Leadership by example in the weak-link game.
Cartwright, Edward ; Gillet, Joris ; Van Vugt, Mark 等
I. INTRODUCTION
The weak-link game was first introduced by Hirshleifer (1983) as a
stylized way to capture the private provision of many public goods. As
an illustration, Hirshleifer tells the story of Anarchia, a low lying
island protected from flooding through a network of interconnected
dikes. The crux of the story is that each citizen makes a private
decision about how strong a dike to build on their land, yet the island
will be flooded if the weakest dike breaks. Most relevant, therefore, is
not the average or total contributions to the public good but the
minimum contribution. The same could be said for the production of any
good, public or private, where output is determined by the weakest
component of production. Consequently, the weak-link game is of much
applied interest in understanding the performance of groups,
organizations and nations (e.g., Brandts and Cooper 2006b; Knez and
Camerer 1994). For example, it can help explain the high wage and
productivity differentials between rich and poor countries (Kremer
1993).
Hirshleifer argued that production will be efficient in a weak-link
game. The basic reasoning is that a person cannot free-ride in the game
and so there is an incentive to contribute an efficient amount to the
public good. This hypothesis was confirmed in two-player games (e.g.,
Harrison and Hirshleifer 1989), and also fares well in three-player
games (e.g., Knez and Camerer 1994; Weber, Camerer, and Knez 2004). It
soon became clear, however, that in games with more than three players
things are different (Isaac, Schmidtz, and Walker 1989; Van Huyck,
Battalio, and Beil 1990). What we typically observe is considerable
coordination failure with contributions rapidly falling to the minimum
level (Camerer 2003). (1) The common explanation for this is that to
contribute an efficient amount requires trust in others, because the low
contribution of one player will make any high contribution redundant and
costly for that contributor, and in games with more than three players
any trust quickly disappears (Yamagishi and Sato 1986).
How can such coordination failure be avoided? Various solutions
have been considered in the literature (Devetag and Ortmann 2007). For
instance, coordination failure is less following a temporary increase in
the gains of coordinating (Brandts and Cooper 2006b), if there is
pre-play communication (Blume and Ortmann 2007; Brandts and Cooper 2007;
Chaudhuri, Schotter, and Sopher 2009), and if players opt in to play the
game (Cachon and Camerer 1996). Generally speaking, however, these
solutions may not always be practical. For example, pre-play
communication may be unwieldy in large groups, and many of the solutions
rely on the full distribution of contributions being known rather than
just the minimum (a point taken up by Brandts and Cooper 2006a). (2)
The basic objective of this paper was to ask whether leadership
reduces coordination failure in the weak-link game. Leadership evolved
to solve coordination problems between individuals and is common in all
social species (Van Vugt 2006; Van Vugt, Hogan, and Kaiser 2008). Our
main hypothesis, therefore, is that leadership can help individuals
coordinate in the weak-link game. By leadership we shall mean that one
player can lead by publicly choosing a contribution before all other
players. Our focus is thus on leadership by example. (3) Various
experimental studies have already demonstrated the positive effect of
this kind of leadership on cooperative behavior in public good and
public bad games (Guth et al. 2007; Pogrebna et al. 2008; Van der
Heijden and Moxnes 2003). (4) It remains to be seen whether it also
works in the weak-link game.
Some evidence on the effectiveness of leadership by example in the
weak-link game is provided by Weber, Camerer, and Knez (2004) and Li
(2007). They analyze a three-player weak-link game in which choices are
made sequentially according to some exogenous order. The sequential
nature of choice means that there is leadership but of a different form
to the one we shall consider. It also means that the efficient Nash
equilibrium is the unique sub game perfect Nash equilibrium of the game
and so there are strong reasons to expect less coordination failure.
Consistent with this both Weber, Camerer, and Knez (2004) and Li (2007)
do find that coordination failure is less, even if some failure remains.
(5) Leadership, therefore, partially worked. The interpretation of this
result is not, however, clear cut because efficiency is relatively high
in three-player games even when there is not leadership. A bigger
challenge is to avoid the extreme coordination failure typically
observed in games with four or more players.
In order to see whether leadership by example can meet this
challenge we first develop a simple model of behavior that allows us to
distinguish different reasons why leadership may work. We then report on
experiments with both exogenous and endogenous leadership in a repeated
four-player weak-link game. Overall our results are somewhat mixed. In
some groups we observe successful leadership in which efficiency is high
because leaders contribute a lot and followers respond to this. In other
groups, however, leadership is less successful and efficiency is no
better than we would expect without leadership. At the aggregate level,
therefore, leadership does make a difference but considerable
inefficiency still remains. We shall argue that it is primarily the
fault of leaders rather than followers that leadership does not prove
more successful.
Interestingly the absolute increase in efficiency we observe from
leadership is very similar to that of Weber, Camerer, and Knez (2004)
and Li (2007). Relatively speaking things look different because our
benchmark of comparison is a four-player game with relatively low and
declining efficiency while theirs is a three-player game with relatively
high and stable efficiency. This is an important distinction, because
escaping from the "trap" of low and declining efficiency in
the weak-link game is very difficult to achieve but crucial for the
group (c.f. Chaudhuri, Schotter, and Sopher 2009; Crawford 2001). Our
results suggest that leadership can help groups escape this trap, and
that is an encouraging finding. Leadership proves, however, far from a
panacea.
We proceed as follows: in Section II we introduce the weak-link
game and in Section III we develop a simple model of leadership and
state our hypotheses. Section IV describes our experimental design and
Section V contains the results. Section VI concludes.
II. THE WEAK-LINK GAME
The weak-link game is a stylized representation of any situation
where members of a group can contribute to some group project and the
outcome depends on the contribution of the least contributing member. We
adopt the standard payoff structure used by Van Huyck, Battalio, and
Beil (1990). In this version n players simultaneously pick a whole
number between 1 and 7 and the payoff of a player is given by the
formula
u(k, m) = 0.6 + 0.2m - 0.1k
where k denotes the player's own choice and m denotes the
minimum choice of all n players. Table 1 describes the payoff of a
player for every potential combination of their own choice and the
minimum choice.
Every outcome in which all players choose the same number is a Nash
equilibrium. Clearly Nash equilibria on higher numbers are preferred to
those over lower numbers, so the Pareto optimum is for every player to
choose 7. (6) Note, however, that higher numbered Nash equilibria
involve a degree of strategic uncertainty. Picking the highest number is
the best strategy only if all other players also pick the highest
number. This means that there are two notions of coordination in a
weak-link game. We can think of players as coordinating if they all
choose the same number and so are coordinating on a Nash equilibrium.
Alternatively we can think of players as coordinating if they all choose
high numbers and so are coordinating on the most efficient Nash
equilibria. Throughout the following we shall focus on the later notion
of coordination. We, thus, say that there is increased coordination and
efficiency if the minimum number increases, and there is coordination
failure and inefficiency if the minimum number chosen is low.
Our objective in this paper is to contrast the standard weak-link
game, in which all players choose simultaneously, with a version in
which one individual, the leader, makes a choice before the remaining
players. To do this, we shall distinguish three games, all sharing the
payoffs given in Table 1, but differing in the dynamics of play:
Simultaneous game: All n players in the game simultaneously and
independently of each other chose a number.
Exogenous leader game: The game consists of two stages. In the
first stage, one of the n players is randomly selected to be a leader,
and chooses a number. In the second stage, the choice of the leader is
made public, and the remaining n - 1 players simultaneously and
independently of each other choose a number.
Endogenous leader game: The game consists of two stages. The first
stage lasts at most T seconds and at any point during this time any of
the n players can choose a number. As soon as one player has chosen a
number the stage ends. We rule out the possibility that two players
choose at the same time and the player who chooses first is called the
leader. In the second stage, the choice of the leader is made public,
and the remaining n - 1 players simultaneously and independently of each
other choose a number.
In both the exogenous and endogenous leader game there is one
player, the leader, who chooses before the remaining players, the
followers. The choice of the leader is known by the followers before
they make their choice, resulting in leadership by example. In the
exogenous leader game the leader is chosen randomly and thus
exogenously. In the endogenous leader game the leader is the first
player to choose a number and so is chosen endogenously.
III. HYPOTHESES ON LEADERSHIP
In the standard, simultaneous, weak-link game we expect to see
significant coordination failure. What difference will leadership make?
By choosing a high number the leader can signal or communicate to others
in the group that it is good to choose high numbers. The choice of the
leader also provides a natural focal point around which others can
coordinate. Our basic hypothesis, therefore, is that leadership can help
groups avoid coordination failure. To develop this idea more formally we
shall work through a simple but relatively general model of how
leadership may affect a player's behavior. More specifically, we
shall consider some player i and contrast what number player i will
choose in a simultaneous game with what he will choose in a game with
leadership.
We begin by focusing on a simultaneous game with n players. Suppose
that player i believes every other player will independently choose
number k with probability [f.sup.n.sub.i](k). We impose that
[[summation].sup.7.sub.h=1] [f.sup.n.sub.i](h) = 1 and, with a slight
abuse of notation, shall denote by [F.sup.n.sub.i](k) =
[[summation].sup.7.sub.h=k] [f.sup.n.sub.i](h) the probability of a
player choosing k or above. Of primary interest in the weaklink game is
the expected minimum choice of others. From beliefs [f.sup.n.sub.i] we
can derive player i's inferred beliefs over what this minimum
choice will be. For example, if the number of players is four, we get
that
[m.sup.n.sub.i](k) = [f.sup.n.sub.i][(k).sup.3] + 3
[f.sup.n.sub.i](k) [F.sup.n.sub.i](k) x ([F.sup.n.sub.i] (k) -
[f.sup.n.sub.i] (k))
is the probability with which player i should expect the minimum
number chosen by others to be k. Let [M.sup.n.sub.i](k) =
[[summation].sup.7.sub.h=k] [m.sup.n.sub.i](h) be the probability with
which he should expect the minimum number to be k or above.
Given his beliefs, the expected payoff of player i if he chooses k
can be written
[[pi].sup.n.sub.i](k) = 0.6 + 0.2 ([[k-1.summation over (h=1)]
[hm.sup.n.sub.i](h) + k[M.sup.n.sub.i](k)) -0.1k.
We will assume that every player chooses k so as to maximize his
payoff given his beliefs. Let [k.sup.S,n.sub.i] denote the number that
would be chosen by player i. For any k there is a set of beliefs such
that it is optimal for a player to choose k. (7) What he does will,
therefore, depend on his beliefs, and without imposing any more
structure on these beliefs we cannot predict what the player will
choose. This, however, is not a problem because we do empirically
observe players choosing all the possible seven numbers. Of more
interest to us is to question how leadership changes the incentives of
the player.
A. Leadership and Strategic Uncertainty
In order to see how leadership changes incentives it is informative
to first of all contrast a simultaneous game with n players to one with
n - 1 players. An informative way to do this is to compare the relative
payoff gain (or loss) from choosing a number one higher. So, let
[[DELTA].sup.n.sub.i](k) = [[pi].sup.n.sub.i](k) - [[pi].sup.n.sub.i](k
- 1), for all k > 1, be the relative payoff gain in a simultaneous
game with n players. Extending the notation introduced above in an
obvious manner, let [[DELTA].sup.n-1.sub.i](k) = [[pi].sup.n-1.sub.i](k)
- [[pi].sup.n-1.sub.i](k - l), for all k > 1, be the relative payoff
gain in a simultaneous game with n - 1 players. It is simple to show
that, (8)
(1) [[DELTA].sup.n-1.sub.i](k) - [[DELTA].sup.n.sub.i](k) =
0.2([M.sup.n-1.sub.i](k) - [M.sup.n.sub.i](k))
for all k. The relative incentive to choose a number one higher
will thus depend on player i's inferred beliefs on the likely
minimum choice of others.
The crucial thing to now recognize is that a reduction in the
number of players should make player i more optimistic about the minimum
choice of others. This is because of reduced strategic uncertainty;
player i is uncertain about the choices of only n - 2 other players
rather than n - 1. For instance, even if player i's beliefs are the
same in a game with n players as in a game with n - 1 players,
[f.sup.n.sub.i](k) = [f.sup.n - 1.sub.i](k) for all k, it will be the
case that [M.sup.n - 1.sub.i](k) [greater than or equal to]
[M.sup.n.sub.i](k) for all k. This motivates assumption 1, that
(2) [F.sup.n - 1.sub.i](k) [greater than or equal to]
[F.sup.n.sub.i](k)
for all k. It immediately follows from Equation (1) and assumption
1 that
[k.sup.S,n - 1.sub.i] [greater than or equal to] [k.sup.S,n.sub.i].
Thus, player i would choose at least as high a number in a game
with n - 1 players as he would do in a game with n players. This effect
has been observed experimentally (Camerer 2003; Van Huyck, Battalio, and
Beil 1990; Van Huyck, Battalio, and Rankin 2007).
Consider now an exogenous leadership game, with n players, and
suppose that player i is a follower. One would expect that the beliefs
of followers will be conditional on the choice of leader. So, let L
denote the choice of leader and let [f.sub.i](k|L) and [F.sub.i](k|L) =
[[summation].sup.7.sub.h=k] [f.sub.i](h]L) denote the beliefs of player
i given the leader's choice. There are two key things to now
recognize. First, the choice of the leader reduces strategic uncertainty
because player i is uncertain about only n - 2 other players rather than
n - 1. Second, the choice of the leader may serve as a focal point that
influences others because of signaling or reciprocity. On this basis we
suggest our main assumption, assumption 2, that
(3) [F.sub.i](k|L) [greater than or equal to] [F.sup.n -
1.sub.i](k)
for all k [less than or equal to] L and any L. Assumption 2
complements assumption 1 by suggesting that followers in a leadership
game will at least take account of the reduced strategic uncertainty
caused by the leader's choice. This assumption appears relatively
mild, particular given the evidence for signaling and reciprocity in
public good and public bad games (Guth et al. 2007; Moxnes and Van der
Heijden 2003).
Given the beliefs [f.sub.i](k|L) we can derive inferred beliefs on
the minimum choice of others [M.sub.i](k|L) and expected payoff
[[pi].sub.i](k|L). With this we can compare incentives with and without
leadership by letting [[DELTA].sub.i](k|L) = [[pi].sub.i](k|L)
-[[pi].sub.i](k - 1|L). It is simple to show that, if assumptions 1 and
2 hold, (9)
(4) [[DELTA].sub.i](k|L) - [[DELTA].sup.n.sub.i](k) =
0.2([M.sub.i](k|L) - [M.sup.n.sub.i](k)) [greater than or equal to] 0
and
(5) [[DELTA].sub.i](k|L) - [[DELTA].sup.n - 1.sub.i](k) =
0.2([M.sub.i](k|L) - [M.sup.n - 1.sub.i](k)) [greater than or equal to]
0
for all k [less than or equal to] L. The incentives to choose a
number one higher are, therefore, at least as great with leadership as
without, and at least as great with leadership as with reduced strategic
uncertainty.
To summarize what we have shown so far, let [k.sup.F.sub.i] (L)
denote the choice player i would make in a game with exogenous
leadership if he is a follower and the leader has chosen L. The
following result follows immediately from Equations (1), (4), and (5).
PROPOSITION 1. Assumptions 1 and 2 imply that (i)
[k.sup.F.sub.i](L) [greater than or equal to] [k.sup.S,n - 1] [greater
than or equal to] [k.sup.S,n.sub.i] if [k.sup.S,n - 1.sub.i] < L, and
(ii) [k.sup.F.sub.i](L) = L if [k.sup.S,n - 1.sub.i] [greater than or
equal to] L.
Player i could, therefore, choose more or less in a game with
leadership compared to a simultaneous game. It depends on what the
leader does. To progress further we need to think about what the leader
may choose.
B. Hypotheses on Leadership
Suppose now that player i is the leader in an exogenous leadership
game. It is likely that player i would expect the choice of others to
depend on his choice. Let [f.sup.D.sub.i](k|L) and [F.sup.D.sub.i](k|L)
= [[summation].sup.7.sub.h=k] [f.sup.D.sub.i](h|L) denote the beliefs of
player i if he leads and chooses L. It is very mild to assume,
assumption 3, that
(6) [f.sup.D.sub.i](k|L) = [f.sub.i](k|L)
for all k and any L. All this assumption imposes is that followers
are expected to react to the choice of the leader, and not his identity.
Given [f.sup.D.sub.i] we can derive inferred beliefs on the minimum
choice of others [M.sup.D.sub.i](k|L). The important thing to recognize
here is that because player i leads he remains uncertain about the
choices of n - 1 other players. Recall, that when he is a follower he is
uncertain about the choices of only n - 2 players. Thus, player i should
be more pessimistic about the minimum choice of others when he leads
than when he is a follower. In particular, assumption 3 implies that
[M.sup.D.sub.i](k|L) [less than or equal to] [M.sub.i](k|L) for all k
and L.
Let [[rho].sup.D.sub.i](k) denote the expected payoff of player i
if he leads and chooses k and let [[DELTA].sup.D.sub.i](k) =
[[pi].sup.D.sub.i](k) - [[rho].sup.D.sub.i](k - 1) be the incentive to
choose a number one higher. Using the now familiar arguments,
assumptions 1, 2, and 3 imply that [[DELTA].sup.D.sub.i](k) -
[[DELTA].sup.n.sub.i](k) = 0.2([M.sup.D.sub.i](k|k) -
[M.sup.n.sub.i](k)) [greater than or equal to] 0 for all k. The
incentives to choose a number one higher are, therefore, at least as
high in a game with leadership as in a simultaneous game with n players.
Assumptions 1, 2, and 3 are not, however, enough for us to say anything
about
(7) [[DELTA].sup.D.sub.i](k) - [[DELTA].sup.n - 1.sub.i](k) =
0.2([M.sup.D.sub.i](k|k) - [M.sup.n - 1.sub.i](k)).
This is because the leader remains uncertain about the choices of n
- 1 players. Plausibly, therefore, one can get [M.sup.D.sub.i](k|k) [??]
[M.sup.n - 1.sub.i](k) depending on whether reduced strategic
uncertainty is expected to have a bigger or smaller effect than
signaling and reciprocity.
To summarize what we have learnt about leader choice, let
[k.sup.D.sub.i] denote the choice player i would make in a game with
exogenous leadership if he is a leader.
PROPOSITION 2. Assumptions 1, 2, and 3 imply that [k.sup.D.sub.i]
[greater than or equal to] [k.sup.S,n.sub.i].
What we cannot say anything about, on the basis of assumptions 1,
2, and 3, is the relationship between [k.sup.D.sub.i] and [k.sup.S,n -
1.sub.i]. As just discussed, this will depend on the relative effects of
reduced strategic uncertainty versus signaling and reciprocity. We have
done enough, however, to motivate our first two hypotheses. Before
stating these hypotheses we briefly note that assumptions 3 and 4, and
proposition 2 can easily be rephrased in terms of a game with endogenous
leadership (we shall discuss this issue in more detail shortly).
Hypothesis 1: There is less coordination failure in a weak-link
game with leadership than in a standard weak-link game.
Hypothesis 2: Coordination failure in a weak-link game with
leadership and n players is less than in a standard weak-link game with
n - 1 players.
Hypothesis 1 follows directly from Propositions 1 and 2. Hypothesis
2 is more speculative and asks relatively a lot of leaders. In
particular, Proposition 1 suggests that we can reasonably expect
followers to choose higher numbers than they would have done in a
simultaneous game with n - 1 players if the leader chooses a high enough
number. Less clear, as we have seen, are the incentives for the leader
to choose a high enough number. Equation (7) suggests that the leader
will only choose a high number if he expects it will cause others to
choose an equally high number. It is an empirical question whether
leaders do choose high numbers, and whether followers do reciprocate.
C. Endogenous Versus Exogenous Leadership
In the preceding analysis we focused on an exogenous leadership
game. This was appropriate given that we were asking what player i would
do if he were a follower and what he would do if he were a leader. In an
endogenous leadership game we need to look, in addition, at whether
player i would want to lead or follow. This requires comparing his
expected payoff if he leads to that if he follows.
For notational simplicity we shall assume that player i has the
same beliefs in a game with endogenous leadership as with exogenous
leadership. (10) We need to supplement this with player i's beliefs
over the choice a leader would make, if the leader were not him. Suppose
that he believes the probability that a leader will choose number L is
[g.sub.i](L) for all L. We can then compare the expected payoff of
player i from leading and following. Player i will want to lead if and
only if
(8) [[pi].sup.D.sub.i]([k.sup.D.sub.i]) [greater than or equal to]
[7.summation over (L=1)]
[g.sub.i](L)[[rho].sub.i]([k.sup.F.sub.i](L)|L).
Informally, there are two basic scenarios where this expression
will be satisfied. If player i intends to choose the lowest number,
[k.sup.D.sub.i] = [k.sup.F.sub.i](1) = ... = [k.sup.F.sub.i](7) = 1,
then condition (8) is trivially satisfied because his payoff will be 0.7
whether he leads or follows. Alternatively, if player i is confident
that others will respond positively to a high leader choice but is not
confident that another leader will choose a high number then condition
(8) is also satisfied. To illustrate this latter possibility, suppose
that [k.sup.D.sub.i] = 7, [M.sup.L.sub.i](7|7) = 0.9 and [g.sub.i](1) =
0.9. Then, player i's expected payoff from leading is at least
1.18, while his expected payoff from following is at most 0.76.
With this in mind we can now briefly compare endogenous and
exogenous leadership. In the first scenario alluded to above, where
player i intends to choose a low number, the minimum choice will be one
in both cases and so there is coordination failure irrespective of
whether leadership is endogenous or exogenous. In the second scenario,
where player i intends to choose a high number, coordination failure
should be no more in the game with endogenous leadership than exogenous
leadership because of the high leader choice. This suggests that
voluntary leadership may be more effective than exogenous leadership.
Hypothesis 3: Coordination failure in a weak-link game with
endogenous leadership is less than in a weak-link game with exogenous
leadership.
Empirical support for this hypothesis comes from the public good
literature. For example, Van Vugt and De Cremer (1999) and Arbak and
Villeval (2007) find that imposed leaders contribute less to a group
than voluntary leaders. Similarly, Rivas and Sutter (2008) find a
positive effect of leadership on cooperation but only with voluntary
leaders. Gachter et al. (2012) also found that reciprocally oriented
leaders contribute more.
IV. EXPERIMENTAL METHOD
To test our hypotheses we performed a laboratory experiment in
which we compared four different versions of the weak-link game: a
simultaneous three-player game (Sim3), a simultaneous four-player game
(Sim4), an exogenous four-player leadership game (Exo), and an
endogenous four-player leadership game (End). In each case the payoff
structure in Table 1 was used and the game was as described in Section
II.
Each experimental session consisted of three distinct parts. In
each part participants were grouped into groups of three or four, as
appropriate, and played ten rounds of either Sim3, Sim4, Exo, or End.
Note that within these ten rounds the game and groups did not change.
Between parts of the session the groups and possibly the game did
change. We ran seven sessions in all, each with four groups. In one
session participants played Sim3 in all three parts of the experiment.
(11) In the other six sessions, participants played each of Sim4, Exo,
and End in varying order. That we had six sessions allowed us to
consider all possible permutations of Sim4, Exo, and End as detailed in
Table 2. To control for any potential order affects that may result from
subjects playing three different games we shall, in the following,
include part dummies in all regressions and provide statistical tests
that use only data from part 1 of a session. We shall see, however, that
there is no evidence of an order affect, and so we will group the data
from all parts unless otherwise stated.
Participants were told at the start of the experiment that they
would play "a number" of games (of ten rounds each).
Participants were only given the instructions to a particular game
before they played that game. It was also emphasized to participants
that they would be playing in a totally new group in each part of the
experiment. For the conditions with a leader we deliberately avoided
terms like "leaders" and "followers" and instead
used more neutral descriptions like "the person choosing
first" and "the other players." The instructions are
available in the supporting information Appendix S1.
After each round participants were told their earnings and the
minimum, and only the minimum, number chosen in the group. Announcing
the full distribution of choices, rather than just the minimum, has been
shown to make it easier to coordinate (Berninghaus and Ehrhart 2001;
Brandts and Cooper 2006a). (12) We provide, therefore, a relatively
tough test of leadership. This approach also allows us to more clearly
distinguish how much the benefits of leadership are due solely to
players seeing the choices of two others, the leader's choice and
minimum choice, rather than seeing just one choice, as in a simultaneous
game.
[FIGURE 1 OMITTED]
The experiment was programmed and conducted with the software
Z-tree (Fischbacher 2007) and run at the University of Kent in 2009.
Afterwards participants were paid the earnings of one randomly selected
game. Participants were recruited via the university-wide research
participation scheme and were randomly assigned to the different
conditions and to their respective groups. In total 108 subjects
participated, who earned on average 8.82 [pounds sterling]. The
experiment took about 45 minutes.
V. RESULTS
To give a first snapshot of the results, Figure 1 plots the average
minimum choice by group in each treatment and each round and Figure 2
plots the average choice. In the Sire4 treatment, as we would expect, we
see large coordination failure with a minimum choice of 1 in over half
the groups. Things are much better in the Sim3 treatment, illustrating
how important group size can be, but significant coordination failure is
still observed. The key question for us is whether leadership helped
groups avoid such failure. We clearly see that leadership had at best a
limited success. Coordination failure appears less in the leadership
treatments than in Sire4 but remains high and as high as in the Sim3
treatment. Indeed, we find that in round 1 there is nothing to
distinguish choices in the leadership conditions from those in Sim4 or
Sim3 (p = .91, Kruskal-Wallis test). By round 10 we do find a
significant difference in choices between the leadership conditions and
Sim4 but not Sim3 (p = .00 all treatments, Kruskal-Wallis test, p = .20
excluding Sim4). (13)
[FIGURE 2 OMITTED]
The one positive sign in Figures 1 and 2 is a possible dynamic
consequence of leadership. This does show up in simple trend terms:
Choices decline in the Sire4 treatment (with coefficient of -0.15, p =
.00) but remain relatively stable in the other treatments, including the
leadership treatments (Sim3: -0.051, p = .17, Exo: -0.00, p = .98, End:
p = .90). Furthermore, minimum choices are relatively stable in the
simultaneous treatments (Sim4: -0.023, p = .10, Sire3: 0.03, p = .35)
but increase in the leadership treatments (Exo: 0.10, p = .02, End:
0.08, p = .07). There is, therefore, some evidence of a dynamic benefit
of leadership. The suggestion would still be, however, that efficiency
is essentially catching up with that in Sim3.
This is also the picture we get from average payoffs, summarized in
Table 3. We find no significant difference between the payoffs of
leaders or followers across leadership treatments (e.g., leaders: p =
.53 in round 1, p = .27 in round 10, Mann-Whitney test, followers: p =
.46 and .43). We also find no significant difference between the payoffs
of leaders and followers (e.g., p = .36 in round 1, p = 1.00 in round
10). Aggregating the data from the leadership treatments we find that
subjects in the leadership treatments do earn significantly more than
subjects in Sim4 in all rounds (e.g., p = .00 in round 1, p = .00 in
round 10). When compared to Sim3 they earn less in round 1 but have
caught up by round 10 (p = .01 in round 1, p = .11 in round 10).
We can begin to summarize our findings.
RESULT 1. Overall efficiency is higher in the leadership treatments
compared to the Sim4 treatment but not the Sim3 treatment. Initial
choices in the leadership treatments appear similar to those in the
simultaneous treatments. There is evidence of a dynamic improvement in
efficiency in the leadership treatments but not the simultaneous
treatments.
This is supportive of Hypothesis 1 but not of Hypotheses 2 or 3. To
explore this further we shall look in more detail at the choices of
followers and leaders, starting with the choice of followers.
A. Follower Choice
Figures 3 and 4 plot the average and minimum choice of followers as
a function of the leader's choice (when averaging over all ten
rounds). We clearly see evidence that follower choice is positively
correlated to leader choice. The Pearson correlation is 0.87 (p <
.001) in the exogenous condition and 0.82 (p < .001) in the
endogenous condition. We also see that followers pick a significantly
lower number than the leader. The average difference between leader
choice and (average) follower choice is 0.54 (p = .001) for exogenous
leaders and 0.38 (p = .001) for endogenous leaders.
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
Of particular relevance to us is whether a high leader choice
causes followers to choose higher numbers than those chosen in Sire3.
This would be evidence that leadership has a benefit beyond reducing
strategic uncertainty. Figures 3 and 4 suggest that it may. To pursue
this in more detail Table 4 gives the average choice of followers if the
leader chooses 7. In order to try and avoid any self-selection bias
(that may exist because only some leaders choose 7) we have included the
averages for round 1 and for round 1 of part 1 of a session. We see in
Table 4 that choices are consistently higher in the leadership
treatments than in the Sim3 and Sim4 treatments if the leader chooses 7.
These differences are statistically significant, even if we restrict
attention to round 1 or round 1 of part 1 of a session. More
specifically, we do not observe any difference in the Exo and End
treatments in the average choice of followers in round 1 or round 1 of
part 1 (p = .14 and .25, respectively, Mann-Whitney test). Pooling the
data from the leadership treatments we do find a significant difference
compared to Sim3 (p = .02 and .08) and Sim4 (p = .00 and .10). A similar
story holds for all other rounds.
We do observe, therefore, subjects choosing higher numbers when
following a leader who chooses 7 than they do in simultaneous games,
even in round 1. This is consistent with Proposition 1 and evidence that
leadership does more than reduce strategic uncertainty. To put all this
in some context Table 5 presents the results of a random effects
generalized least squares (GLS) regression and three ordered probit
regressions with choice as the dependent variable. The regressions
exclude the choice of leaders and so allow us to compare the behavior of
followers with that of players in a simultaneous game. The Sim3
treatment is used as the comparator. Columns 1 and 2 focus on rounds 1
and 10, respectively, and include the choice of the leader and dummy
variables to capture treatment and the order of the game in the session
as independent variables. (14) Columns 3 and 4 report results using data
from all rounds. To capture potential dynamic treatment effects we
include as independent variables an interaction term between the round
number and treatment. To capture potential dynamic choice effects we
include the minimum choice in the previous round. (15) The
"threshold to choose x" parameter indicates the size of
dependent variable required in order that a player is predicted to
choose more than x. For example, the results in columns 1 and 2 imply
that the average player in the Sim3 and Sim4 treatments is predicted to
choose 4.
In comparing follower behavior in the leadership and simultaneous
treatments we need to take account of the dummy variable together with
leader choice. Doing so, we see that choices are expected to be higher
in the leadership treatments than in Sim3 if and only if the leader
chooses 6 or 7. For example, using the results in column 1, the net
effect in the End treatment compared to the Sim3 treatment is -1.69 +
0.34 x L, where L is the choice of the leader. Thus, followers are
expected to choose high numbers if and only if the leader chooses more
than 5. This fits exactly with the earlier analysis and leads to our
second result.
RESULT 2. If the leader chooses 7 then followers choose higher
numbers than can be explained solely by reduced strategic uncertainty.
That is, they choose higher numbers than do subjects in the Sim3
treatment.
Note that this result does not, in itself, imply that it is in the
interest of leaders to choose 7, a point we return to in Section V.C.
B. Group Dynamics
In columns 3 and 4 of Table 5 we see a clear relationship between
choice and what happened in the previous round. This is no surprise
(Crawford 2001). The possibility we want to explore here is whether
leadership can help groups overcome coordination failure. The dynamic
benefit of leadership picked up in Result 1 suggests that it may, and
this is an interesting possibility because escaping from the inefficient
equilibrium typically proves impossible in the standard weak-link game
(e.g., Brandts and Cooper 2006a, 2007; Chaudhuri, Schotter, and Sopher
2009; Weber et al. 2001). The results in Table 5 predict that leadership
can help a group to escape from coordination failure. (16) To back this
up we can provide some direct evidence of leadership working.
The first thing we can do is look at specific group dynamics. We
shall say that there was persistent coordination failure (CF) in a group
if the minimum was 1 in all 10 rounds. By contrast, we shall say that
there was a reversal of coordination failure to x (Rx) if there was one
round with a minimum of 1 and a later round with a minimum of x. Table 6
details how many groups fit into each category. As we would expect in
the Sim3 and Sim4 conditions there is little evidence that groups can
overcome coordination failure. In the leadership conditions we do get a
more positive picture. For example, in none of the 26 groups without
leadership did we see a minimum of 5 or more after there had been a
round with a minimum of 1. In groups with leadership this happens in 12
of the 37 groups. (17)
In all the 12 groups where the minimum did increase to 5 it did so
because a leader chose 6 or 7. This clearly fits with the idea that
leaders can make a difference. To further back this up, Table 7 looks at
the average choice of followers if the leader chooses 7 and there has
been a previous round with a minimum choice of 1. For illustration we
have provided the data for rounds 2 and 10, but the picture is similar
in all rounds. Round 2 is of particular note because we should avoid
self-selection issues (although there is a lack of data for the Sim3
treatment). We find no significant difference between choices in the
leadership treatments (p = .67 in round 1, p = .19 in round 10,
Mann-Whitney) but do find a significant difference between choices in
the leadership and simultaneous treatments (p = .00 and p = .14 compared
to Sim4 and Sim3 in round 1, p = .00 and p = .00 in round 10).
The key thing here is that we see evidence of followers responding
to the leader choice even if there has been previous experience of
coordination failure.
RESULT 3. In simultaneous games one instance of coordination
failure typically leads to persistent coordination failure. In games
with leadership we see that coordination failure need not be persistent.
In a significant number of groups leadership helped overcome
coordination failure.
[FIGURE 5 OMITTED]
Results 2 and 3 suggest that followers do respond to leader choice.
In reconciling this with the lack of success of leadership at the
aggregate level it is natural to question the choices of leaders.
C. Leadership
Figure 5 plots the average choice of leaders in each round. Of
interest to us, given Proposition 2, is whether leaders choose higher
numbers than in the simultaneous treatments. For comparison we therefore
plot average choices in the simultaneous treatments. The clear
suggestion in Figure 5 is that leaders choose higher numbers than
subjects in Sim4 but not those in Sim3. There is no evidence that leader
choices in the Exo and End treatments differ (e.g., p = .53 in round 1,
p = .33 in round 10, Mann-Whitney). There is also no evidence that
leader choices in the leadership treatments differ from choices in the
simultaneous treatments in round 1 whether using the data from all parts
of a session (p = .17 compared to Sim4, p = .32 compared to Sim3) or
only part 1 (p = .18 and .10). By round 10 there is evidence that leader
choices differ from choices in Sim4 but not those in Sim3 whether using
all parts (p = .00 and .29) or only part 1 (p = .00 and .37).
Recall that Hypothesis 1 and Proposition 2 said that leaders should
choose a number at least as high as they would have done in Sim4. The
evidence is consistent with this. Hypothesis 2 reflected an expectation
that leaders might choose higher numbers than would players in Sim3. The
evidence suggests they do not.
RESULT 4. In the early rounds we do not observe any significant
difference between the choice of leaders in the leadership treatments
and that of subjects in the simultaneous treatments. In the later rounds
we do observe leaders choosing a higher number than subjects in the Sim4
treatment but find no difference compared to the Sim3 treatment.
This, together with results 2 and 3, suggest that the overall lack
of success of leadership comes more from the behavior of leaders than
that of followers. Clearly, not all the blame should be put on leaders
because there were groups with persistent coordination failure in which
leaders chose 7 several times. (18) It seems, however, that leaders
simply did not choose high enough numbers often enough in order that
leadership would lead to a significant overall increase in efficiency
beyond that obtained in the Sim3 treatment.
To understand why this happened we note that results 2 and 3, while
showing a higher leader choice can lead to increased group efficiency,
leave open the question of whether choosing a high number pays off for
the leader. This is far from clear because a leader can guarantee a
payoff of 0.7 by choosing 1 and needs the minimum choice of followers to
be at least 4 in order to get a payoff of 0.7 if he chooses 7. Table 8
provides some aggregated data on whether choosing a high number did pay
off for leaders. The payoffs of both leaders and followers are typically
higher if the leader chooses a higher number. The increase, though, is
small and arguably not enough to motivate giving up the sure payoff of
0.7. While choosing a high number can pay off for the group it does not,
therefore, necessarily pay off for the leader. This was captured in the
discussion of Proposition 2 and may explain why leaders did not choose
high enough numbers often enough.
D. Endogenous Versus Exogenous Leadership
One interesting consequence of Result 4 is that there is every
opportunity for the distinction between endogenous and exogenous
leadership to matter. You may already have noticed, however, that the
type of leader appears to have little effect. We have already noted that
there is no apparent difference in follower and leader choice in the two
treatments. There is also no significant difference in minimum choice or
the difference between leader and follower choices. Table 9 provides
more evidence by giving the results of a random effects GLS regression
and ordered probit regressions with leader choice as the dependent
variable. The endogenous treatment is used as the comparator. In this,
and Table 5, we see little consistent evidence that the distinction
between exogenous and endogenous leadership matters.
RESULT 5. We find no significant difference between the endogenous
and exogenous leadership treatments.
Result 5 is clearly contrary to Hypothesis 3. Recall that
Hypothesis 3 was motivated by the observation that a player will lead in
an endogenous game if the player (1) is confident that others will
respond positively to a high leader choice but (2) is not confident that
another leader will choose a high number. Thus, a possible explanation
for Result 5 is a lack of players satisfying these two criteria. In the
Exo treatment we do see players that appear to satisfy criteria (1).
This is evidenced by many leaders choosing high numbers and, in Table 9,
the lack of any correlation between leader choice and what happened in
the previous round in the Exo treatment. We see many exogenously
determined leaders choosing a high number despite previous leaders
choosing a low number, presumably because they think leadership by
example can work.
In the End treatment a player satisfying criteria (1) and (2) will
look to lead and choose a high number. This can only be good for the
group. A player satisfying criteria (1) but not (2) will, however, wait
for someone else to lead. They will prefer to wait and gain from the
reduced strategic uncertainty, hopeful that someone else will lead and
choose a high number. This is potentially not good for the group, if the
eventual leader chooses a low number. In Table 9, we see much more
persistence of leader choice in the End treatment. Overall, therefore,
we suggest that the lack of difference between the End and Exo
treatments can be explained by there being insufficient players who
satisfied criteria (1) and (2). Many of those who think that leadership
by example can work may prefer to wait for someone else to lead. In the
Exo treatment they do not have such an opportunity, but in the End
treatment they do. In other contexts we see people preferring to wait
and see rather than lead (Nosenzo and Sefton 2009) and that appears to
be the case here, too.
VI. CONCLUSIONS
The provision of many public and private goods hinges on the
actions of the weakest link, that is the lowest contributor (Camerer
2003; Hirshleifer 1983). The evidence suggests that in such cases the
likely outcome is coordination failure. Our objective in this paper was
to see whether leadership by example could help groups avoid such
coordination failure.
We find that leadership had a positive but somewhat limited effect.
We argue that the reason it was not more successful is due more to the
actions of leaders than of followers. In particular we do see evidence
of followers responding positively if the leader contributes a lot. We
see, however, little evidence of leaders contributing a lot. So, in some
groups there is successful leadership in which efficiency is high
because leaders contribute a lot and followers respond to this, but in
other groups leadership is less successful and efficiency no better than
we would expect without leadership. Our main conclusion, therefore, is
that leadership can work if leaders persistently set a good example. We
found no discernible difference between voluntary and imposed leaders.
Our results add to a general literature on whether communication
can make a difference in weak-link games. Several studies have shown the
benefits of both costless and costly communication (Blume and Ortmann
2007; Cachon and Camerer 1996; Cooper 2006; Cooper et al. 1992; Van
Huyck et al. 1993). Costless communication has, however, proved less
effective if only one player can communicate (Weber et al. 2001),
primarily because signals are ignored. (19) Costly communication has
also proved ineffective if players avoid the cost of signaling (Manzini,
Sadrieh, and Vriend 2009). Our results are broadly consistent with the
latter observation in that leaders may be unwilling to signal by
choosing a high number. They are also consistent with findings in the
public good literature that leaders may have little incentive to lead by
example (Cartwright and Patel 2010).
To finish we can briefly revisit the comparison made in the
introduction between our results and those of Weber, Camerer, and Knez
(2004) and Li (2007). Recall that they compare sequential to
simultaneous choice in a three-player weak-link game. There are clear
similarities between our findings and theirs. They find no difference
between sequential and simultaneous choice in round 1, but do find a
difference over time that ultimately amounts to an increase of around
one in average choice. This fits exactly with our findings. The key
difference is the benchmark of comparison. We show that leadership can
be of some benefit against a backdrop of low and declining efficiency
while they do so against a backdrop of relative high and stable
efficiency.
The consistency of these results is in contrast to results of
on-going work by Coelho, Danilov, and Irlenbusch (2009). They consider a
ten-player weak-link game in which a leader, the person in the group
with the highest criterion-referenced test (CRT) score, leads by
example. The most significant differences with our approach are that the
leader remains the same throughout the rounds and is selected on
ability. They find that leadership leads to immediate and sustained
efficiency if all players observe the minimum choice of previous rounds
but immediate and declining efficiency if the minimum choice of previous
rounds is not observed. These results suggest that more work on the
consequences of leadership, and in particular the consequences of
different types of leadership--appointed or elected, democratic or
autocratic, selfish or servant--would be desirable (Gillet et al. 2011).
ABBREVIATIONS
CRT: Criterion-Referenced Test
GLS: Generalized Least Squares
SUPPORTING INFORMATION
Additional Supporting Information may be found in the online
version of this article:
Appendix S1. Instructions given to subjects.
Appendix S2. Additional group level data.
Table S1. The minimum choice by round and group in the Sim3
condition.
Table S2. The minimum choice by round and group in the Sim4
condition.
Table S3. The leader choice and minimum choice by round and group
in the Exo condition.
Table S4. The leader choice and minimum choice by round and group
in the End condition.
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(1.) There are some notable exceptions including Bortolotti,
Devetag, and Ortmann (2009) who find higher effort levels in a real
effort weak-link game. See Devetag and Ortmann (2007) for a survey of
the literature.
(2.) To put these issues in some context: In the dike example, with
which we began this paper, the full distribution of contributions would
be observable (a person can just go around the island and look) but
communication (e.g., each landowner saying how high a dike they plan to
build) could be unwieldy. Next consider authors submitting articles to a
special issue of a journal or contributed book. Here, only the minimum
(i.e., slowest) contribution is likely to be observable and
communication between authors may or may not be possible.
(3.) Different types of leadership have been studied in the
weak-link game and closely related turnaround game. Weber et al. (2001)
consider a setting where one player, the leader, reads out a prepared
statement, after the second period, encouraging coordination. The speech
was effective for groups of size 2 but not for groups of size 10. Cooper
(2006) and Brandts and Cooper (2007) consider a setting where a manager
can communicate a message to players while also changing the incentives
to coordinate. Communication was relatively effective.
(4.) The public good literature has also shown that contributions
may be lower if they are made sequentially rather than simultaneously
(Varian 1994; Gachter et al. 2009).
(5.) Note that the focus of Weber et al. (2004) and Li (2007) was
on virtual observability and not leadership.
(6.) This makes the weak-link game a coordination game with
Pareto-ranked equilibria. This class of coordination game can be
distinguished from games with asymmetric players, such as the battle of
sexes. Evidence on leadership in such games includes Cooper et al.
(1989), Rapoport, Seale and Winter (2002), and Cartwright, Gillet and
van Vugt (2009). See Camerer (2003) for a survey of the literature.
(7.) For instance, if [f.sub.i](k) [??] 1 then it is optimal for
player i to choose k.
(8.) To derive this it is useful to use that [[pi].sup.n.sub.i](k)
= [[pi].sup.n.sub.i](k - 1) - 0.1 + 0.2 [M.sup.n.sub.i](k), and
[[pi].sup.n-1.sub.i](k) = [[pi].sup.n-1.sub.i](k - 1) - 0.1 +
0.2[M.sup.n-1.sub.i](k).
(9.) See footnote 8 for the derivation of the first equality. The
reduction in strategic uncertainty implies that [M.sub.i](k|L) [greater
than or equal to] [M.sup.n.sub.i](k) and assumption 2 implies that
[M.sub.i](k|L) [greater than or equal to] [M.sup.n - 1.sub.i](k).
(10.) If the decision to lead is not expected to be random then
beliefs could be different in a game with endogenous leadership.
Formally, one should also allow for the possibility that beliefs in an
endogenous leadership game depend on the time spent waiting for someone
to lead. A-priori, however, it is not clear in which way beliefs would
differ in a game with endogenous or exogenous leadership, and so we
focus on the more important issue of player i deciding whether or not to
lead. Note also, that assumptions 2 and 3 remain appropriate with
endogenous leadership.
(11.) We did not combine Sim3 with any of the other treatments
because the lab could not accommodate 24 subjects and 12 subjects is
insufficient to maintain random matching between parts of a session.
(12.) Basically, if the distribution of choices is observed then
players can signal through repeated interaction that higher numbers
could be chosen to mutual benefit. Observed coordination failure is,
thus, typically less. A similar effect is seen by Blume and Ortmann
(2007) in a setting where only the minimum choice is made public but in
a pre-play communication stage all players can send a signal of what
they intend to do.
(13.) Pairwise Mann-Whitney tests by treatment give the same
conclusion.
(14.) To allow an easier comparison between the Exo and End
treatments we use a leader treatment dummy (which takes value 1 for both
the Exo and End treatments) and an Exo dummy (which takes value 1 for
the Exo treatment). Differences between the Exo and End treatments and
the Sim4 or Sim3 treatments should show up in the leader treatment dummy
and differences between the Exo and End treatments in the Exo dummy.
(15.) In order to model all rounds it is necessary to try and
capture dynamic effects. This, however, does create potential
econometric concerns, particularly in regressing choice on the minimum
choice in the previous round. The results are, however, robust to
different specifications, such as only including subjects who chose more
than the minimum choice in the previous round. Note that we also
included, where relevant, the difference between the leader's
choice and the minimum choice in the previous round. This, however,
proved insignificant and so is omitted.
(16.) For example, suppose the minimum choice is 1 in round 1 and a
subsequent leader chooses 7. Applying the results from column (4) we get
a prediction that the average choice with endogenous leadership will be
4 in round 2, 6 in round 3, and 7 in all subsequent rounds. With
exogenous leadership the prediction is 5 in round 2 and 7 in all
subsequent rounds.
(17.) There were 26 and 37 groups respectively where the minimum
was 1 at some point. In all other groups the minimum choice was always
above 1 and so there is no possibility to overcome coordination failure
as we have defined it.
(18.) One should not necessarily read too much into this because
there were also groups in the Sim4 treatment with persistent
coordination failure despite some choosing 7.
(19.) Costless communication has also proved less effective if
there is not common knowledge of what has been communicated (Chaudhuri,
Schotter. and Sopher 2009).
EDWARD CARTWRIGHT, JORIS GILLET and MARK VAN VUGT *
* Financial support from the Economic and Social Research Council
through Grant number RES-000-22-1999, "Why some people choose to be
leaders: The emergence of leadership in groups and organizations"
is gratefully acknowledged. We also thank two referees for their
constructive criticism of an earlier version and helpful suggestions for
improvement.
Cartwright: Senior Lecturer in Economics, Department of Economics,
University of Kent, Canterbury, Kent, CT2 7NP, UK. Phone +44 1277
823460, Fax +44 1227 827850, E-mail E.J.Cartwright@kent.ac.uk
Gillet: Postdoctoral Student, Department of Economics, Universitat
Osnabrtick, Rolandstr. 8, 49069 Osnabruck, Germany. Phone +49 (0)541 969
2732, Fax +49 (0)541 969 2705, E-mail Jgillet@uni-osnabrueck.de
Van Vugt: Professor of Psychology, Department of Social and
Organizational Psychology, VU University Amsterdam, 1081 BT Amsterdam,
The Netherlands; Department of Psychology, University of Kent,
Canterbury, Kent, CT2 7NP, UK. Phone +31 205985323 (8700), Fax +31
653853831, E-mail m.van.vugt@psy.vu.nl
doi: 10.1111/ecin.12003
TABLE 1
Payoff Table
Own choice
Minimum
Choice 1 2 3 4 5 6 7
1 0.7 0.6 0.5 0.4 0.3 0.2 0.1
2 0.8 0.7 0.6 0.5 0.4 0.3
3 0.9 0.8 0.7 0.6 0.5
4 1.0 0.9 0.8 0.7
5 1.1 1.0 0.9
6 1.2 1.1
7 1.3
TABLE 2
Summary of Sessions
Session Participants Part 1 Part 2 Part 3
1 16 Exo End Sim4
2 16 End Sim4 Exo
3 16 Sim4 Exo End
4 16 Exo Sim4 End
5 16 Sim4 End Exo
6 16 End Exo Sim4
7 12 Sim3 Sim3 Sim3
TABLE 3
Average Payoffs by Treatment
Overall Round 1 Round 10
Overall Sim4 0.69 0.55 0.69
Sim3 0.87 0.77 0.89
Exo 0.74 0.61 0.83
End 0.80 0.68 0.86
Leaders Exo 0.69 0.53 0.79
End 0.76 0.65 0.80
Followers Exo 0.76 0.64 0.85
End 0.81 0.70 0.88
TABLE 4
Average Choice of Subjects in the Sim4 and Sim3 Treatment Compared to
Those Following a Leader Who Chose 7 in the Exo and End Treatments
All parts
All rounds Round 1
No leader Sim4 3.01 (960) 4.23 (96)
Sim3 4.23 (360) 4.33 (36)
Leader chooses 7 Exo 5.24 (234) 4.81 (36)
End 5.70 (252) 5.71 (24)
Exo + End 5.48 (486) 5.17 (60)
Part 1
All rounds Round 1
No leader Sim4 2.86 (320) 4.03 (32)
Sim3 3.84 (120) 3.50 (12)
Leader chooses 7 Exo 4.77 (66) 4.40 (12)
End 3.95 (57) 5.50 (6)
Exo + End 4.39 (123) 4.78 (18)
Note: The number of observations are given in parentheses.
TABLE 5
Results of a GLS Random Effects Regression
(3) and Ordered Probit Regressions (1), (2),
and (4) with Choice as the Dependent Variable
Round 1 Round 10 All Rounds
Variable (1) (2)
(3) (4)
Leadership -1.69 ** -1.96 ** -1.34 ** -1.01 **
treatment (0.32) (0.37) (0.34) (0.30)
Exo treatment -0.24 -0.40 -0.05 0.11
(0.41) (0.40) (0.31) (0.32)
Sim4 treatment -0.07 -0.75 * -0.19 -0.07
(0.22) (0.24) (0.21) (0.12)
Round -- -- -0.09 ** -0.07 **
(0.02) (0.01)
Round x leader -- -- 0.09 ** 0.07 **
treatment (0.02) (0.02)
Round x Exo -- -- -0.03 -0.03
(0.03) (0.03)
Leaders choice 0.34 ** 0.41 ** 0.46 ** 0.32 **
(0.06) (0.06) (0.05) (0.05)
Leaders choice x -0.01 0.07 0.08 0.07
Exo (0.08) (0.08) (0.05) (0.04)
Min choice in last -- -- 0.78 ** 0.65 **
round (0.05) (0.05)
Min choice last -- -- -0.43 ** -0.29 **
round x leader (0.06) (0.06)
Min choice last -- -- -0.04 -0.08
round x Exo (0.06) (0.04)
Round 1 -- -- 2.34 ** 1.83 **
(0.18) (0.14)
Round 1 x leader -- -- -1.16 ** -0.69 **
treatment (0.33) (0.25)
Round 1 x Exo -- -- -0.78 * -0.71 *
(0.38) (0.29)
Part 2 of session 0.04 0.04 0.02 -0.04
(0.14) (0.14) (0.09) (0.07)
Part 3 of session 0.13 0.17 0.05 -0.07
(0.14) (0.17) (0.08) (0.06)
Constant -- -- 2.12 ** --
(0.27)
Threshold to -1.11 -0.82 -- 0.23
choose 2
Threshold to -0.85 -0.42 -- 0.72
choose 3
Threshold to -0.31 -0.08 -- 1.25
choose 4
Threshold to 0.12 0.34 -- 1.80
choose 5
Threshold to 0.52 0.59 -- 2.29
choose 6
Threshold to 0.81 1.02 -- 2.75
choose 7
No of obs. 276 276 2760 2760
Notes: We include only the choices of subjects who were
not leaders. The cluster corrected standard errors are given
in parentheses.
* Significant at 5%; ** significant at 1%.
TABLE 6
Characterizing Group Dynamics by Leadership
Condition
R7 R6 R5 R4 CF
Exogenous (n = 20) 1 3 9 10 6
Endogenous (n = 17) 2 2 3 7 4
Sim4 (n = 19) 0 0 0 1 10
Sim3 (n = 5) 0 0 0 2 0
Note: The number of groups that fit into each category
is given in parentheses.
TABLE 7
Average Choice of Subjects in the Sim4 and
Sim3 Treatment Compared to Followers in the
Exo and End Treatments if There Had Been a
Previous Round with a Minimum Choice of 1
Round 2 Round 10
No leader Sim4 2.68 (56) 1.94 (72)
Sim3 2.67 (3) 2.60 (15)
Leader chooses 7 Exo 5.13 (15) 5.28 (18)
End 3.89(9) 4.33 (18)
Exo + End 4.67 (24) 4.81 (36)
Note: The number of observations is given in parentheses.
TABLE 8
Average Payoffs of the Leader and Follower by Leader Choice
Exo End
Leader's Choice Leader Followers Leader Followers
1 0.70 (59) 0.65 (177) 0.70 (57) 0.63 (171)
2 0.65 (13) 0.67 (39) 0.67 (15) 0.62 (45)
3 0.62 (17) 0.67 (51) 0.68 (21) 0.74 (63)
4 0.67 (27) 0.74 (81) 0.88 (23) 0.89 (69)
5 0.75 (28) 0.82 (84) 0.65 (16) 0.76 (48)
6 0.93 (18) 0.99 (54) 0.78 (24) 0.88 (72)
7 0.64 (78) 0.81 (234) 0.82 (84) 0.95 (252)
1-7 0.69 (240) 0.76 (720) 0.76 (240) 0.81 (720)
Leader's Choice Sim4 Sim3
1 0.70 (344) 0.70 (37)
2 0.69 (125) 0.76 (45)
3 0.69 (116) 0.80 (80)
4 0.78 (144) 0.76 (39)
5 0.70 (94) 0.89 (38)
6 0.65 (60) 1.00 (39)
7 0.43 (77) 1.05 (82)
1-7 0.69 (960) 0.87 (360)
Note: For comparison we give the average payoffs in Sim4 and Sim3 of
subjects who choose the same number.
TABLE 9
Results of a GLS Random Effects Regression
(3) and Ordered Probit Regressions (1) (2) and
(4) With Leader Choice as the Dependent
Variable
Variable (1) (2) (3) (4)
Exo treatment 0.19 -0.36 0.86 0.82 *
(0.32) (0.33) (0.49) (0.25)
Round -- -- -0.02 -0.01
(0.05) (0.03)
Round x Exo -- -- -0.02 -0.02
(0.07) (0.04)
Min choice in last -- -- 0.60 ** 0.48 **
round (0.08) (0.07)
Min choice last -- -- -0.17 -0.18 *
round x Exo (0.09) (0.07)
Difference -- -- 0.09 0.14 **
between leader (0.08) (0.05)
choice and min
in last round
Difference last -- -- -0.12 -0.13 *
round x Exo (0.10) (0.06)
Round I -- -- 1.86 ** 1.59 **
(0.44) (0.34)
Round 1 x Exo -- -- -0.58 -0.60
(0.73) (0.46)
Part 2 of session -0.05 0.33 0.04 -0.07
(0.39) (0.40) (0.29) (0.17)
Part 3 of session -0.19 0.00 -0.34 -0.25
(0.39) (0.40) (0.29) (0.15)
Constant -- -- 2.78 ** --
(0.44)
Threshold to -0.88 -0.58 -- 0.52
choose 2
Threshold to -0.67 -0.46 -- 0.76
choose 3
Threshold to -0.42 -0.40 -- 1.04
choose 4
Threshold to -0.20 -0.30 -- 1.38
choose 5
Threshold to 0.06 -0.14 -- 1.66
choose 6
Threshold to 0.22 -0.08 -- 1.94
choose 7
No of obs. 48 48 480 480
Note: The cluster corrected standard errors are given in
parentheses.
