Behavioral Ecology Vol. 12 No. 6: 753–760
Dangerous games and the emergence of social
structure: evolving memory-based strategies
for the generalized hawk-dove game
Philip H. Crowley
Center for Ecology, Evolution and Behavior and T.H. Morgan School of Biological Sciences,
University of Kentucky, Lexington, KY 40506, USA
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 memorybased 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. [Behav Ecol 12:753–760 (2001)]
S
ocial 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, fitnessrelated 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 resourceholding 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
Address correspondence to P.H. Crowley. E-mail: pcrowley@pop.
uky.edu.
Received 5 March 2000; revised 9 December 2000; accepted 29 January 2001.
2001 International Society for Behavioral Ecology
‘‘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 cooperation (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; May-
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Behavioral Ecology Vol. 12 No. 6
nard Smith and Price, 1973). Next, the size-structured situation and some key strategies are presented. Then a classifiersystem 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).
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;
Figure 1
(A) Payoff matrix for a focal individual in the classical hawk–dove
game, which is based on simultaneous binary responses by two
players (Maynard Smith and Price, 1973). V is the gain from
winning, and C is the cost of losing a fight. When dove meets dove,
the winner is assumed to be determined by some arbitrary but
nonaggressive means. Dove immediately concedes the field to hawk,
but an encounter between two hawks results in combat. Letting h
be the proportion of individuals in the population who have
adopted the hawk strategy (or h is the probability that each
individual will play ‘‘hawk’’ in a particular game) and 1 ⫺ h be the
proportion (or probability of) playing dove then allows the
evolutionarily stable strategy to be determined. This is accomplished
by equating the expected payoffs for hawk and dove players,
yielding h ⫽ min{ 1, V/C}. (B) The focal individual’s payoff matrix
for the generalized hawk–dove (GHD) game to be considered in
the present analysis (see Crowley, 2000). To avoid confusion with
the classical game, dove and hawk are replaced by C (careful) and
D (daring), respectively, and the cost of losing a fight C by L. The
cost, T, of settling a contest between individuals playing C (doves)
by conventional means is also taken into account, where T includes
the opportunity costs and predation risks that may be associated
with display (Hammerstein, 1981); and I incorporate the cost W of
injury and other risks and lost opportunities associated with mutual
daring (fighting) that must be paid by the winner. The chance that
the focal wins in combat is p, and 1 ⫺ p is the chance that the
other wins. With Pij as the payoff to the focal individual when it
plays i against j, the GHD game is characterized as PDC ⬎ PCC, PDC
⬎ PCD, PDC ⬎ PDD, and PDC ⫹ PCD ⬎ 2PCC. When the two players
are the same size, then p ⫽ 1/2, and the logic yields d ⫽ min{1, (V
⫹ 2T)/(W ⫹ L ⫹ 2T)}, where d replaces h as the frequency of
daring behavior.
Crowley • Dangerous games and social structure
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.
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
755
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.
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 ⱕ3 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
Behavioral Ecology Vol. 12 No. 6
756
Figure 2
(A) Probability of winning in combat versus relative size. This relationship should generally be sigmoid because incremental increases in
relative size can have little effect on the fighting effectiveness of individuals very different in size—but should more strongly influence the
chances of winning for similar-sized individuals (see Crowley, 1984; Mesterton-Gibbons, 1994). Relationships shown here are hypothetical and
are expressed as the arithmetic difference between size-category numbers in a population that includes five discrete sizes. Most analyses for
this article are based on relationship 2, but 1 and 3 are also spot-checked for comparison. (B) Probability of winning in combat versus own
size. Here, each individual is assumed to know only its own size and the distribution of sizes in the population, from which it can determine
the probability p of winning against the average opponent. (The underlying relative size relationship is curve 2 in panel A.) Results of such
calculations are shown for (1) a population composed of 45% size 1, 25% size 2, 15% size 3, 10% size 4, and 5% size 5; (2) a population
composed of 20% in each size category; and (3) a population composed of 5% size 1, 10% size 2, 15% size 3, 25% size 4, and 45% size 5.
(Strictly, these percentages indicate the frequencies of interacting with individuals in these size categories, which may primarily reflect activity
levels and could differ from relative abundances in the population.) Most analyses for this article are based on relationship 2, though the
other two cases were also examined.
Table 1
Parameter magnitudes for the classifier system
Parameter
Magnitudes
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).
V, gain from winning a fight (fitness)
W, cost of winning a fight (fitness)
L, cost of losing a fight (fitness)
T, cost of display (fitness)
p(rF), focal’s chance of winning a fight
vs its relative size
Number of size categories
Size distribution among categories
Population size (individuals ⫽
algorithms)
Number of games per sequence
Sequences played as focal per
individual per generation
5
1
2, 10, 40
0.5
Generations per run
Replicate runs per parameter
combination
Number of rules per algorithm
Number of previous games
remembered
1000
RESULTS
25
10
Size information
Crossover probability at a rule locus
Mutation probability—rule
component or rule length
Mutation probability—sizeconditional index n
Equivalent, own, relative
0.21
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
Curve 2 of Figure 2
5
4 ‘‘others’’ in each category
21
50
100
0, 1
0.002
0.005
See Crowley (1996) and the text for more on the model and its
parameters.
Crowley • Dangerous games and social structure
757
Figure 4
Strategies that emerged from the classifier-system analysis. Results
are shown for the 18 combinations of three information states
(equivalent size, own size, and relative size), three fitness costs of
losing in combat (L ⫽ 2, 10, and 40), and two memory types
(memory, no memory). Symbols used to identify strategies and
combinations of strategies produced in the runs are shown at the
top. For the careful-and-daring mixed strategy and for cases in
which different stable patterns were found in replicate runs, the
approximate frequency of each component strategy is indicated
diagrammatically. Results for each of the six equivalent-size cases are
illustrated within a single square because they are not conditional
on size. In the own-size cases, strategies are conditional on the focal
individual’s absolute size (1 to 5), and in the relative-size cases,
strategies depend on the focal’s relative size (⫺4 to 4), so strategies
are shown by size category.
Figure 3
Behavioral results of the classifier system analysis. The upper graph
summarizes patterns from the no-memory cases, and the lower
graph presents patterns from the memory cases. Each set of three
adjacent bars corresponds to a particular cost of losing in combat
(L ⫽ 2, 10, or 40). Each individual bar applies to the availability of
unknown, own, or relative size information and indicates the
percentages of all games resulting in D/D (black), C/C (white), or
complementarity (C & D, cross-hatched). Standard errors around
these mean percentages, taken over means from replicate runs,
ranged from 0.0 to 2.4 in the no-memory cases and from 0.2 to 9.7
in the memory cases. Numbers beneath the bars are corresponding
mean fitnesses; SEs around these means ranged from 0.00 to 0.03
(no-memory) and from 0.03 to 0.22 (memory). Using twice the
larger standard error as a rule-of-thumb measure of the confidence
interval around the mean indicated that almost all of the
percentage patterns and the fitness values are statistically
distinguishable from each other.
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.
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 noncomplementarity; 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
Behavioral Ecology Vol. 12 No. 6
758
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.
DISCUSSION
memory of previous interactions in responding within a contest.
The present analysis generates testable predictions for these
animals:
(1) 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.
(2) 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.
(3) 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).
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
CAD and DorC
When responses are based on memory, the analytical and simulation results of the present study show that the ‘‘takingturns’’ 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 com-
Crowley • Dangerous games and social structure
bat 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 sizeconditional) 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).
I acknowledge with appreciation the hospitality of Martin Nowak and
759
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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