = A0 0 + A1 1 , ψB = B0 0 + B1 1 . If a game starts with a | i | i | i | i | i | i general entangled state ψ = ψAψB A B between Alice and Bob | i K | i ∈ H ⊗ H X X † ψ = Kij ij , KijKij = 1, (2.3) | i i,j∈{0,1} | i ij the initial state ψ 0 0 eventually becomes | i | iA | iB

K01a1b1 10 1 1 + , (2.4) | i | iA | iB ··· 2 where terms which do not contribute to trade are omitted. So the success probability of the trade is K01a1b1 . | | Now we consider a trade among the three persons. In a classical system, they have an indivisible quantum commodity called goods 1,2 or 3. Two of the three meet and decide whether to exchange their commodities. If both two persons want to trade, they exchange their commodities. We write each i in the binary form | i 1 = 01 , 2 = 10 , 3 = 11 . (2.5) | i | i | i | i | i | i

Suppose each of the three, Alice, Bob and Charlie has a quantum commodity ψi = i1 1 + i2 2 + i3 3 (i = | i | i | i | i A, B, C). Two of them meet and trade pure goods state, as in the classical system. We assume a person meets another with the equal probability 1/3 and introduce W = √1 ( 110 + 101 + 011 ) to describe a person who | i 3 | i | i | i someone meets. Here 110 means that Alice meets Bob, 101 says that Alice meets Charlie, and 011 indicates | i | i | i that Bob meets Charlie. We illustrate the quantum circuit for a three-person trade in Fig.4. The introduction of W is one of the most significant differences from the two-person trade diagram (Fig.3). For example, Alice’s | i ⊗4 2 ⊗2 ⊗2 3 i strategy is a sum of tensor products UA = 1 1 I + 2 2 U I + 3 3 I U , where U is her | i h | ⊗ | i h | ⊗ A ⊗ | i h | ⊗ ⊗ A A strategy for trading a good i when she has i . | i

– 4 – Figure 4: Quantum circuit for exchanging quantum states among three persons. Alice, Bob and Charlie exchange 1 , 2 and 3 , when they have. creates entanglement among states ψA , ψB and ψC . yields entanglement | i | i | i K | i | i | i J among quantum strategies UA, UB and UC . Pij is a gate for consuming i and producing j . | i | i

2.3 Entangled Quantum Goods, Strategies and Quantum Money as a Medium of Exchange We first address the case where quantum goods are entangled ( = I⊗3) and strategies are not entangled K 6 ( = I⊗12). Fig.5 presents the diagram of exchange. J

Figure 5: A sequence of numbers in a circle means a state: 231 means Alice, Bob, and Charlie have 2 , 3 | i | i and 1 . A 3-tuple of numbers outside a circle indicates that of the number of persons who have the consuming | i quantum commodities. When 211 is shown in a circle, (2,1,0) expresses that two persons have Alice’s consuming C commodity 1 , only Alice has Bob’s consuming good 2, and no one has the state Charlie desires. 231 211 | i −→ means that it is Charlie who gains a reward by an exchange. From the state 231 to 211 , Charlie receives 3 | i | i | i from Bob, consumes it and produces 1 . | i

– 5 – It is convenient to consider a ρ that describes entangled goods, instead of ψAψBψC . With | i respect to a given initial state ψAψBψC , we define ρ by | i † ρ = ψAψBψC ψAψBψC End( Commodity) (2.6) KX | i h | K ∈ H = Cabc,ijk abc ijk , (2.7) abc,ijk | i h | where Cabc,ijk is a coefficient. In order to discuss an equilibrium of their business, we define an operator acting on the space of quantum commodities T : End( Commodity) End( Commodity) H → H † T (ρ) = TrAncillaE (ρ Ancilla Ancilla ) E (2.8) ⊗ | i h | Ancilla = W 0000 0000 0000 0 0 0 (2.9) | i | i | iA | iB | iC | iA | iB | iC where E : Commodity Ancilla Commodity Ancilla represents all single-round quantum gate operations in H ⊗ H → H ⊗ H Fig.4. We trace out ancilla qubits after all operations are made in a single round, in order to see a transition of quantum commodities. Definition 2.1. we call ρ a steady state if it obeys

T ρ = ρ. (2.10)

If a steady state ρ is used for an initial state, the total profit πi (2.1) converges into

Vi(ρ) lim πi(t) = , (2.11) t→∞ 1 δ − where Vi is a certain function of ρ. To give a concrete detail, we first derive Alice’s VA. Let ρ be a quantum state 1 computed on the basis 211 , 212 , 231 , 232 , 311 , 312 , 331 , 332 . Let vi be the storage cost that i {| i | i | i | i | i | i | i | i} should pay and wi be the possibility that i receive i. Each vi can be expressed as

vA = (c2, c2, c2, c2, c3, c3, c3, c3) vB = (c1, c1, c3, c3, c1, c1, c3, c3) vC = (c1, c2, c1, c2, c1, c2, c1, c2) (2.12)

j and wi depends on and U . Using vi and wi, we find fi(t)(2.1) becomes J i t−1 fi(t) = wi T (ρ) · X t (2.13) cijpij(t) = vi T (ρ), j · where we denote by the inner product of vectors. We may simply put T −1(ρ) = 0,T 0(ρ) = ρ. Then the total · profit of a player i is ∞ X  τ−1 τ  τ lim πi(t) = uwi T (ρ) v·T (ρ) δ t→∞ τ=0 · − ∞ (2.14) X τ τ = ( vi + δuwi) T (ρ)δ . τ=0 − ·

If ρ is a steady state T ρ = ρ, we find Vi = ( vi + δuwi) ρ from the equation − · ∞ X τ lim πi(t) = ( vi + δuwi) ρδ t→∞ − · τ=0 (2.15) ( vi + δuwi) ρ = − · 1 δ − 1Here we identify |ψi hψ| with |ψi.

– 6 – Note that Vi dose not explicitly depend on vj and wj (j = i), but it is related with them via ρ. 6 Before we describe a generic equilibrium and the flow of entangled quantum goods, let us start with an example which includes a review on the original Kiyotaki-Wright model [38, 43]. More general case is given in Sec.2.4, where Theorem 2.2 summarizes our main result here. In what follows we assume Alice,Bob and Charlie prefer the following strategies (2.16)

2   3   U = Y √sAI + i√1 sAY U = Y √1 sAI + i√sAY A ⊗ − A ⊗ − 1   3   U = Y √sBI + i√1 sBY U = √1 sBI + i√sBY Y (2.16) B ⊗ − B − ⊗ 1   2   U = √sC I + i√1 sC Y YU = √1 sC I + i√sC Y Y C − ⊗ C − ⊗

Indeed the classical results [38, 43] are reproduced when parameters si (i = A, B, C) are chosen appropriately. As the quantum game must reproduce the classical results, we generally require

[ ,U i] = 0 (2.17) J j when the strategies are restricted to classical strategies. For this purpose, we use = exp iθY ⊗12 throughout J this article. It will be yet another interesting question to study our model when a different is used. The J strategy is called fundamental when si is limited to 1 and speculative when si = 0 [43]. Then we identify the concrete form of wi as follows 1 wA = (2 sC , 1, 1 sC , 0, 2 sB, 1 sB, 1, 0) 3 − − − − 1 wB = (1, 1 + sC , 1 sA, 2 sA, 0, sC , 0, 1) (2.18) 3 − − 1 wC = (0, 0, sB, 1, 1, sA, 1 + sB, 1 + sA). 3

For example, the first component of wA can be derived from 211 as follows. Alice and Bob exchange 2 1 | i | i ↔ | i with the probability 1 when they meet and, Alice and Charlie exchange 2 1 with the probability 1 sC | i ↔ | i − when they meet. Therefore Alice obtains 1 with the probability 2−sC on 211 . | i 3 | i One can find that there is a unique steady state for each of sA = sB = sC = 1 (the fundamental strategy) and sA = 0, sB = sC = 1 (the speculative strategy). Those results are perfectly consistent with the Kiyotaki-Write case [38]. The steady state at sA = sB = sC = 1 is 1 1 ρ = 211 211 + 231 231 , (2.19) 2 | i h | 2 | i h | 2 and the steady state at sA = 0, sB = sC = 1 is 3 2 1 1 ρ = 211 211 + 231 231 + 311 311 + 331 331 . (2.20) 7 | i h | 7 | i h | 7 | i h | 7 | i h |

Moreover there is additional equilibrium at sA = sB = 1, sC = 0, in which 1 and 2 can be media of | i | i exchange. The corresponding steady state is 1 1 2 3 ρ = 211 211 + 212 212 + 231 231 + 232 232 . (2.21) 7 | i h | 7 | i h | 7 | i h | 7 | i h | Those three equilibira are probabilistic mixture of pure states, which are mixed state without entanglement [55]. The flow of goods are exhibited in Fig.6. √ √ 2  2  In [38], the steady state of speculative equilibrium is characterized by (p12, p23, p31) = 2 , 2 − 1, 1 = (0.707 ··· , 0.414 ··· , 1). 5 3  In our setup, they are (p12, p23, p31) = 7 , 7 , 1 = (0.714, 0.428, 1).

– 7 – Alice Alice Alice

2 1 2 1,3 3 1 2 1 2 1

1 1 1,2 Bob Charlie Bob Charlie Bob Charlie 3 3 3

(a) sA = 1, sB = 1, sC = 1 (b) sA = 0, sB = 1, sC = 1 (c) sA = 1, sB = 1, sC = 0

Figure 6: The flow of goods. (a) 1 is a medium of exchange in the fundamental equilibrium. (b) 1 and 3 | i | i | i are media of exchange in the speculative equilibrium. (c) 1 and 2 are media of exchange in the equilibrium of | i | i (2.21)

It is instructive to observe a transition process of an entangled density matrix ρ by T . As an example we address ρ = 311 211 to help readers follow the computation (2.33). By exchange, consumption and production, | i h | the state 211 becomes | i 211 W 0000 0000 0000 0 0 0 | i | i | iA ⊗ | iB ⊗ | iC ⊗ | iA | iB | iC 1 231 110 1 (√sA 0 √1 sA 1 ) 00 1 (√sB 0 √1 sB 1 ) 00 0000 1 1 0 → − | i √3 | i | iA | iA − − | iA | iA ⊗ | iB | iB − − | iB | iB ⊗ | iC | iA | iB | iC 1 211 101 1 (√sA 0 √1 sA 1 ) 00 0000 √sC 0 100 0 0 0 − | i √3 | i | iA | iA − − | iA | iA ⊗ | iB ⊗ | iC | iC | iA | iB | iC 1 + 212 101 1 (√sA 0 √1 sA 1 ) 00 0000 √1 sC 1 100 1 0 0 | i √3 | i | iA | iA − − | iA | iA ⊗ | iB ⊗ − | iC | iC | iA | iB | iC 1 211 011 0000 1 (√sB 0 √1 sB 1 ) 00 (√sC 0 √1 sC 1 ) 100 0 0 0 − | i √3 | i | iA ⊗ | iB | iB − − | iB | iB ⊗ | iC − − | iC | iC | iA | iB | iC (2.22)

Similarly, the state 311 is mapped to | i 311 W 0000 0000 0000 0 0 0 | i | i | iA ⊗ | iB ⊗ | iC | iA | iB | iC 1 311 110 001 (√1 sA 0 √sA 1 ) √sB 1000 0000 0 0 0 → − | i √3 | i | iA − | iA − | iA ⊗ | iB ⊗ | iC | iA | iB | iC 1 + 231 110 001 (√1 sA 0 √sA 1 ) √1 sB 1100 0000 1 0 0 | i √3 | i | iA − | iA − | iA ⊗ − | iB ⊗ | iC ⊗ | iA | iB | iC 1 211 101 001 (√1 sA 0 √sA 1 ) 0000 (√sC 0 √1 sC 1 ) 100 1 0 1 − | i √3 | i | iA − | iA − | iA ⊗ | iB ⊗ | i − − | i | iC | iA | iB | iC 1 311 011 0000 1 (√sB 0 √1 sB 1 ) 00 (√sC 0 √1 sC 1 ) 100 0 0 0 − | i √3 | i | iA ⊗ | iB | iB − − | iB | iB ⊗ | iC − − | iC | iC | iA | iB | iC (2.23)

By tracing out the ancilla qubits, we obtain 1 T ( 311 211 ) = 311 211 (2.24) | i h | 3 | i h | and lim T t( 311 211 ) = 0. (2.25) t→∞ | i h | By repeating the same calculation, one can find (2.25) is true for any entangled state abc ijk ( abc = ijk ). | i h | | i 6 | i Therefore it is sufficient to consider the abc = ijk cases. Then the main components of the transition matrix | i | i

– 8 – T can be represented with basis 211 , 212 , 231 , 232 , 311 , 312 , 331 , 332 in such a way that {| i | i | i | i | i | i | i | i}

 1+sC sB 1 sA  3 0 3 0 3 3 0 0  1−sC 2−sC   0 0 0 0 0 0   3 3   1 sC sA+1−sB +sC 1 1−sB 1 sA   3 3 3 3 3 0 3 3   1 1−sC sA+1 1−sB   0 0 0 0  T =  3 3 3 3  . (2.26)  1+sB sB   0 0 0 0 3 0 3 0   2−sA+sB −sC   0 0 0 0 0 0 0   3   1−sA sC 2−sB 1   0 0 3 0 0 3 3 3  1−sA 2−sA 0 0 0 3 0 0 0 3

For example, 211 is mapped into 212 and 231 with the probability 1−sC and 1 , respectively. 312 is mapped | i | i | i 3 3 | i to itself by the T operation, so T6i = 0 unless i = 6. From the Perron-Frobenius theorem, the maximal eigenvalue of the non-negative matrix T is 1 and the components of the corresponding eigenvector, which may not be uniquely determined, is non-negative. Now let us discuss equilibria of quantum strategies. We assume sB = 1 because of the inequality VB(sA, 0, sC ) < VB(sA, 1, sC ) for all sA, sC [0, 1]. Let ρ0, ρ1 be steady states at sB = 0, 1, respectively. Bob keeps 3 if sB = 0. ∈ | i Therefore we obtain

vB ρ0 = c3 vB ρ1 (2.27) · ≥ · and

VB(sA, 1, sC ) VB(sA, 0, sC ) = vB (ρ1 ρ0) + δuwB (ρ1 ρ0) (2.28) − − · − · − = (c3 vB ρ1) + δuwB (ρ1 ρ0). (2.29) − · · − c3 vB ρ 0 and the second term is grater than 0 for sA, sC [0, 1] as Fig.7 shows. Therefore VB(sA, 1, sC ) > − · ≥ ∈ VB(sA, 0, sC ) is satisfied.

Figure 7: The (sA, sC )-dependence of wB (ρ1 ρ0), which is larger than 0 for any pair of sA, sC [0, 1]. · − ∈

An optimal pair of sA, sC depends on c2 c1, c3 c2 and δu. Equilibrium strategies of Alice and Charlie are − − shown in Fig.8. (sA, sC ) = (1, 1), (0, 1), (1, 0) are the equilibria in the red, orange and blue domains, respectively. In the white region, at least one of Alice and Charlie uses a mixed strategy si = 0, 1 for an equilibrium. Note 6

– 9 – that there might be multiple equilibria near a boundary. For example, as illustrated in Fig.9, there are three c2−c1 c3−c2 equilibria at a point uδ = 0.19, uδ = 0.23 near the boundary between orange and white.

c3 c2 uδ−

0.4

0.3

0.2

0.1

0 0.1 0.2 0.3 0.4 c2 c1 uδ−

Figure 8: Equilibrium strategies of Alice and Charlie when sB = 1.

c2−c1 c3−c2 Figure 9: Equilibrium strategies of Alice and Charlie at a point uδ = 0.19, uδ = 0.23 near the boundary between the red and the white domains in Fig.8. The blue line in the right figure corresponds to the optimal sA against sC and the red line indicates the optimal sC against sA. There are three crossing points. Charlie desires his best situation (sA, sC ) = (0, 1), which is not best for Alice. The right figure also suggests that sA = 1 is Alice’s optimal strategy if sC [0, 0.9]. ∈

– 10 – 2.4 Emergence of Commodity Money from Quantum Exchange In this section we generalized our discussion in Sec.2.3. In addition to entangled quantum goods, we also consider quantum strategies that do not always commute = exp iθY ⊗12, hence strategies of Alice, Bob and Charlie are J j j generally entangled. For strategies in Fig.4, we impose 00 U 00 = 11 U 11 = 0 (i = A,B,C, j 1, 2, 3 ) in h | i | i h | i | i ∈ { } order to secure the independence of strategies for different goods. The strategies consist of

U¯ 2 = U 2 I⊗2, U¯ 3 = I⊗2 U 3 A A ⊗ A ⊗ A U¯ 1 = U 1 I⊗2, U¯ 3 = I⊗2 U 3 (2.30) B B ⊗ B ⊗ B U¯ 1 = U 1 I⊗2, U¯ 2 = I⊗2 U 2 C C ⊗ C ⊗ C We first consider = I⊗12. With respect to a state abc of quantum goods, the strategies act on the ancillary J | i term as

1 a b ⊗4 W 0000 0000 0000 110 (U¯A U¯B I ) 0000 0000 0000 | i | iA | iB | iC →√3 | i ⊗ ⊗ | iA | iB | iC 1 a ⊗4 c + 101 (U¯A I U¯C ) 0000 0000 0000 (2.31) √3 | i ⊗ ⊗ | iA | iB | iC 1 ⊗4 b c + 011 (I U¯B U¯C ) 0000 0000 0000 √3 | i ⊗ ⊗ | iA | iB | iC By definition, each player has 4 qubits. If two persons meet, at least one of the first half two qubits or the second half two qubits becomes 1 . If a person meets no one, all of his/her 4 qubits are 0 . For example, if Alice | i | i 2 meet someone, one of her first two qubits is 1 . Similarly if she has 3 and meet someone, one of her second | i | i | i half two qubits becomes 1 . Therefore T ( abc ijk ) = 0 obeys if and only if one of the following conditions | i | i h | 6 satisfies b = j , c = k | i | i | i | i a = i , c = k (2.32) | i | i | i | i a = i , b = j | i | i | i | i In addition, if only one player has a different good, say a = i , b = j , c = k , non-zero terms are given | i 6 | i | i | i | i | i when Bob and Charlie meet α 1 α T ( abc ibc ) = abc ibc + − abc ibc , (2.33) | i h | 3 | i h | 3 | i h | where abc means the state after the exchange b c and α [0, 1] is the success probability of the exchange. | i | i ↔ | i ∈ 1 t As a general result, any entangled term becomes smaller with the factor , hence limt→∞ T ( abc ibc ) = 0 holds. 3 | i h | Now let us consider the case = exp iθY ⊗12. The ancillary term is mapped to J 1 † a b ⊗4 W 0000 0000 0000 110 (U¯A U¯B I ) 0000 0000 0000 | i | iA | iB | iC →√3 | i J ⊗ ⊗ J | iA | iB | iC 1 † a ⊗4 c + 101 (U¯A I U¯C ) 0000 0000 0000 (2.34) √3 | i J ⊗ ⊗ J | iA | iB | iC 1 † ⊗4 b c + 011 (I U¯B U¯C ) 0000 0000 0000 √3 | i J ⊗ ⊗ J | iA | iB | iC The terms sandwiched by †, can be decomposed into a sum of terms which commute or anti-commute with J J J †((commutative) + (anti-commutative)) = (commutative) (anti-commutative) 2. (2.35) J J − J

– 11 – The "(commutative)" part produces the same states of = I⊗12 and the "(anti-commutative)" part gives different J states which consist of the following terms: if a player meets someone, at least one of the first or second half two qubits is 0 and, if a player dose not meet anyone, all of his/her 4 qubits are 1 . Therefore T ( abc ijk ) = 0 | i | i | i h | 6 if and only if one of the conditions (2.32) is satisfied. Transition of states before tracing out the corresponding ancilla qubits is summarized in Table.1.

Goods Encounter 110 101 011 \ | i | i | i ?? 00 ?? 00 0000 ?? 00 0000 ?? 00 0000 ?? 00 00 ?? 211 | iA | iB | iC | iA | iB | iC | iA | iB | iC | i 11 11 1111 11 1111 11 1111 11 11 |∗ ∗ iA |∗ ∗ iB | iC |∗ ∗ iA | iB |∗ ∗ iC | iA |∗ ∗ iB | ∗ ∗iC ?? 00 ?? 00 0000 ?? 00 0000 00 ?? 0000 ?? 00 ?? 00 212 | iA | iB | iC | iA | iB | iC | iA | iB | iC | i 11 11 1111 11 1111 11 1111 11 11 |∗ ∗ iA |∗ ∗ iB | iC |∗ ∗ iA | iB | ∗ ∗iC | iA |∗ ∗ iB |∗ ∗ iC ?? 00 00 ?? 0000 ?? 00 0000 ?? 00 0000 00 ?? 00 ?? 231 | iA | iB | iC | iA | iB | iC | iA | iB | iC | i 11 11 1111 11 1111 11 1111 11 11 |∗ ∗ iA | ∗ ∗iB | iC |∗ ∗ iA | iB |∗ ∗ iC | iA | ∗ ∗iB | ∗ ∗iC ?? 00 00 ?? 0000 ?? 00 0000 00 ?? 0000 00 ?? ?? 00 232 | iA | iB | iC | iA | iB | iC | iA | iB | iC | i 11 11 1111 11 1111 11 1111 11 11 |∗ ∗ iA | ∗ ∗iB | iC |∗ ∗ iA | iB | ∗ ∗iC | iA | ∗ ∗iB |∗ ∗ iC 00 ?? ?? 00 0000 00 ?? 0000 ?? 00 0000 ?? 00 00 ?? 311 | iA | iB | iC | iA | iB | iC | iA | iB | iC | i 11 11 1111 11 1111 11 1111 11 11 | ∗ ∗iA |∗ ∗ iB | iC | ∗ ∗iA | iB |∗ ∗ iC | iA |∗ ∗ iB | ∗ ∗iC 00 ?? ?? 00 0000 00 ?? 0000 00 ?? 0000 ?? 00 ?? 00 312 | iA | iB | iC | iA | iB | iC | iA | iB | iC | i 11 11 1111 11 1111 11 1111 11 11 | ∗ ∗iA |∗ ∗ iB | iC | ∗ ∗iA | iB | ∗ ∗iC | iA |∗ ∗ iB |∗ ∗ iC 00 ?? 00 ?? 0000 00 ?? 0000 ?? 00 0000 00 ?? 00 ?? 331 | iA | iB | iC | iA | iB | iC | iA | iB | iC | i 11 11 1111 11 1111 11 1111 11 11 | ∗ ∗iA | ∗ ∗iB | iC | ∗ ∗iA | iB |∗ ∗ iC | iA | ∗ ∗iB | ∗ ∗iC 00 ?? 00 ?? 0000 00 ?? 0000 00 ?? 0000 00 ?? ?? 00 332 | iA | iB | iC | iA | iB | iC | iA | iB | iC | i 11 11 1111 11 1111 11 1111 11 11 | ∗ ∗iA | ∗ ∗iB | iC | ∗ ∗iA | iB | ∗ ∗iC | iA | ∗ ∗iB |∗ ∗ iC Table 1: Correspondence between quantum goods and the terms associated with 110 , 101 and 011 after | i | i | i exchange, consumption and creation. Terms contained in (commutative) are shown in the upper lows and terms contained in (anti-commutative) are shown in the lower and upper lows. ?? means at least one of the two qubits is 1 and means at least one of the two qubits is 0. For example, looking at (2.22) we find that 211 ∗∗ | i is mapped to 1000 1000 0000 , 1000 1100 0000 , 1100 1000 0000 , 1100 1100 0000 | iA | iB | iC | iA | i | iC | iA | iB | iC | iA | i | iC after all possible exchange, consumption and creation of goods that may happen when Alice and Bob meet ( 110 ). | i This situation is simply expressed as ?? 00 ?? 00 0000 . Moreover terms in a lower low appear if θ = 0. | iA | iB | iC 6 See (2.43) for example.

We give a more general argument which assures quantum states can be a medium of exchange. We find entangled stats and classical states have a different durability. Our main goal in this section is to show the following statement. j j Theorem 2.2. Suppose = exp iθY ⊗12 and 00 U 00 = 11 U 11 = 0. For any θ [0, 2π] and any J h | i | i h | i | i ∈ quantum commodity state ρ, there is a classical mixed state ρc such that t lim T ρ = ρc (2.36) t→∞ . Proof. We consider transition of a state abc ijk by T . Consulting Table.1, one can verify the following formula | i h | one-by-one for any abc = ijk | i 6 | i   1 abc ? ijk ? (a = i, b = j, c = k), (a = i, b = j, c = k), or (a = i, b = j, c = k) T ( abc ijk ) = 3 | i h | 6 6 6 (2.37) | i h | 0 otherwise

– 12 – where abc ∗ ijk ∗ is a certain normalized state. Important fact is that operating T to any entanglement term | i h | 1 t makes it smaller with the factor at most. Therefore we conclude limt→∞ T ( abc ijk ) = 0 for any abc = ijk . 3 | i h | | i 6 | i When we restrict ourselves to the cases of abc = ijk , then T ( ijk ijk ) is a non-negative matrix. Note | i | i | i h | that T is a stochastic matrix. If two persons have the same goods, which corresponds to a vertex state of the hexagon in Fig.5, the state is mapped to itself with the probability 1/3. Suppose all three persons have different goods. Based on their strategies, it turns out 312 is mapped to itself with the probability 1, in which case 312 | i | i is a fixed point, or it is mapped to another vertex state. If 312 is mapped to another state, then it never returns t | i to 312 . Therefore we find it vanishes limt→∞ T ( 312 312 ) = 0. Regarding 231 , it can be mapped to itself | i | i h | | i and other states can be mapped to 231 . Based on strategies, the 7 states or the 6 vertex states in the hexagon | i in Fig.5 form a closed, ergodic and reducible Markov chain. For such a Markov chain, we can find a family of closed, ergodic and irreducible subchains as follows. Up to similarity transformations, our transition matrix for a given Markov chain can be written as   T11 0  ∗   ..   .     0 T   1k     T21   ∗   ..   .  T =   (2.38)  T   0 2l     ..   .     T   N1   . ∗   ..    0 0 TNm

where Tij are transition matrices of an irreducible Markov subchain in a given reducible Markov chain and the block matrices in boxes are transition matrices of closed Markov subchains. In particular, Ti (i = 1, ,N) are 1 ··· transition matrices of closed irreducible and ergodic Markov subchains. According to the convergence theorem for

Markov chains, each Ti1 has a stationary ρi. By repeating the deals, the initial state ρ converges into a sum of the stationary distributions

ρ =a1ρ T ,··· ,T + a2ρ T ,··· ,T + + aN ρ T ,··· ,T (2.39) | 11 1k | 21 2l ··· | N1 Nm N X aiρi = ρc (2.40) → i=1

N PN where ai is a certain family of non-negative numbers that obey ai = 1 and ρ T ,··· ,T is a projection { }i=1 i=1 | i1 in state of ρ onto a state transited by the corresponding ith block matirx consisting of Ti1 , ,Tin . t ··· Therefore for any ρ, one can find a fixed state ρc such that limt→∞ T ρ = ρc. Note that such a fixed point is probabilistic mixture of pure states, thereby classical state. Therefore, in general, repetitions of an exchange of quantum commodities make entangled states gradually disappear and, thereby only a classical mixed state ρc survives. This completes the proof of Theorem 2.2.

2.5 Emergence of Coalition and Quantum Effects on Deals Morgenstern and Neumann assert that a three-person decision making should be completely different from a two-person decision making [23]. In a three-person game, a reciprocal relationship among agents could emerge and a coalition for which they take cooperative behavior helps them pursuit their own benefit. This leads to the

– 13 – concept of a cooperative game, in which players act in deference to an agreement. In our previous discussion in Sec. 2.3, when quantum strategies are not entangled, all players have a chance to increase their own profits by themselves, by exchanging quantum states appropriately. So the game is completely written in terms of a non-cooperative . In this section, we investigate an economy where quantum strategies are generally entangled. Our main here can be stated in a twofold way: 1) Can a quantum state become a medium of exchange? 2) Can a player find an optimal quantum strategy when strategies are entangled? In the previous section, we showed that a quantum state can be a medium of exchange in the sense that the associated classical term becomes a steady state. Again, readers will find the first question can be affirmatively solved (see Table2, for example). A remarkable result in this section is the existence of an optimal quantum strategy that allows Bob to obtain his consuming good 2 from Alice regardless of her inclination (2.42). Alice has no way | i to decline this trade with Bob and may be forced to have a state that she does not desire, hence she may suffer a loss. To compensate her loss, Alice responds to Bob’s attack by counter-attacking, but it turns out that her action may cause damage not only to Bob but also to Charlie due to entanglement. So Bob and Charlie share a mutual interest and they have a motivation to form a coalition. Indeed Bob and Charlie can find an amicable solution that does not decrease their profits (Fig:12). This novel property only becomes evident after quantum goods and entangled strategies are introduced, and cannot be explained by conventional classical games or the corresponding quantum variants [18–20]. Those salient features of our quantum extension of the Kiyotaki-Wright model strongly suggests that general quantum extensions of conventional games, including the and mechanism design, offers novel and various possibilities that cannot be discussed in the framework of the traditional economics. In other words, economics in the quantum era will become very different from economics today. This incentivizes us to study Quantum Economics. In the previous section, we consider a trade of entangled quantum goods, and we see the quantum entan- glement gradually disappears through repetitions of a trade. Now we discuss the case of unentangled quantum goods and entangled quantum strategies. Following our previous discussions, we again employ = exp iθY ⊗12 Jπ for entanglement among strategies. Quantum strategies are maximally entangled when θ = 4 , which we use throughout this section. We are interested in a game where a quantum strategy that do not commute is used c3−c2 c2−c1 J for a deal. Suppose Alice, Bob and Charlie choose the following strategies when δu and δu are in the red region of Fig.8

2 3 p U = Y IU = 1 qAY Y + √qAY X A ⊗ A − ⊗ ⊗ 1 3 p U = Y IU = 1 qBY Y + √qBZ Y (2.41) B ⊗ B − ⊗ ⊗ 1 2 p U = I YU = 1 qC Y Y + √qC Z Y C ⊗ C − ⊗ ⊗

By choosing qB = 0 Bob can reproduce his classical strategy (sB = 1 at (2.16)). An important difference from Sec.2.3 is that Bob may use quantum strategy Z when Bob has 3 . If there is no entanglement, Bob’s state does | i not change from 3 to 1 . In this case, however the strategy has other implications. Bob desires to exchange his | i | i state 3 for Alice’s 2 , by which Alice may suffers a loss. If Alice, Bob and Charlie have 2 , 3 and 1 , then | i | i | i | i | i

– 14 – their initial state 231 is mapped to | i 231 W 0000 0000 0000 0 0 0 | i | i | iA ⊗ | iB ⊗ | iC ⊗ | iA | iB | iC i√qB 331 110 0111 1110 1111 0 1 0 → √3 | i | i | iA ⊗ | iB ⊗ | iC ⊗ | iA | iB | iC i√1 qB − 231 110 1000 0011 0000 0 0 0 − √3 | i | i | iA ⊗ | iB ⊗ | iC ⊗ | iA | iB | iC 1 (2.42) 231 101 1000 0000 0100 0 0 0 − √3 | i | i | iA ⊗ | iB ⊗ | iC ⊗ | iA | iB | iC i√qB + 211 011 1111 1110 1011 0 0 1 √3 | i | i | iA ⊗ | iB ⊗ | iC ⊗ | iA | iB | iC i√1 qB − 211 011 0000 0011 0100 0 0 1 . − √3 | i | i | iA ⊗ | iB ⊗ | iC ⊗ | iA | iB | iC It is important to note that 231 is mapped on to 311 if Alice and Bob meet. This implies that, regardless of | i | i Alice’s strategy qA, Bob creates a new 3 by exchanging his 3 for Alice’s 2 . This happens since Bob uses a | i | i | i quantum strategy that does not commute with and Alice and Bob are entangled. This is a key characteristic J of our quantum game in the following two senses: 1) classical games does not allow those solutions 2) this occurs even for a single stage game. The first property is clearly important since it distinguishes a quantum game from a classical one. Alice receives 3 in a different logic that she accepts it in the classical speculative equilibrium. | i Apart from that, here we would like to emphasize the second characteristic. As explained in [21, 22], the EWL- types of single stage quantum games [18] are entirely non-quantum mechanical and there are no radical solutions created thus far. Therefore, to our best knowledge, our work is the first model that reaches a milestone for this problem. Note that the basic part of our quantum circuit (Fig.3) is the EWL protocol for the prisoner’s dilemma [18]. Quantum properties of the infinitely repeated quantum prisoner’s dilemma are reported in [33, 34] Similarly we find 311 is mapped on to 231 if Alice and Bob meet (2.43). This implies that, regardless of | i | i Bob’s strategy qB, Alice creates a new 1 by exchanging her 3 for Bob’s 1 . | i | i | i 311 W 0000 0000 0000 0 0 0 | i | i | iA ⊗ | iB ⊗ | iC ⊗ | iA | iB | iC i√qA 231 110 1100 0111 1111 1 0 0 → − √3 | i | i | iA ⊗ | iB ⊗ | iC ⊗ | iA | iB | iC i√1 qA − 311 110 0011 1000 0000 0 0 0 − √3 | i | i | iA ⊗ | iB ⊗ | iC ⊗ | iA | iB | iC 1 (2.43) 311 011 0000 1000 0100 0 0 0 − √3 | i | i | iA ⊗ | iB ⊗ | iC ⊗ | iA | iB | iC i√qA 211 101 1100 1111 1011 1 0 1 − √3 | i | i | iA ⊗ | iB ⊗ | iC ⊗ | iA | iB | iC i√1 qA − 211 101 0011 0000 0100 1 0 1 − √3 | i | i | iA ⊗ | iB ⊗ | iC ⊗ | iA | iB | iC

– 15 – The transition matrix T corresponding to the strategies (2.41) is

 2 1 1 1  3 0 3 0 3 3 0 0  1  0 0 0 0 0 0 0  3   1 1 2−qB 1 qA 1 1   3 3 3 3 3 0 3 3   1 2−qB 1  0 0 0 0 0 T =  3 3 3  . (2.44)  2−qA 1  0 0 0 0 3 0 3 0   0 0 0 0 0 0 0 0    qB 1 1 1  0 0 3 0 0 3 3 3  qB 1 0 0 0 3 0 0 0 3

Even if Charlie has a quantum good 2 initially, he loses it in the next step and never receives 2 again. He | i | i eventually gets 1 . So we do assume Charlie has the good 2 . Therefore the corresponding fixed point is | i | i (2 + 2qA + 2qB) 211 211 + (2qA + 2) 231 231 + qB 311 311 + qB(1 + qA) 331 331 ρ = | i h | | i h | | i h | | i h |. (2.45) 4(1 + qA) + qB(3 + qA)

So the state 211 seems most likely to appear. Let us discuss equilibrium strategies. The expected total profit | i Vi(i = A, B, C) (2.11) of Alice, Bob and Charlie can be expressed as   2 + 2qA + qB     uδ 1  2 + 2qA  VA = (c2, c2, c3, c3) + (1, 0, 1 + qA, 1)   − q q q  q  3 4(1 + A) + B(3 + A)  B  qB(1 + qA)   2 + 2qA + qB     uδ 1  2 + 2qA  VB = (c1, c3, c1, c3) + (1, qB, 0, 0)   (2.46) − q q q  q  3 4(1 + A) + B(3 + A)  B  qB(1 + qA)   2 + 2qA + qB     uδ 1  2 + 2qA  VC = (c1, c1, c1, c1) + (0, 1, 1, 2)   , − q q q  q  3 4(1 + A) + B(3 + A)  B  qB(1 + qA) where each vector is represented with the basis 211 , 231 , 311 , 331 , that constitute (2.45). Consulting {| i | i | i | i} (2.15), we find 1 vA = (c2, c2, c3, c3), wA = (1, 0, 1 + qA, 1) 3 1 vB = (c1, c3, c1, c3), wB = (1, qB, 0, 0) (2.47) 3 1 vC = (c1, c1, c1, c1), wC = (0, 1, 1, 2) 3

– 16 – c2−c1 c3−c2 Figure 10: Total expected profit of Alice, Bob and Charlie when uδ = 0.4, uδ = 0.4.

For any qB (0, 1], Alice’s optimal strategy which maximizes VA is qA = 1 since ∈ 2 ∂VA (c3 c2)qB(4 qB) + δuqB = − − 2 0. (2.48) ∂qA (4(1 + qA) + qB(3 + qA)) ≥

The inequality saturates if and only if qB = 0. So any qA can be her optimal strategy when qB = 0. Similarly, for any qA [0, 1], Bob’s optimal strategy is also qB = 1 since ∈ 2 ∂VB (c3 c1)2(1 qA)(1 + qA) + δu2(1 + qA) = − − 2 > 0 (2.49) ∂qB (4(1 + qA) + qB(3 + qA))

Therefore qA = qB = 1 is an equilibrium. Alice gains more rewards compared with the case when she chooses the fundamental strategy (qA = 0). Bob and Charlie succeed in increasing their rewards. By substituting qA = qB = 1 into (2.45), we find the corresponding steady state 5 4 1 2 ρ = 211 211 + 231 231 + 311 311 + 331 331 . (2.50) 12 | i h | 12 | i h | 12 | i h | 12 | i h |

Alice

2,3 1,3 3 1

1 Bob Charlie 3

Figure 11: The flow of goods associated with the steady state (2.50). The difference from the classical case is that 3 is communicated between Alice and Bob, which happens when the game is in a state 231 or 311 . | i | i | i

Bob’s strategy U 3 = Z Y decreases Alice’s profit. As her counter strategy, she may use U 2 = X I when B ⊗ A ⊗ she has 2 , in order to prevent Bob from exchanging his 3 for her 2 . In addition, if Charlie plays U 1 = I Y , | i | i | i C ⊗ then Alice can receive 1 and increase her profit by exchanging her 2 for 1 . Quantum strategies and the flow | i | i | i of quantum goods are summarized in Table.2. It is important that there are several strategies that make 1 , 2 | i | i or 3 media of exchange. | i

– 17 – A:Y I A:X I Alice⊗ Alice⊗ B:Y Y ⊗ 2 3 2 1 C:I Y ⊗ 1 1 Bob Charlie Bob Charlie 3 3 Alice Alice B:Z Y ⊗ 2 3 2 1 C:I Y ⊗ 1 1 Bob Charlie Bob Charlie 3 3

B:Y Y ⊗ C:Z Y ⊗

Alice Alice

B:Z Y ⊗ 2 3 2 1 C:Z Y ⊗

Bob Charlie Bob Charlie

Table 2: Quantum strategies and the flow of quantum goods when 231 is an initial state of Alice, Bob and | i Charlie who are allowed to decide U 2 , U 3 and U 1 independently. If Alice uses X I, she and Charlie can exchange A B C ⊗ 2 1 and she prevents Bob from using his previous strategy Z Y for exchanging 2 3 . However if Bob | i ↔ | i ⊗ | i ↔ | i uses Y Y , the trade 2 3 can be realized. Nothing is mediated in the bottom right case. ⊗ | i ↔ | i

As Charlie gets his consuming good, a trade 1 3 between Bob and Charlie is the best trade for Charlie | i ↔ | i and, as Bob reduces his storage cost, it is also a good trade for Bob. So they may want to trade. However if Bob and Charlie use Z Y and Z Y respectively, they do not reach a mutually acceptable agreement on the ⊗ ⊗ trade 1 3 . To solve the problem, they cooperate and try to increase their profits by means of the following | i ↔ | i strategy. Suppose both of Bob and Charlie choose the strategy (2.51) with the probability p and choose the strategy (2.52) with the probability 1 p. − ( 1 3 UB = Y I UB = √1 qBY Y + √qBZ Y 1 ⊗ 2 − ⊗ ⊗ (2.51) UC = I Y UC = Y Y  ⊗ ⊗ 1 3  UB = Y I U = Y Y ⊗ B 1 q 0 q 0 2 ⊗ (2.52)  U = 1 q I Y + q Z Y U = Y Y C − C ⊗ C ⊗ C ⊗ Bob and Charlie make an arrangement and decide a single p [0, 1]. Alice may want to use ∈ q q U 2 = 1 q0 Y I + q0 X IU 3 = Y X (2.53) A − A ⊗ A ⊗ A ⊗ 0 If Alice uses qA = 0, Bob and Charlie can maximize their profits by choosing p = 1, qB = 1. Similarly, if she uses 0 qA = 1, they can maximize their profits by choosing p = 0, qC = 1. Let T1 and T2 be the transition matrices associated with strategies (2.51) and (2.52). The total transition matrix is T = pT1 + (1 p)T2. Their explicit −

– 18 – matrix representations are

 0  2−qA 1 1 1 3 0 3 0 3 3 0 0  q0   A 1  3 3 0 0 0 0 0 0  0 0 0   1 1 2−qB (1−qA)−(1−qB )qA−qA 1 1 1 1   3 3 3 3 3 0 3 3   0 0 0   q 2−qB (1−q )−(1−qB )q   0 1 A A A 0 1 0 0 T1 =  3 3 3 3  (2.54)  1 1   0 0 0 0 3 0 3 0    0 0 0 0 0 0 0 0  0 0   qB (1−qA)+(1−qB )qA 1 1 1   0 0 3 0 0 3 3 3   0 0  qB (1−qA)+(1−qB )qA 1 0 0 0 3 0 0 0 3  0 0 0 0  2−qA(1−qC )−(1−qA)qC 1 1 1 3 0 3 0 3 3 0 0  q0 (1−q0 )+(1−q0 )q0   A C A C 1   3 3 0 0 0 0 0 0  2−q0 −q0 (1−q0 )−(1−q0 )q0   1 1 A A C A C 1 1 1 1   3 3 3 3 3 0 3 3   q0 (1−q0 )+(1−q0 )q0 2−q0   0 1 A C A C A 0 1 0 0 T2 =  3 3 3 3  (2.55)  1 1   0 0 0 0 3 0 3 0    0 0 0 0 0 0 0 0  0   qA 1 1 1   0 0 3 0 0 3 3 3   0  qA 1 0 0 0 3 0 0 0 3

It is only the pair of Bob and Charlie who can establish the coalition as described. Both of them can increase their profit without loss. This is not true for Alice and Bob, or for Alice and Charlie. If Alice and Bob exchange 2 3 , then Bob gains but Alice suffers a loss. Similarly if Alice and Charlie trade 2 1 , then Alice | i ↔ | i | i ↔ | i benefits but Charlie incurs a loss. The emergence of such a coalition is based on a typical quantum phenomena that has not been reported in the past studies on the Kiyotaki-Write model. The flow of quantum commodities under those strategies are summarized in Fig. 12.

Alice

2,3 1,3 2,3 1,2

1,2 Bob Charlie 1,3 Coalition

Figure 12: The flow of goods. The difference from the classical case is that 3,2,1 are communicated between Alice and Bob, Bob and Charlie, and Charlie and Alice, respectively. Bob and Charlie can cooperate that does not decrease their profits. This does not happen in a classical situation.

– 19 – 3 Conclusion and Discussion

The main contributions of our article are summarized as follows. In Sec. 2.1, we introduced our quantum game model which describes the emergence of quantum commodity money as an equilibrium strategy of quantum exchange. In Sec. 2.2 we presented quantum circuits for consensus building and exchanging quantum states. In Sec. 2.3 we addressed generic entangled states and clarified the mechanism of the quantum economy leading a quantum state to a medium of exchange. In Sec. 2.5 we studied some equilibria of entangled quantum strategies and demonstrated how a coalition can emerge from a non-cooperative game. We propose several research directions for readers interested in quantum games and quantum economics. First of all, it is straightforward but important to explore an economy that allows quantum fiat money to be a medium of exchange. For this purpose one should simply add 4 := 00 as a quantum object with no intrinsic | i | i value. Moreover it will be a good exercise to implement our algorithms (Fig.3 and Fig.4) with a quantum computer or a to study a of consensus building and verify our results numerically. In addition, it is also interesting to study the quantum model of the type B economy, where Alice, Bob and Charlie produce 3 , 1 and 2 , respectively. As shown in the original Kiyotaki-Wright model [38], the dominant | i | i | i equilibrium strategy depends on models and it will be also true for our quantum extension. Indeed, as discussed in Sec. 2.5, a choice of initial states is crucial for the optimal strategies. Moreover it is exciting to address generic entangled strategies. In our analysis we focused on the simplest case = exp iθY ⊗12. Our three-person J quantum game with generic entangled strategies is very complicated and there are many open questions left. Furthermore, it is also possible to analyse equilibrium strategies of our economy where quantum walkers [56–58] enjoy business. In this article, we used W to decrease complexity, but it is yet another important question | i whether a quantum state can be a medium of exchange even if players encounter in a different way. For this purpose, it may be helpful to use the following matrices to introduce quantum walk       a11 a12 a13 0 0 0 0 0 0        0 0 0  , a21 a22 a23 ,  0 0 0  (3.1) 0 0 0 0 0 0 a31 a32 a33

2 2 2 where ai1 + ai2 + ai3 = 1 for i = 1, 2, 3. | | | | | | For advanced readers, it is meaningful to investigate a general quantum theory of economics, including market design and contracts, so that we can provide efficient quantum information and communication markets in the upcoming era of quantum technology. For example, one can refer to our algorithm to consider many-to-many matching. Moreover, our study suggests that entangled strategies case a situation that is not predetermined in an agreement, thereby it will be complex and costly for the parties to make their contract complete. That means quantum theory also sheds new light on incomplete contracts [59–61]. Quantum theory of economics will also shed new light on . In particular, we guess different countries will start using different quantum . It is an open problem to study how quantum currencies circulate in the global economy. A formal model for a classical setup is provided in [62]. The more quantum technologies develop, the more important quantum economical perspectives will become. In addition, it will contribute to development of legal systems in the quantum era, since economics are inextricable from jurisprudence.

Acknowledgement

We thank Avi Beracha for carefully reading the manuscript.

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