Groups in many species of mammals and birds form hierarchies: these hierarchies ... Most of the data cited are from studies of dominance in chickens, but the ...
MODELS OF HIERARCHY FORMATION I N ANIMAL SOCIETIES’ by Ivari D. Chase Department of Sociology, Dartmouth College Groups in many species of mammals and birds form hierarchies: these hierarchies are frequently linear or near linear in moderate size groups. Two models have been proposed, both implicitly and explicitly, in the literature to explain the process of hierarchy formation. One explanation contends that animals win their places in a hierarchy through a round robin competition, and the other states that there i s a high statistical correlation between position and rank on some trait or composite of traits thought to predict dominance. I t is demonstrated here that both models require stringent mathematical conditions to predict linear and near linear hierarchies and that available data indicate that these conditions are not met. Most of the data cited a r e from studies of dominance in chickens, but the same mathematical conditions a r e necessary to generate linear and near linear hierarchies by the round robin and correlational models in any species. Thus, the findings presented here have application to any species of animals forming linear and near linear hierarchies. w
The race is not to the swift, nor the battle to the strong, . . . but time and chance haopeneth to them all. (Ecclesiastes) INTRODUCTION
species of animals, nhen scvcral unacquaintcd animals are placed togcxthtlr in a group, they will erigagc in competitions, usually pairwise, for dominance. Some of the competitions result in violent fights, some in less furious actions, and still others in the passive recognition of a supcrior and a subordinate. For an initial period ranging from hours to we&s, depending upon such factors as group sizc arid species of animal, aggressive interactions will be high in frequency. In time the frequency and intensity of aggressivc interactions subside as the dominance-submission relations among animals become stable. For inany species of animals, groups of three or four members usually settle their relations in a day or so; for larger groups, around 10 members, scvcral weeks may be required. Oncr t hc dominance-submission rclations :ire initially settled, the positions of animals in a hierarchy usually remain unchanged for
relatively long periods of timci-on t h r order of months; as a rule there arc relatively few rebellions against superiors by subordinates, and nearly all rebellions that do occur are unsuccessful. Hierarchies in chiclicns and other sprcirs of birds were first systematically invcstigated and described by Schjeldcrup-Ebbe (1935). \;I ith the influence of his seminal work, sociobiologists have discovered and described hierarchy behavior in many spwii2s of birds that forage in flocks or roost cornrnunally and many species of mammals that live in group situations. Hierarchy bchavior has even been observed for various insect and reptile species (Wilson, 1973). Hictrarchy is a ccntral concern in sociotiology because it has a crucial organizing function in a wide range of behaviors. For (bxamplc, hierarchy behavior influences u hich a n i n d s become mates and therefore influmccs the population gcrietics of groups, the acc(’s5 of group members to food arid thus thcir individual fitness, and the division of latlor
‘Much of the work for this paper was done during the tenure of a Social Science Research Council Postdoctoral Fellowship, and the Council’s support is gratefully acknowledged. The
author is particularly indebted to Edward h1 Brouri and E. 0. Wilson and t o Hans Follnier .tnd E. B Hale for their discussion and comments on this paper.
I
N VAXY
374 Behavioral Science, Volume 19. 1974
R~ODELS OF HIERARCHY FORMATION arid thus thc individual roles that animals play, c.g., predator lookout, boundary patroller, and organizer for group travel. I n spite of the numcrous empirical dcscriptions and central importance of hierarchy behavior in animal groups, littlc thcoretical work has been done to discover and explain the bchavioral processes through which hierarchies are established and maintained. One typc of hierarchy which has rcc-eivcd somc theoretical attention is thc linear or near linear hicrarchy frcqurntly fourid in small groups among many spc’cies of animals. I n a linear hierarchy, an animal A dominates all members of the group, an animal B dominates all but A , and so forth to the last member of the group which dominates no o m . Such hierarchies have bcen observed, for example, in wasps (\%ilson, 1971), chickens (Guhl, 1933), chaffinches (Rlarler, 1955), cows (Schein K- Fohrman, 1955), and ponies (Tyler, 1973). Two explanations have been suggestrd to cxplain thc occurrence of linear and w a r linear hierarchies; these explanations havr a common sense appeal and are proposed both explicitly and implicitly in the 1iter:tture of animal behavior (Guhl, 1933; Guhl k Fischcr, 1969; Beilharz c! Cox, 1967; Warren & AIaroney, 1958; Collias, 1943). The mathematical criteria necessary for the acceptance of each model will be formulated brlow. The formulations will show that the criteria nrcessary for the acceptance of the models are most stringent, and available data indicate that these criteria are not met. TOURNAMENT MODEL
Considcr a moderate number of adult animals of the same sex placed together for thc first time. The group is obsrrved a short period aftrr being formed and, as is common, a linrar or near linear hierarchy is found. Could this hierarchy possibly be the result of a round robin tournamcrit among the mcmhers of the. group? That is, could a strong, i.e., a linear or near linear, hierarchy bc f o r n d if carh animal fought or just simply
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375
compared itself to every other animal successively in pairwise contests, and then each animal continued to dominate all those animals it had beaten in its competitions? The answer is that it is very doubtful that a linear or near linear hierarchy could come about in this way. I shall explain why this model is doubtful first in intuitive tcrms and then more rigorously by means of a mathematical model. I n a group of animals, it would be expected that some individuals would have qualities that would give them a high probability of being very successful in pairwise competitions, while others would have qualities that would give them a low probability of being very suecessful, but most animals would have attributes that would enable them t o have moderate probabilities of success in competitions. Or, in other words, it would be expected that the distribution of competitive ability would have many individuals near the mean, but many fewer individuals with either very high or very low ability. Given a distribution of probabilities for winning of this sort, a few animals would win many, perhaps most of their contests, a few would lose many, and the majority of the animals would win a moderate number of their contests. If the hierarchy in such a group were formed so that each animal dominated those it had beaten in pairwise competitions, the result would probably not be a hierarchy close to linearity. To have a strong hierarchy, the results of the competitions would have to be near those of a linear arrangement: A beats all members of the group, B beats all members but A, and so forth t o the last animal which beats no one. The large number of animals which would win only a moderate number of competitions would destroy the possibility of a strong hierarchy. I l a ~ i yof these animals would win the same number of contests, arid they could iiot be ranked successively by the number of foes dominated as are most animals in an ordering that approaches linearity. The following tournament model of hicr-
IVAN D. CHASE
3 76
TABLE 1 measures greater than .9 are found for one VALUESOF THE HIERARCHY MEASUREFOR ONE and two intransitive triads. On the basis of A N D Two INTRANSITIVE TRIADS BY GROUPSIZE this information, a strong hierarchy can now -
Group Size
5 8
7
X 9 10 15
20 25 30
Values of h One Intransitive Two Intransitive Triad Triads ,800 ,886 ,929 .952 .967 ,976 ,993 ,997 ,998 .999
.77l ,857 .905 ,933 .952 ,988 .994 .997 .99x
archy formation is a simplification and extension of a model developed by Landau (1961). Landau formulated a hierarchy mcasure
where n is the number of animals in a group forming a hierarchy and v, is the number of animals that the ath individual dominates. Thc measure is a multiple of the variarire of v, since ( n - 1)/2 is the mean of the possible values of v,; the mean takes an integer value only for odd 71. The variance of v, is at a maximum with h equal to unity when there is a lincar hierarchy (v, = n - 1,n - 2, . . ., 0) arid is zero with h equal to zero when each animal dominates an equal number of other animals, v,s all equal to ( n - 1)/2. I n a linear hierarchy all possible groups of three animals form a transitive triad-an animal A dominates animals B and C, arid B dominates C. If a hierarchy is not linear, then it contains one or more intransitivc triads-A dominates B, B dominates C, arid C doniinates, rather than is dominatcd by, A . Table 1 gives values for h in a variety of group sizes for hierarchies with one and two intransitive triads; each triad is composed of individuals at the samc level, dominating an equal numbcr of animals, in the group structure. 111 a group of size seven, onc intransitive triad gives a hierarchy measure of .929, and in groups of size cight and Iargcr, hirrarchy
Behavioral Science, Volume 19, 1974
be defined more exactly, albeit somewhat arbitrarily, as a hierarchy giving an h measure of .9 or higher. The tournament model assumes that for all possible pairs, a and j , of individuals in a group, there is a probability pa, that a beatsj in pairwise competition and a probability p,, that j beats a. One individual must bt3at the other, so pa, plus p,,, equals one. Expressions for the expected value and variance of h can be derived in terms of the prob abilities for winning in pairwise contests. The expected value of h is given bclow; the formula for the variance of h is quite lengthy and spacc limitations prohibit its inclusion here. (The derivation for the expected value of h and the formula for the variance of h are available from the author upon request .)
- 3(?L - 3 ) / ( n
+ I).
If all pU7sare equal to ..5, cwh individual has the same probability to win a contest 21s its foes, and the expected valuc. of h is 3/(12 1) and the variance is
+
lS(n -
2)/Yl(!L
-
1)(1L
+ 1)'.
Thus, hicrarchies that givc II nieaburcs far from thc strong level of .9 arc expected evt'n in relatively small groups of animals \\here all individuals have equal probabilities of winning contests. If an animal A has a probability of 1 for winning all its competitions, an animal B has a probability of 1 ior winning all its competitions other than with A , arid so forth to thc last animal nhich has a probability of 0 for winning any of its contests, thcri the expected value of 11 for the hierarchy so formcd is 1 and thc variaiicc is 0, i.e., a linear hierarchy is expectrd. Table 2 gives the expected valuc and variance of h for various probability distributions in different size groups. One word of explanation about the table: all distrihutionr~
MODELSOF HIERARCHY FORMATION EXPECTED VALUESAND VARIANCE OF h ~
_
_
FOR
-
_
377
TABLE 2 VARIOUSPROBABILITY DISTRIBUTIONS B Y GROUPSIZE _ _ _ _ ~
-
Probability Distributiun .6.4
5 6 7 8
9 10 12 15 20 25 30
.a-.2
.?-.3
__..__
~
.500 ,075 ,429 ,049 ,375 ,033 ,333 ,024 ,300 ,018 .273 .0:3 ,231
.520 ,079 ,451 .053 .400 .037 ,360 .027 ,328 ,021 ,302 .015 ,262
.580 ,087 ,520 ,061 ,475 ,044 ,440 ,034 .412 ,027 ,389 ,021
,188
.220
.143 ,115 ,097
,177 ,151 .133
,317 ,280 ,257 ,241
,354
_.
.680 ,088 ,634 ,065
.600 ,050 ,573 ,039 .552 ,032 ,535 ,025 ,508 .480 ,451 ,434 .423
.9-.1
.95-.05
.96-.04
.97-.03
.98-.02
.905 ,041 ,891 .032 ,881 ,025 .a73
,923 .034 .912 .027 .904 .021 .a98 .018 .892 .014 .888 .006 ,882 .875 .868 .864 .861
.942
.961 ,019 ,955 ,014 ,951 .011 ,948 .009 .945
.99-.01
1.0-u
__ ..a20 ,067 ,794 .052 ,775 ,040 ,760 ,033 ,748 ,027 ,738 ,025 ,723 ,707 ,691 ,682 .675
.020 ,867 .015 ,862 ,017 ,854 .846 .a37 ,832
.828
,027 ,933 .021 ,927 ,017 ,922 .013 .919 ,010 .915 .012 ,910 .905 .goo
.897 ,895
.080
1 0
.010 ,977 ,008 ,975 ,006 .974 ,005 ,972 ,002 .971 ,001 ,970
.009 .943 ,012 .940 ,936 ,933 .931 ,929
1 0 1 0 1 0 1 0 1 0 1 1
.968
1
.966 .965 ,964
1 1
Note: The variance of h appears in the second line of figures under each group size entry of five through ten. When the group .size exceeds ten, the amount of computer time required to calculate the variance of h becomes prohibitive. The variance of h goes nversely with group size, so the variance of h for groups greater than ten is very small.
follow a standard form, for example, .(i-.4 indicates that A has a .6 probability of winning all its contests, B has a .ti probability of winning all its contests except with A where the probability is .4 and so forth to the last member which has a .4 probability of ninning any of its contests. Probability distributions other than those in the table can give identical expected values and variances for h , hut thc standard form distributions used h t w give the maximum valucs for h while miriimizirig the values of any p a , greater than .5. Table ‘2 indicates that probability distributions of .95-.05 or more extrrme are required in order for the expected value of h to be greater than .9. It is intuitively doubtful that t.mpirica1 distributions of probabilities ninning as extremc as these would be expected in groups of animals. Actual cstimates of distributions of pairwise competitive ability can be made by matching each prospective group member with every member, in turn, of a fixcd panel of animals. The panel could be composed of the other members of the prospective flock or a group of scparatc animals. Each prospcvtive group
Behavioral Science, Volume 19, 1974
member is given a score sa(a = 1, 2, . . n) which is the number of panel animals that it beats, and a’s probability of beating j can be estimated as a ,
p a , z=
sa/(sa
+
~ 1 ) .
AII expected value for h can then be calculated using these empirically estimated probabilities for winning. Guhl (1953; 1968) arid IIcBride (1968) provide data on the number of pairwise contests won for groups of chickens, and these data can be used for estimating probabilities. Guhl’s data result from a round robin tournament among chickens that were later placed together to form a ffocli, and RlcBride’s data are derived from compcltitions of a group of chickens with a fixed panel of other individuals. RlcBride points out that there is a tendency toward runs of wins and losses in the bout records of his chickens. He suggests that this arises because some of thc bouts werc in quick succession, and the chickens were inff ucnced by memorim of their previous successes o r failures. The c.xpccted values of h in the flocks for probabilities estimated by t h e
IVAN D. CHASE
37s
TABLE 3 necessary complication. Second, Landau EXPECTED VALUEFOR h WITH EXPERIMENTALLY used various functional forms to generate the p,js ESTIMATED paJs,while in the present modcl the pals are __
___.__
Flock Size
9 9 9 9 11 11
12 12
Expected h .59 .53 .61 .53 .56 .60 .40 .28
relative proportion of pairwise encounters won are given in Table 3. The first six entries in the table are from Guhl’s data and the last two from McBride’s. h’one of the flocks would bo expected to form strong hierarchies, with h 5 at or above the .9 level, on the basis of the estimated probabilities for winning. For both theoretical and experimental considerations, it is unlikely that strong hierarchies can be formed on the basis of pairwise competitions ainong members of animal groups. The tournament model indicatcs that extreme probability distributions are required to yield strong hierarchies from round robin competition, and the available data indicate that these probability distributions are not met. It should be noted that these extreme probability distributions are required t o predict a strong hierarchy from a tournament no matter what the specks of animal, the method of pairwise competition, or the individual traits that are conceived to determine probability of dominance. I n addition, it is shown below that the statistical correlation between results in round robin competition and position in actual hierarchies is only moderate. Finally, it should be rioted that the w r l i here diflers from Landau’s (1951) study in two ways. First, Landau uses a continuous model, while the one here is discrete. Since the sizes of animal groups in which strong hierarchies are found are small and relatively fe~virt number-groups from three members t o about 2&a continuous model is an un-
Behavioral Science, Volume 19, 1974
taken directly and not assumed to be generated by any particular functional form. For example, Landau assumed the pairwise probabilities could be generated by a linear or normal function on distributions of iiidividual abilities for animals. However, what abilities criable animals to win contests is not clear, and what functional transformations of ability ranltings yield reliable estimates of probabilities for winning contests is less understood. Therefore, Landau’s use of specific functional forms to generate the probabilities is mislcadirig and not inclusive of inariy probabilistic situations. The one essential feature of ;1 tournament model is the probabilities for succeeding in corites ts. That feature is retained in the work here, arid the probabilities are not constrained to particular distributions but arise directly froin data sources or hypothetical considerations. CORRELATIONAL MODEL
The second explanation for the occurrerice of strong hierarchies is the correlatioiial model. The correlational model does not propose a specific mechanism for the establishment of hierarchies; it simply posits that there is a high statistical correlation between some trait or composite of traits thought tc predict dominance and the actual positiom of animals in a hierarchy, In other words, tht animal scoring highest on thc dominance trait or composite measure is expected to dominate almost all othcr animals, the, animal scoring next highest is expected t o dominate nearly all but the first animal, :ind so on to the lon-cst scoring animal uhicli i8j expected t o dominate 110 one. Among thosI> individual traits which have received most attention are relative aggressiveness Incatured by success in round robin c o m p e t i h n and body weight. Place in the hierarchy has been most frequently nieasured by the nuniber of animals dominated, and t d a
hlODELS OF
HIEIZARCHY FORMATION
lesser extent by variables based upon the proportion of encounters won with group members. Let me make the correlational model explicit. First, the members of a hypothetical group of animals are ranked successively from highest to lowest score on a single or composite of traits assumed to be associated with dominance position in a hierarchy. The ranking can be thought of as determined by any means on any trait or traits. Second, each animal is given a dominance score using one of the possible vectors of v,s, starting with the highest dominance score for the highest ranking individual on the trait score and so on to the lowest dominance score for the lowest ranking animal on the trait score. For each possible vector of v,s there is an associated h measure, For example, as was mentioned above, the v,s associated with a linear hierarchy give an h measure of unity, and those associated with the case where each animal dominates an equal number of other animals give an h measure of zero. Third, a correlation coefficient is calculated between the trait and dominance scores of the animals, and the calculation is repeated for different vectors of v,s. Since we know the h measure for each possible vector of dominance scores, we arc now able to find the range in the h measure associated with particular levels of correlation between trait and dominance scores. I n Table 4 are given the product moment correlation coefficients required to predict individuals' dominance scores, u,s, from a ranking of individuals where h is as small as possible but not 0; even to predict minimal hierarchies substant,id correlation coefficients are required. The ranking used in the table starts a t n fnr the highest ranking animal and goes to 1 for the lowest animal, rather than 1 to n, in order t o make the correlation Coefficient between individual score and hierarchy score positive: the magnitude of the coefficient is the same as in a 1 to 7z ranking. I n groups of even size, the weakcs:, possible, nonzero hierarchy is found
Behavioral Science, Volume 19, 1974
379 TABLE 4
CORRELATIONCOEFFICIENTSI~EQUIREDTO PREDICTDOMINANCE SCORES I N WEAKEST, NONZERO HIERARCHIES FROM A RANKING OF INDIVIDUALS Group Size
Required Correlation Coefficient
5 6 7 8 9 10 15 20 25 30
.89 .88 .80 .87
.73 .87 .59 .87 .47' .87
* Note: This correlation coefficient is quite low because in odd size groups all dominance scores but two are identical, and consequently the coefficient varies inversely with group size.
when half of the dominance scores are (n - l ) / 2 3. 1/2 aridhalf arc ( n- l)/2 - l/2. For groups of odd size, thc weakest possible hierarchy, h = 0, is found when all dominance scores are (n - 1)/2. But in this case the correlational coefficient is undefined since the standard deviation of the dominance scores is 0. The weakest possible nonzero hierarchy in odd size groups is found when all dominance scores are (n - 1)/2 but two; one of these is (n - 1)/2 1 and the other is (n - 1)/2 - 1. The correlation coefficient does not increase monotonically with increasing h, and for groups larger than six, hierarchy structures that yield equal h measures do not necessarily give equal correlation coefficients with a rank order for individuals. However, a correlation coefficient of unity is required between a linear hierarchy and a ranking of individuals, and correlation coefficients of greater than .9, accounting for more than SO percent of the variance, are required to predict a strong hierarchy from a ranking of individuals. Therefore, in order to support the correlational model, experimentally measured coeffirients must meet or exceed the target level of .9. Guhl (1953) raised a flock of 11 pullets together, observed the peck order, h = .91,
+
IVAN D. CHASE
350
isolated the birds, put them through two scparate round robin tournaments, and then noted the peck order, h = .91, after the birds were placed togethcr following another period of isolation. He reports correlation coefficients between the number of animals animals dominated in the t n o peck ordcrs and the number of pairwise contests won; his results are found in columns 1 and 2 , lines 1 through 4 of Table 5 (Guhl, 1953, pp. 7-8). In another article, Guhl (1968, pp. 228-229) gives the riuniber of chickens dominated in the pcck order and the number of coritrsts won in each of two tournaments for tno flocks. The correlations bctween the n u m b ~ rdominated in the peck orders and th(. nuniber of contcsts won arc found in linw .? through S of Table 3 . His flocks D2 arid C2 each had nine chickens with h measurw of .37 and .93, respectivcxly. In Table 5 , a correlation coefficient bet n w n the number of chickens dominatcd in t h r x peel; orders and the number of pairwise TABLE 5 COKRELATIONS BETWEEN POSITION IN PECK ORDER A N I ) SLJCCESS I N I ~ O U X DROHIN TOURNAMENTS
__ -
- -- _ __ ____ __ __
Correla Percent Correla- Percent tion for \ arition for taw Scores ance Ranks ance 1. initial perk order
2. 3.
4.
5.
;Ind round robin 1 lnitinl peck order irnrl ruund robin I1 Reassembled peck order nnd round robin I Reassembled peck c d e r and rrrund ruhin I1 1’w.k order in flock 1J2 and round robin
.77
SO
.7?
so
.58
34
-*
-
.63
40
-*
-
.72
52
.72
52
.53
28
.60
36
.32
10
.43
I8
.7Y
62
.7Y
62
.74
55
.i9
62
r
6. Peck (irder in flock 1X and round robin 11 7. 1’ei.k order in Nock CZ and round robin
1 8. Pe1.k order in flock
C2 itnd round robin 11
* Not?: Data were not available in a form which allowed a cxl
