We quantise the generalised Hawk-Dove Game. By restricting the strategy space
available to the players, we show that every game of this type can be extended
into the quantum realm to produce a Pareto optimal evolutionarily stable
strategy. This equilibrium replaces the inefficient classical one when the
entanglement prepared in the game exceeds a critical threshold value, which we
derive analytically.Comment: 7 pages, 2 figures, submitted for review, comments welcom

Lindsell, E. A. B.

Wiesner, K.

English

arXiv

We quantise the generalised Hawk-Dove Game. By restricting the strategy space available to the players, we show that every game of this type can be extended into the quantum realm to produce a Pareto optimal evolutionarily stable strategy. This equilibrium replaces the inefficient classical one when the entanglement prepared in the game exceeds a critical threshold value, which we derive analytically

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                          Lindsell, E. A. B., & Wiesner, K. (2014). Entanglement gives rise toPareto optimality in the generalised quantum Hawk-Dove Game.arXiv.Peer reviewed versionLink to publication record in Explore Bristol ResearchPDF-documentThis is the author accepted manuscript (AAM). The final published version (version of record) is available onlinevia arXiv at https://arxiv.org/abs/1410.3631. Please refer to any applicable terms of use of the publisher.University of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only thepublished version using the reference above. Full terms of use are available:http://www.bristol.ac.uk/red/research-policy/pure/user-guides/ebr-terms/Entanglement gives rise to Pareto optimality in thegeneralised quantum Hawk-Dove GameE A B Lindsell1‡ and K Wiesner11 School of Mathematics, University of Bristol, University Walk, Clifton, Bristol,BS8 1TW, UKE-mail: eablindsell@gmail.com, k.wiesner@bristol.ac.ukAbstract. We quantise the generalised Hawk-Dove Game. By restricting thestrategy space available to the players, we show that every game of this type canbe extended into the quantum realm to produce a Pareto optimal evolutionarily stablestrategy. This equilibrium replaces the inefficient classical one when the entanglementprepared in the game exceeds a critical threshold value, which we derive analytically.PACS numbers: 02.50.Le, 03.67.-a, 03.67.BgKeywords: Hawk-Dove Game, entanglement, quantum games, ESS, Pareto effieciency‡ Present address: 2 Crabtree Cottages, Marlborough, Wiltshire, SN8 3HP, UK.arXiv:1410.3631v1  [quant-ph]  14 Oct 2014Pareto optimality in the quantum Hawk-Dove Game 21. IntroductionVon Neumann & Morgenstern (1947) introduced a theory of games in their paperThe Theory of Games and Economic Behaviour. Although their intention was tobuild a theory for predicting economic variation, the field of game theory has sincefound a diverse range of unforeseen applications from political science (Boyd &Richerson 1992) to evolutionary biology (Nowak & Sigmund 1999). Games such asthe Prisoner’s Dilemma (PD) (Poundstone 2011) and the Hawk-Dove Game (HDG)(Maynard Smith 1982) have proved successful models in evolutionary biology to explainthe growth of selfish and aggressive behaviours within a population of a speciesrespectively.Meyer (1999) inspired the extension of this theory by asking, “What happens ifthese games are generalised into the quantum realm?” The field of quantum gametheory was consequentially established by Eisert et al. (1999). They reasoned thatquantum games may already be played on a molecular level where quantum mechanicsdictates the rules. As game theory involves the communication of information (such asa player communicating their choice of strategy to the game’s arbiter), we can thinkabout this information as quantum information (as we live in a quantum world), thusproviding a natural link between game theory and quantum information theory. Theyalso highlight how the presence of phenomena such as entanglement and superpositioncan drastically change the observed dynamics in this new framework. Examples includethe original dilemma present in the classical PD being removed under the quantisationof the game when the two players have access to a specific two-parameter strategy space(Eisert & Wilkens 2000). Du et al. (2001) have since demonstrated the first physicalrealisation of a quantum game using a nuclear magnetic resonance quantum computer.It is these types of new results that spur the continued interest in the field of quantumgame theory.In this paper we quantise the generalised Hawk-Dove Game (HDG). By consideringthe entanglement of the initial state we show that a unique threshold of entanglementcan be derived for the onset of a Pareto optimal evolutionarily stable strategy (ESS) forevery game of this type. We start with an introduction to the classical HDG, followedby a quantisation of the generalised HDG following the work of Eisert et al. (1999).2. The Classical Hawk-Dove GameThis game was originally formulated by Maynard Smith (1982) to model the evolution ofaggressive genes within a species. He imagined a species who meet pairwise to competeover a resource of value, v. In this contest an individual has the choice of two actions.It may either be aggressive (hawk), or non-aggressive (dove).When hawk is played against dove, the dove will flee before risking an injury, leavingthe resource, v, to the hawk. If both choose dove they display to each other with a smallcost, d (effort or time), and one takes the resource. If both choose hawk they attackPareto optimality in the quantum Hawk-Dove Game 3each other, one sustaining a large injury, i, with the other taking the resource, v. Thisleads to the symmetric pay-off matrix shown in table 1.H DH (v2− i2, v2− i2) (v, 0)D (0, v) (v2- d, v2- d)Table 1. The pay-off matrix for the Hawk-Dove Game. The expected pay-offs for the“row player” are the left-hand entries, the expected pay-offs for the “column player”are the right-hand entries.As is standard, and indeed intuitive, we assume the hierarchy of values to be,0 < 2d < v < i. (1)These conditions ensure hawks gain a negative expected pay-off when competing againstother hawks, while doves gain a positive expected pay-off when competing against otherdoves.The classical game gives rise to a mixed ESS (p∗, 1−p∗) (Maynard Smith 1982). Thismixed strategy is to play hawk with probability p∗ = v+2di+2d. This equilibrium is inefficienthowever, as the average pay-off received per round to an individual in a population withthis resident strategy is(i−vi+2d) (v2− d)which is less than the non-aggressive, cooperativepay-off of v2− d.2.1. QuantisationA few quantisations of this game have already been examined, including models byNawaz & Toor (2010) and Hanauske et al. (2010). Nawaz & Toor (2010) use a densitymatrix quantisation approach to examine the conditions required on the initial statefor a pure strategy NE to exist, but do not investigate the entanglement of these initialstates. Hanauske et al. (2010) also use the model of Eisert et al. (1999) to show thatvarying the entanglement in the system can produce non-aggressive ESSs. They onlyconsider three specific HDGs however, and only a few fixed values of entanglement. Weconsider both generalised entanglement, and the most general HDG.To quantise this game we use an analogous process to Eisert et al. (1999), shown infigure 1. The process is as follows. We start by associating the states |0〉 and |1〉 withFigure 1. The process I use to implement the quantum Hawk-Dove game.the strategies dove and hawk respectively. We then assign one qubit in C2 to each playerPareto optimality in the quantum Hawk-Dove Game 4in the dove state. Hence we describe the state of the game at this point as |DD〉. Thisallows us to associate quantum strategies with classical strategies. As a player wantingto use the dove strategy does not want the state of their qubit to change, and a playerchoosing the hawk strategy wishes to rotate the state of their qubit from |0〉 to |1〉, thenatural association of operators is,D̂ ∼(1 00 1), Ĥ ∼(0 1−1 0). (2)To generalise the entanglement in this game we use the entangling gate Ĵ(γ) =eiγĤ⊗Ĥ , where we have absorbed the usual power of 1/2 into gamma for simplicity. Thisproduces the initial state |ψi〉 = cos(γ)|DD〉+ i sin(γ)|HH〉.Here γ ∈ [0, π/4] can be viewed as a parameter of entanglement. If γ = 0the initial state of the game is |ψi〉 = |DD〉, a product state. If γ = π/4, |ψi〉 =1√2(|DD〉+ i|HH〉) which is a maximally entangled state. Thus varying γ between 0and π/4 allows us to control the prepared entanglement in the initial state.Once the qubits have been entangled the players select a unitary operator, ÛR ∈SR ⊂ C2 × C2 (ÛC ∈ SC ⊂ C2 × C2) to apply to their qubit. Here SR (SC) is calledthe row (column) player’s strategy space. After the application of these operators thequbits are disentangled using the gate Ĵ† and the final state, |ψf〉, is measured. Bymaking a measurement in the computational basis the measured eigenvalues will beeither zero or one. A zero corresponds to the measurement of the state |0〉 = |D〉, i.e.the corresponding player receives the pay-off for the classical dove strategy. Similarlythe eigenvalue one corresponds to the classical hawk strategy.For simplicity we rewrite the pay-off matrix as,H DH (−a, −a) (b, 0)D (0, b) (c, c)Table 2. The simplified version of the pay-off matrix for the HDG shown in table 1,where a = i2 −v2 , b = v and c =v2 − d, so a, b, c > 0.2.2. Two-Parameter Strategy SpaceWe consider the specific case where each player has access to the two-parameter strategyspace, S. Thus SR = SC = S, where S contains all operators of the form,Û(θ, φ) =(eiφ cos(θ) sin(θ)− sin(θ) e−iφ cos(θ))(3)with 0 ≤ θ ≤ π/2 and 0 ≤ φ ≤ π/2. Thus the row (column) player’s choice of strategyis characterised by their choice of θR and φR (θC and φC).This strategy space was originally constructed by Eisert et al. (1999) to show theexistence of a Pareto optimal NE for a maximally entangled game in the quantumPareto optimality in the quantum Hawk-Dove Game 5Prisoner’s Dilemma. We now extend this to the quantum HDG to show the analogousPareto optimal ESS exists for all HDGs when the entanglement in the initial state, |ψi〉,exceeds a threshold value which we determine.It is important to observe that the classical game is necessarily contained withinthis quantisation. By restricting the players to the strategies characterised by φ = 0,a player’s choice of θ ∈ [0, π/2] is a bijection to the choice of classical mixed strategiescharacterised by p ∈ [0, 1].We calculate the expected pay-off for the row player for generalised entanglementin this two-parameter strategy space using the equation,$R(θR, φR, θC , φC , γ) = cPDD + bPHD − aPHH , (4)where Pσσ′ = |〈σσ′ |ψf〉|2 is the joint probability of measuring σ, σ′ ∈ {D,H}.These equations yield the expected pay-off to the row player as,$R(θR, φR, θC , φC , γ) = c| cos(θR) cos(θC)[cos(φR + φC) + i cos(2γ) sin(φR + φC)]|2+b|2 cos(γ) sin(γ) sin(φR) cos(θR) sin(θC)− cos(φC) sin(θR) cos(θC)[1 + i cos(2γ)]|2−a| sin(θR) sin(θC) + 2 cos(γ) sin(γ) sin(φR + φC) cos(θR) cos(θC)|2.(5)As this game is symmetric, the column player’s pay-off can be found byinterchanging the subscripts “R” and “C”. Note that these pay-offs depend on theentanglement parameter γ.We now show that (Q̂, Q̂) becomes an ESS iff, γc ≤ γ ≤ π/4, where,Q̂ = Û(0, π/2) =(i 00 −i). (6)I.e. this strategy Q̂ becomes the game’s new unique ESS when the entanglement in thegame exceeds the critical value, γc, which will be determined.As this is a symmetric game, we can show (Q̂, Q̂) is a Nash equilibrium by showingQ̂ is in the set of best responses to itself, B(Q̂), when the entanglement in the gameexceeds γc.Calculating the pay-off received against the strategy Q̂ using equation 5 and lookingfor the critical values of φ and θ that maximise/minimise this pay-off gives the threestrategies D̂, Ĥ and Q̂. From equation 5 the expected pay-offs these strategies yieldagainst Q̂ are,$R(D̂, Q̂, γ) = c− 4a cos2(γ) sin2(γ), (7)$R(Q̂, Q̂, γ) = c, (8)$R(Ĥ, Q̂, γ) = b cos2(2γ). (9)Note that Q̂ is a best response to itself iff the pay-off,$R(Q̂, Q̂, γ) ≥ $R(ÛR, Q̂, γ), ∀ÛR ∈ S. (10)Pareto optimality in the quantum Hawk-Dove Game 6Figure 2. The expected pay-off to the row player when playing the strategiesD̂ = Û(0, 0) (dotted line), Ĥ = Û(π/2, 0) (dashed line) and Q̂ = Û(0, π/2) (solidline) against a column player playing Q̂, shown as a function of the entanglementparameter γ. Here we have used the specific values v=50, i=100 and d=10, ensuring0 < 2d < v < i. D̂, Ĥ and Q̂ have been chosen because these are the strategies thatmaximise/minimise a player’s pay-off against Q̂. This graph shows that Q̂ becomesthe unique best response to itself when γc < γ ≤ π4 , and is hence an ESS.We now see that the best response to Q̂ depends on the prepared entanglement in thesystem, as demonstrated in figure 2. As D̂, Ĥ and Q̂ were found from the turning pointsof θ and φ for all γ, we need only compare these three strategies. Clearlyc ≥ c− 4a cos2(γ) sin2(γ) ∀γ (11)⇒ $R(Q̂, Q̂, γ) ≥ $R(D̂, Q̂, γ) ∀γ (12)as c, a, cos2(γ) and sin2(γ) are all non-negative. However, we only have$R(Q̂, Q̂, γ) ≥ $R(Ĥ, Q̂, γ) (13)iff,c ≥ b cos2(2γ). (14)This gives the critical point, γc, when c = b cos2(2γc). So,γc =12cos−1(√cb)=12cos−1(√12− dv). (15)Note that under our original formulation we required 0 < 2d < v to ensure thegame remained intuitive, which implies,π8< γc <π4. (16)As γ ∈ [0, π/4], this implies (Q̂, Q̂) is only an NE iff,γc ≥ γ ≥π4. (17)Note that γc always lies strictly within the range of γ, and hence can always be exceeded.This is shown in figure 2 where we see that playing Q̂ against Q̂ is only the best choiceof strategy when γc ≤ γ ≤ π4 .Pareto optimality in the quantum Hawk-Dove Game 7By the conditions stated by Maynard Smith & Price (1973), (Q̂, Q̂) is also an ESSin the region γc ≤ γ ≤ π4 . Indeed, when γ > γc, Q̂ becomes the unique best response toitself, and is hence an ESS.As the pay-off received from playing Q̂ against Q̂ is c (the cooperative, non-aggressive pay-off), this ESS is also Pareto optimal.3. Summary and conclusionsWe have shown that for every intuitive HDG as defined by the conditions 0 < 2d < v < i,there exists a critical threshold of entanglement, γc ∈ (π8 ,π4), such that when theentanglement prepared in the initial state exceeds this value, a new Pareto optimal ESSreplaces the old, inefficient, classical ESS when the players are restricted to a specifictwo-parameter strategy space.In other words, for high enough levels of entanglement, populations of self-interestedagents will benefit from being non-aggressive when competing over resources if theirstrategy space is restricted to the two-parameter set of unitary operators defined inequation 3.Few applications of the quantum HDG are currently known, although Hanauskeet al. (2010) speculate that the quantum HDG may be used in aid of the prevention ofthe type of economic crisis observed in 2008.No known applications of this two-parameter strategy space are known. Henceit may prove more relevant for practical application to examine this game when bothplayers are free to apply any quantum operator of their choosing. Under these conditionsit would be interesting to investigate whether we still observe an increase in expectedpay-off at the game’s ESS when compared to the classical result.This original result gives further insight into the intriguing power of entanglementand the effect of expanding the strategy spaces available to the players in quantumgames.BibliographyBoyd R & Richerson P J 1992 Ethology and Sociobiology 13(3), 171–195.Du J, Li H, Xu X, Shi M, Wu J, Zhou X & Han R 2001 Physics Review Letter 88(137902).Eisert J & Wilkens M 2000 Journal of Modern Optics 47(14-15), 2543–2556.Eisert J, Wilkens M & Lewenstein M 1999 Physical Review Letters 83(15).Hanauske M, Kunz J, Bernius S & König W 2010 Physica A: Statistical Mechanics and its Applications389(21), 5084–5102.Maynard Smith J 1982 Evolution and the Theory of Games Cambridge University Press.Maynard Smith J & Price G R 1973 Nature 246, 15.Meyer D A 1999 Physical Review Letters 82(5), 1052.Nawaz A & Toor A 2010 Chinese Physics Letters 27(5), 050303.Nowak M A & Sigmund K 1999 Nature 398(6726), 367–368.Poundstone W 2011 Prisoner’s dilemma Random House LLC.Von Neumann J & Morgenstern O 1947 Princeton University Press, Princeton .

Entanglement gives rise to Pareto optimality in the generalised quantum Hawk-Dove Game

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