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1、Dyn Games Appl (2014) 4:407431 DOI 10.1007/s13235-014-0111-5 The Iterated HawkDove Game Revisited: The Effect of Ownership Uncertainty on Bourgeois as a Pure Convention Mike Mesterton-Gibbons Tugba Karabiyik Tom N. Sherratt Published online: 2 April 2014 Springer Science+Business Media New York 2014
2、 Abstract Classicalevolutionarygametheoryshowsthatrespectforownership(“Bourgeois” behavior)canariseasanarbitraryconventiontoavoidcostlydisputes,butthesametheoryalso predicts that a paradoxical disrespect for ownership (“anti-Bourgeois” behavior) can evolve under the same conditions. Given the rarity
3、 of the latter strategy in the natural world, it is clearthattheclassicalmodelislackinginsomeimportantbiologicaldetails.Forinstance,the classical model assumes that roles of owner and intruder can be recognized unambiguously. However, in the natural world there is often confusion over ownership, med
4、iated for example by the temporary absence of the owner. We show that if intruders sometimes believe them- selves to be owners, then the resulting confusion over ownership can broaden the conditions under which Bourgeois behavior is evolutionarily stable in the one-shot HawkDove game. Likewise,intro
5、ducingmistakesoverownershipintoamorerealisticgamewithrepeatedinter- actions facilitates the evolution of Bourgeois behavior where previously such a result could arise only if owners are intrinsically more likely to win fi ghts than intruders. Collectively, therefore,wefi ndthatmistakesoverownershipf
6、acilitatetheevolutionofBourgeoisbehavior. Nevertheless, relaxing the assumption that ownership is unambiguously recognized does not appear to completely explain the extreme rarity of anti-Bourgeois behavior in nature. Keywords Resource-holding potential Animal confl ict Evolution of private property
7、 M. Mesterton-Gibbons (B) T. Karabiyik Department of Mathematics, Florida State University, 1017 Academic Way, Tallahassee, FL 32306-4510, USA e-mail: mestertomath.fsu.edu T. Karabiyik e-mail: T. N. Sherratt Department of Biology, Carleton University, 1125 Colonel By Drive, Ottawa, ON K1S 5B6, Canad
8、a e-mail: sherrattconnect.carleton.ca 408Dyn Games Appl (2014) 4:407431 1 Introduction The starting point for almost every game-theoretic analysis of confl ict over a territory or any other resource is the now classic, symmetric, two-strategy HawkDove game 24. The fi rst strategy is Hawk or H, which
9、 is aggressive; the other is Dove or D, which is peaceable. In a single contest between Hawk and Dove, H will prevail without a fi ght, thereby increasing its reproductive fi tness by V, the value of the resource; whereas D will lose, thereby increasing its fi tness by zero. In a single contest betw
10、een two Hawks, each has an equal probability of winning and thereby increasing fi tness by V Ce, where Ceis the cost of engagement for either animal, or losing and thereby decreasing fi tness by Ce+Ci, where Ciis the additional cost of injury for the loser; and so the payoff is 1 2(V Ce) + 1 2(Ce Ci
11、) = 1 2(V C), where C = 2Ce+Ci . These assumptions are unambiguous, and together determine the fi rst three entries of the payoff matrix A = 1 2(V C) V 0 1 2V = 1 2V 1 2 01 (1) inwhichmatrixelementaijdenotesthepayofftoani-strategistinapopulationof j-strategists and = C V (2) is a cost-value ratio. B
12、ecause C denotes the average cost of a pair of fi ghts, however, it lacks biological appealdespite being a convenient mathematical notationunless Ceis so small compared to Cithat we can set Ce = 0 and interpret C as the cost of losing a fi ght, which is both how A was initially framed by 24, p. 12 a
13、nd how it is usually interpretedincluding here. Depending on what is meant by not escalating, however, there are three approaches to obtain the payoff 1 2 V for a contest between two Doves. The fi rst is to suppose with 24, p. 13 that the resource is shared, as recently discussed in the context of f
14、i ghts over food by 31. The secondand most commonapproach is to suppose with 25, p. 161, 17, p. 58, 6, p. 70, and 26, p. 89 that the territory is not shared, but that the two contestants are equally likely to be fi rst to retreat in a war of attrition; in which case, the payoff to each is 0 or V, wi
15、th equal probability. The third approach is to suppose that the territory belongs to the individual who gets there fi rst, which is equally likely to be either; and because they arrivesimultaneouslywithzeroprobability,thepayoffisagain V or0,withequalprobability. Following 28 we will say that the Dov
16、eDove interaction is between intrusive individuals accordingtothesecondapproach,andbetweenunintrusiveindividualsaccordingtothethird. We will not concern ourselves further with the fi rst approach; rather, we will assume that territories are indivisible and of lasting value. Because all three approac
17、hes to DoveDove interaction lead to the same expected payoff, 1 2V, the differences between them in the symmetric game are inconsequential. Nevertheless, these differences become real when asymmetry arises through segregation of roles. Maynard Smith 24, pp. 9495 supposes in this context that each of
18、 two contestants is equally likely to be an owner or an intruder, and that each knows for certain which role it is in. He then introduces two role-dependent strategies B (Bourgeois) and X (anti-Bourgeois); B plays Hawk if owner but Dove if intruder, whereas X plays Dove if owner but Hawk if intruder
19、. The game now has four strategies, namely, H, B, X, and D, with H and D correspondingly reinterpreted to signifystrategies ofadoptingHawk andDovebehaviorin either role,respec- tively. A key parameter now arising is the resource-holding potential or RHP 25,33. In Dyn Games Appl (2014) 4:407431409 Ta
20、ble 1 Payoffs with correlated asymmetry and unintrusive Doves, where denotes an owners probability of winning an escalated contest against an intruder HBXD H 1 2(V C) (1 1 2)V 1 2C 1 2(1 + )V 1 2(1 )C V B 1 2V 1 2(1 )C 1 2V 1 2V 1 2(1 )C 1 2V X 1 2(1 )V 1 2C (1 1 2)V 1 2C 1 2V V D0 1 2V 0 1 2V This
21、payoff matrix agrees with Table 3 of 28, p. 200 for = 1 2 Table 2 Payoffs with correlated asymmetry and intrusive Doves, agreeing with Table 2 of 28, p. 200 for = 1 2 HBXD H 1 2(V C) (1 1 2)V 1 2C 1 2(1 + )V 1 2(1 )C V B 1 2V 1 2(1 )C 1 2V 1 2( 1 2 + )V 1 2(1 )C 3 4V X 1 2(1 )V 1 2C 1 2( 3 2 )V 1 2C
22、 1 2V 3 4V D0 1 4V 1 4V 1 2V our analysis, differences in RHP refl ect a correlated asymmetry in ownership (with owners, for example, tending to be better fi ghters), as opposed to uncorrelated variation in fi ghting ability 24. To incorporate this effect of RHP, we denote the probability that an ow
23、ner wins a fi ght by (hence the probability that an intruder wins by 1). It is then straightforward to show that the new payoff matrix A is Table 1 or 2, according to whether Doves are unin- trusive or intrusive. For example, the payoff to B against X in an unintrusive population is 1 2V+(C)(1)+ 1 2
24、0 = 1 2V 1 2(1)C becausethe B-strategistisequallylikelyto beanescalatingBourgeoisownerthatwinswithprobabilityandloseswithprobability1 againstanescalatinganti-Bourgeoisintruderoranon-displayingBourgeoisintruderagainsta displayinganti-Bourgeoisowner,yieldinga23inTable1.Bycontrast,thepayoffto B against
25、 X inanintrusivepopulationis 1 2V +(C)(1)+ 1 2 1 2V = 1 2( 1 2+)V 1 2(1)C, because the B-strategist is equally likely to be an escalating Bourgeois owner against an escalating anti-Bourgeois intruder or a displaying Bourgeois intruder against a display- ing anti-Bourgeois owner, yielding a23in Table
26、 2. The other matrix entries are similarly obtained. Note that 10 of the 16 entries are the same in both matrices. The ones that dif- fer are those corresponding to interaction between two different strategies from the subset B, X, D. Of course, this approach also assumes that an individuals role is
27、 independent of its strategy; thus, for example, Bourgeois are no more likely to be owners than are anti- Bourgeois.20,pp.2224hascriticizedthisassumption,notingthatitisunlikelytobesatis- fi ed “except, perhaps, in relatively unusual situations;” however, we relax the assumption in Sect. 3. Strategy
28、k is an evolutionarily stable strategy or ESS sensu 24 if the entries of the payoff matrix satisfy akk aikfor all i and either akk aik(strong ESS) or aki aii(weak ESS) for all i = k. Thus conditions for a strategy to be an ESS are readily obtained by inspection of Tables 1 and 2. Although whether H
29、or X is an ESS is unaffected by whether Doves are 410Dyn Games Appl (2014) 4:407431 H No ESS X 01 0.5 1 UNINTRUSIVE CASE H B B or X 01 0.5 1 INTRUSIVE CASE Fig. 1 Monomorphic ESS diagram in the plane for certain ownership, where = C/V is the cost-value ratio and is RHP or owner advantage. The right-
30、hand panel is a pictorial summary of results fi rst presented by 25, p. 166, left-hand column intrusive or unintrusive, the results for B are quite different, as shown by Fig. 1, where the curve = max(1,) 1 + (3) forms the lower boundary of the region in which either B is the ESS or none of the four
31、 strategies is an ESS. These two diagrams make clear what is puzzling about anti-Bourgeois behavior: even if is quite close to 1, X remains an ESS as long as a fi ght would be suffi ciently costly (specifi cally, as long as /(1 ); moreover, in the unintrusive case, X is then the only ESS. Thus, alth
32、ough there is never a fi ght, resources are routinely ceded to intruders by owners who would be exceedingly likely to succeed in defending their resources if only they escalated; whereas under B, the animals who refrain from escalating are those who would have little chance of winning a fi ght in an
33、y case. Given the relative biological plausibility of the strategies, 24, p. 102 referred to X as a “paradoxical” strategy and to B as a “common-sense strategy.” Although fi ghts are avoided by any population that adopts either strategy, B respects property, whereas X does not. That either B or X ca
34、n be an ESS is especially puzzling when = 1 2, so that owners are not favored in the event of an actual fi ght, and the asymmetry of ownership is said to be uncorrelatedwithRHP25,p.159.Therearetheninprinciplealwaystwostrategicallystable conventions through which fi ghting costs can be avoided by an
35、intrusive population, namely, ownership-respecting Bourgeois and ownership-disrespecting anti-Bourgeois. In theoryin terms of the basic modelthere is no reason why either is more likely to arise than the other. In practice, however, the ownership-respecting convention appears to be extremely common
36、innature,whereastheownership-disrespectingconventionappearstobeextremelyrare21. We seek to explain why both arise so readily in principle, yet only one is readily observed in practice.Inparticular,howcanpopulationsconsistingofindividualswhoinvariablyfi ghtover territories evolve to non-fi ghting pop
37、ulations that almost invariably observe the ownership- respectingconventionattainedthroughBourgeois,andonlyinrarecircumstancesifever observetheownership-disrespectingconventionattainedthroughanti-Bourgeois?Ofcourse, other tie-breaking conventions could also evolve to reduce confl ictsuch as who has
38、the longer thumb 7but it is hard to envisage a more straightforward and less ambiguous trait to evaluate than occupancy. Mesterton-Gibbons 28 suggested one such pathway using a model of competition over sites. The RHP , defi ned above, is one of two key parameters for this model, which is a specialc
39、aseoftheonetobedescribedinSect.3below.Thesecondparameteristheprobability Dyn Games Appl (2014) 4:407431411 Hawk H Bourgeois B Bourgeois B or anti Bourgeois X H or X X HB HX SURVIVAL PROBABILITY RESOURCE HOLDING POTENTIAL PROBABILITY THAT OWNER WINS predation high predation low w 00.51 0.5 1 Fig. 2 C
40、artoon of ESS diagram as a function of survival probability w and RHP for the intrusive case. Redrawn after 30 of surviving the current period, denoted by w, which is inversely related to predation. An ESS diagram for the model is cartooned in Fig. 2, where the environment facing a population is rep
41、resented by a point with coordinates (w,). Predation is high to the left of the diagram, and low to the right; and in Maynard Smith and Parkers 25 terms, the asymmetry of ownership is uncorrelated with RHP when = 1 2 (the base of the rectangle) and correlated if 1 2 (the interior). When interpreting
42、 such adiagram, we must keep in mind that starting conditionscaninfl uencewhatESSisarrivedat:whenthepopulationpointshiftsintoaregion where there are two ESSs, we expect the population to remain at the ESS that it was at before the shift. So, as conditions change, they can infl uence not only the cur
43、rent ESS, but also which ESS is evolved toward when conditions change even further. This point is particularly relevant to the far right of the diagram where B and X are both evolutionarily stable. Although some aspects of the true ESS diagram (e.g., the size of the H-or-X region) have been exaggera
44、ted or suppressed for the sake of clarity in Fig. 2, the overall topology is correct, and what it shows is the following. If initially high predation is associated with low RHP so that is not much bigger than 1 2, and if subsequently predation decreases and if the population tracks the ESS, then it
45、will evolve from H to X. Moving to the right below the dashed line in Fig. 2, when the population moves into the H-or-X region it will stay at H, because it was at H; but it will switch to X in the X-only region, and it will stay at X in the B-or-X region, because it was at X. Likewise, above the da
46、shed line, the population will shift from H to B under decreasing predation. Thus B can arise under decreasing predation only if 1 2; if = 1 2, then X will arise instead. Thus 28 predicts that respect for private property cannot arise unless the asymmetry of ownership is correlated with RHP, in whic
47、h case anti-Bourgeois will be extremely rare. By contrast, anti-Bourgeois will readily arise if the asymmetry is uncorrelated. Nevertheless, these results are predicated on certainty of ownership. Our purpose here is to explore whether ownership uncertainty can alter the above conclusions. Mistakes
48、over ownership may be relatively commonplace in nature. Such instances could include times when two individuals arrive almost simultaneously at a valuable resource, so thatitisnotentirelyclearwhogottherefi rst.Forexample,13observedanumberofnatural 412Dyn Games Appl (2014) 4:407431 territorial contes
49、ts between male tarbrush grasshoppers (Ligurotettix plunum) in which the two individuals arrived at the encounter site so close together in time that it prevented the observerfromunequivocallyidentifyingtheresidentandintruder.Perhapsmoreimportantly, the current owner may temporarily leave the defended resource, only to fi nd it occupied on its return. For example, 38 regularly observed instances of male speckled wood butterfl ies (Pararge aegeria) settling in territories during a temporary absence of the original occupant. I