BELOW TOP STATUS INDICATES MEDIOCRITY WHEN
SOCIAL INFLUENCE IS STRONG
JERKER DENRELL AND CHENGWEI LIU
Abstract. Is higher status necessarily a signal of higher quality? Prior research on how status and quality can become decoupled has shown that social
influence processes can lead to a low correlation between status and quality.
Here we show that having high but not top status can be an indication of
medium rather than high quality, even if top status indicates top quality. A
formal model shows that such a non-monotonic pattern occurs when social
influence in status attributions is non-linear and sufficiently strong. A close
examination of the outcome of the Salganik, Dodds, and Watts (Science, 2006)
music download experiment illustrates the underlying mechanism. We discuss
implications for evaluations, imitation, and reactions to success.
1. Introduction
The relation between status and underlying quality is a central concern within
the social sciences. Do the best get ahead (Jencks, 1979; Rosenbaum, 1979)?
Will the most efficient organizational form or the highest quality product dominate (Arthur, 1989; Podolny, 1993; Carroll & Harrison, 1994; Perrow, 2002)? Are
the most cited researchers the most impressive (Merton, 1968; Allison et al., 1982;
Simonton, 2003)?
Prior research has explored several reasons for why status may be decoupled from
underlying quality (Gould, 2002; Lynn et al., 2009). Status and quality are conceptually distinct, with quality referring to unobservable skills or traits while status
denotes the social position of an actor. Because high quality often increases chances
of status attainment, high status should be an indication of relatively high quality.
But status is not necessarily a reliable indicator of quality. Luck or inheritance can
lead to a low correlation between status and quality (Jencks, 1979). Social influence
processes, including the Matthew Effect, may amplify initial status differences and
distort the association between quality and obtained status (Arthur, 1989; Chase
et al., 2002; Merton, 1968; Simcoe & Waguespack, 2011), even to such extent that
the highest status actors may not have the highest expected quality (Denrell & Liu,
Date: April 22, 2015.
We are grateful for ideas and input from Ron Burt, Nick Chater, Thomas House, Stefan Jonsson,
Thorbjørn Knudsen, Balázs Kovács, Gaël Le Mens, and seminar participants at seminars at
University of Chicago Booth School of Business, Stern School of Business, New York University
and Goizeta Business School, Emory University.
1
BELOW TOP STATUS INDICATES MEDIOCRITY
2
2012). The implication is that when social influence processes are significant and
initial status assignments are noisy eventual status will be an unreliable indicator
of quality (Salganik et al., 2006; Lynn et al., 2009).
Here we argue that certain status positions are particularly poor indicators of
quality because of the way in which social influence processes work. We show that
positions of below top status can indicate lower expected quality than positions of
medium status when social influence in status attributions is sufficiently strong and
non-linear. Many social scientists have noted that status attainment is often subject
to social influence effects: the probability that an individual defers to another
depends on the number of others who do so (Asch, 1955; Gould, 2002; Salganik
et al., 2006; van de Rijt et al., 2014). Our model shows that when such social
influence in status assignment is non-linear and sufficiently strong, higher status
will not always indicate higher expected quality. Rather, expected quality first
increases with status, then starts to decrease at a relatively high level of status,
but eventually quality increases again with status. That is, the function relating
quality and market share will ”dip” at a relatively high, but not top, level.
The dip implies that actors with higher status can have systematically lower
expected quality than actors of lower status. Such a dip represents a more fundamental decoupling of status and quality than what prior work has demonstrated.
Most prior work on status-quality decoupling has focused on how processes of cumulative advantage can lead to a low but still positive correlation between status
and quality (Lynn et al., 2009). Our model shows that strong social influence can
generate a negative correlation between status and quality for some status levels.
The intuition for why a dip occurs is that strong social influence in status attainment implies that initial good luck can have substantial long-term effects. As a
result, outcomes are bimodal. Actors will either reach relatively low status (most
likely, occurs when the actor lacks initial good luck) or high status (less likely,
occurs when the actor benefits from initial good luck). Good initial luck is not
sufficient to propel actors of low or medium quality to top status because status
depends on quality in addition to social influence. Only actors of top quality who
benefit from initial good luck can reach top status. Actors with lower quality who
benefit from initial good luck will reach high but not top status. It is thus as if
actors become separated into two different playing fields: with or without the extra boost from social influence due to initial good luck. Within each playing field,
relative performance depends on quality and higher status indicates higher quality.
The dip occurs because the actors with the highest status among those who did not
receive the extra boost are of higher quality than the actors with the lowest status
among those who received an extra boost from social influence.
We first develop our argument formally using a simulation model in which the
status of an actor is determined by this actor’s relative performance in a population
BELOW TOP STATUS INDICATES MEDIOCRITY
3
of competing actors. We then show a dip also occurs in other models that share
the same mechanism, including models in which status depends on absolute rather
than relative performance. A close examination of the outcome of the Salganik,
Dodds, and Watts (Science, 2006) music download experiment illustrates that the
conditions required for a dip are empirically plausible. Section five compares our
model and result to prior literature on status-quality decoupling. In the concluding
section we discuss how reward systems, reactions to success, and imitation processes
are impacted when a dip exists and higher status does not indicate higher quality. We also explore implications for the phenomenon of the 41st Chair (Merton,
1968) and how a status-weighted voting count can overcome some of the problems
associated with a dip.
2. Model
We formalize our argument using a simulation model of a stochastic status accumulation process. Our model considers a set of actors who compete for being
selected by evaluators. Following Gould (2002) and Lynn et al. (2009) we assume
that evaluators consider both the perceived quality of an actor and the proportion
of past evaluators who have selected an actor (social influence) when deciding about
whether to select an actor. We assume that an actor selected by a larger number of
evaluators has higher status (Goode, 1978), consistent with the definition of status
proposed by Sauder et al. (2012) as ”...the position in a social hierarchy that results
from accumulated acts of deference” (p. 268). Accordingly, we measure the status
of actor i by the share of evaluators who select actor i (Gould, 2002). More refined
status measures exist in which status depends not only on the number of selections
but also on the status of those making the selections (Katz, 1953; Bonacich, 1987;
Bothner et al., 2010). Using such ”centrality” measures of status would not change
the results of our model because evaluators do not differ in status in our model. In
the discussion section we explore when centrality measures of status would change
the results.
2.1. Set-up. Our model is very simple. There are n actors in competition for
status (actors can be individuals, organizations, websites, journal articles, songs).
The quality of actor i is qi , where qi is drawn from a uniform distribution between
zero and one.
In each period, t, an evaluator arrives and selects one actor. The probability that
the evaluator in period t selects actor i, Pi,t , depends on actors i’s attractiveness
in period t, Ai,t , as follows
(1)
ecAi,t
Pi,t = Pn
.
cAj,t
j=1 e
BELOW TOP STATUS INDICATES MEDIOCRITY
4
Here c is a parameter that regulates the extent to which actors with higher attractions are more likely to be selected. When c is large the actor j with the highest
attraction is very likely chosen (Pj,t is close to one).
Actor i’s attractiveness in period t, Ai,t , is a weighted average of actor i’s quality
and the market share of actor i in period t − 1, mi,t−1 :
(2)
Ai,t = wqi + (1 − w)mi,t−1 ,
Here w ∈ (0, 1) is the weight evaluators give to quality and mi,t−1 is the proportion
of the periods 1, ...t−1 in which actor i was selected. In the first period when mi,t−1
is not defined we set Ai,1 = wqi . Note that while we assume that attractiveness
depends on quality this does not imply that evaluators necessarily know the qualities
of the actors. Attractiveness may depend on perceived quality and perceived quality
may in turn be a function of quality. A more detailed model could specify all the
steps involved in such this process.
Figure 1 illustrates how the probability of being selected, Pi,t , varies with the
market share for different values of c (Figure 1 assumes that w = 0.2 and n = 10
and that all ten actors have quality equal to 0.5). When c is relatively low (c = 3),
the probability of being selected is not much influenced by market share. Moreover,
the probability of being selected increases almost linearly with the market share.
When c is relatively high (c = 7) there is stronger social influence: the probability
of being selected is strongly influenced by market share. Moreover, the probability
of being selected increases in a non-linear way with market share.
2.2. Results. We simulate the above stochastic process to examine the association
it implies between market share and quality. In the simulations reported here there
are n = 10 actors and we simulate each process for 50 periods (equivalent to 50
evaluators). We initially set the weight evaluators give to quality (w) to a relatively
low value: w = 0.2.
Figure 2 shows how average quality varies with market share for different values
of c. Higher market share indicates higher average quality only when c = 3. For
values of c above 3 higher market share does not indicate higher average quality.
Rather, average quality first increases with the market share, then decreases, and
finally increases again. That is, the function relating market share and quality
”dips”. The dip implies that actors with a lower market share can have higher
average quality. For example, when c = 5 actors with a market share of 50% have
an average quality equal to 0.51 while actors with a market share of 17.5% have an
average quality equal to 0.66.
Figure 3 shows more detail about the case when c = 5. The first graph shows
how average quality varies with market share. The second graph shows how the
proportion of wins - the proportion of simulation runs in which an actor obtained
the highest market share among the n competing actors - varies with market share.
BELOW TOP STATUS INDICATES MEDIOCRITY
5
1
c=7
c=5
c=3
0.9
0.8
Probability Selected
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Market Share
Figure 1. How the probability of being selected varies with the
market share and the parameter c which regulates the tendency to
choose the most attractive actor.
The final graph shows a histogram of market shares across all simulation runs. The
second graph illustrates that only actors who win, i.e., obtain the highest market
share among all n actors in their simulation run, reach a market share of about
50% where the dip occurs. Moreover, the dip occurs at a market share that is
among the lowest market shares achieved by winners (winners obtain a market
share from about 30% to 100%). Finally, the histogram in the third graph shows
that the distribution of market shares is bimodal. Market shares tend to be below
0.3 (most common) or above 0.5 (only about 8% of all actors obtain a market
share this high). The bimodality in market shares reflects the bimodal nature of
the competition when there is strong social influence. Actors who are chosen early
benefit from social influence and can reach a high market share. Actors who are
not chosen early on will likely only reach a low market share even if they are of
high quality.
Why does the dip occur then? To examine the immediate cause of the dip, we
first examine the quality levels associated with market shares around the dip when
c = 5. Figure 4 plots histograms of qualities for three market share intervals: i)
market shares around the first local maximum (between 0.1 and 0.3), ii) market
BELOW TOP STATUS INDICATES MEDIOCRITY
6
1 0.9 0.8 Average Quality 0.7 0.6 0.5 0.4 c = 3 0.3 c = 4 c = 5 0.2 c = 6 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Market Share Figure 2. How average quality varies with market share for different values of c which regulates the tendency to choose the most
attractive actor (each line is based on 500 000 simulations of 50
periods in which w = 0.2 and n = 10).
shares around the dip (between 0.4 and 0.6), iii) high market shares (between 0.8
and 1.0). As shown, top qualities (close to 1) are relatively common for market
shares between 0.1 and 0.3 as well as for market shares between 0.8 and 1.0. In
contrast, top quality levels are less common for market shares between 0.4 and 0.6
(where the dip occurs).
The immediate reason for why the dip occurs is thus clear: top quality actors
seldom have market shares in the interval around the dip. Rather, actors of top
quality either have high or low market shares. The reason is that when c is high,
market shares for top quality actors bifurcate: a high quality actor will either obtain
a high market share (which happens when this actor is selected initially) or obtain
a comparatively low market share (when some other actor is selected first). Such a
bimodal distribution of market shares, for top quality actors, does not occur when
c is low, such as when c = 3. To illustrate this, Figure 5 shows the distribution of
market shares for an actor with quality 0.9. The distribution is bimodal when c = 5
but unimodal when c = 3. The intuitive reason why the market share distribution
is unimodal when c = 3 is that being selected first does not matter as much when
c = 3 as when c = 5. When c = 3 actors with a high market share will only be
BELOW TOP STATUS INDICATES MEDIOCRITY
7
Average Quality
1
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.7
0.8
0.9
1
0.7
0.8
0.9
1
Market Share
Proportion Win
1
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Market Share
4
9
x 10
8
Frequency
7
6
5
4
3
2
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Market Share
Figure 3. How average quality, the proportion wins, and frequency vary with market share. Based on 10 000 simulations of 50
periods each where c = 5, n = 10, and w = 0.2
slightly more likely to be selected than actors with a low market share (see Figure
1). Initial good luck is thus unlikely to propel an actor to a high market share.
When c is large it is as if actors become separated into two different playing fields.
One - ’elite’ - playing field consist of sufficiently high quality actors who benefitted
from initial good fortune. The extra boost they receive from being selected early
propels them to high market shares. How high their market share becomes depends
on their quality: only top actors can reach the highest market shares. The reason
is that both quality and market share impact selections. Because market shares
within the ’elite’ playing field depends on quality, higher market share indicates
higher quality. The second playing field consists of medium to high quality actors
who were unlucky and were not selected in the first few periods. The second playing
field also consists of low quality actors: these actors cannot reach a high market
share even if they are initially lucky. Among the actors in the second playing
field, higher market share again indicates higher quality. The dip occurs because
BELOW TOP STATUS INDICATES MEDIOCRITY
0.1 < Ms < 0.3
Frequency
2500
0.4 < Ms < 0.6
2500
2000
2000
1500
1500
1500
1000
1000
1000
500
500
500
0
0
0.5
Quality
1
0
0
0.5
0.8 < Ms < 1.0
2500
2000
1
0
8
0
Quality
0.5
1
Quality
Figure 4. Histograms of qualities for three market share intervals: i) market shares around the first local maximum (between
0.1 and 0.3), ii) market shares around the dip (between 0.4 and
0.6), iii) high market shares (between 0.8 and 1.0). Based on 10
000 simulations in which c = 5, w = 0.2, and n = 10.
the actors with the highest market share in the second playing field are of higher
quality than the actors with the lowest market shares in the ’elite’ playing field.
2.3. Bifurcation Analysis. Our model belongs to the class of ’non-linear Polya
urns’ analyzed by Arthur et al. (1987). To gain further insight into why market
shares bifurcate when c is large, we follow Arthur et al. (1987) and analyze the
change in the expected market share. Let Bi,t+1 be an indicator variable equal to
one if actor i is chosen in period t + 1 and equal to zero otherwise. Let ki,t be the
number of periods in which actor i has been selected during periods 1, ..., t. The
market share of actor i in period t + 1 is mi,t+1 = (ki,t + Bi,t+1 )/(t + 1). By adding
and subtracting mi,t this can be written as
(3)
mi,t+1 =
1
1
(mi,t t + mi,t + Bi,t+1 − mi,t ) = mi,t +
(Bi,t+1 − mi,t )
t+1
t+1
By adding and subtracting Pi,t /(t + 1) this can be written in terms of Pi,t as follows
(4)
mi,t+1 = mi,t +
1
1
(Pi,t+1 − mi,t ) +
(Bi,t+1 − Pi,t )
t+1
t+1
Equation (4) shows that the change in market share is the result of a deterministic
term (Pi,t+1 −mi,t ) and a stochastic noise term (Bi,t+1 −Pi,t ). Because E[Bi,t+1 ] =
Pi,t , the noise term does in fact have zero expected value.
By taking expectations, we find that the expected market share in period t + 1,
conditional upon the market share in period t, is
(5)
E[mi,t+1 |mi,t ] = mi,t +
1
(Pi,t+1 − mi,t ).
t+1
BELOW TOP STATUS INDICATES MEDIOCRITY
9
0.5 0.45 c = 5 c = 3 0.4 Propor%on 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 0.2 0.4 0.6 0.8 1 Market Share Figure 5. The distribution of market shares after 50 periods for
actors with a quality equal to 0.9 when c = 5 and c = 3. Based on
50 000 simulations of 50 periods each in which w = 0.2 and n = 10;
in these simulations the quality of the first actor was fixed while
the quality of the others were drawn from a uniform distribution.
This analysis shows that the change in the market share of actor i depends on
whether the probability of selection (Pi,t+1 ) is larger or smaller than the current
market share (mi,t ). Whenever Pi,t+1 > mi,t , the expected market share of actor i
increases. Whenever Pi,t+1 < mi,t , the expected market share of actor i decreases.
The probability of selection, in turn, depends on the current market share since
evaluators are more likely to select actors with higher market shares. A graph
plotting how Pi,t+1 varies with mi,t can thus helps us to understand how market
shares evolve over time and when bifurcation is possible. Figure 6 provides this
plot for two values of c and three quality levels (0.1, 0.5, and 0.9). Note that
Pi,t+1 depends on the market shares and qualities of all n actors. In Figure 6 it is
assumed that the qualities of all actors other than the focal one is 0.5. Moreover, it
is assumed that all other actors share the remaining market equally: their market
shares each equal (1 − mi,t )/(n − 1). Other assumptions give similar results. Figure
6 also plots the 45-degree line. Whenever Pi,t+1 crosses the 45-degree line, Pi,t+1
equals mi,t . Such a crossing represents a ’fix-point’, i.e., a point at which there is no
expected change in market share: the expected market share tomorrow is today’s
market share.
Figure 6 illustrates why the market share for high quality actors (quality equal to
0.9) is rarely around 0.5 when c = 5. Consider first market shares above 0.55. The
BELOW TOP STATUS INDICATES MEDIOCRITY
c=5
q = 0.9
q = 0.5
q = 0.1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
Market Share
c=3
1
Probability Selected
Probability Selected
1
10
0.8
1
q = 0.9
q = 0.5
q = 0.1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
Market Share
Figure 6. How the probability of being selected in the next period
(Pi,t+1 ) depends on current market share (mi,t ) for different quality
levels and different values of c.
probability of selection exceeds the current market share for market shares above
0.55 and below 0.74. Whenever the market share is just above 0.55 the market share
thus tends to increase until it reaches 0.74 (where Pi,t+1 = mi,t ). If the market
share is above 0.74, the probability of selection is below the current market share,
which implies that the market share tends to decrease back to 0.74. Consider next
market shares below 0.55. The probability of selection is lower than the current
market shares for market shares between 0.55 and 0.22. Whenever the market
share is just below 0.55 the market share thus tends to decrease until it reaches
0.22 (where Pi,t+1 = mi,t ). If the market share is below 0.22, the probability of
selection is above the current market share, which implies that the market share
tends to increase back to 0.22. In other words, when c = 5 the market shares of
high quality actors tend to converge to either 0.22 or 0.74; these are two fix-points
at which Pi,t+1 = mi,t . A market share of 0.55 is also a fix-point but it is unstable:
any deviation will lead to further deviations away from this point. More generally,
because there are two stable fix-points of the mapping, the distribution of market
shares will be bimodal and each mode will be centered on one of the stable fixpoints (Arthur et al., 1987; Pemantle, 2007; Borkar, 2008). Moreover, the reason
that there are two stable fix-points is that Pi,t+1 is non-linear when c is high.
The situation is very different when c = 3. In this case, there is only one fixpoint, even for actors of high quality. As a result, the market share distribution is
unimodal, centered on the only fix-point. For example, the market share of high
quality actors tends to converge to a value around 0.13. Figure 6 also shows why
1
BELOW TOP STATUS INDICATES MEDIOCRITY
11
the market shares of lower quality actors (0.1) seldom bifurcate even when c = 5.
For such quality levels Pi,t+1 is low whenever mi,t is low. Even if c is high the low
starting point implies that Pi,t+1 only crosses the 45-degree line once, implying a
unimodal distribution of market shares.
The reason why a large value of c is necessary for a dip is hence that a large
value of c implies that Pi,t+1 increases rapidly with mi,t for intermediary levels
of market shares but then levels off. This non-linearity implies that small chance
events can have large long-term consequences (Arthur et al., 1987) but only for
actors of sufficiently high quality. Actors of sufficient quality will either reach a
high or a low market share. Low quality actors, in contrast, will almost always
obtain a low market share.
2.4. When the dip occurs. There are four parameters in the model: c, w, n,
and the number of periods. We have seen that a dip requires a relatively high value
of c. The value of c cannot be too high, however. If c is very high, evaluators
almost always choose the actor with the highest attraction. In this case, any actor
with sufficient quality who is selected in the first period will be selected in almost
all future periods. Such an actor will very likely obtain a market share equal to
100%. It follows that nearly all actors who are lucky initially and selected early
will reach the same market share (100%) independent of their quality. As a result,
a dip cannot occur.
The magnitude of the dip, and whether a dip exists, also depends on the value of
w, the weight evaluators give to quality. Larger values of w reduce the magnitude of
the dip and, for a given value of c, may eliminate the dip altogether. For example,
when c = 5 there is no dip if w ≥ 0.4 (when n = 10). But a dip can still occur when
w is high if c is sufficiently large. For example, a dip occurs when w = 0.4 if c = 7
(when n = 10). The impact of a high value of w is not difficult to understand.
If w = 1 only quality matters and medium quality objects are very unlikely to
obtain a high market share. More generally, higher values of w reduce the impact
of social influence. In terms of Figure 6, the effect is to flatten the relationship
between market share and the probability of being selected: the probability of
being selected is not much higher for actors with high market share than for actors
with low market share. As a result, the curve is likely to cross the 45-degree line
only once, preventing market share bifurcation.
The weight given to quality cannot be too low either. For example, when c = 5
there is no dip if w ≤ 0.02 (when n = 10). The reason why w cannot be too low is
that a dip only occurs if market share is informative about quality. When w = 0
quality does not matter and market share will be independent of quality.
Changing n does not change the basic result much if the other parameters are
suitably rescaled. In particular, c needs to be increased when n is increased because
otherwise the probability of being selected in the next period will be low even for
BELOW TOP STATUS INDICATES MEDIOCRITY
12
the actor with the highest market share. Changing n also changes the percentile
at which a dip occurs. As a rule of thumb, the dip occurs at a market share level
reached by the top 1/n actors. The reason is that a dip occurs, roughly, at the lowest
market share reached by the winner in a simulation run (the actor who receives the
highest market share among all competing n actors). The rule of thumb follows
because the proportion of winners is 1/n.
The bifurcation analysis suggests that the dip is not much affected by the number
of periods examined. Once the dip has emerged, it tends to persist. The reason is
that the stochastic process will converge to a (stochastic) equilibrium in which the
market shares of the actors stabilize (Arthur et al., 1987). Simulations confirm this
expectation. For example, suppose c = 5 and w = 0.2. The dip continues to exist,
and looks about the same, if the system is simulated for 50, 500, or 5000 periods.
Finally, the dip continues to exist if we change the distribution of quality. Rather
than being uniform it is perhaps more realistic that the distribution of quality is bell
shaped. For example, quality might be drawn from a symmetric beta distribution
with parameters (3,3). This distribution has mean 0.5 but in contrast to the uniform
distribution it is unimodal and centered around 0.5. When c = 5, w = 0.2, and
n = 10, there is still a dip which looks very similar to when qualities are drawn
from a uniform distribution.
3. A Model Without Competition
Is the reason for the dip the assumption that status depends on relative attraction in a competition between a small number of actors? It is intuitively plausible
that a medium quality actor who is selected first could end up being the most attractive when there are few competing high quality actors available. This reasoning
might suggest that competition between a small number of actors is central to the
mechanism behind the dip. Nevertheless, a dip can occur even in a model without
competition. What is essential is that social influence is sufficiently strong and nonlinear so to generate a bimodal distribution of outcomes for actors of sufficiently
high quality.
To illustrate that non-linearity in social influence is sufficient to generate a dip
we here develop a model without competition. Like the model described in section
2.1 we here assume that status depends on choices by evaluators. Specifically, there
is a sequence of evaluators who each chooses whether to defer to actor i or not.
In contrast to the model described in section 2.1 there is no explicit comparison
between actor i and other actors in the current model. In the current model evaluators only consider the quality of actor i and the judgment of past evaluators about
actor i when deciding whether to defer to actor i or not.
3.1. Model. Formally, the model is specified as follows. The quality of actor i, qi ,
is drawn from a uniform distribution. In each period t an evaluator arrives and
BELOW TOP STATUS INDICATES MEDIOCRITY
13
1
S = 15
S = 10
S=5
0.9
0.8
Choice Probability
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Choice Proportion
Figure 7. How the choice probability (Pi,t ) varies with the past
choice proportion (di,t−1 ) for different values of S (for an actor of
quality equal to 0.5 when w = 0.2 and h = 0.4.
decides whether to defer to actor i or not. The probability that the evaluator in
period t will choose to defer to actor i is
(6)
Pi,t = wqi + (1 − w)Ii,t .
Here Ii,t is the social influence function which depends in a non-linear way on the
proportion of past evaluators who chose to defer to actor i:
(7)
Ii,t =
1
,
1+
where di,t−1 is the proportion of evaluators in periods 1, ..., t − 1 who chose to defer
to i. Moreover, S is a parameter regulating how sensitive Ii,t−1 is to whether the
choice proportion di,t−1 is above the threshold, h.1 As Figure 7 shows, when S is
high the choice probability (Pi,t ) increases rapidly for choice proportions (di,t−1 )
above h.
In this model there is no explicit competition. Status does not depend on a
comparison with other actors. Status only depends on the quality of the focal actor
and the proportion of past evaluators who chose to defer to this actor. Hence,
e−S(di,t−1 −h)
1In period 1 where d
i,t−1 is not defined we set di,0 = 0.
BELOW TOP STATUS INDICATES MEDIOCRITY
(A)
(B)
1
0.9
0.9
0.8
0.8
Choice Probability
Average Quality
1
0.7
0.6
0.5
0.4
0.3
0.6
0.5
0.4
0.3
0.2
0.1
0.1
0
0.2
0.4
0.6
Choice Proportion
0.8
1
q = 0.8
q = 0.5
q = 0.2
0.7
0.2
0
14
0
0
0.2
0.4
0.6
0.8
1
Choice Proportion
Figure 8. A) How average quality varies with the choice proportion in model without competition described in section 3.1.
Based on one million simulations of 50 periods each when S = 10,
w = 0.2, and h = 0.4. B) How the choice probability varies with
the choice proportion for actors of different qualities together with
the 45-degree line (when S = 10, w = 0.2, and h = 0.4).
an actor cannot be evaluated favorably because the competing actors are of low
quality. Still, the choice probability depends in a non-linear way on the past choice
proportion. This non-linearity is sufficient to generate a dip.
3.2. Result. Figure 8 (A) plots how average quality varies with the choice proportion reached after 50 periods (when S = 10, w = 0.2 and h = 0.4). As shown,
there is a dip. Simulations show that a dip occurs whenever S ≥ 6 (when w = 0.2
and h = 0.4). The reason for the dip is the same as in the model developed in
section 2.1. Strong and non-linear social influence generates a bimodal distribution
of choice proportions. Moreover, only top quality actors can reach choice proportions close to one because the choice probability depends on quality as well as
past outcomes. To illustrate the importance of quality, Figure 8 (B) plots how the
choice probability varies with the choice proportion for actors of different qualities
together with the 45-degree line (when S = 10, w = 0.2 and h = 0.4). In this model
even actors of low quality could end up converging to a high choice proportion: the
choice probability function crosses the 45-degree line at a high choice proportion
even for actors with quality equal to 0.2. Still, if the choice proportion converges
to a high level the level it reaches depends on quality: actors with higher quality
converge to higher choice proportions than actors of lower quality do. This explains
why there is a dip: high but not top choice proportions indicate that the actor is a
low quality actor who happened to be lucky initially.
BELOW TOP STATUS INDICATES MEDIOCRITY
15
3.3. Discussion. An essential property of the model without competition described
in section 3.1 is that social influence (the past choice proportion) cannot compensate for low quality. As Figure 8 (B) shows, low quality effectively sets an upper
limit on the choice proportion an actor can converge to. Models without this property will not generate a dip at high but not top status levels even if these models
generate a bimodal distribution of choice proportions.
To illustrate this, consider the following exponential choice model without competition. The quality of actor i, qi , is drawn from a uniform distribution. In
each period t an evaluator arrives and decides whether to defer to actor i or not.
The decision to defer to actor i or not depends on actors i’s attractiveness, Ai,t .
Actors i’s attractiveness is a weighted average of her quality and past outcomes:
Ai,t = wqi + (1 − w)di,t−1 , where w is the weight evaluators give to quality and
di,t−1 is the proportion of evaluators in periods 1, ..., t − 1 who chose to defer to
i. The probability that the evaluator in period t will choose to defer to actor i
depends on attractiveness as follows
1−r
,
1 + e−S(Ai,t −b)
where S is a parameter regulating how sensitive the choice probability is to how
attractive the actor is, r is the baseline probability of deference and b is a normalizing constant ensuring that the probability of deference is not too high when
attractiveness is low.
Social influence (the past choice proportion) can compensate for low quality in
this model. To explain this, note that the choice probability will be close to one
whenever Ai,t − b is positive and S is very high. Moreover, Ai,t − b can be positive
even when quality is low, indeed even if qi = 0, as long as (1 − w)di,t−1 > 0, i.e.,
when di,t−1 is high and w is low. It follows that when w is low any actor, even
low quality actors, who happen to be selected initially can be propelled to a choice
proportion at or close to one.
Because quality does not impact the choice proportion an actor obtains, if he
or she is lucky initially, a dip of the kind shown in Figures 2 or 8 does not occur.
Rather, as Figure 9 (A) shows, the association between market share and quality
implied by the exponential choice model without competition is inverted u-shaped:
average quality first increases with market share, reaches a maximum, and then
decreases with market share. The reason for this pattern is precisely that a high
past choice proportion can compensate for low quality in this model. Figure 9
(B) illustrates this by plotting how the choice probability varies with the choice
proportion for actors of different qualities together with the 45-degree line. As
Figure 9 (B) shows, quality does not impact the choice proportion actors converge
to if they happen to be lucky initially and are chosen in the first periods. Even
actors of low quality who are lucky initially will reach a choice proportion close
(8)
Pi,t = r +
BELOW TOP STATUS INDICATES MEDIOCRITY
(A)
(B)
1
0.9
0.9
0.8
0.8
Choice Probability
Average Quality
1
0.7
0.6
0.5
0.4
0.3
0.6
0.5
0.4
0.3
0.2
0.1
0.1
0
0.2
0.4
0.6
Choice Proportion
0.8
1
q = 0.8
q = 0.5
q = 0.2
0.7
0.2
0
16
0
0
0.2
0.4
0.6
0.8
1
Choice Proportion
Figure 9. A) How average quality varies with the choice proportion in the exponential choice model described in section 3.3. Based
on one million simulations of 50 periods when S = 15, w = 0.2,
r = 0.1, and b = 0.5. B) How the choice probability varies with
the choice proportion for actors of different qualities together with
the 45 degree line (when S = 15, w = 0.2, r = 0.1 and b = 0.5).
to 100%. It follows that a choice proportion close to one is not informative about
quality. Why, then, are choice proportions around 60% indicative of high quality?
The brief answer is that quality has most impact on the choice probability at a
choice proportion around 60%. In terms of Figure 9 (B): the choice probabilities of
actors of different qualities differ most for intermediary choice proportions.
4. Empirical Plausibility
The two most central assumptions of the models that generate a dip for high
but not top outcomes (the models described in section 2.1 and 3.1) are: 1) Even an
actor or object of medium quality can occasionally reach a high outcome if social
influence is strong and the actor or object is lucky enough to be chosen early; 2)
Only top quality actors / objects can reach a top outcome because social influence
cannot compensate for low quality. How realistic are these assumptions?
Experimental research has convincingly demonstrated that social influence exists (Salganik et al., 2006; van de Rijt et al., 2014). But is such social influence
sufficiently strong to generate a dip? The impact of social influence on outcomes
has been debated in the literature (Tellis et al., 2009; Azoulay et al., 2014). Some
empirical studies have shown that, when quality and alternative explanations are
properly controlled for, social influence only has a small effect on outcomes (Van den
BELOW TOP STATUS INDICATES MEDIOCRITY
17
Bulte & Lilien, 2001; Azoulay et al., 2014) while other studies have found large effects (Iyengar et al., 2011; Simcoe & Waguespack, 2011). Few studies, however,
have examined the relative importance of social influence versus quality.
An exception is Salganik et al. (2006) experimental study of social influence in
music downloads. To study how quality (song specific traits) versus social influence
impact choices and inequality, Salganik et al. (2006) set up a website with 48 songs
that participants could listen to and eventually download. In one condition - the No
Social Influence condition - participants independently listened to and downloaded
songs, without any information about the activities of other participants. The
market share of downloads in this condition provides an estimate of the ’quality’ of
a song, i.e., the song specific traits that influence the market share a song receives.
Here we use their data to examine whether social influence can propel a medium
quality objects to a high market share and whether social influence can compensate
for low quality.
Social influence was experimentally manipulated in this study by adding information about the downloading activities of other participants. Two different
manipulations of social influence were used, corresponding to weak and strong social influence. In the Weak Social Influence condition information about how the
number of others who had downloaded a particular song was available but not
particularly salient. In the Strong Social Influence condition information about
downloads were made salient. Specifically, the songs were presented in order of
popularity: the song that had been downloaded the most by prior participants was
presented first on the webpage, the second most popular was presented second etc.
It follows that if a song became one of the most popular in the Strong Social Influence condition it was displayed prominently and many participants were likely to
listen to it and eventually download it.2
The experiment was repeated with eight different participant pools, generating
data from eight different realizations of the process (eight ’worlds’). The data shows
that market shares of songs varied quite a bit across worlds. Interestingly, the
data does show that even objects of medium quality can reach high market shares.
In particular, in world 3 in the strong social influence condition song number 31
obtained the highest share of downloads of all songs in this world (12.8%) despite
having a below medium quality (the 26th lowest quality level out of 48 songs).
This data provides existence proof for assumption (1) that social influence in a
setting similar to recommendation websites can propel an object of medium quality
to a high market share. The data from the strong social influence condition is
also consistent with assumption (2) that reaching top market shares required high
2Song position did influence listening and downloading decisions. In an additional experiment,
songs were listed in random order but with download information. The results showed that
participants were more likely to listen to and download songs listed earlier (Hendricks et al.,
2012).
BELOW TOP STATUS INDICATES MEDIOCRITY
18
quality in addition to good initial luck. Specifically, the market share reached by
a song that became the most popular in its world depended strongly on quality.
For example, when the below medium quality song 31 became the most popular in
world 3 it reached a market share of 12.8%. When the highest quality song (song
number 25) became most popular (which happened in five out of the eight worlds)
it reached a market share of 16% on average (and once reached 19.4%). More
generally, quality of a song explains much of the variation in the average market
share reached by a song when it became the most popular song in a world: the
correlation between the average market share reached and quality is 0.88.
Interestingly, there is also evidence in the Salganik et al (2006) data that the distribution of market shares is bimodal for objects of sufficient quality. Specifically,
Hendricks et al. (2012) show that a bimodal distribution fits downloads of medium
to high quality songs significantly better than a unimodal distribution. Such bimodality in market shares suggests that a reinforcement mechanism similar to that
in the model we described in section 2.1 is the reason for why a song of medium
quality could end up being the most popular in one world: by being chosen early
on this song was propelled to a high market share.
Do the result also support the existence of a dip in the market share-quality
association? Figure 10 shows the association between market share and quality in
the two social influence conditions. Following Salganik et al (2006), the quality
of song i was measured by the market share of song i in the no social influence
condition. The solid lines are the best fitting smothing splines (see the Appendix
for how the splines were computed). The dashed lines represent the 95% confidence
limits based on 1,000 bootstrap simulations. As Figure 10 shows, the association
between quality and obtained market share is monotonic in the weak social influence
condition: higher market share indicates higher quality. However, in the strong
social influence condition the association between quality and obtained market share
is not monotonic. Rather, there is a dip: average quality increases with market
share for market shares up to about 9%. After that, average quality decreases with
market share until about 13%. After 13% average quality increases again with
market share.
The fact that we observe a dip is intriguing but its existence is entirely due to
the influence of one observation: song number 31 with below medium quality that
reaches a high market share in world 3. Because there were only eight worlds,
corresponding to eight simulation runs in the model developed in section 2.1, the
large impact of one outlier is expected. Even if our model were correct, we would
only expect to see occasional instances of songs of medium quality being propelled
to high market shares. The dip would only be significant if data from a large
number of simulation runs (a large number of ’worlds’) was available.
BELOW TOP STATUS INDICATES MEDIOCRITY
Strong social influence
quality
0.020
0.005
0.01
0.010
0.015
0.02
quality
0.03
0.025
0.04
0.030
0.035
0.05
Weak social influence
19
0.00
0.02
0.04
0.06
market share
0.08
0.10
0.00
0.05
0.10
0.15
market share
Figure 10. The association between market share and quality
(market share in the independent conditions) in Salganik et al
(2006) experimental data. The solid lines represent the best-fitted
smoothing spline function and the dashed lines are the 95% confidence limits.
In the case of the experimental data, it is perhaps a coincidence that a song of
medium quality won and that song 31 that won in world 3. However, the rank song
31 reaches - it reaches a market share among the top 2.08% of all market shares
- is consistent with expectations based on the model developed in section 2.1. In
a competition between n = 48 songs, our model predicts that a dip will occur at
the market share corresponding to the 1/48 = 2.08% highest market share (see
the discussion of the impact of n in section 2.4). The intuition is that if a song of
medium quality becomes the most popular in a simulation run, the market share
it reaches is among the lowest of any song that becomes most popular. This is
consistent with the experimental data from the strong social influence condition:
song 31 did reach the lowest market share among any winner in a world, giving it
an overall rank of 8 out of 384.
An important caveat is that social influence is not always as strong as it appears
to be in Salganik et al. (2006) experimental study of social influence in music
downloads. In some context, social influence appears to be of marginal importance
(Azoulay et al., 2014) or swamped by other forces (Van den Bulte & Lilien, 2001).
van de Rijt et al. (2014) also showed that social influence often has a rapidly
declining marginal effect, implying that the run-away process required for a dip
may be unlikely to occur. Finally, in some settings concerns for reciprocity (Gould,
0.20
BELOW TOP STATUS INDICATES MEDIOCRITY
20
2002) and status diffusion (Bothner et al., 2010) imply that market share is unlikely
to accumulate without bound.
5. Relation to prior work on status and quality decoupling
Our focus is on when the association between status and quality reverses sign,
implying that higher status does not indicate higher quality. Most prior work on
status-quality decoupling has instead focused on the strength of the status-quality
association. For example, Lynn et al. (2009) show that strong social influence can
lead to a low rank correlation between status and quality when initial status assignments are subject to noise. The key mechanism is that social influence preserves
rather than corrects errors in initial status assignments. Because actors who were
assigned high status last period are more likely to be assigned high status in the
next period, actors with low quality who happened to obtain high initial status will
likely continue to have high status.
Our model can replicate the finding that stronger social influence leads to a
lower rank correlation between status and quality. Specifically, reducing w in our
model will reduce the weight evaluators give to quality and this will reduce the rank
correlation between status and quality. For example, the Spearman rank correlation
between market share (our measure of status) and quality generated by our model
is 0.84 when w = 0.8, 0.72 when w = 0.5, while it is 0.29 when w = 0.2 (based on
5000 simulations of 50 periods where c = 5 and n = 10).
In contrast to prior work our model also shows that when w goes from 0.5
to 0.2 something new happens: a dip occurs which implies that the sign of the
association between status and quality reverses. Such a reversal of the sign of
the association between status and quality illustrates a more fundamental type of
decoupling between status and quality than merely a low correlation. A low but
positive correlation is still consistent with status being a signal of quality. That is,
even if the correlation between status and quality is low (but positive) higher status
may still indicate higher expected quality. A low but still positive correlation only
implies that the association between status and quality is not very strong: higher
status actors may only be of marginally higher quality than lower status actors.
Our model, in contrast, shows that when social influence is strong and non-linear
the association between status and quality can reverse in a certain status interval,
implying that higher status systematically indicates lower expected quality in this
interval.
Our results similarly differ from other accounts of how low quality actors could
end up dominating as a result of processes of cumulative advantage or positive feedback (Merton, 1968; Chase, 1980; Carroll & Harrison, 1994; Perrow, 2002; DiPrete
& Eirich, 2006). These accounts focus on when and why low quality actors could
BELOW TOP STATUS INDICATES MEDIOCRITY
21
end up dominant. The implication is that the association between status and quality is not as strong as one might have imagined: high status does not guarantee
high quality. These accounts usually remain silent, however, about whether the
possibility that low quality actors can end up dominant implies that dominance
indicates low expected quality.
For example, Carroll & Harrison (1994) show that inefficient organizational forms
could end up dominant due to positive feedback resulting from non-monotonic density dependence. In their simulations, a less efficient organizational form that develops earlier than a more efficient organizational form can remain the most popular
form. The reason is that density impacts both adoption of organizational forms
and survival among adopters. Their simulations demonstrate that an organizational form with a large market share is not necessarily the more efficient than an
organizational form with lower market share. Carroll & Harrison (1994) do not
claim that this demonstration implies that higher market share generally indicates
lower expected efficiency. Whether this is the case or not is an open question: Carroll & Harrison (1994) do not examine the association between market share and
efficiency implied by their model.
Our model shows that the possibility that low quality actors may end up dominant for long does not necessarily imply that higher status indicates lower quality.
For example, an actor of low quality could end up dominant for a long period in
our model even when c is low. When c is low, however, there is no dip and higher
status always indicates higher expected quality. For a concrete illustration, suppose
that c = 2 and w = 0.2 and consider a competition between n = 2 actors. Suppose
that the quality of the first actor is 0.32 while quality of the second actor is 0.75.
Simulations show that if the lower quality actor is selected in the first period his
market share can remain higher than the market share of the higher quality actor for more than 2000 periods. This simulation shows that domination by a low
quality actor is possible. Nevertheless, when c = 2 and w = 0.2 a dip does not
occur. Rather, higher market share always indicates higher expected quality. This
example illustrates that the fact that a model implies that a lower quality actor
could end up being dominant for long does not imply that the model also implies
that higher market share indicates lower expected quality.
Our model also differ in emphasis from prior discussions of how ’winner-take-itall’ dynamics (Frank & Cook, 1995), processes of cumulative advantage (DiPrete &
Eirich, 2006), and social influence (Gould, 2002) can magnify the rewards received
by high status actors. Such processes change the scale of rewards but not necessarily
the rank correlation between status and quality (Lynn et al., 2009).
The model closest to ours is the Denrell & Liu (2012) model of how rich-get-rich
dynamics imply that top performance is an unreliable indication of quality. In their
model, the association between performance and quality reverses at very high levels
BELOW TOP STATUS INDICATES MEDIOCRITY
22
of performance: the expected quality of top performers is lower than the expected
quality of performers with high but not top performance. They get this result in
a model where the probability of a success depends on quality and the outcome in
the previous period. In contrast to our model, Denrell and Liu assume that the
impact of past outcomes is not known by observers. Past outcomes could have a
strong or a weak impact on future outcomes. The intuition for the reversal of the
performance-quality association is that extreme levels of performance indicate that
past outcomes strongly impact future outcome: extreme levels of performance are
more likely then. If past outcomes strongly impact future outcomes, however,
performance is an unreliable indicator of quality because even an actor of low
quality who is lucky initially can obtain extremely high performance. It follows
that extreme performance indicates unreliability and that not much can be learned
about the quality of actors who achieved it. Moderately high levels of performance
indicate less strong path dependence and may be a more reliable indication of high
quality than top performance.
We can replicate the Denrell and Liu (2012) finding that top performance indicates unreliability and medium quality by making the importance of social influence
random in our models. Specifically, consider the model described in section 2.1.
Suppose the weight on quality, w, is not a fixed constant but a random variable
drawn from a uniform distribution between zero and one. As figure 11 shows, this
modification of the model implies that the association between market share and
quality changes to become inverted u-shaped: average quality first increases with
market share, reaches a maximum, and then decreases with market share. Moreover, market shares around 50%, which previously were associated with a dip, are
now associated with the highest level of quality. The reason why top market shares
now indicate medium quality is that obtaining top market shares (close to one) is
much more likely for actors with a low value of w: when w is low social influence
is maximized and actors with initial good luck can reach very high market shares.
Such low level of w, however, generate very noisy outcomes that are unreliable
indicators of quality.
The Denrell and Liu (2012) mechanism is realistic when comparing outcomes
across a large class of settings, in which the strength of social influence may vary
substantially (because the nature of the task differs or because the social dynamics
and network configurations may differ). The mechanism we develop in this paper
is more applicable when social influence is known to be quite strong but is unlikely
to vary much.
Moreover, an essential assumption in the Denrell and Liu (2012) model is that
social influence could compensate for low quality. That is, even low quality actors
who are lucky and selected early on can end up with top outcomes. Specifically,
Denrell and Liu (2012) assume that the weight on past outcomes (w in their model)
BELOW TOP STATUS INDICATES MEDIOCRITY
23
1
0.9
0.8
Average Quality
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Market Share
Figure 11. How average quality varies with market share in a
model with a random weight on quality. Based on 50 000 simulations of 50 periods when c = 5 and n = 10
may be close to one and the weight on quality close to zero. In this case, even a
low quality actor who is lucky initially could reach top performance (a success
proportion close to one). Similarly, the model described in the above paragraph,
which replicates the Denrell and Liu (2012) finding, also allows for the possibility
that social influence can compensate for low quality. If the weight w evaluators give
to quality is very low and the weight given to social influence is very high, even a
low quality actor who is selected in the first period might reach choice proportion
close to one. In contrast, the models developed in this paper that generate a dip
for high but not top outcomes (the models described in section 2.1 and 3.1) assume
that only top quality actors can reach top outcomes. This is the reason why top
market shares indicate top quality in the models described in sections 2.1 and 3.1.
6. Discussion of the mechanism
6.1. When social influence cannot compensate for quality. A dip occurs
for high but not top status in our model when social influence cannot compensate
for quality and hence top quality is necessary to reach top status. The model
described in section 3.1 showed that a dip occurs if this assumption was satisfied.
The model described in section 2.1 showed that this assumption is satisfied when
there is competition between actors. Rivalry from other actors, possibly of high
quality, limits the market share achieved by a medium quality actor who is selected
early on and benefits from social influence.
What are other social processes that imply that social influence cannot compensate for quality? One example is a two-stage adoption process (Rogers, 2003)
BELOW TOP STATUS INDICATES MEDIOCRITY
24
in which social influence first determines awareness and quality then determines
adoption (Lieberson, 2000; Hendricks et al., 2012). Specifically, suppose that adoption proceeds in two stages: 1) awareness and 2) adoption. Suppose awareness
depends on social influence but not on quality: the probability that an actor i is
aware of object j only depends on the proportion of other actors who are aware
of object j. Conditional on awareness, however, adoption only depends on quality:
the probability that an actor i who is aware of object j will adopt object j is equal
to the quality of object j. Social influence cannot compensate for low quality in
this model: high quality as well as awareness is necessary to obtain a large number
of adoptions.
Simulations show that a two-stage adoption model of this kind generates a dip
if social influence in the awareness stage is sufficiently strong and non-linear so to
generate a bimodal awareness distribution. The bimodal distribution of awareness
generates two playing fields: high and low awareness. Within each field, high quality
objects will get more adoptions than low quality objects. A dip occurs because the
objects with the highest number of adoptions in the low awareness field are of
higher quality than the objects in the high awareness field with the lowest number
of adoptions.
Another process that implies that the impact of social influence is limited is
when social influence mainly operates within a clique. Specifically, suppose that
the strength of social influence depends on the proportion of others in the same
clique who has adopted a given object. If several others within the same clique
have adopted an object, other members of this clique are likely to adopt the object
even if it is low quality. Nevertheless, an individual would not adopt an object of
low quality just because a large proportion of individuals outside his or her own
clique have adopted it.
The assumption of local social influence implies that social influence can compensate for low quality within a clique but not between cliques. If a central member
of a clique has created an object many others in the same clique are likely to hear
about it and adopt it, even if the object is of relatively low quality. But the fact
that a high proportion of individuals within the clique have adopted it will not ensure that this object becomes adopted more widely in the network. An object only
becomes widely diffused if individuals outside the initial clique adopt the object.
Individuals outside the clique, however, are unlikely to adopt a low quality object
even if many inside the clique have adopted it. It follows that only top quality
objects are likely to diffuse widely. Simulations show that this kind of process also
generates a dip in the association between adoptions and average quality: average
quality initially increases with the number of adoptions, then starts decreasing with
adoptions, and eventually increases again. The dip occurs at a level of adoptions
corresponding to the average size of a clique: the objects associated with the dip
BELOW TOP STATUS INDICATES MEDIOCRITY
25
are lower quality objects that became popular within a clique but did not manage
to diffuse more widely in the network.
6.2. When quality evolves as a function of status. Our model assumes that
status does not impact quality: quality is a fixed individual attribute. In some
settings, however, it is natural to expect that quality evolves with status (Merton,
1968; Podolny, 1993; Sauder et al., 2012). One possible scenario is that increases
in status lead to a reduction in quality. High status might breed complacency and
distraction (Bothner et al., 2012). Because the demands on their time and because
higher status actors feel secure high status actors might have both less time and less
motivation to engage in the practice that is required to sustain high quality. This
scenario offers an alternative explanation for a dip in the status-quality association.
Whether and where the dip occurs depends on how much status impacts quality.
The dip predicted by this alternative explanation can be distinguished from the dip
predicted by our model if data is available about past and future levels of quality. If
increases in status reduce quality, the expected future quality of high status actors
may be low. Our model, in contrast, predicts that the past quality of below top
status actors was also low. The reason is that the dip occurs, in our model, because
below top status is an unreliable signal of high quality, not because status causally
impacts quality.
Another possible scenario is that increases in status lead to higher quality. Higher
status might provide actors with the resources necessary to improve (Merton, 1968)
High status actors might also have stronger incentives to invest in quality (Podolny,
1993; Sauder et al., 2012). How does the possibility that increases in status leads to
improved quality change the result of our model? Does the dip in the association
between status and quality disappear?
The answer depends on how much status impacts quality and whether the effect
of status depends on an actor’s initial level of quality. The dip disappears if status
leads to high quality for any actor that obtains high status but the dip survives
if status only has a minor effect on quality. Even if status has a significant effect
on quality the dip can survive if only sufficiently high quality actors can benefit
from status. For example, only sufficiently qualified actors may be able to benefit
from the opportunities provided to high status actors. To formalize this, suppose
the quality of actor i increases in every period actor i gets selected but only if
the initial quality of actor i is above 0.5 (i.e., in the top 50%). Simulations show
that the dip survives even if status substantially increases the quality of actors of
sufficient initial quality. The intuition is that those low quality actors who happen
to achieve below top status are not able to benefit from this status to increase their
quality.
Finally, even if the dip in the association between final quality and obtained
status disappears because status leads to increased quality, the dip predicted by
BELOW TOP STATUS INDICATES MEDIOCRITY
26
our model can still be identified if data is available on initial quality levels. Our
model predicts that the initial quality levels of those who achieved below top status
is especially low.
6.3. Social influence and cumulated disadvantage. Our model focuses on
how good luck may be boosted by social influence and when this can lead to a
dip - at below top status - in the status-quality association. Prior studies have
also examined how bad luck and disadvantages can be amplified by social influence
(Elman & Angela, 2004; Schafer et al., 2011; Maroto, 2012). Poor performance,
perhaps due to bad luck, can increase chances of future problems because evaluators
treat actors with past poor performance less favorably. Is our mechanism also
applicable to this scenario? If so, does our model predict a dip in the association
between status and quality for ’above-bottom’ statuses?
Technically, our model could be applied as-is with ’selections’ being reinterpreted
as ’being singled out for denigration’ and ’quality’ being reinterpreted as a trait that
increases the chances of being denigrated. In this interpretation, the proportion of
periods in which actor i has been selected would be a measure of negative status. In
this model an early shock can have enduring effect on status attainment process if
social influence is sufficiently strong. The model predicts a dip in the status-quality
association for actors with ’above-bottom’ status, i.e., actors with negative status
just below the maximum. But the model also predicts that obtaining maximum
negative status is a reliable indicator of high quality, i.e., a reliable indicator that
the actor posesses traits that lead to denigration. The intuition is the same as in
our original model: the non-linear effect of social influence separates actors into
two playing field: with or without disadvantages due to early bad luck. And in
the playing field where all actors have suffer from cumulated disadvantage relative
status will mainly be determined by relative quality. Formulated differently, our
model assumes that cumulative disadvantage cannot overwhelm quality: only actors
with individual traits that dispose them to be denigrated will obtain the maximum
negative status.
The assumption that cumulative disadvantage cannot overwhelm quality may
not be appropriate in some contexts. Rather, it is conceivable that cumulated
disadvantages can overwhelm quality to such an extent that any actor which is
unlucky initially may end up achieving very poor performance. An argument of
this kind has been advanced in the literature on how disasters happen (Perrow,
1984; Rudolph & Repenning, 2002) where it has been argued that catastrophic
failure is often due to minor accidents or errors that are amplified in a tightly
coupled system. In particular, when interruptions are created at a rate higher than
the rate at which system operators can remove them, the system may be doomed
to fail regardless of the skill of operators (Rudolph & Repenning, 2002). In terms
of our model setup this argument implies that the probability of being selected goes
BELOW TOP STATUS INDICATES MEDIOCRITY
27
to one for all actors, independent of their quality, when the choice proportion is
close to one. A model with this feature is the exponential choice model without
competition developed in section 3.3 as Figure 9 (B) shows. For this model, the
dip occurs at the end rather than in the middle (see Figure 9 A) That is, obtaining
maximum negative status would be an indication of lower average quality (less likely
to have individual traits that lead to denigration) than obtaining middle status. In
summary, whether the lowest status actors have the lowest expected quality depends
on how the process of cumulative disadvantage unfolds. In particular, it depends
on whether cumulative disadvantages can overwhelm individual differences or not.
6.4. Constraints on status accumulation. In this paper we assume that status
is a function of ’accumulated acts of deference’ (Sauder et al, 2012, p. 268). Status
increases whenever an actor is selected by an evaluator and evaluators only consider
the quality and prior share of selections when making a decision. In reality, status
is determined by several other factors, such as the status of those one affiliates with
(Bonacich, 1987; Gould, 2002; Bothner et al., 2010) and the status of the categories
one belongs to (Sharkey, 2014). When these determinants of status have significant
impact the runaway process leading to the dip (medium quality actors who are
selected early on continue to get selected due to social influence) may be less likely
to occur.
Suppose the status of an actor depends on who she affiliates with as well as
selections by evaluators. Suppose affiliations have been developed over a long period
and do not change quickly. An actor of medium quality may be lucky initially and
selected by an evaluator. This act of deference does not imply that the actor reaches
high status if the actor is, and continues to be, affiliated with several other medium
status actors. If evaluators consider status in addition to prior share of selections
an initial selection does not raise the probability of future selections much and a
dip is unlikely to occur. A similar dampening effect occurs when status depends
on (difficult to change) category membership (Sharkey, 2014) and when a need for
reciprocity implies that evaluators only select actors with status levels similar to
them (Gould, 2002).
The run-away process critical for a dip is also less likely to occur when high
status actors are constrained in the set of opportunities they can exploit without
losing their status (Podolny, 1993; Bothner et al., 2010). We have assumed that
medium quality actors can be propelled by social influence to a high market share.
In markets where high status actors cannot profitably exploit certain low status
segments (Podolny, 1993) there is a natural limit to the market share of high status
actors.
6.5. Alternative explanations of a dip. A dip in the status-quality association
may occur as a result of mechanisms other than non-linear social influence emphasized in this paper. We have assumed that evaluators are more likely to select
BELOW TOP STATUS INDICATES MEDIOCRITY
28
actors with higher quality, but this may not always be true, especially when evaluators are concerned about status diffusion (Bothner et al., 2010). For example,
a top status actor may only want to endorse those who are not a real threat. A
strong Chief Executive Officer may deliberately select a Chief Operating Officer of
medium rater than high quality. At lower status levels, competition may lead to a
strong positive correlation between position and quality. The overall result is a dip
in the position-quality association at below top status.
We have assumed that all actors seek status. In reality, actors may deliberate
select a lower status position if this position has other compensating attributes such
as high pay. A dip can occur if it makes sense for medium to low status organizations
to attract high quality actors with high pay. It is possible, for example, that a high
quality actor is much more valuable to a medium to low status organziation than to
a below top status organization. High quality actors, trying to break through, may
also position themselves in currently peripheral niches that may become dominant
in the future. Most evaluators might assign low status to such actors (Cattani et al.,
2014). Moderate quality actors, in contrast, may be more likely to conform and
focus on established genres (Phillips & Zuckerman, 2001) thus achieving middle
status. Such strategic positioning dependent on quality can also generate a dip.
7. Implications
7.1. Reward systems and reactions to success. The dip implies that outcome
based metrics, such as the market share, will be systematically misleading indictors
of quality. Not only will the rank correlation between the metric and quality be
relatively low, the association between the metric and quality will even be negative
for some market share intervals. If rewards were based on outcomes, the dip would
imply that higher rewards would not systematically indicate higher quality. In
contrast, if rewards were not based on outcomes but on inferred quality, i.e., rewards
were based on expected quality given the observed market share, then rewards
would be a non-monotonic function of outcomes. In this case, actors might have
incentives to deliberate reduce or hide outcomes. For example, no one would want
to be associated with market share in the region of the dip; it would be better to
destroy or hide outcomes in order to be associated with a lower market share.
The discrepancy between rewards based on outcomes and expected quality also
likely creates resentment and suspicions of favoritism. Observers who do not understand that a non-linearity in social influence can generate the dip might instead
falsely attribute it to personal or organizational characteristics. For example, consider a system in which a dip occurs and suppose observers note that high status
actors are of medium rather than high quality. Such observers might conclude that
the success achieved by these high status people cannot possibly be due to good
luck alone but requires explanation. One possible explanation might attribute their
BELOW TOP STATUS INDICATES MEDIOCRITY
29
high status to unfair advantages such as favoritism. A different explanation might
attribute their high status to exceptional personal qualities, such as persistence or
ambition. If qualities represent initial advantages (such as socioeconomic status)
the possibility of succeeding with mediocre rather than high quality may seem encouraging to observers who seek success themselves. People who have achieved high
status without initial advantages might seem to be especially good role models and
targets for imitation.
A dip can also complicate interpretations of the effect of status, possibly even
for researchers. Suppose data on two independent outcomes in two periods is available. In a setting where a dip occurs actors with high, but not top, status in the
first period will only reach medium status in the second period. Observers who
are unaware that this dip in performance is expected might falsely attribute it to
personal or organizational traits. For example, actors with high but not top status
might be argued to be most insecure and afraid of downward status mobility. The
resulting status anxiety, it might be argued, leads to poor subsequent performance.
7.2. Status-Weighted Recommendation Systems. Our results show that a
system with votes or recommendations can fail systematically. A system with votes
by a large number of evaluators can sometimes achieve a high level of accuracy
even when individual evaluators are error prone due to the ’wisdom of crowds’
(Surowiecki, 2004; Hastie & Kameda, 2005). It is well-known, however, that social
influence can undermine the accuracy of group decision-making (Asch, 1955; Lorenz
et al., 2011). The finding of a dip adds to this stream of literature by showing that
when social influence is strong and non-linear a system based on counts of votes
can lead to systematic mistakes.
Specifically, suppose voters behave as the evaluators in the model described in
section 2.1. That is, voters select whom to vote for based on the qualities of actors
but also on the proportion of votes an actor has obtained so far. Our results show
that such behavior implies that the association between votes and quality is not
monotonic. That is, higher votes do not imply higher expected quality. Rather, a
high number of votes indicate medium expected quality rather than high expected
quality. Thus, votes will not reliably signal quality.
If relying on votes does not work, what does? One approach is to rely on an
adjusted vote count; adjusted to remove, as much as possible, the systematic bias.
For example, votes could be weighted by the status of the voter, as in centrality
models of status (Katz, 1953; Bonacich, 1987). This would not change the result
in the model described in section 2.1 because evaluators were all assumed to have
the same status in that model. Suppose, however, that evaluators do differ in
status. Moreover, suppose that higher status evaluators are less likely than lower
status evaluators to rely on social influence. Specifically, suppose that higher status
evaluators place a higher weight (w) on quality. If status of evaluators correlates
BELOW TOP STATUS INDICATES MEDIOCRITY
30
with relying more on quality and less on social influence, using an adjusted vote
count in which higher status voters are weighted more can remove the dip.
Relying on evaluators who care more about quality is of course always useful
to improve accuracy. Our point is that this way of improving accuracy might be
especially important in a system in which a dip occurs. The reason is that when
a dip occurs the association between outcomes and quality is not monotonic and
expected quality will sometimes deviate systematically from observed outcomes.
In particular, lower status actors might have systematically higher quality than
higher status actors. To ensure accuracy evaluators must thus disregard status
when evaluating actors. Only high status evaluators might feel comfortable to
do so and claim that an actor with a lower status is more capable than an actor
with higher status (Hollander, 1958; Phillips & Zuckerman, 2001). Lower status
evaluators might make the same inference but might be hesitant to speak up out
of fear of appearing foolish or offensive (Asch, 1955).
Instead of always weighting high status evaluators more, a more refined approach
would be to weight high status evaluators more when evaluating actors with outcomes in some intervals. In particular, the model developed in section 2.1 implies
that there are two market share intervals when more accurate quality judgment
is especially needed: i) the highest market shares for actors who did not win a
competition (market shares associated with the local maximum, see figure 3); ii)
the lowest market shares for actors who did win a competition (market shares associated with the dip, see figure 3). Evaluators need to assess whether actors in
the first range are in fact high quality actors who have been unlucky and whether
actors in the second range are low to medium quality actors who have been lucky.
Systems of rites of passage at career transitions into the elite, such as tenure, which
combine outcome metrics with evaluations by high status experts may reflect these
concerns.
7.3. The 41st Chair. Our model provides new insight into the phenomenon of
the 41st chair that motivated Merton to write about the Matthew Effect (Merton,
1968). Merton notes that the French academy limited the number of members to
40. As a result, many famous authors and researchers were excluded. Merton went
on to discuss how such distinctions could have important long-term effects because
well-known scientists receive disproportionate credit for their work. His emphasis
was on the self-fulfilling nature of scientific fame and productivity. He noted that
scientists who were excluded from the highest ’class’ would suffer in productivity
but might nevertheless be equally talented.
Our model suggests that social influence in scientific reputations will imply that
highly talented scientists will have a bimodal outcome distribution. They will either
reach the elite or obtain relatively low status. The implication is that those who
narrowly missed being admitted to the elite, i.e., the holders of the ’41st Chair’,
BELOW TOP STATUS INDICATES MEDIOCRITY
31
are not equally talented as the elite. Rather, our model suggests that they will be
of mediocre rather than high talent. Scientists with the same talent as the elite will
instead have medium to low status. If our model is true, it follows that scientific
talent scouts on the look out for ’diamonds in the rough’ should search among
medium to low status scientists.
BELOW TOP STATUS INDICATES MEDIOCRITY
32
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Appendix
To examine whether a linear or a non-linear function best fits how average quality
varies with market share in data from the Salganik et al (2006) experiment, we fitted
a smoothing spline to the data. Specifically, we searched for a function m(x) that
minimized the following objective function:
Z
n
1X
(9)
L(m(x), λ) =
(yi − m(xi )2 + λ m00 (x)2 dx
n i=1
The first term in the spine objective function is the mean squared error of using
the curve m(x) to predict y, where x is the independent variable such as market
share and y is the dependent variable such as quality. The second term is a penalty
function for the curvature of function m(x), as measured by the squared value of the
second derivative of m(x) with respect to x, m00 (x)2 . Note that the second derivative
of m(x) would be zero if m(x) were linear. If the function is more curved, such as
convex of concave, m00 (x)2 would be larger than zero. We integrate m00 (x)2 over all
values of x to measure the average curvature. The parameter λ regulates the penalty
for non-linearity, i.e., the reduction in the objective function L(m(x), λ) that follows
from using a more curved function m(x). When λ approaches infinity, only linear
functions are allowed. When λ approaches zero, curvature is maximized and the
function will pass through every data point. We apply a cross validation method
in R (a statistical package) to find the value of λ that minimizes the objective
function, L(m(x), λ).
Warwick Business School, University of Warwick. Email: jdenrell@gmail.com.