Quantum Barro--Gordon Game in Monetary Economics
Ali Hussein Samadi, Afshin Montakhab, Hussein Marzban, Sakine Owjimehr
aa r X i v : . [ q -f i n . E C ] A ug Quantum Barro-Gordon Game in MonetaryEconomics
Ali Hussein Samadi a , Afshin Montakhab b , Hussein Marzban a ,Sakine Owjimehr a a Department of Economics, College of Economics Management and SocialSciences, Shiraz University, Shiraz 71946-85111, Iran b Department of Physics, College of Sciences, Shiraz University, Shiraz71946-84795, Iran
Abstract
Classical game theory addresses decision problems in multi-agent environmentwhere one rational agent’s decision affects other agents’ payoffs. Game theory haswidespread application in economic, social and biological sciences. In recent yearsquantum versions of classical games have been proposed and studied. In this paper,we consider a quantum version of the classical Barro-Gordon game which capturesthe problem of time inconsistency in monetary economics. Such time inconsistencyrefers to the temptation of weak policy maker to implement high inflation whenthe public expects low inflation. The inconsistency arises when the public punishesthe weak policy maker in the next cycle. We first present a quantum version ofthe Barro-Gordon game. Next, we show that in a particular case of the quantumgame, time-consistent Nash equilibrium could be achieved when public expects lowinflation, thus resolving the game.
Key words:
Barro-Gordon Game, Quantum Game Theory, Time Inconsistency
PACS: • We reformulate Barro-Gordon Game using quantum game theory. • We use Marinatto-Weber approach for quantization of game theory. • We find that the well-known time inconsistency in the classical game is removedafter quantization.
Email address: [email protected] (Afshin Montakhab).
Preprint submitted to Elsevier 21 August 2017
Introduction
Game theory is a mathematical formulation of situations where, for two ormore agents, the outcome of an action by one of them depends not only on theparticular action taken by that agent but on the actions taken by the other (orothers)[1]. Game theory has gained much attention in recent years as indicatedby the many texts written on it [2,3,4,5]. After Neuman and Morgenstern’sbook [6], which was the first important text in game theory, John Nash madeimportant contributions to this theory [7,8]. However, application of gametheory to different areas such as economics, politics, biology, started in 1970’sand has been growing ever since [1].In recent years, game theory has attracted the attention of many physicistsas well. Among the many contributions that has come along from the physicscommunity is the inclusion of the rules of quantum mechanics in game the-ory where quantum effects such as superposition and entanglement can play arole [9,10]. Quantum game was introduced by Eisert, Wilkens, and Lewenstein(EWL) [11], where the role of entanglement was considered first. Subsequently,Marinatto and Weber (MW) [12] offered a more general scheme for quanti-zation of games based on Hilbert space approach. It is now believed thatquantization of games can offer interesting situations where various dilemmapresent in classical games can be removed. As a result of the above-mentionedworks, many authors have employed the quantization of various games. Forexample, Cheon and Tsutsui [13], Fliteny and Hollenberg [14], Makowski [15]and Landsburg [16] have used EWL approach. On the other hand, Arfi [17],Deng et. al [18] and Frackiewicz [19], based their approach on MW scheme.Much of the attention in the above-mentioned works has been paid to well-known games such as prisoner’s dilemma game [11,13,14,17], while others haveconsidered various other games with social implications.However, despite the growing interest in econophysics [20], little attention hasbeen paid to quantization of classical games in finance and economics. Thepresent work offers a step in this direction. Here, we intend to quantize theclassical game of Barro and Gordon (BG) in monetary economics employingthe MW scheme. We then study some specific cases of the quantum gamein order to find Nash equilibrium and the advantages quantization may of-fer. BG game, to be explained in details in the following, is a classical gamewhich illustrates the problem of time inconsistency in monetary policy. Timeinconsistency was first introduced by Kydland and Prescott [21] and later byBarro and Gordon [22]. The main idea is that when the output is inefficientlylow, policy maker can increase it by applying discretionary policy, causingsurprising inflation. In this situation, although the output increases which isbeneficial, it causes inflation, which is costly. Therefore, we encounter an infla-tionary bias. Since BG introduced a noncooperative game between public and2entral bank, many researchers [23,24,25,26,27] have been using game theoryto study time inconsistency. All of these studies are based on the classicalgame theory, while the principles of quantization has not been applied so far.There are, however, a variety of reasons to apply the roles of quantum me-chanics to various disciplines outside of traditional physics. For example, inpsychology and decision making theory, quantum cognition has gained muchattention [28,29]. In most such approaches, quantum probabilities are consid-ered in order to represent certain uncertainty in decision making process. Fur-thermore, a quantum Hamiltonian approach to various fields of social scienceshas also been considered [30,31,32,33,34,35] where operator-valued dynamicalvariables are governed by a general Hamiltonian which includes all possibleinteraction. However, we must note that while our approach in this paper issimilar in spirit, it is different in its basic assumption and methodology. Theproblem of strategy selection which is the essence of game theory could be for-mulated using the laws of quantum mechanics instead of classical probabilitytheory. In fact, it is the purely quantum mechanical concept of superposition(of strategies) which provides the key ingredient in our approach to gametheory here.Furthermore, micro-evolution teaches us that our selfish (microscopic) genesmay make our decision on a fundamental level, where quantum mechanicsmay be relevant. Perhaps from a more practical point of view, recent advancesin quantum computational and quantum communication technology[36] mayhelp us in creating quantum devices which must take on quantum strategiesin order to solve problems [11]. It is with such motivations in mind that onemay consider quantum game theory in various fields of social sciences, andconsequently monetary economics in our present case.The rest of this paper is organized as follows: In Section 2, we briefly describethe problem of time inconsistency in monetary policy. The classical BG gameis then described in details in Section 3. In Section 4, we present our mainresults of quantization of BG game and consider some specific cases of thequantum game. The last section offers concluding remarks.
A policy is time inconsistent or dynamic inconsistent when it is considered asthe best policy for particular time in the future, but it does not remain so,when that particular time actually arrives. There are two possible mechanismsthat have been considered for such a time inconsistency. (i) Strotz [37] explainsthat time inconsistency is because of changing preferences, (ii) Kydland andPrescott [21] consider another explanation that is based on agent’s rational3xpectation. The main idea is, when people expect low inflation, central bankfinds the incentive for high inflation. If the public understand this incentiveand predict high inflation, the central bank finds it optimal to deliver thepublic’s expectation and therefore implements high inflationary policy. There-fore, while low inflation is the optimal policy for both banker and the publicto begin with, high inflation is eventually implemented. We use this secondmechanism in this paper as it is the concept of time inconsistency associatedwith the Barro-Gordon game.Barro and Gordon [22] have explained time inconsistency of monetary policyas follow: in a discretionary regime, central banker can print more money andmake more inflation than people’s expectations. Benefits of this surprisinginflation might provide more economic activities or reduce government’s debt.However, if people, due to their rational expectation, understand it and adjusttheir expectations with it, then policy maker will not reach his goal at all.This is simply due to the fact that inflationary advantages is best when itis unanticipated. Besides that, due to increased money supply, the level ofprices will grow, which will have negative consequences for the policy maker.The classical Barro-Gordon game captures the essence of this type of time-inconsistency in the context of decisions (strategy) that a policy maker mustmake and the expectations that the public can have. The actual game isrepresented in two different formats where the strong policy maker implementslow inflation which is time consistent, while in the case of weak policy makera time inconsistent strategy is the alternative to a Nash equilibrium.
In BG game, similar to prisoners’s dilemma, there are two players; public andcentral bank. In this game, one assumes that the public has rational expec-tations. The public then predicts inflation by solving out the policy maker’soptimization problem. On the other hand, the policy maker selects inflationpolicy by considering the public’s inflationary expectations. In this paper, byfollowing Backus and Driffill [38] and Storger [39], we use a special versionof BG game. This version is easier to convert to quantum game due to hav-ing a definitive payoff matrix. In this version, there are two types of policymaker: weak policy maker, which uses discretionary policies and gains benefitby making unanticipated inflation. In the other words, a weak policy makercan cheat the public when they formed their low inflationary expectation atthe start of the period. Strong policy maker, on the other hand, commits tozero inflation and is not interested in unanticipated inflation.4he utility function of these policy makers is as follows: U polt = θb ( π t − π et ) − aπ t π t and π et are actual inflation and expected inflation rates, respectively.Inflationary cost is assumed to be proportional to the square of inflation andtherefore aπ t is the cost of inflation where a is an arbitrary cost parameter. θ is a dummy variable that is equal to 1 for weak policy maker and 0 forstrong policy maker and b is a coefficient for benefit of inflation term with b > π t > π et then policy maker can decrease unemployment (accordingto Philips curve ) and gain benefit using the first term in Eq.(1). Public’sutility function is as follows: U pubt = − ( π t − π et ) . (2)This function shows that every deviation from expected inflation causes disu-tility for the public.We next briefly review the payoff matrix for weak and strong policy maker asobtained by [38,39]: First we use weak policy maker optimization. The weakpolicy maker optimizes Eq.(1) without any constraint. By taking derivativeof Eq.(1) with respect to π t , optimal inflation will be ˆ π t = ba . If π et = ba ,replacing it in Eq.(1) and (2) will result in U polt = − b a < U pubt = 0. Therefore,in this case, weak policy maker cannot gain any benefit and this strategy willnot be chosen. On the other hand, optimizing unconstraint Eq.(2), with theassumption of public rational expectations, results in π t = π et . If weak policymaker commits to zero inflation, both players will receive zero payoff. However,if the public expects zero inflation, and the weak policy maker implements π t = ba , then he can gain some benefit (equal to b a ) and the public will incurlosses of − ( ba ) . Therefore, we have U polt = b a > U pubt = − ( bb ) and the weakpolicy maker therefore prefers this strategy. Even if the public expects π et = ba and the policy maker chooses π t = ba , he can get more payoff than choosingzero inflation. Thus, π t = 0 is a dominated strategy and will never be selected.Following [38], normalization condition ( a = b = 2), leads to a simple payoffmatrix for the weak policy maker as shown in Table 1. Note that the caseof π t = 1 , π et = 1 is a Nash equilibrium in this case. However, the actualequilibrium is the case of π t = 1 , π et = 0 if the policy maker is successfulin cheating the public . The key point here is that this equilibrium ( π t = 1, π et = 0) is time inconsistent because π et = 0 is announced but π et = 1 isimplemented.In the case of strong policy maker ( θ = 0 ), π t = ba is never chosen as it is adominated strategy. Strong policy maker will incur a loss equal to − b a in both Philips curve shows an inverse relation between the unemployment and inflation. = 1 Public π et = 0 π et = 1Weak policy maker π t = 0 (0,0) (-2,-1) π t = 1 (1,-1) (-1,0)Table 1Weak policy maker payoff cases (either the public expects zero inflation or ba ). Therefore, π t = 0 is thebest policy for strong policy maker and therefore always commits to it. In thissituation, there would be no problem of time inconsistency. Therefore, ( π t = 0, π et = 0) is the Nash equilibrium for this case and it is time consistent. θ = 0 Public π et = 0 π et = 1Strong policy maker π t = 0 (0,0) (0,-1) π t = 1 (-1,-1) (-1,0)Table 2Strong policy maker payoff Barro and Gordon [22] showed that the weak policy maker loses his reputationfor cheating the public. In fact, in the next period, public plays “tit for tat”game and punish the weak policy maker by adjusting their expectations. Inother words, if π t − = π et − then π t = π et = 0 ; otherwise, π t = π et = ba .Therefore, weak policy maker compares marginal cost and benefit of cheatingthe public, and then decides to make unanticipated inflation. Consequently,classical BG game needs two time periods to solve the game between the publicand weak policy maker. In this paper we generalize this classical game to aquantum framework and ask if it can be made more efficient. There is essentially two different ways to quantized classical games in theliterature. EWL [13,14,15,16] took the original steps in this regard. However,the approach of MW [17,18,19] has been more widely used recently and weintend to use their approach in quantizing BG game.6rackiewicz [19] argued that, in EWL method, the result of the game dependson many parameters because each player’s strategy is a unitary operator.Therefore it has cumbersome calculation. But in MW, player’s local operatorswere performed on some fixed entangled state | ψ i . It seems that MW is sim-pler than EWL [40]. We therefore use the MW method to quantize the gamebetween weak policy maker and the public. In this game, there are two play-ers: weak policy maker (M) and public (U). Each player has two strategies:high inflation (H) and low inflation (L). Consider a four- dimensional Hilbertspace, H , as the strategy space in ket notion: H = H M ⊗ H U = {| LL i , | LH i , | HL i , | HH i} (3)where the first qubit is related to the state of the policy maker and the secondone to that of public. Kets show a given strategy in strategy space which inquantum version is a Hilbert space. Therefore, we use an arbitrary quantumstrategy as a normalized state vector, | ψ i i . | ψ i i = α | LL i + γ | LH i + δ | HL i + β | HH i (4)Where | α | , | β | , | γ | , | δ | are probability of observing the strategies of (L, L),(H, H), (L, H) and (H, L), respectively, with | α | + | γ | + | δ | + | β | = 1. Densitymatrix is written as ρ i = | ψ i i h ψ i | . Let C be a unitary Hermitian operator (i.e, C † = C = C − ), such that C | H i = | L i and C | L i = | H i and I is the identityoperator such that I | H i = | H i and I | L i = | L i . In the game, policy makerand [public] use operators I and C with probabilities of p, (1 − p ) , [ q, (1 − q )].Final density matrix for this system is as follows: ρ f = pq h ( I M ⊗ I U ) ρ i ( I † M ⊗ I † U ) i + p (1 − q ) h ( I M ⊗ C U ) ρ i ( I † M ⊗ C † U ) i +(1 − p ) q h ( C M ⊗ I U ) ρ i ( C † M ⊗ I † U ) i +(1 − p )(1 − q ) h ( C M ⊗ C U ) ρ i ( C † M ⊗ C † U ) i (5)and the two payoff operators are given as follows: P M = 0 | LL i h LL | − | LH i h LH | + | HL i h HL | − | HH i h HH | (6) P U = 0 | LL i h LL | − | LH i h LH | − | HL i h HL | + 0 | HH i h HH | . (7)Finally, payoff functions are calculated according to:¯$ M ( p, q ) = T r ( P M ρ f ) (8)¯$ U ( p, q ) = T r ( P U ρ f ) . (9)7his can be written as: ¯$ M ( p, q ) = ΦΩ γ TM (10)¯$ U ( p, q ) = ΦΩ γ TU (11)where Φ = [ pq, p (1 − q ) , (1 − p ) q, (1 − p )(1 − q )] (12) Ω = α γ δ β δ β α γ γ α β δ β δ γ α γ M = [0 , − , , − γ U = [0 , − , − , M ( p, q ) = 2 p ( α − β + δ − γ ) + q ( δ − α − γ + β ) − α + γ − δ (13)¯$ U ( p, q ) = (1 − δ + γ ))( q (2 p − − p ) − ( δ + γ ) (14)In order for Nash equilibrium to exist one needs to implement the followingconditions [12]: ¯$ M ( p ∗ , q ∗ ) − ¯$ M ( p, q ∗ ) ≥ , ∀ p ∈ [0 ,
1] (15)¯$ U ( p ∗ , q ∗ ) − ¯$ U ( p ∗ , q ) ≥ , ∀ q ∈ [0 ,
1] (16)which in our case lead to:2( p ∗ − p )( α − β + δ − γ ) ≥ − δ + γ ))( q ∗ − q )(2 p ∗ − ≥ . (18) Equation (13-14) and (17-18) are our main results. Following MW’s approach,we consider the validity of three possible situations below:(a) p ∗ = q ∗ = 1In this case the payoffs are as follows:¯$ M (1 ,
1) = − β − γ + δ (19)¯$ U (1 ,
1) = − γ − δ (20)8nd Nash equilibrium conditions are:2(1 − p )( α − β + δ − γ ) ≥ − δ + γ ))(1 − q ) ≥ ⇒ γ + δ ≤ / . (22)If α + δ > β + γ , then the first Nash equilibrium condition will be satisfied.In fact, this condition will most likely be satisfied for the weak policy makerwhich we are considering here, since according to Table (1), he prefers (L,L) or(H,L) strategies rather than (H,H) or (L,H) strategies. We thus assume thatfor weak policy maker the condition of α + δ > β + γ is always true. Thesecond Nash equilibrium condition may or may not be satisfied depending onthe choice of the quantum strategy. We will return to this point later on inthis paper.(b) p ∗ = q ∗ = 0In this case the payoffs are as follows:¯$ M (0 ,
0) = − α + γ − δ (23)¯$ U (0 ,
0) = − γ − δ (24)and Nash equilibrium conditions are calculated as: − p ( α − β + δ − γ ) ≥ − δ + γ )) q ≥ ⇒ γ + δ ≤ / α + δ < β + γ which isexactly the opposite of the previous case. Again, since we are considering aweak policy maker here (see above), we will consider this condition as unac-ceptable. Therefore, we do not consider Nash equilibrium to hold for the weakpolicy maker in the case of p ∗ = q ∗ = 0.(c) p ∗ = q ∗ = 1 / M (1 / , /
2) = − / U (1 / , /
2) = − / / − p )( α − β + δ − γ ) ≥ − δ + γ ))(1 / − q )(1 −
1) = 0 (30)The second Nash equilibrium condition is trivially satisfied. However, the firstNash equilibrium condition is clearly violated for p > / α − β + δ − γ > α, β, δ, γ . In the following we consider two different quantum strategies ofweak policy maker where Nash equilibrium could potentially exist:(i) Suppose that the public has false prediction about inflation. It means that,quantum strategy is a superposition of two strategies, (L, H) and (H, L) where α = β = 0: | ψ i i = γ | LH i + δ | HL i . (31)Therefore, payoff functions will be as follows:¯$ M (1 ,
1) = − γ + δ (32)¯$ U (1 ,
1) = − γ − δ = − − p )( δ − γ ) ≥ γ + δ ≤ / δ > γ . However,this scenario cannot imply a stable situation due to Eq. (35) which indicatethat Nash equilibrium can never be obtained since γ + δ = 1. This result isreminiscent of the classical version of the game where the weak policy makercan earn positive payoff by cheating the public for just one cycle. Afterwards,the public will punish him and correct their expectation. Here, we showed thata quantum strategy that is superposition of false prediction strategies is nota Nash equilibrium. In other word, it is not a sustainable equilibrium.(ii) Suppose that public has correct prediction about the inflation. Therefore,quantum strategy is a superposition of two strategies, (L, L) and (H, H) where γ = δ = 0: | ψ i i = α | LL i + β | HH i (36)¯$ M (1 ,
1) = − β (37)¯$ U (1 ,
1) = 0 (38)Thus Nash equilibrium conditions in this case are:2(1 − p )( α − β ) ≥ − q ) ≥ α > β which is an acceptable conditionfor the weak policy maker. This shows that the larger the share of (L, L)strategy is chosen ( α ) the smaller the negative payoff of policy maker becomes( β ). The important point here is that a Nash equilibrium exist for the weakpolicy maker where the public is guaranteed not to lose and the policy maker’sloss can be minimized by reducing β . In fact, in an extreme case β →
0, wherethe quantum strategy converges to a (non-superposition) single strategy (L,L),the payoff of both players will be zero as in the classical case (see Table(1)). However the important difference is that Nash equilibrium is satisfiedhere, where in the classical version it is not. Therefore, the quantum versionof the game in this scenario offers a time consistent Nash equilibrium. Thisconstitutes our main result.
Barro and Gordon proposed a game between the public and policy makerbased on the theory of time inconsistency. In this game, a weak policy makercan earn some benefits in short time by cheating the public about inflation.However, the public will punish him in the next period. Therefore, inflationincreases and policy maker will lose his benefits. Thus, in this classical versionof BG game, the implementation of low inflation by the weak policy maker isnot a Nash equilibrium. In this paper we generalized the BG game by usingthe quantum game scheme according to Marinatto and Weber. We consideredthe quantum game as a superposition of four classical strategies, and Nashequilibrium conditions were subsequently calculated. The results showed thatamong the three possibilities we have considered, the first scenario was moreacceptable. Then we considered two different quantum strategies of weak pol-icy maker where Nash equilibrium could potentially exist:(i) public has falseprediction and (ii) public has a correct prediction. It was shown that Nashequilibrium was not satisfied when the public has false prediction. However,we obtained a Nash equilibrium that is time consistent in the second scenariowhere the public has a correct prediction about inflation. Our result is impor-tant since it shows that in the quantum version of BG game, unlike its classicalversion, the low inflation policy is a Nash equilibrium when the public expectslow inflation thus removing the time inconsistency and therefore solving thegame. We emphasize that the purely quantum effect of superposition entan-gled states was the key ingredient in solving the game and removing the timeinconsistency present in the classical version.We next briefly comment on some issues regarding our results. The relevanceof quantum game may at first glance seem a bit peculiar despite the motiva-11ions provided in the Introduction (Section 1) above. Ever since 1935 whenSchrodinger introduced what is now known as the Schrodinger cat, the possi-bility of macroscopic superposition states was debated in the literature. How-ever, quantum technology has provided for macroscopic superposition states[41], and one can imagine that with sufficiently advanced technology, futuremachines could employ strategies that could benefit from quantum game the-ory. Furthermore, one may ask if our results would be different if we had em-ployed other methods besides MW for game quantization. Arfi[17] has shownthat one would obtain the same results for prisoner’s dilemma game regardlessof the method of quantization. We chose the MW method since it suited ourgame in a more straightforward way. However, one can imagine that employingthe method of EWL would lead to essentially the same results. The impor-tant point that seems to be the common point of most game quantization isthat quantum games offer an advantage over their classical version becausethey employ superposition principle and are thus able to resolve the conflictexisting in the classical version. We have also obtained the same essential re-sults here, and suspect that our result would be independent of the methodof quantization.Our aim here has been to provide an example of quantum game theory ineconomics and how the rules of quantum mechanics may offer advantages inthis regard. However, one might consider further work along the same line pre-sented here. For example, one can consider the possibility of other equilibriathat might exist for the case of various other choices of p and q besides theones considered here (which were purely motivated by previous studies). An-other interesting avenue would be to consider a Hamiltonian formulation andthus the time evolution of various operators along the line of [31,34,35]. Thismight be interesting as dynamical evolution would become quantum mechan-ical and one might consider the different evolution of an initially (quantum)superposition state vs. its classical analog of a mixed state. Acknowledgements–
Grants from Research Council of Shiraz University is kindlyacknowledged. This paper has also benefited from constructive criticism of re-spected (anonymous) referees.
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