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HIERARCHY FROM
EXCHANGE
Joseph M. Whitmeyer
UNC Charlotte
August, 2003
Hierarchy is ubiquitous
Anthropological evidence on human societies (D. Brown)
Discussion Groups (Bales) Someone talks most (usually 50%) Big talker deemed to have most
influence
Theories of Hierarchy Exchange of prestige for ...
(Homans, Coleman) Expectation States Theory (Berger,
Ridgeway, Webster) In process of social learning
(Henrich and Gil-White) Reciprocal influence of status and
exchange (Eckel, Thye)
My Study Task-focused group
Achievement (get prize) Choice (choose between options) Both (win, but with your candidate)
Only concerns selection of leader Computer simulation
3 Assumptions Human behavior: consistent,
similar, motivated Humans value prestige (Veblen) Leaders provide collective benefit
to group
The Simulation Model Each iteration each member
decides whom to support Support member giving highest
payoff for YOU Know how much support each
member had in previous iteration
Payoff components Perceived ability of member Amount of support member has Value of collective benefit Rivalness of collective benefit If supporting self, value of prestige
Parameters and variables 1
Group size n Collective benefit: B > 0 Rivalness: 0 λ 1 Individual share:
S = [(n - n + )/n]B Net value of prestige
Normally distributed Usually mean > 0, s.d. = 1/2 mean
Parameters and variables 2
Ability to deliver B: 0 Ai 1 Spread of ability perception
0 U 1 U = 0: complete agreement about
abilities U = 1: maximal divergence For choice, spread includes
heterogeneity of values
0
0.2
0.4
0.6
0.8
1
1.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
SPREAD OF ABILITY PERCEPTION
CO
RR
ELA
TIO
N
n = 10
n = 30
n = 50
Figure 1. Correlation Between Actual and Perceived Abilityas Function of U, for 3 Group Sizes.
Parameters and variables 3
Production function for leader effectiveness linear accelerating decelerating S-shaped (logistic) inverse S-shaped
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
PROPORTION BACKING LEADER
PR
OP
OR
TIO
N O
F B
EN
EFI
T
decel
linear
accel
logistic
inv. logistic
Figure 2. Production Functions for Leader Effectiveness.
Equations Let sj be j’s share of support, Pij be i’s
perception of j’s ability i’s perception of j’s effectiveness:
eij = sja Pij
a = 1: linear a > 1: accelerating a < 1: decelerating
S: eij = Pij / (1 + exp[10(.5 - sj)])
inverse S: eij = {.5 - .1ln([1/sj] - 1)}Pij
Equations (cont.)
Let uij be i’s net benefit from supporting j
Let vi be the net value of prestige for i
uij = eij S for i j
uij = eij S + vi for i = j
Results - Consensus
Consensus is quick or not at all 1-6 iterations for n = 3 to 10 More iterations for larger groups
Consensus is stable: no one ever switches from leader
Like discussion groups
Results - By p.f.
To standardize magnitude of results, collective benefits (Bs): 70 for decelerating 95 for linear, inverse S-shaped 395 for S-shaped 495 for accelerating
U set at 0.2. Ability spread evenly from .8 to .2.
0
0.2
0.4
0.6
0.81
3 6 912 15 18 21 24 27 30 33 36 39 42 45 48
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PROPORTION FULL
RIVALNESSGROUP SIZE
Figure 3. Production Function: Linear
0
0.2
0.4
0.6
0.81
3 6 9
12
15
18
21
24
27
30
33
36
39
42
45
48
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PROPORTION FULL
RIVALNESSGROUP SIZE
Figure 4. Production Function: Accelerating
00.
20.
40.
60.
81
3 6 9
12 15 18 21 24 27 30 33 36 39 42 45 48
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PROPORTION FULL
RIVALNESSGROUP SIZE
Figure 5. Production Function: Decelerating
00.
20.
40.
60.
81
3 7
11 15 19 23 27 31 35 39 43 47
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PROPORTION FULL
RIVALNESS
GROUP SIZE
Figure 6. Production Function: S-shaped
00.
20.
40.
60.
81
3 7
11 15 19 23 27 31 35 39 43 47
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PROPORTION FULL
RIVALNESSGROUP SIZE
Figure 7. Production Function: Inverse S-shaped
Group size & rivalness:
For all p.f.s, proportion reaching consensus is lowest with high group size and low rivalness
With n < 6, rivalness 0, all reach consensus
With n > 25, rivalness 1, none reach consensus
Interactions:
Rivalness 0: For linear, decelerating, inverse S-
shaped, increase in group size has NO effect
For accelerating, S-shaped, increase in group size means consensus goes to 0
Interactions (cont.) For linear, accelerating, and S-
shaped: negative effect of rivalness on proportion reaching consensus increases as group size increases.
For decelerating and inverse S: group size does not affect negative
effect of rivalness group size has little direct effect on
proportion reaching consensus.
Implications
Crucial period is when support low Since
prestige is awarded consensually in large groups
and prestige is awarded automatically, then
prestige is awarded especially for nonrival goods.
Spread of ability perception
U has strong negative effect on proportion of groups that reach consensus on leader
Its effects become stronger as group size increases.
So: to reach consensus, some agreement among group members about relative ability is necessary.
00.20.40.60.81
3 7
11 15 19 23 27 31 35 39 43 47
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
CONSENSUS
SPREAD
GROUP SIZE
Figure 8. Effect of Spread of Ability Perception (U) and Group Size.Rivalness = 0; Decelerating P. F.
Mean value of prestige
Question: Suppose two populations differ in mean value of prestige. Will leaders tend to come from one?
Answer: No.
Mean val. of prestige (cont.)
Suppose As have higher mean than Bs, s.d.s same (5). Then: Chances of random A having
higher preference for prestige than random B are greater than the chances that an A will be leader.
Difference of 1: .556 vs. < .52 Difference of 5: .76 vs. < .57
Summary
1. Stability is reached quickly.
2. Full consensus is permanent.
3. Increasing spread of ability perception diminishes proportion of groups reaching consensus.
4. Production function probably decelerating or inverse S-shaped, perhaps linear.
Summary (cont.)5. Critical period for deciding group leader is initial phases of
gathering support.
6. Prestige awarded mostly for nonrival or nearly nonrival benefits.
7. Populations with different mean net values of prestige produce similar numbers of leaders.
Next Steps Items 1 - 3 parallel experimental
findings Items 4 - 7 are predictions to be
tested
On nonrival benefits (#6): Exs.: Leadership, entertainment,
protection or defense, and information of some types.
Suggests theory can be applied to socio-emotional groups too.
Apply to different historical periods, with different non-rival benefits.
