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Behavioral Ecology Vol. 12 No. 6: 753-760
© 2001 International Society for Behavioral Ecology
Dangerous games and the emergence of social structure: evolving memory-based strategies for the generalized hawk-dove game
Center for Ecology, Evolution and Behavior and T.H. Morgan School of Biological Sciences, University of Kentucky, Lexington, KY 40506, USA
Address correspondence to P.H. Crowley. E-mail: pcrowley{at}pop.uky.edu
Received 5 March 2000; revised 9 December 2000; accepted 29 January 2001.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONRESULTSDISCUSSIONREFERENCES |
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How can stable relationships emerge from repeated, pairwise interactions among competing individuals in a social group? In small groups, direct assessment of resource-holding potential, which is often linked to body size, can sort individuals into a dominance hierarchy. But in larger groups, memory of behavior in previous interactions may prove essential for social stability. In this study, I used a classifier-system model (similar to a genetic algorithm) to evolve strategies that individuals play in pairwise games that are potentially dangerous (i.e., fitness benefits of winning are outweighed by losing costs that result mainly from risk of injury). When the two possible responses by each player in a single interaction are designated C (= careful) and D (= daring), the average gain is highest if responses are complementary (i.e., one plays C and the other plays D). Stable dominance relationships, which depend on such complementarity across a sequence of interactions, are more common when both sizes are known to the contestants, when strategies are based on memory, and when combat is especially dangerous. Two key memory-based strategies generated by the classifier system (DorC and CAD) are particularly adept at achieving and maintaining complementarity; these strategies, which represent building blocks from which social structure can arise, are linked here with pairwise contests for the first time. When most individuals in the group differ in size, stable dominance relationships generally yield transitive hierarchies consistent with size. Empirical tests of these predicted patterns are proposed.
Key words: body size, complementarity, game theory, resource-holding power, social behavior.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONREFERENCES |
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Social group structure is often thought to arise from pairwise contests for resources. Dominant—subordinate relationships established within pairs can generate dominance hierarchies (Landau, 1951a
,b),
territorial systems (Davies and Houston,
1984; Maynard Smith,
1976), and other important social phenomena. Here I address
repeated, pairwise, fitness-related interactions among individuals in a
population or social group when these interactions can result in dangerous
combat (i.e., combat in which costs of losing, which are primarily associated
with risk of injury, exceed the benefit of winning). Of fundamental importance
in understanding such interactions are the extent to which individuals may
differ in their chances of prevailing in combat and the information they have
about those chances. Implications of these two types of information for
potentially agonistic pairwise interactions and social structure that may
arise from them are the focus of this analysis.
Individuals in a population may differ widely in their likelihood of
prevailing in combat (i.e., they differ in resource-holding power, or RHP;
Parker, 1974). In this
article, I use body size as the relevant indicator of RHP, but this is mainly
for clarity and convenience; the analysis should also apply to many systems
for which other features are better RHP indicators.
Some animals are capable of obtaining reliable sensory information outside
of combat range and may accurately estimate the opponent's size
(Clutton-Brock and Albon, 1979,
Smith et al., 1994) and thus
RHP before risking a fight (Parker,
1974). The analysis to follow is based on three alternative
"information states" that characterize different social systems in
nature: (1) Interacting individuals may know enough about their own body size
and the opponent's to accurately determine their relative size and to directly
estimate the chance of winning in combat (i.e., the relative size case). (2)
Individuals may know their own size in relation to sizes in the population but
not know the opponent's size (or conceivably vice versa), providing less
information about combat effectiveness (i.e., the own size case). (3) No
reliable information about body size may be available or no differences may be
detectable—the classical hawk—dove scenario (i.e., the equivalent
size case). The implications of such differences in the information state of
interacting individuals have received little direct attention in the
literature.
Animals that do and those that do not remember previous encounters with
specific individuals differ in access to another important source of
information. The role of memory in repeated social interactions is pivotal in
the emergence of co-operation (e.g.,
Crowley et al., 1996;
Dugatkin, 1997a) but has
seldom been related directly to strategies for agonistic interactions. In this
study, memory-based strategies entail the ability of the focal individual to
base its response to an opponent on the responses of the focal and the
opponent in their previous contest.
Specific goals of this study were to (1) obtain an overview of the behavior patterns associated with different memory and RHP information states in repeated, pairwise interactions; (2) identify conditions in which dominant-submissive relationships linked or unlinked to size predominate, and clarify how this comes about; (3) understand how fighting is avoided in situations where fights are especially costly; and (4) generate testable hypotheses for social groups that differ in ability to use memory-based strategies.
To address these issues and their implications, I first characterize the
generalized hawk—dove game (Crowley,
2000; based on the hawk—dove game of
Maynard Smith, 1974;
Maynard Smith and Price,
1973). Next, the size-structured situation and some key strategies
are presented. Then a classifier-system model is introduced, modified to
include size-conditional logic, and used to evolve strategies for playing the
iterated and noniterated games in the three information states. Finally, I
interpret the results, present some testable hypotheses, and summarize ways to
follow up.
Generalized hawks and doves, memory, and symmetry
To incorporate differences in size (and thus RHP) into pairwise contests, I
used a variant of the classical formulation
(Maynard Smith and Price,
1973; Figure 1A),
the generalized hawk—dove game (GHD;
Crowley, 2000;
Figure 1B). Repeated
interactions between players that remember previous behavior are known as
iterated games. In this study, the iterated GHD is associated with
memory-based strategies in social populations. When such memory is lacking,
interactions with a particular opponent are independent of each other (or
noniterated), though they may be repeated. Information states are assumed to
be established before the interaction sequence begins (but see the
Discussion).
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GHD games between players aware that they differ in size and capable of
assessing their chances of prevailing in combat are asymmetric, and only pure
strategies can be evolutionarily stable strategies (see
Selten, 1980, on bimatrix
games). The larger individual ordinarily dominates (i.e., plays D, daring, the
aggressive response), and the smaller is ordinarily submissive (i.e., plays C,
careful, the appeasement response generally featuring nonaggressive display)
when combat is sufficiently dangerous
(Crowley, 1984;
Hammerstein, 1981;
Hammerstein and Parker, 1982;
the dominant/larger and submissive/smaller pattern is abbreviated D/s.) When
individuals cannot distinguish their sizes, the situation reverts to the
symmetric scenario of the classical hawk—dove game. When individuals
know their own size but not their opponent's (or vice versa), evolutionarily
stable strategies can be mixed for at most a single size category; larger
individuals are daring, and smaller individuals are careful (i.e., mixed
symmetry; see Crowley, 2000).
Here I address sequences of GHD interactions in the memory-based case
(iterated games) and the memory-independent case (noniterated games) for
equivalent size (symmetric), relative size (asymmetric), and own size (mixed
symmetry) information states.
Sizes, strategies, and the chance of winning in combat
In this study, I assumed that the population of interest is composed of
individuals in five discrete size categories, with the corresponding sizes
represented in increasing order by the numbers 1-5. Discrete size categories
can correspond to biologically realistic situations (e.g., interactions
between arthropods with intermolt intervals at consistent body sizes). Also,
the more analytically tractable discrete formulation may adequately
approximate the continuous case for many biological systems of interest. In an
interaction between a focal individual (F) and another (O), the five size
categories imply that there are nine possible relative sizes for the focal
(rF, ranging from -4 to +4), determined by subtracting the
size of O from the size of F.
A further assumption is that the chance of winning a fight, p, is
a function of relative size, whether or not sizes are known by the interacting
individuals. Fights are taken always to have exactly one winner and to be all
or nothing (e.g., as in Hammerstein,
1981; Mesterton-Gibbons,
1994)—not gradually escalated (e.g.,
Enquist and Leimar, 1983) or
based on attrition without risk of injury (e.g.,
Hammerstein and Parker, 1982;
Maynard Smith, 1974). (See
Hammerstein and Parker, 1982;
Riechert, 1998, on these
distinctions.) These assumptions imply that the probability of victory by the
focal individual pF = p(rF) =
p(-rO) = 1 —
p(rO). The function p(r) should
generally be sigmoid in shape (see Figure
2A, curve 2), as in the analyses presented below. But the extent
to which interacting individuals can assess and use information about their
chances of winning in combat depends on information state of the population,
as described in the Introduction. Costs of assessment are ignored.
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Two strategies are clearly important in the GHD game, with or without
memory: (1) AllD, aggressive responses (daring = D) in every interaction, and
(2) AllC, submissive responses (careful = C) in every interaction. But when
individuals can remember responses from the previous game, it becomes possible
to adjust behavior in response to the opponent's behavior as the interaction
proceeds. Moreover, in doing so, individuals can exploit an important feature
of the payoff matrix: that the mean of the off-diagonal elements exceeds the
payoff for either both careful or both daring. Thus, if two players using the
same strategy can achieve and maintain complementarity (i.e., C by one player
and D by the other), then their expected payoff thereafter is V/2.
This is the highest possible expected payoff for a symmetric iterated GHD. The
only two strategies that can accomplish this, based on memory of the previous
game alone, are DorC and CAD (Crowley et
al., 1998): (3) DorC, probabilistic responses initially, and
following non-complementarity, but repetition of the preceding response
following complementarity; (4) CAD, probabilistic responses initially, and
following noncomplementarity, but alternation of responses following
complementarity.
DorC and CAD are memory dependent because they respond in a contingent way
to their own and their opponent's previous behavior. Though logically simple,
they recognize and lock onto the complementary pattern against an opponent
playing the same strategy, perhaps after one or more noncomplementary
interactions. (See Crowley et al.,
1998, for more on these strategies and the general significance of
complementarity.) DorC does this by attempting consistently to award the
benefits of winning to one member of the interacting pair. This establishes a
clear dominant-submissive relationship, though not necessarily the
size-related D/s pattern. In contrast, CAD attempts to share the benefits of
winning between the two players by alternating winners, the antithesis of a
dominant-submissive relationship.
Classifier-system analysis
Here the classifier-system model EvA (see
Crowley, 1996, for technical
details of the baseline model), tailored to the situation at hand, is used to
simulate the evolution of strategies effective in playing the GHD game across
a range of size-assessment and memory capabilities and costs of losing in
combat. Classifier systems evolve algorithms composed of rules that specify an
action to be taken in response to particular conditions
(Holland, 1992;
Holland et al., 1986). In the
present application, the algorithms that constituted the population competed
by playing a round-robin tournament of GHD games in each generation. Overall
performances were scored and ranked, worst to best, and an individual's
probability of parenthood in the production of each offspring algorithm was
directly proportional to its rank. (This resembles standard replicator methods
but achieves higher resolution among strategies when their absolute fitness
magnitudes are similar; Davis,
1991.) Reproduction was biparental, with more successful
algorithms likely to become parents more often in forming the next generation
of algorithms. Offspring were a mixture of their parents' rules via crossover,
and proto-offspring were then subjected to mutation. Once all offspring
algorithms had thus been formed, the new generation replaced the old, a new
tournament was conducted, and so forth. Starting with random rules, this
process continued for at least the number of generations required to generate
repeatable results among runs. Because evolutionary algorithms depend on
mutation and recombination to generate novelty, these results are not
identical. But the strategies that emerge usually differ only in minor details
relative to the consistency of the main patterns. Algorithms within runs are
almost always more than 80% functionally identical (see
Crowley, 1996, for more on
consistency of results).
In the present analysis, the population was assumed to consist of 21 individuals, a number chosen to maintain variation within the population of evolving strategies while keeping the numerical analyses manageable. When each individual took its turn as focal, the remaining 20 were randomly subdivided into the 5 equally abundant size categories (Figure 2). Becoming each of the five sizes in turn, each individual interacted in an equally long series of games with each other individual, under the assumption that the focal's performance at each size weighs equally in the overall performance evaluation that determines its contributions to future generations. Parameter values used in the runs presented here are summarized in Table 1.
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Algorithms for playing the GHD game were composed of rules that differed slightly in form for size-independent and size-dependent cases. First, consider the size-independent case (i.e., equivalent size). Here each rule took the form F1/O1:F0, where F1 stands for C or D, the response by the focal individual one game ago; O1 indicates the response by the other individual in that previous game; and F0 is the response by the focal in the present game. The rule C/D:D is an example, in which the focal played C and the other played D in the previous game, so the focal plays D in the present game. F1, O1, or both could be absent in more general rules. For example, /C:C means that the focal plays C in the present game if the other played C in the last game; /:D means the focal plays D regardless of what happened in the last game.
For the size-dependent cases (i.e., relative size and own size), each rule
took the form
F1/O1:F0f0S. Here, the
focal's response in the present game was conditional on size S. The response
is F0 only if size exceeds S; otherwise, the response is
f0. The rule D/D:CD3 is an example, in which the response to both
daring in the previous game is D if size is 
3 and C otherwise. When the
rules were relative-size conditional, S ranged from -5 to +4, and the
focal's relative size was compared to S to determine whether
F0 or f0 was the relevant response. When the rules were
own-size conditional, S ranged from 0 to 5, and the focal's absolute
(own) size was compared to S to determine the response.
Rules within an algorithm interact to determine behavior as follows (see
Crowley, 1996, for more
details): (1) more specific rules (i.e., those based on more previous
responses) override more general rules when both fit the relevant history
(e.g., the third rule in the CAD example [next paragraph] overrides the first
two if the focal played C and the other played D in the previous game,
resulting in D by the focal in the present game). (2) When two or more equally
specific rules fit the previous responses, then these rules are equally likely
to be invoked (e.g., in either CAD or DorC below, if both were careful in the
previous game, then the first two rules together specify being careful in the
present game with probability 0.5). (3) Every algorithm must contain at least
one maximally general rule (e.g., the first two rules in both examples).
Note that AllC and AllD can each be specified most simply by a single rule (/:C and /:D, respectively). A careful—daring mixed strategy can be expressed with a combination of these simple rules, such as the 10-rule example (/:C; /:C; /:C; /:D; /:D; /:D; /:D; /:D; /:D; /:D), for which the probability of playing careful is 0.3. Four-rule, size-independent versions of CAD and DorC, with rules separated by semicolons, are (/:C; /:D; C/D:D; D/C:C) [CAD] and (/:C; /:D; C/D:C; D/C:D) [DorC].
To characterize the implications of size assessment and memory in the GHD
game, I focused on strategies and behavior patterns that evolved at three
different magnitudes of the cost of losing in combat (i.e., 2, 10, and 40). A
few additional runs were conducted to determine whether the general pattern of
results was especially sensitive to some of the more arbitrary assumptions
about parameter magnitudes. Extensive sensitivity analysis previously
conducted on most of the other parameters revealed no problematically strong
sensitivities (Crowley,
1996).
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RESULTS |
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TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONREFERENCES |
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Results of the analysis are summarized in Figures 3 and 4. Unsurprisingly, fighting is much more prevalent where losing is less costly, whereas mutual submissiveness or appeasement appears more often as the cost of losing rises (Figure 3). Memory-dependent strategies produced much higher frequencies of complementarity overall than did memory-independent strategies. When the cost of losing in combat was relatively low (L = 2), there was more fighting in the memory-independent scenario than with memory dependence; when the cost of losing in combat was relatively high (L = 10 and L = 40), there was considerably more mutual submissiveness with memory-independent than with memory-based strategies. Focusing on the sequence equivalent size
own size relative size indicates that increasing size information
reduces fighting and increases mean fitness for all three costs of losing in
the memory-independent case, but the situation is more complex when strategies
invoke memory. Note that when combat is extremely dangerous (L = 40),
the more frequent complementarity associated with the use of memory seems to
make size information more valuable in avoiding combat.
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Figure 4 summarizes the strategies evolved by the model, which can account for the observed patterns of behavior. As expected, the strategy patterns show that greater relative or absolute (own) size tended to produce greater aggressiveness. The increased complementarity in memory-dependent versus memory-independent cases was largely explained by the substitution of the memory-dependent strategies DorC and CAD for AllC and AllD in the absence of detectable differences in relative size—especially as losing in combat became more costly. These substitutions are unlikely to have strongly influenced the frequency of the size-linked D/s dominance pattern. (Only DorC vs. DorC in the own-size case has the potential to increase D/s via size-linked aggressiveness in response to non-complementarity; see Discussion.) Overall, Figure 4 implies that size information was much more important than memory in establishing D/s relationships. When losing in combat was not very costly, D/s appeared only when the size difference was great enough; increasing the cost of losing had little effect on the prevalence of D/s, as long as the cost of losing a fight exceeded the cost of yielding to larger individuals (i.e., L > 4).
The equivalent-size case generated the well-known hawk—dove pattern in the absence of memory: AllD for L < V — W (i.e., L = 2) and a probabilistic mixture of C and D otherwise, with C increasingly prominent in the mixture as the cost of losing increased. In the presence of memory, equivalent size produced alternative stable patterns increasingly dominated by DorC at higher costs of losing. Results for a relative size of zero (i.e., equal size) were similar to those for equivalent size, as they should be, since the chances of winning in combat are 0.5 in both cases.
With memory-dependent strategies, DorC and CAD became increasingly prominent as the cost of losing increased: both appeared over a greater range of relative sizes, and DorC extended larger absolute (own) sizes, but CAD became more frequent across all absolute size categories equally. The relative sophistication of these strategies and the dependence of their frequencies on information, size, and cost represent the greater overall strategic complexity to be expected from populations of individuals capable of remembering previous contests.
To probe the sensitivity of the classifier-system results to a few of the parameters, I used results for the relative-size case with memory and L = 10 as a baseline. I examined the effects of population size, the size distribution in the population, the p(r) function, and the number of previous games remembered on the percentages of mutual submissiveness, combat, and complementarity, and on mean fitness. With twice the larger standard error as a rule-of-thumb confidence interval, there were negligible effects of shifting population size from 21 to 11 or 41, or of switching the p(r) function from curve 2 in Figure 2A to curve 1 or 3. The only effects of major shifts in the population size distribution from uniform to strongly skewed were on fitness: when the number of "others" in size categories from 1 to 5 were, respectively, 9, 5, 3, 2, and 1, fitness accrued by being in each category for equal numbers of encounters was sharply higher; when the distribution was reversed, the corresponding fitness was sharply lower. This difference reflects the different numbers of encounters as a relatively large or relatively small individual. Strategies were influenced little or not at all. Perhaps most suggestive were runs in which strategies were based on memory of three or of five previous games, instead of the single previous game remembered in the main set of runs. The additional memory progressively increased the extent of complementarity and reduced the frequency of mutual cooperation, suggesting that more sophisticated strategies were operating. Exploration of this effect must await future work.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONREFERENCES |
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Placing this study into context
Results of the present study based on relative size in the absence of memory are in good agreement with those of previous analyses based on similar assumptions (e.g., Crowley, 1984
,
2000;
Hammerstein, 1981;
Hammerstein and Parker, 1982;
Matsumura and Kobayashi, 1998;
Maynard Smith and Parker,
1976). But this study is apparently the first to evaluate the
implications of remembering responses in the previous game for the stable
pattern of behavior resulting from the iterated hawk—dove game (cf.
Houston and McNamara, 1990, a
repeated hawk—dove analysis without memory). Because memory of previous
encounters is important in establishing relationships among individuals in
social populations (Brown et al.,
1982), and because the generalized hawk—dove game may
adequately characterize the nature of interactions among individuals in at
least some such populations (see Maynard
Smith, 1982), the patterns found here may help clarify the
relationship between behavior and social structure in natural systems. For
example, maximal group sizes expected to be consistent with the formation of
transitive dominance hierarchies (e.g.,
Mesterton-Gibbons and Dugatkin,
1995) increase when the higher frequencies of D/s patterns
resulting from memory are taken into account. Moreover, differences that arise
from whether or not the opponent's size is known have not previously been
considered in the context of the hawk—dove game. Cross-classifying
information about sizes with the presence or absence of memory provides a
useful overview of information effects on strategies that emerge.
Crayfish and larval odonates—testable predictions
Crayfish seem to be a good example of a group that uses memory and can
accurately assess relative size before engaging in potentially dangerous
combat (Bovbjerg, 1953;
Pavey and Fielder, 1996). But
accurate assessment is probably much more difficult for blind cave-crayfish,
or for any individual faced with an unfamiliar conspecific at night. Larval
odonates can assess relative size in the light but not in the dark
(Crowley et al., 1988;
Hopper et al., 1996;
McPeek and Crowley, 1987), yet
they do not appear to recognize or interact extensively with individual
conspecifics and thus are unlikely to use memory of previous interactions in
responding within a contest.
The present analysis generates testable predictions for these animals:
- Hierarchies will be less transitive and less stable and may take longer to
establish for blind cave-crayfish (having limited ability to assess relative
size) than for visual crayfish.
- Both blind and sighted crayfish will use chemical and tactile cues to
respond appropriately to familiar individuals in the dark. These cues could be
blocked experimentally, forcing crayfish to respond in own-size mode as if to
a new intruder. This should produce far more wariness (mutual submissiveness)
and less complementarity than when cues are available.
- Regardless of size, dragonfly larvae will be more likely to disengage
immediately without aggression on contact in the dark than in the light, where
larger individuals will dominate smaller ones. Yet even in the light, a
transitive dominance hierarchy is unlikely to form in these larvae, except
perhaps for very small groups differing extensively in size.
Empirical studies of these and other systems should lead to improvements in the models that generated these initial predictions (also see below).
CAD and DorC
When responses are based on memory, the analytical and simulation results
of the present study show that the "taking-turns" strategy CAD can
predominate if interacting individuals do not differ too much in size or
cannot assess the opponent's size. In contrast, the
"dominant-submissive" strategy DorC mainly appears in symmetrical
contests and as a replacement for AllC in own-size (mixed-symmetry) contests,
where the possibility of meeting another DorC provides a chance of achieving
dominance. CAD is egalitarian and inconsistent with the formation of dominance
hierarchies. DorC forms stable dominance-submissive relationships, yet does
not necessarily lead to transitive dominance hierarchies because interacting
individuals are equally likely to become dominant by this mechanism in any
particular series of interactions, regardless of their sizes. (Note that this
conclusion is based on the assumption that the probabilities of playing C and
D following noncomplementarity are independent of size, which may not always
be true; see below.) But in groups of individuals differing enough in size
that each is recognized as being larger or smaller by each other individual, a
strategy of AllD when larger and DorC (or AllC) when smaller yields
transitivity. Moreover, adding the proviso that DorC is also played against
opponents of the same size (as in the relative-size case with memory and
dangerous combat in Figure 4)
means that groups containing no more than two individuals of the same size
will always have transitive hierarchies.
CAD-like behavior patterns are found in mating systems based on egg
swapping and in grooming sequences (see
Crowley et al., 1998).
Additional work is needed to identify conditions in which the CAD strategy can
become common. Perhaps situations involving memory of more than just the
previous interaction can generate patterns of CAD-like reciprocity more
sophisticated than strict alternation.
Probabilities of careful or daring behavior by either DorC or CAD before complementarity is established can be optimized. Setting this "transient probability" at 0.5 minimizes the expected number of noncomplementary interactions. But as the cost of fighting increases, this probability should generally be shifted toward careful to reduce the chance of combat before complementarity is achieved. When individuals using DorC interact, however, the individual whose opponent plays C first becomes dominant and achieves a higher fitness, suggesting that transient probabilities should generally be more daring in DorC than in CAD for any particular cost of losing. Exploration of this biased transient behavior is warranted (Crowley, in preparation).
By incorporating the flexibility to become either dominant or submissive,
DorC-like behavior patterns can address asymmetries that cannot be assessed
before an interaction is initiated. Possible examples include RHP-related
features such as size, fighting skill, confidence level (e.g., through winner
and loser effects; Dugatkin,
1997b), and physical condition; a possibility unrelated to RHP is
the value of the contested resource to the interacting individuals (e.g.,
Maynard Smith and Parker,
1976; Parker,
1974). In the own-size case, when size is the relevant feature,
larger individuals may bias the transient more toward being daring than
smaller individuals do because larger individuals are penalized less than
smaller ones for combat that may ensue before complementarity is established.
Thus, larger individuals playing DorC would be more likely to dominate
interactions with smaller individuals also playing DorC (Crowley, in
preparation). Note that the (possibly size-conditional) behavioral flexibility
characteristic of DorC is fundamentally different from actually altering the
strategy (i.e., the rules themselves) in response to characteristics or
behavior of opponents, though the distinction may be difficult or impossible
to detect from observations of behavior alone.
Some ways to build on this approach
The present analysis represents a dynamic approach to identifying
behavioral strategies expected to evolve under the conditions of interest.
Such evolutionary models cannot generate true evolutionarily stable strategies
because of the relatively high levels of randomness (i.e., mutation,
recombination, and demographic stochasticity) used to produce variation
(Boyd and Lorberbaum, 1987).
Yet closely related evolutionarily stable strategy (ESS) studies can and
should be conducted as a check on the present results and to address the
larger question of consistency between dynamic models like this classifier
system and static approaches like classical ESS analysis. Strangely, these two
complementary approaches have rarely been tightly linked, despite the clear
dependence of the ESS concept on some kind of evolutionary model
(Thomas, 1985). Future work
must strengthen this linkage and account for discrepancies (Crowley, in
preparation).
Some additional ways to improve and extend the present analysis include:
- constructing strategies based on remembered responses from two or more
previous games (see Axelrod,
1987; Crowley,
1996; Crowley et al.,
1996; Lindgren,
1991);
- devising and implementing more biologically realistic ways of expressing
payoffs (e.g., see Grafen,
1987; Houston and McNamara,
1990; Korona,
1991; and the fully parameterized payoff matrices of
Hammerstein and Riechert,
1988).
- introducing rare errors into both the responses and perception of responses
(i.e., the "trembling hand" of
Selten, 1980; see
Johnstone, 1994;
Nowak and Sigmund, 1992);
- when assessment is costly (rather than free, as here), considering when
assessment or nonassessment strategies are evolutionarily stable (cf.
Parker and Rubenstein, 1981);
and
- permitting additional conditions in rules, such as winner/loser effects and
within-series adjustments (as for Pavlov; see
Kraines and Kraines,
1993).
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ACKNOWLEDGEMENTS |
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I acknowledge with appreciation the hospitality of Martin Nowak and the Mathematical Biology Group and the Department of Zoology at Oxford University during June 1998, when this project began to develop. I also thank Michael Fishman, Mike Mesterton-Gibbons, Martin Nowak, and Rob Ziemba for discussions of these ideas, and Mike Mesterton-Gibbons, Andy Sih, and Rob Ziemba for helpful comments on earlier drafts of the manuscript.
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TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONREFERENCES |
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