| Synopsis: This section assumes that you have thoroughly investigated
the Hawks and Doves game, that you understand those strategies ,that you
can calculate payoff matrices, and that you understand pure and mixed ESS.
Building on this foundation, we will now consider a new strategy, Bourgeois, whose central feature is that ownership of a resource determines the behavior used in a particular contest. If a Bourgeois strategist owns, it will defend its ownership with hawk-like ferocity; if Bourgeois does not own, it will attempt to obtain the resource using display but it will not escalate to fighting. You will get a chance to calculate a payoff matrix for a three strategy game and then use this to consider whether Bourgeois is stable against either Hawk or Dove. This will prepare you to use to the next simulation which looks at evolution in a population containing these three strategies. |
On the previous hypertext page and using the Hawks and Doves simulation you examined the solutions to a simple game involving two strategies. You have learned that this game usually yields a mixed ESS, but under certain circumstances can give a pure ESS for Hawk. On the other hand, pure Dove ESSs involved unacceptable assumptions.
In the Hawks and Doves game, each contest involved situations where the competing individuals either:
(Recall that the concept of ownership was not part of the definition of either strategy). Now, there are certainly situations where ownership is irrelevant for a number of reasons. However, in numerous cases animal do possess resources which others may sometimes attempt to wrest from them. In other cases, animals seem to respect ownership of a resource -- they do not bother to attempt to take it from another individual. Are they simply being nice or is this respect the result of an evolutionary calculation based on the benefits and costs of respect for ownership?
We can certainly use game theory to see if there are situations where strategies that respect ownership are stable or form a mixed ESS with other strategies such as Hawk and Dove. In this section we will define such a strategy, which has been named Bourgeois by Maynard Smith, and we will consider it in games with Hawk and Dove. This addition to our Hawks and Doves population will :
Does ownership imply
territoriality? |
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Bourgeois is a strategy associated with respect for "ownership" (i.e., possession) of a resource. Bourgeois strategists fight to 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 simple example, we will therefore define Bourgeois as having either hawk-like or dove-like behavior contingent on whether it or the other contestant owns the resource. To recapitulate, if Bourgeois then:
This is in contrast with both Hawk and Dove strategists who always play the same strategy regardless of whether or not they or their opponent own a resource.
| Does Bourgeois seem to you like a behavioral
strategy that an animal might really employ? Critique the strategy. Think about this and then press here for discussion. |
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We will define B very simply in light of H and D strategies. We will assume 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 continue to keep things simple, we will assume that as in the H and D contest, you don't know who you are playing against until the game starts (otherwise Hawks 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 same default payoffs to Hawks and Doves as were used on the previous page into the equations in the matrix above, then the payoff matrix for our three strategy 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 the rules we learned earlier for comparing the payoffs of different encounters were for two-strategy games. However, a three-strategy game can be broken down into simpler two-strategy contests. 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 in both of these separately, then it is reasonable to conclude that B is a pure 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 B are better than what H receives when H interacts with B (both of which interactions will 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 against H.
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 confirm this result for these benefits and costs by using a simulation in the next section. You will also have the chance to try to find sets of benefits and costs where B is not a pure ESS. And you will be able to deepen your understanding of how a population might evolve when three strategies are present.
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Copyright © 2000 by Kenneth N. Prestwich
About Fair Use 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 predominant use (something less than ownership) of some physical space of the environment. It also implies ownership or predominant use of at least some of the resources in this space (resources are defined with reference to individuals needing to make use of them). However, an animal can "own" a resource without holding what is usually construed as a territory. One example might be a male guarding its mate through prolonged copulation as is common in insects.
When discussing the Bourgeois strategy it is common to use the words territory and ownership interchangeably although they are not exactly the same thing.
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Are the Bourgeoisie sophisticated? With apologies to Virginia Woolf and leaving discussions of high, low and middle brow aside, we can ask 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, is a rather simple-minded strategy. For instance, an individual practicing Bourgeois decides whether or not to fight entirely on whether or not it owns the resource under contention. Bourgeois has the property of being an UNCORRELATED ASYMMETRY. The asymmetry in a contest traces to the fact that the player either owns or does not own (the opponent owns) a resource. The strategy is uncorrelated since condition, fighting ability, and likelihood of 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 the resource.
This is fine as far as it goes -- animals that hold a resource are more likely to fight. But it is not uncommon for an animal to consider the likelihood that it will prevail in a given contest. Fights are most often escalated affairs that follow considerable amounts of assessment -- it has been repeatedly documented that the most serious fights occur when the parties are evenly matched. Thus, a more sophisticated treatment would correlate likelihood of escalation to a fight and continued fighting with factors like the value of the resource and likelihood of injury (which is determined in large part by assessment of the fighting abilities of both contestants).
Nevertheless, even though it is simple-minded, it still is an advance over simple H and D strategies - animals that own resources often are more likely to fight and those that do not are often more likely to display or avoid an escalated fight with an owner.
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