| Synopsis: This section assumes that you have thoroughly investigatedthe Hawks and Doves game, that you understand those strategies ,that youcan calculate payoff matrices, and that you understand pure and mixed ESS. Building on this foundation, we will now consider a newstrategy, Bourgeois, whose central feature is that ownership of a resourcedetermines the behavior used in a particular contest. If a Bourgeois strategistowns, it will defend its ownership with hawk-like ferocity; if Bourgeoisdoes not own, it will attempt to obtain the resource using display but itwill not escalate to fighting. You will get a chance tocalculate a payoff matrix for a three strategy game and then use this toconsider whether Bourgeois is stable against either Hawk or Dove. This willprepare you to use to the next simulation which looks at evolution in apopulation containing these three strategies. |
On the previous hypertext pageand using the Hawks and Doves simulation you examined the solutions to asimple game involving two strategies. You have learned that this game usuallyyields a mixed ESS, but under certain circumstances can give a pure ESSfor Hawk. On the other hand, pure Dove ESSs involved unacceptable assumptions.
In the Hawks and Doves game, each contest involved situations where thecompeting individuals either:
(Recall that the concept of ownership was not part of the definitionof either strategy). Now, there are certainly situations where ownershipis irrelevant for a number of reasons. However, in numerous cases animaldo possess resources which others may sometimes attempt to wrest from them.In other cases, animals seem to respect ownership of a resource -- theydo not bother to attempt to take it from another individual. Are they simplybeing nice or is this respect the result of an evolutionary calculationbased on the benefits and costs of respect for ownership?
We can certainly use game theory to see if there are situations wherestrategies that respect ownership are stable or form a mixed ESS with otherstrategies such as Hawk and Dove. In this section we will define such astrategy, which has been named Bourgeois by Maynard Smith, and wewill consider it in games with Hawk and Dove. This addition to our Hawksand Doves population will :
Does ownership implyterritoriality? |
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Bourgeois is a strategy associated with respect for "ownership"(i.e., possession) of a resource. Bourgeois strategists fightto hold on to resources they already own (i.e., act like a hawk)and they display over resources that they do not own. In our simpleexample, we will therefore define Bourgeois as having either hawk-like ordove-like behavior contingent on whether it or the other contestant ownsthe resource. To recapitulate, if Bourgeois then:
This is in contrast with both Hawk and Dove strategists who always playthe same strategy regardless of whether or not they or their opponent owna resource.
| Does Bourgeois seem to you like a behavioralstrategy that an animal might really employ? Critique the strategy. |
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We will define B very simply in light of H and D strategies. We willassume that B has a 50% chance of owning a resource in any contest. Thus,in any contest with B, there is a 50% chance that it will act like Hawk(owns) and a 50% chance that it will be a Dove (doesn't own). To continueto keep things simple, we will assume that as in the H and D contest, youdon't know who you are playing against until the game starts (otherwiseHawks could avoid Hawks, for instance, and they would be a pure ESS). Thus,the payoff matrix (with links to explanations):
Hawk | Dove | Bourgeois | |
| Hawk | |||
| Dove | |||
| Bourgeois |
If we insert the samedefault payoffs to Hawks and Doves as were used on the previous pageinto the equations in the matrix above, then the payoff matrix for our threestrategy game is:
Hawk | Dove | Bourgeois | |
Hawk | -25 | +50 | +12.5 |
Dove | 0 | +15 | +7.5 |
Bourgeois | -12.5 | +32.5 | +25 |
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Notice that there is no simple way to answer this question since therules we learned earlierfor comparing the payoffs of different encounters were for two-strategygames. However, a three-strategy game can be broken down into simpler two-strategycontests. Since we already know the outcome of H vs. D (for these payoffs),the contests of interest are H vs. B and D vs. B. If B is a pure ESSs inboth of these separately, then it is reasonable to conclude that B is apure ESS vs. a mix of Hawk and Dove because it can invade both.
If we follow the rules
For B vs. H:
E(B,B) is greater than E(H,B) (that is, +25 is greater than +12.5). Thus,for any frequency when B interacts with B the fitness consequences to Bare better than what H receives when H interacts with B (both of which interactionswill occur at the same frequency). If, for completeness, we turn this around:E(H,H) = -25, which is less than E(B,H) = -12.5. Thus B is stable againstH.
For B vs. D:
E(B,B) = +25, which is greater than E(D,B) = +7.5; if turned around,E(D,D) = +15, which is less than E(B,D) = +32.5.
Thus, bourgeois is stable against both Hawk and Dove.
You'll be able to confirmthis result for these benefits and costs by using a simulation in the nextsection. You will also have the chance to try to find sets of benefitsand costs where B is not a pure ESS. And you will be able to deepen yourunderstanding of how a population might evolve when three strategies arepresent.
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Copyright © 2000 by Kenneth N. Prestwich About FairUse of these materials Last modified 11 - 10 - 2000 |
Ownership is a broader and different concept than is territory.The exact meaning of territory is muddied and beyond the scope of this discussion.For our purposes, let's just say that territory implies ownership or predominantuse (something less than ownership) of some physical space of the environment.It also implies ownership or predominant use of at least some of the resourcesin this space (resources are defined with reference to individuals needingto make use of them). However, an animal can "own" a resourcewithout holding what is usually construed as a territory. One example mightbe a male guarding its mate through prolonged copulation as is common ininsects.
When discussing the Bourgeois strategy it is common to use the wordsterritory and ownership interchangeably although they are not exactly thesame thing.
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Are the Bourgeoisie sophisticated? With apologies to VirginiaWoolf and leaving discussions of high, low and middle brow aside, we canask the question of whether or not Bourgeois represents a reasonably sophisticated(i.e., realistic) behavioral strategy.
Bourgeois, like the behaviors on which it is based, Hawk and Dove, isa rather simple-minded strategy. For instance, an individual practicingBourgeois decides whether or not to fight entirely on whether or not itowns the resource under contention. Bourgeois has the property of beingan UNCORRELATED ASYMMETRY. The asymmetry in a contest traces to thefact that the player either owns or does not own (the opponent owns) a resource.The strategy is uncorrelated since condition, fighting ability, and likelihoodof victory have nothing to do with the decision of whether or not to fight.This decision is based entirely on whether or not the individual owns theresource.
This is fine as far as it goes -- animals that hold a resource are morelikely to fight. But it is not uncommon for an animal to consider the likelihoodthat it will prevail in a given contest. Fights are most often escalated affairs that followconsiderable amounts of assessment -- it has been repeatedly documentedthat the most serious fights occur when the parties are evenly matched.Thus, a more sophisticated treatment would correlate likelihood of escalationto a fight and continued fighting with factors like the value of the resourceand likelihood of injury (which is determined in large part by assessmentof the fighting abilities of both contestants).
Nevertheless, even though it is simple-minded, it still is an advanceover simple H and D strategies - animals that own resources often are morelikely to fight and those that do not are often more likely to display oravoid an escalated fight with an owner.
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