One Step at a Time: Does Gradualism Foster Group Coordination? * Abstract This study is based on a framed field experiment conducted in China and the study examines how the pattern of varying threshold levels influences group coordination at high-threshold levels. Of primary interest in varying the threshold level successively is the role of gradualism. We define gradualism as the hypothesis that proposes that allowing agents to coordinate first on small and easy-to-achieve goals and then increasing the level of goals slowly over the course of a game facilitates subsequent coordination on otherwise hard-to-achieve outcomes. We find that successful coordination at a high-stakes level in the Gradualism treatment group was very high. Our findings suggest that for a group to establish successful coordination at a high level, it is better to begin at a low-stake level and, equally important, to increase the stake level slowly. The paper sheds light on how to foster group coordination in a context where it is not clear how to structure incentives for individual players. * Joint with Sam Asher (Oxford) and Maoliang Ye (Harvard). We would like to thank Alberto Alesina, Jim Alt, Nejat Anbarci, Abhijit Banerjee, Max Bazerman, Iris Bohnet, Hannah Bowles, David Canning, Gary Charness, Raj Chetty, David Cutler, Sreedhari Desai, Ernst Fehr, Daniel Friedman, John Friedman, Roland Fryer, Francis Fukuyama, Edward Glaeser, Francesca Gino, Torben Iversen, Garett Jones, Yuichiro Kamada, Lawrence Katz, Judd Kessler, David Laibson, David Lam, Randall Lewis, Jeffrey Liebman, Jaimie Lien, Erzo F. P. Luttmer, Brigitte Madrian, Juanjuan Meng, Louis Putterman, Alvin Roth, Jason Shachat, Kenneth Shepsle, Andrei Shleifer, Monica Singhal, James Snyder, Dustin Tingley, Yan Yu, Tristan Zajonc, Richard Zeckhauser, Yao Zeng. I have been financially supported by the following sources: The National Science Foundation Graduate Research Fellowship in the Economics Program under Grant No. 1227274, The Harvard Institute for Quantitative Social Studies and The CEPR AMID Marie Curie Initial Training Network Grant.
Transcript
One Step at a Time: Does Gradualism Foster Group
Coordination?*
Abstract
This study is based on a framed field experiment conducted in China and the
study examines how the pattern of varying threshold levels influences group coordination
at high-threshold levels. Of primary interest in varying the threshold level successively is
the role of gradualism. We define gradualism as the hypothesis that proposes that allowing
agents to coordinate first on small and easy-to-achieve goals and then increasing the level
of goals slowly over the course of a game facilitates subsequent coordination on otherwise
hard-to-achieve outcomes. We find that successful coordination at a high-stakes level in
the Gradualism treatment group was very high. Our findings suggest that for a group to
establish successful coordination at a high level, it is better to begin at a low-stake level
and, equally important, to increase the stake level slowly. The paper sheds light on how to
foster group coordination in a context where it is not clear how to structure incentives for
individual players.
* Joint with Sam Asher (Oxford) and Maoliang Ye (Harvard). We would like to thank Alberto Alesina, Jim
Alt, Nejat Anbarci, Abhijit Banerjee, Max Bazerman, Iris Bohnet, Hannah Bowles, David Canning, Gary
Charness, Raj Chetty, David Cutler, Sreedhari Desai, Ernst Fehr, Daniel Friedman, John Friedman, Roland
Fryer, Francis Fukuyama, Edward Glaeser, Francesca Gino, Torben Iversen, Garett Jones, Yuichiro Kamada,
Lawrence Katz, Judd Kessler, David Laibson, David Lam, Randall Lewis, Jeffrey Liebman, Jaimie Lien, Erzo
F. P. Luttmer, Brigitte Madrian, Juanjuan Meng, Louis Putterman, Alvin Roth, Jason Shachat, Kenneth
Shepsle, Andrei Shleifer, Monica Singhal, James Snyder, Dustin Tingley, Yan Yu, Tristan Zajonc, Richard
Zeckhauser, Yao Zeng. I have been financially supported by the following sources: The National Science
Foundation Graduate Research Fellowship in the Economics Program under Grant No. 1227274, The Harvard
Institute for Quantitative Social Studies and The CEPR AMID Marie Curie Initial Training Network Grant.
1
1 Introduction
Successful coordination is at the core of a wide variety of economic and political
situations (Schelling, 1960; Arrow, 1974). Nonetheless, coordination failure is common in
the real world (Van Huyck et al., 1990; Cooper et al., 1990; Knez & Camerer, 1994, 2000;
Cachon & Camerer, 1996). Such failures significantly influence social welfare in many
ways, from causing setbacks in economic development (Ray, 1998; Kaul & Stern, 1999;
Bardhan, 2005; UNIDO, 2008), to creating disruptions in business cycles (Cooper & John,
1988) and defects in international monetary policies (Krugman & Obstfeld, 2009), to
influencing the building and promotion of the rule of law (Weingast, 1997).
We focus on a coordination mechanism related to varying the thresholds, which are
monetary points that a group must reach with the total of all group members’
contributions for a public (i.e., group) good to exist. Both in the lab and in the real world,
many privately provided public goods make use of a threshold to determine whether the
good is produced.1 Key questions related to a coordination game in which only the
threshold patterns can be varied are as follows:
1 We define a successful outcome as the sum of all individuals’ contributions reaching a minimum predefined
monetary threshold.
2
1. What buy-in strategy should a social planner choose if he or she wants to target
high levels of voluntary group coordination?
2. Should the social planner choose a series of successive low-threshold levels followed
by high-threshold levels, a slow and gradual increase of threshold levels, or
immediately introduced high-stake patterns?
Using a framed field experiment2 conducted in China, we address how the pattern
of varying threshold levels influences group coordination at high-threshold levels.3 Of
primary interest in varying the threshold level successively is the role of gradualism. We
define gradualism as the hypothesis that proposes that allowing agents to coordinate first
on small and easy-to-achieve goals and then increasing the difficulty of goals slowly over
the course of a game facilitates subsequent coordination on otherwise hard-to-achieve
outcomes.
To test the gradualism hypothesis, we conducted a computer-based study. Players
within a group were able to interact repeatedly. This feature mimicked a typical
coordination setting in the real world, but it did not replicate it exactly. In each period, we
endowed the participants with points (i.e., monetary units in the laboratory) and asked
each person to contribute a given amount (stake) to a hypothetical group project. Each
2 As per the taxonomy put forth in Harrison and List (2004), a framed field experiment is a conventional lab
experiment with field context in either the commodity, the task, or the information set that the subjects can
use. 3 Our paper focuses on a fixed-size group. For gradual organizational growth, see Weber (2006). In Section 2,
we compare and contrast Weber’s study to our study.
3
person had two options: (1) to contribute the pre-set amount exactly or (2) to contribute
nothing. In each period, members earned a profit only if all group members contributed to
the project. Otherwise, each person ended up with only the points that he had left from
the original sum he was given. 4,5,6
We assigned the participants to four main treatment groups of stake patterns,
which differed in the first six periods but featured an identical stake for the final six
periods. The first treatment, labeled Big Bang, featured a constant high stake for all 12
periods. The second treatment, labeled Semi-Gradualism, featured a constant low stake for
the first six periods and then a high stake for the final six periods. In our third and key
treatment, termed Gradualism, we increased the stake in each of the first six periods by
small amounts until it reached the highest stake in Period 7.7 The final treatment, which
we call the High Show-up Fee treatment, was a variant of the Big Bang treatment. The
High Show-up Fee treatment featured the same constant high stake for all 12 periods as
the Big Bang treatment but offered a higher show-up fee.
4 The setup we chose is generally referred to as the minimum-effort or weakest-link coordination game: The
payoff depends on each individual’s effort and the minimum effort of group members. Our setting simplifies the
payoff function. 5 Generally, the minimum-effort coordination game entails a complex payoff matrix: Several action choices are
available, and payoffs depend on both an individual’s actions and the minimum action of all other players’
actions. See Van Huyck et al. (1990), Knez and Camerer (1994, 2000), Cachon and Camerer (1996), Weber
(2006), and Chaudhuri et al. (2009). 6 Because of the binary choice feature (i.e., to contribute or not to contribute) available to each player in a
given period, our game is a multiperiod stag hunt game. Our game is a standard discrete public good game,
but no opportunity for individual free-riding is available. Our setup also relates the “weakest-link” public goods
game featured in several theoretical and experimental papers (see Hirshleifer, 1983; Harrison & Hirshleifer,
1989; Cornes & Hartley, 2007). 7 Note that the Semi-Gradualism treatment fell between the Big Bang and Gradualism treatments: The stake
in this case started and remained at a low level but then suddenly increased to the high value in Period 7.
4
Notes: The vertical line between Periods 6 and 7 separates the two halves of the first stage;
coordination performance of different treatments in the second half (periods 7–12) is the
main interest of this study.
Our first main finding is that successful coordination at a high-stakes level in the
Gradualism treatment group was very high. At the end of our experiment, 61.1 percent of
people in the Gradualism groups successfully coordinated, whereas only 16.7 percent and
33.3 percent, respectively, of those in the Big Bang and Semi-Gradualism groups did so.
Strikingly, the Semi-Gradualism group failed to foster high group coordination in
comparison to the Gradualism treatment group. Our findings suggest that for a group to
establish successful coordination at a high level, it is better to begin at a low-stake level
and, equally important, to increase the stake level slowly.
05
1015
Sta
ke in
the
Firs
t Sta
ge (
poin
ts)
0 5 10 15Period
BigBang SemiGradualismGradualism HighShowupFee
Figure 1: Stake Patterns of the Treatments in the First Stage
5
Our second main finding is that individuals in the Gradualism category were about
10 percentage points more likely to cooperate upon entering a new group.8 However, when
they found that their cooperation was not rewarded in the new environment (because the
new group members may have been treated differently and/or had different coordination
outcomes previously), these subjects tended to become less cooperative.
This paper makes two key contributions. First, the paper contributes to the
coordination literature by examining how stake path dependence, as opposed to other
contextual factors, fosters successful group coordination. Economists have addressed ways
to promote successful coordination via various mechanisms: (1) introducing either
monetary or nonmonetary punishment (Fehr & Gaechter, 2000)9, (2) allowing player
communication (Cooper, DeJong, Forsythe, & Ross, 1992), (3) promoting competition
between groups (Myung, 2008), (4) introducing entrance fees (Cachon & Camerer, 1996),
(5) varying group size (Weber, 2006), and (6) allowing accumulation of player
contributions (Dorsey, 1992; Marx & Matthews, 2000; Kurzban et al., 2001; Duffy et al.,
2007). The only other study to examine the effect of path dependence on group
coordination is Romero’s (2011); he generally explores the importance of historical game
8 When a person interacts with others in a group, he or she may develop beliefs about the cooperative
tendencies of an average person from the general population. For example, if a group manages to cooperate
successfully, this outcome may lead to a subsequent tendency toward cooperation among its members. To
explore this potential channel, we introduced a second stage in the experiment: In the second stage, we
reshuffled experimental participants into new groups and observed their actions. 9 Fehr and Gaechter (2000) find that when either monetary or nonmonetary punishment is available to players,
they end up using it, and availability of either kind of punishment increases average player contributions.
6
parameters on game coordination.10,11 The second contribution of this paper is that it sheds
light on how to foster group coordination in a context where it is not clear how to
structure incentives for individual players.
That the use of gradualism results in the highest coordination rates has various
policy implications. Coordination is essential to intra-team collaboration, to domestic
country reforms, and to the success of international agreements. In the context of small
intra-organization employee–employer dynamics, this setup could eventually ensure that
employees coordinate well in large tasks (e.g., employers can provide small initial tasks to
new employees to assist with coordination building). Gradualism can foster intracountry
coordination, for a country transitioning from a planned economy to a market economy, or
a country on the path to democratization (Dewatripont & Roland, 1992, 1995; Wei, 1997;
Weingast, 1997).12 Finally, the gradualism approach could be equally salient to bilateral
investment treaties (Chisik & Davies, 2004) such as the United Nations Framework
Convention on Climate Change and the Kyoto Protocol (Mitchell, 2003), or for arms races
10In game theory, coordination refers to resolving which strategy (associated with an equilibrium) a player will
choose to play when there are multiple equilibrium choices. In this paper, we focus on those coordination
games with Pareto-ranked equilibriums, especially weakest-link (minimum-effort) coordination games. 11Some studies find sanction institutions, social pressure, and reputation to be mechanisms that promote
cooperation (see Olson, 1971; Ostrom et al., 1992; Fehr & Gächter, 2000; Masclet et al., 2003; Gächter &
Herrmann, 2011; Bochet et al., 2006; Carpenter, 2007). Others explore methods to facilitate coordination when
sanctions and social pressure cannot be imposed, such as repetition with fixed group members (Clark & Sefton,
2001), complete information structure (Brandts & Cooper, 2006a), communication (Cooper et al., 1992;
Charness, 2000; Weber et al., 2001; Duffy & Feltovich, 2002; Chaudhuri et al., 2009), and between-group
competition (Bornstein et al., 2002; Riechmann & Weimann, 2008). 12 Coordination among stakeholders is only one aspect of economic and political reforms.
7
as states avoid rapid reductions that diminish their bargaining power (Downs & Rocke,
1990; Kydd, 2000; Langlois & Langlois, 2001).
The rest of the paper is organized as follows. In Section 2, we discuss related
economics literature. In Section 3, we detail the experimental design. In Section 4, we
present results. Section 5 concludes.
2 Coordination Games and Gradualism
The gradualism hypothesis relates to previous research on the dynamics of
coordination games, prisoners’ dilemma games, and public goods games.13
In a laboratory dynamic weakest-link14 coordination experiment, Weber (2006)
studies the dynamics of organizational growth. He finds that for successful coordination in
a large group to occur, it is better to grow the group size gradually than to start with a
large group. Our study differs from Weber (2006) in three major ways: (1) We explore
gradualism in coordination within a given fixed-size group; (2) in our study, the choice set
13 Several studies examine gradualism in the setup of sequential move or trust (investment) games. Pitchford
and Snyder (2004) present a model in which a sequence of gradually smaller investments solves the hold-up
problem, a situation where two parties (e.g., a supplier and a manufacturer or the owner of capital and
workers) may be able to work most efficiently by cooperating, but refrain from doing so due to concerns that
they may give the other party increased bargaining power. In Pitchford and Snyder’s (2004) setup the buyer’s
ability to hold up a seller’s investment is substantial. However, Kurzban et al. (2008) contradict the prediction
of Pitchford and Snyder (2004) by showing that subjects prefer starting with small levels of investment and
subsequently increasing them, rather than the other way around. 14 In a standard weak-link game, participants simultaneously pick a number. The earnings of a particular
player depend on the number they chose and on the lowest number chosen. Usually each individual's payoff
function is positively related to the minimum of all individuals' choices and negatively related to the difference
between their own choice and the lowest choice. A weak-link game is a representation of any situation where
the group output depends on the contribution (or effort) of the least contributing member and contributing is
costly.
8
in each period is binary, and the payoff structure is much simpler; and (3) we have a third
main treatment, Semi-Gradualism, which explores whether a sudden increase of the stake
negatively affects coordination.
Romero (2011) studies the effect of path dependence (past game parameters)
on subsequent weakest-link game coordination and finds that groups coordinate better
with a certain cost when the cost is increasing than when the cost is decreasing to that
level. Our study differs from Romero (2011) in two ways: (1) We change the stake level,
which indicates not only the cost but also the benefit, and (2) we compare a slow increase
of the stake with a sudden increase and with a start at a high stake, while he alternatively
compares an increasing path of the cost with a decreasing one.
A third strand of the literature, related to our research, allows players in a public
goods provision game to accumulate contributions over periods (Dorsey, 1992; Marx &
Matthews, 2000; Kurzban et al., 2001; Duffy et al.,2007). Besides the binary weakest-link
structure that differs from these studies, another feature of our setup that is unique to our
design is that the “public good” in our game is independent from one period to the next.
Contributions cannot accumulate over periods, and each project features its own target
(stake). In the aforementioned studies on dynamic voluntary contribution to a single
public project, players are allowed to contribute whenever and as much as they wish and
accumulate their contributions over the course of the project (there is no objective for each
period before the game's end). Our experimental design examines the causal effects of
9
stake variation in contrast to a design that exogenously varies game period lengths or how
player contributions accumulate. Although the aforementioned studies relate to some real-
world examples (e.g., long-term fund drives), our study is better aligned with other
important real-world factors we mention earlier. In the examples we refer to in Section 1,
the duration of the final high-stake project is relatively short and is not divisible into
subperiods to accumulate effort; regular feedback about what other participants contribute
to the final project is not provided. Players, in our setup, face an independent project with
a clear small-scale objective in each period, and after each period, players assess how they
performed on these small-scale tasks.
We outline more clearly the efficiency gains of gradualism than Andreoni and Samuelson
(2006). Andreoni and Samuelson (2006) examine a twice-played prisoners’ dilemma in
which the total stakes in two periods are fixed, while the distribution of these stakes across
periods can be varied. Both their theoretical and experimental results show that it is best
to “start small,” with bigger stakes in the second period. However, cooperation is low for
the period with a high stake in their experiment.15 One potential explanation of the
advantage of our setup in promoting cooperation over Andreoni and Samuelson’s (2006) is
that our "weakest-link" structure does not allow free-riding, unlike their setup.
15 When the relative stake of Period 2 is high, more cooperation takes place in Period 1 but less cooperation
occurs in Period 2; when the relative stake of Period 2 is low, less cooperation occurs in Period 1 but more
cooperation takes place in Period 2.
10
Offerman and van der Veen (2010) study whether governmental subsidies geared
toward promoting public good provisions should be abruptly introduced or gradually
increased; that is, given the benefit of the public good, whether the individual cost of
providing the public good should be decreased sharply or gradually. Their results favor an
immediate increase of subsidy: When the final subsidy level is substantial, the effect of a
quick increase is much stronger than that of a gradual increase. Our study differs from
Offerman and van der Veen (2010) in three key ways. First, these authors focus on how
the use of subsidies can stimulate cooperation after unsuccessful cooperation at the start of
a game. Because our mechanism focuses on the variation of threshold patterns, it is quite
distinct from their subsidy mechanism. Second, our study manipulates the stake level:
Both the cost and the benefit of the public good could change (whereas in Offerman and
van der Veen’s [2010] setup only the cost changes). A third key distinction relates to the
fact that our study stake paths are non-decreasing, whereas their paths are non-increasing.
Several other studies examine monotone games, multi-period games in which
players are constrained to choose strategies that are non-decreasing over time (i.e., players
need to increase their respective contributions over time) (see Gale, 1995, 2001; Lockwood
& Thomas, 2002; Choi et al., 2008). In contrast to these studies, our experiment employs a
different feature—we enable the stake to be non-decreasing, rather than the contribution.
Watson (1999, 2002) examine theoretically how “starting small and increasing
interactions over time” is an equilibrium for dynamic cooperation. Our setup adopts an
11
empirical, as opposed to theoretical, approach to test this assumption. By determining the
stake path exogenously, we address whether gradualism promotes cooperation at high-
stake levels, rather than whether players themselves choose to adopt a gradualist
approach.
3 Information Structure and Theoretical Framework
In this section, we review what information we provided to each player, what
strategies each player faces in each game period and what the payoffs options are.
3.1 Information Structure
The information structure we provided to each player is as follows: we told
subjects that the game would consist of two stages, although they were not told the exact
number of periods in each stage. Instead, we told players that the game would last
between 30 minutes and 1 hour, including time for sign-up, reading of instructions, and
taking a quiz designed to ensure that subjects understood the experimental guidelines as
well as what rules guide the calculation of their final payment. We note two features of the
game. First, we wanted to reduce the possibility of backward induction16 and a potential
end-of-game effect.17 Second, our study design approximated features of real-world
16 Because the number of periods of a game is unknown but the number of periods is finite, in theory players
could backward induct to some extent. The likelihood of actual backward induction occurring is, however, not
well supported in the empirical literature. 17 In minimum-effort coordination games, the end-of-game effect should be absent or minor because there is no
incentive to free-ride given that others cooperate.
12
situations—in the real world, people do not know ex ante the exact number of
coordination opportunities.
At the beginning of each period, each subject knew the stake of the current period
but not those of future periods. This replicates the circumstances of many real-world cases,
in which people do not know what is at stake in future interactions. At the end of each
period, each player knew whether all four group members (including himself or herself)
contributed the required points for that period but did not know the total number of
group members who contributed (in case fewer than four members contributed). The study
design was consistent with minimum-effort coordination games (see Van Huyck et al.,
1990), in which the only commonly available historical data to players is the minimum.18, 19
In our game, we did not allow communication among players for two reasons.
First, communication among heterogeneous groups, which could benefit from coordination,
is often impossible in the real world. Second, a design that precludes communication makes
coordination among players more difficult.20
3.2 Stages, Payoffs and Nash Equilibria
18 In our setup, if all members cooperate, then the minimum is the stake (or coded in a binary way, “1”);
otherwise the minimum is zero. 19 This feature is popular in the contract theory literature, in which imperfect observation of effort is common. 20 Ostrom (2010), Charness (2000), and Chaudhuri et al. (2009) provide evidence that communication improves
cooperation.
13
We set up the game in two stages: The first stage comprised 12 periods, while the
second one comprised 8 periods.21 In each period, we endowed each subject with 20 points
and asked that each provide a certain number of points to their assigned groups’ common
pools. The required number could vary across periods, and each subject could only choose
either to contribute the exact amount, which we refer to as the stake, or not to contribute
at all. If all members in a group contributed the exact number of points required, then
each member not only received the points he had contributed to the pool, but he also
gained an extra return, which equaled the required number of points (i.e., the stake). If
not all group members contributed (or at least one of the players deviated), then each
member finished the period with his remaining points only (i.e., the initial endowment
minus the player’s contribution during the period).
Players earned according to the following function in each period, conditional on
each player's action:
=≠∃=−=≠∀==+
=DAtsijandCAifTh
DAif
ijCAandCAifTh
Earnings
tjtit
ti
tjtit
it
,,
,
,,
..,20
20
,20
(1)
where Earningsi,t is i ’s payoff in period t , tTh is stake at t . , i tA and , j tA are the actions
of i and j at t , respectively ( i and j are in the same group). C represents “cooperate”
(“contribute”), while D represents “deviate” (“not contribute”).
21 We define a period as each one-shot interaction among all four players. A game stage comprises multiple
periods.
14
Generally, the stag hunt differs from the prisoner's dilemma in that there are two
Nash equilibria: when both players cooperate and both players defect.22
3.3 A Belief-Based Model
The coordination problem at the heart of our experimental design involves multiple
equilibria in each period. Both the stake level at the start of each game (i.e., period) and
the stake path influence how players form their initial beliefs about other players' actions
and how players subsequently update these beliefs. In the framework we develop, we posit
that players' beliefs are central in determining the game equilibrium.
We adopt a belief-based learning framework to generate theoretical predictions
regarding coordination outcomes.23,24 The theory of level-k reasoning, first proposed by
Stahl and Wilson (1995) and Nagel (1995), with further extensions by Ho, Camerer, and
Weigelt (1998), Costa-Gomes, Crawford and Broseta (2001), and Costa-Gomes and
Crawford (2006), can be used to rationalize subject behavior in any coordination context.
The level-k model is based on the presumption that subjects’ behavior can be classified
22 The payoff matrix in Appendix Table A.1 illustrates the payoff structure of a stag hunt game, where a > b
≥ d > c 23 We cannot rule out alternative explanations for our experimental results. Recent economics papers provide
strong evidence that players play consistent with their beliefs, although we do not formally test this assertion
2010). Direct incentive-compatible belief elicitation has become increasingly popular in experimental economics
(see Offerman et al., 1996, 2001; Nyarko & Schotter, 2002; Costa-Gomes & Weizsäcker, 2008; Rey-Biel, 2009;
Hyndman et al., 2009). We ended up not eliciting beliefs in favor of cleanly testing our gradualism hypothesis.
We were concerned that if we were to elicit player beliefs, we likely would have contaminated our results.
Several recent studies (see Rutström & Wilcox, 2004, 2009) support our concern. In particular, Rutström and
Wilcox (2004, 2009) provide strong evidence that belief elicitation results in higher player sophistication and
higher-order rationalities, both of which ultimately influence how player act. 24 Appendix C details formal belief-based learning model with level-k thinking.
15
into different levels of reasoning. The zero level of reasoning, L0, corresponds to
nonstrategic behavior (i.e., when strategies are selected at random without forming any
beliefs about opponents’ behavior). In the literature, L0 is typically considered to be a
person’s model of others in general rather than a specific person. Level-1 players, L1,
believe that all their opponents are L0 and play the best response based on this belief.
Level-2 players, L2, play the best response based on their belief that all their opponents
are L1, and so on.
While level-k thinking is not particularly unique to the gradualism contest (see
Costa-Gomes & Crawford, 2006), the structure of the game and its simplicity are very
conducive to this type of behavior. Success in the coordination game largely depends on a
person’s ability to correctly predict the choice made by others. This explicitly forces
individuals to think about the decisions of other players. Moreover, the symmetry of
information makes this task relatively simple, which can further encourage participants to
focus on the behavior of others.
We posit that non-strategic L0 players have constant “willingness-to-contribute”
and intend to contribute if and only if their “willingness-to-contribute” is larger than the
16
current stake,25 while rational players (i.e., level-k players for any 1k ≥ ) best respond to
level-(k–1) players.
Rational players have existing beliefs regarding successful group cooperation. Based
on what a rational individual observes in each period, he updates his beliefs about other
players in his group. We assume that attempting coordination has a cost. The lower the
stake at the start of a game, the "cheaper" it is for players to coordinate and the stronger
their beliefs are that others will contribute to the common pool. Therefore, the lower the
stake at the start of a game, the higher the success rate of group coordination is at the
start of a game.26 When groups successfully coordinate at a given stake level, players get to
reinforce their beliefs about the likelihood that others will contribute at the same stake
level. Alternatively, cooperation failure at a given stake level causes players to doubt that
other players in their assigned group will contribute later at the same or a higher stake
level.
When stakes increase in two consecutive game periods, successful coordination at
the low stakes may not influence each player’s posterior beliefs regarding other people's
actions at the high-stake levels. In other words, successful group coordination at a low-
stake level may not necessarily imply successful coordination at a high-stake level.
25 The definition of L0 players varies in the literature, as we detail in Appendix C. However, our theoretical
results hold under various definitions. We allow L0 players to make mistakes, and we assume that the belief
updating process of L1 players about L0 players’ actions follows a standard Bayesian rule. 26 We assume risk-averse preferences and a weakest-link payoff structure.
17
Several key predictions emerge from our model (supporting details and proofs can
be found in Appendix C):
• Proposition 1. The lower the 1S (stake at t = 1), the higher the probability that
the coordination at t = 1 will succeed.
• Proposition 2. When players are informed about the number (m) of contributors
at t with a stake tS , the larger the m, the higher the probability that the
coordination at t + 1 will succeed if 1t tS S+ = .
• Proposition 3. No matter whether a group succeeds or fails at t with a stake tS ,
the lower the 1( )t tS S+ ≥ , the (weakly) higher the probability that the coordination
at t + 1 will succeed.
Coordination outcomes in the first stage of the game enable players to form beliefs about
other group members and how likely these other players are to cooperate. Based on his
observations, each player will likely form his own beliefs about properties of the general
18
population regarding cooperative tendency. The first stage of the game could influence how
members play in the second stage.27
Because the Gradualism treatment may promote more group coordination (relative
to other treatment groups) in the first game stage, we propose that conditional on being
placed in the Gradualism treatment during the first stage, players will cooperate more
(relative to players in other treatment groups) in the game’s second stage. Proposition 4
summarizes this prediction.
• Proposition 4: Conditional on being in the Gradualism treatment group in the
first stage, players will contribute more (relative to players in other treatment
groups) in the first period of Stage 2. The higher success rate of Gradualism
treatment (relative to other treatment groups) at the end of Stage 1 drives this
result.
4 Setting, Experimental Design, and Data
4.1 Participants and Payoff Structure
We conducted the lab experiment at Renmin University of China in Beijing,
China, in July 2010 with 256 subjects recruited via the Bulletin Board System and
27 Other studies provide evidence that history can influence subsequent behavior. In a two-stage trust game,
Bohnet and Huck (2004) find that once players get to experience a cooperative environment in the first stage
of a game, they become more trusting (of others) in a new environment in the second stage.
19
posters.28 The majority of subjects were students from Renmin University and universities
nearby. Appendix Table A.2 provides basic summary characteristics of the subject pool:
The average age was 22 years old, 91 percent were college or graduate students, 41 percent
were male, 12 percent majored in economics, 16 percent majored in other social sciences,
27 percent majored in business, and the remaining 45 percent majored in other disciplines.
The subjects’ individual annual income range for 2009 was 5,000 to 10,000 yuan
(approximately U.S. $620 to $1254).
The experiment consisted of 18 sessions, all computerized using the z-Tree software
package (Fischbacher, 2007). Both the instructions and the game information shown on
the computer screen were in Mandarin. In each session, we randomly assigned subjects to
groups of four; 29 our sample consisted of 64 groups in total.
Players earned, as outlined earlier, according to function (1) in each period,
conditional on each player's action. The final total payment to each player equaled the
sum of each period's earnings plus a show-up fee. The exchange rate was 40 points per
yuan.30 Each subject earned approximately 21 to 22 yuan (around U.S. $3 to $4) including
the show-up fee, for the whole experiment, which covered ordinary meals for 1 to 2 days
on campus.
28We conducted a minimum-effort coordination game. Specifically, it was a multi-period stag hunt game due to
the binary choice featured in each period. 29 In coordination games, four is usually a small or moderate group size. Croson and Marks (2000) demonstrate
via a meta-analysis study that in public goods games, group sizes are most often four, five, and seven. 30 The yuan/$U.S.dollar exchange rate was ≈ 6.7.
20
4.2 Treatment Assignments
Our experiment comprised four treatment groups: (1) Big Bang, (2) Semi-
Gradualism, (3) Gradualism, and (4) a variant of the Big Bang treatment, which we call
the High Show-up Fee treatment. All groups in the three main treatments faced the same
stake in the second half (Periods 7 to 12) of the first stage, but stake paths differed for
each treatment group in the first half (Periods 1 to 6). The first half of the experiment
featured different stake paths for each treatment group. The different stake paths
concerned us because high stakes could potentially lead to an income effect, due to
potential earnings differences, for subjects in higher-stakes treatment groups. To isolate
the income effect on participants' contribution from the effect of the three main
treatments in the second half of the first stage, we introduced the High Show-up Fee
treatment, which as mentioned is a variant of the Big Bang treatment. We describe the
High Show-up Fee treatment in more detail in a later section.31 In 8 of the 18 sessions, we
randomly assigned 12 subjects into the three main treatments; in the remaining 10
sessions, we randomly assigned 16 subjects into the four treatments (three main
treatments and one supplementary treatment). In total, we had 18, 18, 18, and 10 groups
in Big Bang, Semi-Gradualism, Gradualism, and High Show-up Fee treatments,
respectively.
31 Experimental studies usually adopt a random period for payment to address the income effect concern. We
did not follow that precedent because we were worried that a random period income provision could make
some subjects less serious in playing the game, and we also wanted to capture how big the income effect was.
21
Appendix Table A.3 displays randomization tests for balancing on observables
across treatment groups. The default category is the Gradualism treatment. The
regressions in Appendix Table A.3 exhibit results without other control variables.32 Only
four characteristics exhibit statistically significant differences across treatment groups.33
Overall, the treatment group characteristics are balanced along most background
characteristics.
Figure 1 shows the game stakes over the 12 periods in the first stage. For the Big
Bang treatment, the stakes were always kept at the highest level (e.g., 14 for 16 sessions
and 12 for two sessions34). For the Semi-Gradualism treatment, we set the stakes at 2 for
the first six periods and then we set them at the highest stake for the next six periods.
Finally, for the Gradualism treatment, we increased the stakes gradually from 2 to 12 with
a jump of 2 for the first six periods, and we kept them fixed at the highest stake for the
next six periods. The show-up fee provided to each individual for these three treatments
was 400 points. The High Show-up Fee treatment was 480 points instead of 400 points.
The extra 80 points sufficiently captured the potential earnings difference accumulated
over Periods 1 through 6; thus, this treatment enabled us to isolate the effect generated by
32 Therefore the constant term indicates the mean value of dependent variables for the Gradualism treatment. 33 Subjects in the Semi-Gradualism group reported higher family economic status than that for the other
treatment groups; Big Bang and High Show-up Fee group members report higher risk aversion indexes than
that of subjects in other treatment groups, and subjects in the Big Bang treatment group are more likely to be
students than subjects in the other treatment groups. 34 We calibrated the highest stake level using 12 and 14, and finally opted for 14 in most sessions. To make full
use of the samples, we pool all 18 sessions together in our analysis.
22
an income effect by comparing the High Show-up Fee to the Big Bang treatment (we
discuss this in detail in Section 5).
When subjects entered the second stage of the game, they were randomly
reshuffled into groups of four. New group members did not necessarily come from the same
treatment group as the first stage; however, we made it clear to all players that new group
members could potentially come from a different treatment group. Within the second stage
of the game, group compositions were fixed, and stakes were all set at the highest stake for
all periods and all groups (i.e., those in different treatment groups in the first stage faced
the same stake in each period of the second stage).
At the start of the second stage, we notified each player that he or she would enter
a new random group. At the end of each stage, we notified each player how many points
he or she had accumulated to date. We asked subjects to complete a brief survey that
collected information on age, gender, nationality, educational level, concentration at
school, working status, income, and risk preferences over various lotteries (see Appendix
B).
5 Results: Impact of Gradualism on Coordination
This section presents our baseline estimates on coordination outcomes. We begin
by focusing our analysis on the following three outcome variables per period: (1) whether a
23
group coordinates successfully or not, (2) whether an individual contributes or not, and (3)
each individual’s net payoff.
5.1 First Stage Result Highlights
We examine outcomes in Periods 7 through 12 of the first stage, when all
treatment categories faced the same high stake.
Main Result 1: The Gradualism treatment recipients significantly
outperformed those receiving alternative treatments, showing that when
groups start at low stakes and face gradual increases, they coordinate more
successfully and they earn more in the high-stake periods (than those in other
treatments).
Figure 2 shows the success rate by treatment. In Period 7, 66.7 percent of
Gradualism groups coordinated successfully (i.e., all four group members contributed),
whereas success rates for the Big Bang, Semi-Gradualism, and High Show-up Fee groups
are only 16.7, 33.3, and 30 percent, respectively. Figure 2 illustrates that success rates for
treatment groups remained stable from Periods 7 to 12. Differences in average success
rates between the Gradualism treatment and Bing Bang, Semi-Gradualism, and High
24
Show-up Fee treatments are all statistically significant.35 This finding is consistent with
Propositions 1–3.
Notes: A group is successful if all four members contribute the stake in that period.
Figure 3 shows average individual earning by treatment. The Big Bang and High
Show-up Fee groups have higher earnings potentials (i.e., higher stakes) from Periods 1 to
6. Yet, on average, individuals in the Big Bang and High Show-up Fee treatments earned
less than individuals in the Gradualism treatment due to the high success rates in the
Gradualism treatment; the Semi-Gradualism groups earned less than the Gradualism
35 The Wilcoxon-Mann-Whitney test: p <0.01, p = 0.06, and p = 0.09, respectively; observations are at the
group level, as coordination success is a group-level outcome.
Figure 2: Success Rates of Groups by Treatment and Period in the First
Stage
25
treatment groups from Periods 2 to 6. These payoff differences persisted from Periods 7 to
12, when all treatment groups experienced the same high stakes. Differences in cumulative
individual earnings over Periods 7 to 12 between the Gradualism, on one hand, and Big
Bang, Semi-Gradualism, and High Show-up Fee treatments, on the other hand, are all
highly statistically significant.36
Figure 3: Average First Stage Individual Earning by Treatment
36 The Wilcoxon-Mann-Whitney test for Period 7: p < 0.0001, p < 0.0001, and p = 0.02; observations are at
the individual level.
1520
2530
Ave
rage
Indi
vidu
al E
arni
ng in
the
Firs
t Sta
ge (
poin
ts)
0 5 10 15Period
BigBang SemiGradualismGradualism HighShowupFee
26
To address the potential "income effect"37 for performance outcomes from Periods 7
to 12, we summarize individual payoffs through Period 6 (i.e., payoff accumulated from
Period 1 to 6, not including the show-up fee) for each treatment group in Figure 3. On
average, subjects in the Gradualism treatment group earned the most through the first six
periods: the average (median) yield by Period 6 was 112.42 (106) for the Big Bang
treatment, 126.31 (130) for the Semi-Gradualism treatment, and 143.94 (162) for the
Gradualism treatment. However, the differences in means (and medians) are dramatically
smaller than 80 points (the difference in show-up fee between the High Show-up Fee
treatment and the other three treatments). This result demonstrates that a show-up fee
difference of 80 points between the Big Bang and High Show-up Fee treatments is large
enough to capture the potential income differences at the start of Period 7 between Big
Bang, Semi-Gradualism, and Gradualism treatments. In fact, even when we added the
show-up fee, subjects in the Gradualism treatment group earned less, on average, than
subjects in the High Show-up Fee treatment group by the end of Period 6. Because the
Gradualism treatment results in better performance than the High Show-up Fee treatment
from Periods 7 through 12 in Stage 1, an income effect from the first six periods could not
account for the difference in performance in subsequent periods.38
37 For instance, individuals treated in the Gradualism treatment may earn more from Periods 1–6, so they are
more likely to contribute in Periods 7–12. 38 Individual wealth levels (in the real world) may influence individual decisions in the lab. However, because
we randomize subjects into treatment groups, the randomization design balances wealth levels (outside of the
lab) across treatment groups.
27
Table 1: Summary of Treatments in the First Stage
Treatment Big Bang Semi-Gradualism Gradualism High Show-up Fee
Endowment in each period 20 20 20 20
Show-up Fee (points) 400 400 400 480
Exchange Rate 40 40 40 40
Stake in Period 1 (points) 14 2 2 14
Stake in Period 6 (points) 14 2 12 14
Stake in Period 7–12 14 14 14 14
Number of groups 18 18 18 10
Number of subjects 72 72 72 40
Average earnings up to
period 6 (points; excluding
show-up fee)
112.42 126.31 143.94 127.35
Median earnings up to
period 6 (points; excluding
show-up fee)
106 130 162 106
Treatment
28
5.2 Coordination Dynamics in the First Stage
To identify why the Gradualism treatment group performs best in Periods 7 to 12,
we examine the coordination dynamics in Figure 2, Figure 4, and Appendix Figure A.1.
Pattern 1: The lower the stake size, the higher the average contribution
and success rates in Period 1.
Figure 4 displays contribution rates for Period 1. The average contribution rate is
above 90 percent for Semi-Gradualism and Gradualism treatments with a low stake, which
is higher than the contribution rate of 60 percent for Big Bang and High Show-up Fee
treatments with a high stake.40
40 The Wilcoxon-Mann-Whitney test between these two categories: p < 0.0001; observations are at the
individual level.
29
Figure 4: Contribution Rate by Treatment and Period in the First Stage
Figure 2 exhibits success rates and outlines stark differences across treatment
groups. Over two-thirds of the Semi-Gradualism and Gradualism groups coordinate
successfully at the low initial stake, whereas only 16.6 percent (or 30 percent) of the Big
Bang (or High Show-up Fee) groups succeed at the high initial stake.41 A weakest-link
structure requires that all four group members contribute at the same time to make the
coordination a success.42 We detect no statistically significant difference in success rates
between Big Bang and High Show-up Fee treatments, ruling out a potential income effect.
Pattern 1 is consistent with Proposition 1 in Appendix C.
41 The Wilcoxon-Mann-Whitney test between these two categories: p < 0.0001; observations are at the group
level. 42 Assuming the probability of contributing is independent across members in a group (which is plausible in
Period 1 because players are randomly assigned to groups and have not interacted with each other), the
success rate should be the biquadrate of the contribution rate. As long as the contribution rates are high
enough, the difference in the success rate exceeds that in the contribution rate.
.2.4
.6.8
1C
ontr
ibut
ion
Rat
e in
the
Firs
t Sta
ge
0 5 10 15Period
BigBang SemiGradualismGradualism HighShowupFee
30
Pattern 2: Conditional on having failed coordination in period t, most
groups fail at the same or a higher stake in period t + 1.
Appendix Figure A.1 details coordination for groups across periods by treatment
type. A group in a given period succeeds in cooperating if and only if the number of
contributors (i.e., the vertical axis) equals four. Appendix Figure A.1 shows that once a
group fails to coordinate, it rarely becomes successful thereafter.43 This pattern is likely
due to players obtaining limited information regarding the group outcome each period:
Each member does not know how many group members contribute or not.44 This finding is
consistent with Weber et al. (2001) and Weber (2006).45
Pattern 3: Conditional on successfully coordinating in period t, most
groups succeed at the same or a slightly higher stake in period t + 1.
However, few groups remain successful with a much higher stake in period t +
1.
43 Only six exceptions (one Big Bang group: group no.101; five Semi-Gradualism groups: groups no. 112, 132,
142, 172, 182) out of 64 groups exist. 44 Berninghaus and Ehrhart (2001), and Brandts and Cooper (2006a) find perfect information feedback
improves coordination. 45 Weber (2001, 2006) find that once groups reach an inefficient outcome, they are unable subsequently to
reach a more efficient outcome. However, changing incentives can improve subsequent coordination
A group remains successful in coordinating46 once its group members successfully
coordinate. Conditional on successful coordination in Period 1, groups in the Big Bang,
Gradualism, and High Show-up Fee treatment groups usually remain successful in
subsequent periods. Groups in the Big Bang and High Show-up Fee treatment groups
exhibit lower success rates in Period 1 than groups in the Gradualism treatment groups
and therefore, on average, they perform worse than groups in Gradualism treatment
groups at a high-stake level.
We note a large gap in success rates between groups in the Gradualism treatment
group and those in the Semi-Gradualism treatment group in Period 7.47 Both treatments
exhibit success rates of approximately 70 percent for the first six periods. However, the
success rate of groups in the Semi-Gradualism treatment falls to 33.3 percent in Period 7,
whereas that of groups in the Gradualism treatment remains at 66.7 percent.48 The success
rates remain constant across time except for a drop from Period 6 to 7 for groups in the
Semi-Gradualism treatment. Patterns 2 and 3 are consistent with Propositions 2 and 3 in
Appendix C.
Pattern 4: The overall decrease in contribution rates can be attributed
to groups failing to reach cooperation across periods; the decrease for the
46 Appendix Figure A.1 shows that there are only seven exceptions (one Big Bang group: group no.101; four
Semi-Gradualism groups: nos. 82, 112, 132, 142); and two Gradualism groups: nos. 33 and 173). 47 When the stake jumps from 2 to 14 for the latter treatment. 48 The Wilcoxon-Mann-Whitney test between these two treatments in period 7: p<0.05; observations are at the
group level.
32
Semi-Gradualism treatment from Period 6 to Period 7 can be attributed to the
groups that fail to cooperate in Period 7 but that had been previously
successful in Period 6.
Figure 4 displays the general downward pattern of the contribution rate. The
decline in contribution rate does not translate to a decrease in success rate,49 suggesting
that individuals who give up contributing are mostly from previously failing groups,
whereas members in successful groups keep contributing. This finding is consistent with
Proposition 2 in Appendix C.50, 51
When we compare the difference between the contribution rate and the success rate
by treatment type, we find that differences in the latter are more pronounced (as Pattern
1 suggests), due to the success rate requirement that all four members contribute at the
same time.
5.3 Second Stage Result Highlights
49 Except for the Semi-Gradualism treatment from Period 6 to 7. 50 However, for the Semi-Gradualism treatment, a moderate 15 percentage-point decrease in the contribution
rate from Period 6 to Period 7 translates to a sharp 40 percentage-point drop in the success rate, suggesting
that a large portion of individuals who give up contributing in Period 7 come from previously successful
groups: An unanticipated big jump in the stake makes some subjects in previously successful groups unwilling
to continue contributing. Previously established coordination established at low-stake periods gets sabotaged
even if only one of the four groups members stops contributing. 51 Appendix Figure A.1 confirms this: Among the eight Semi-Gradualism groups in which the number of
contributors decreases from Period 6 to 7, seven groups (groups no. 32, 52, 112, 142, 172 and 182) are
successful in Period 6 but fail in Period 7 due to one or two “betrayers” in each group, while only one group
(group no. 82) already fails in Period 6.
33
In Table 2, we examine whether the treatment type in the first stage influences
individual behavior and outcomes in the second stage when subjects are placed in a new
group. Note that everyone knows that the new group members may have been exposed to
other stake paths in the first stage (but they do not know the exact stake paths).
34
Table 2: Contribution and Earnings in Each Period of the Second Stage by Treatment
Notes: OLS regression results are reported; when the dependent variable is contribution or success (binary variables), probit regressions have similar results
(available upon request). The default category is three non-Gradualism treatments all together: Big Bang, Semi-Gradualism, and High Show-up Fee; when
we separate these three treatments, the results (available upon request) are similar and we do not detect significant differences among these three treatments.
Robust standard errors in parentheses. Standard errors are all clustered at group level. * significant at 10%; ** significant at 5%; *** significant at 1%.
35
Main Result 2: Individuals exposed to the Gradualism treatment in
Stage 1 are more likely to contribute in Stage 2 (with a new group) than
individuals who were previously exposed to any other treatment type.
Prediction 4 underpins the finding in Main Result 2. We pool all three non-
Gradualism treatments together (Big Bang, Semi-Gradualism, and High Show-up Fee) and
define them as the default category. Therefore, we interpret coefficients on the Gradualism
variable as the difference in contribution rate between the Gradualism treatment and the
other three treatments lumped together.52
As Column 1 shows, individuals in the Gradualism treatment group are 12.2
percentage points (86.1 vs. 73.9; p <0.05) more likely to contribute in the first period of
the second stage.53 We examine if treatment type in the first stage influences the likelihood
of contributing in the second stage. We estimate an OLS regression of the contribution
rate in the first period of Stage 2 on the coordination outcome from the last period of
Stage 1 in Column 2. The results confirm our hypothesis: For individuals who fail to
coordinate in the last period of the first stage, about two-thirds contribute in the first
period of Stage 2. All individuals who belong to a successful group in the last period of
52 When we separate the other three treatments, the results (available upon request) are similar, and we do not
detect significant differences among the other three treatments. 53 The contribution rates for subjects from each treatment are all much higher than those in the last period of
Stage 1, and close to those in the first period of Stage 1, which suggests a restart effect observed in the
literature (see Andreoni, 1988), although in our experiment the restart is anticipated and subjects enter a new
group in Stage 2.
36
Stage 1 contribute in the first period of Stage 2. The difference in the contribution rate in
Period 1 of Stage 2 by success type in the last period of Stage 1 is 35.4 percentage points
and highly statistically significant. In Column 3, we report results with a binary control for
the treatment type. The coefficient on the variable “Success in Previous Period” does not
change, although the coefficient for the treatment dummy becomes insignificant (as
compared with Column 1). This result suggests that the treatment regimen in the first
stage of the experiment influences the contribution rate of individuals once they enter the
first period of Stage 2, mostly through individuals belonging to a successful group in the
last period of Stage 1.
We note that the higher contribution rate of individuals exposed to Gradualism
treatment at the beginning of Stage 2 translates into neither a higher success rate (Column
4) nor higher average earnings (Column 5). On average, subjects in the Gradualism
treatment group earn 1.5 points less than subjects in other treatments (the difference is
statistically insignificant).
In addition, the decrease in contribution rate between Periods 1 and 2 (in the
second stage) for Gradualism subjects is faster than the decrease in contribution rate for
subjects in other treatment groups. The contribution rate of Gradualism subjects in Period
2 of the second stage becomes comparable to the contribution rate of subjects from other
treatment groups and remains comparable throughout Stage 2 (as Columns 6 and 7 show).
This convergence of behavior suggests a possible learning process: Individuals from the
37
Gradualism treatment group realize that their new group partners are not as cooperative
as the ones they partnered with in the first stage of the game, and therefore initial
Gradualism treatment subjects become less cooperative in subsequent periods than they
were in Stage 1 of the game. This learning behavior suggests a potential externality of
coordination building (or collapse) across different social groups.
6 Conclusion
This paper analyzes the effect of gradualism, defined as increasing the stake level
required for group coordination by steps, on successful group cooperation using data from
a randomized experiment in China.54 No previous study has identified what pattern of
successively and exogenously set threshold levels yields cooperation most successfully
among individuals.
Through a framed lab experiment, we find strong evidence that gradualism can
serve as a powerful mechanism for achieving socially optimal outcomes in group
coordination. We show that gradualism significantly outperforms alternative paths to
coordinated behavior. A striking result is that the Semi-Gradualism treatment fails, in
comparison to the Gradualism treatment, to foster high group coordination. That the
Gradualism treatment group outperforms other treatment group shows that starting at a
54 Although we acknowledge that cooperative behavior may relate to the Chinese context, Oosterbeek, Sloof,
and van de Kuilen (2004) find no geographic or country differences in responders’ behavior in trust games.
38
low stake level requirement for group coordination and slowly growing the stake size are
both important for coordination at a later stage and at a higher stake level.
We also find an externality of coordination building (or collapse) across treatment
groups: Individuals treated in the gradualism setting are more likely to cooperate upon
entering a new environment than those treated differently. However, this cooperative effect
is due to having been in the Gradualism treatment group previously, and it quickly
dissipates because the new group exhibits very low successful cooperation.
We note that we focus only on certain properties of what we call gradualism.
Future studies can improve and extend this study in various ways: allowing
communication, adopting a non-weakest-link payoff structure (e.g., allowing free-riding),
changing the group size and the highest stake level, and adopting a different information
structure and a more complex dynamic path of stakes, are all important dimensions that
could enhance our understanding of how these features influence cooperative behavior. An
interesting future research contribution could examine whether gradualism helps rebuild
coordination once it has collapsed; whether an end-of-game effect55 exists, and whether
such an effect enhances the role of gradualism for groups attempting to coordinate
successfully.
55 The end-of-game effect is the phenomenon of individuals interacting in a finite rounds contribution game
who often start out by contributing substantial amounts that decline as the number of rounds increases,
reaching their minimum toward the end of the game.
39
Compared to previous studies on coordination games, we highlight a case in which
limiting individual choices can improve social welfare. Previous studies on coordination
games generally feature a complex payoff structure in which each individual has as many
as seven choices for actions in each period. These studies' findings highlight that after
several periods, groups reach an inefficient outcome. Thereafter, groups in these studies
remain trapped in this state of low cooperation.
This study makes two important contributions. First, the paper contributes to the
coordination literature by examining how stake path dependence, as opposed to another
contextual factor, fosters successful group coordination. Second, the paper provides
evidence on how to foster group coordination in a context in which a social planner does
not know how to structure optimal incentives for each player.
As we suggest in this study, if a social planner limits subject choices in each period
(but without mandatory or semi-mandatory institutions, such as sanction and social
pressure), and designs an appropriate institutional threshold path, he can induce subjects
to reach a socially optimal outcome.56 However, our studies cannot address the issue of
56 In the Gradualism treatment of our experiment, we limit the number of choices in each period to two (i.e.,
each player gets to choose if he wishes to contribute the exact stake point in each period or not to contribute).
Because our experiment involves interactions with other players and therefore generating more obvious
externality of own action on others, it should not be surprising that limiting individual choices is good for
social welfare. What is worth noting in our findings, however, is that individuals are still free to choose
between two options in each period (i.e., they are not forced to cooperate), and no sanction, punishment, or
social pressure exists.
40
what constitutes an optimal path to attain a long-run objective, a research question that
future studies could address.
The results in this paper may have policy implications that go beyond the
particular case of building coordination within small groups. Because we quantify large
positive effects of gradualism on successful coordination, our results can be used to inform
policies on how to promote coordination among individuals, organizations, regions, and
countries.
41
Appendix A: Supplemental Data
Figures
Figure A.1: All Group Coordination Results for Each Treatment
A: “Big Bang” Groups (each subgraph indicates a “Big Bang” group)
adaptive dynamics (e.g., Crawford, 1995; Van Huyck et al., 1997; Weber, 2006). Second,
by assuming myopia, we can focus on belief updating as the key feature influencing the
individual decision to cooperate.
In this model, we assume N periods. In each period, a group of I players gets to interact
towards a coordination task. Each player has to choose either to participate (i.e.,
cooperate, “C”) or not to participate (i.e., deviate, “D”). Each person has an initial point
endowment E.
Each coordination task has a pre-determined stake, ( 0)t tS S > during each game period.
During each period, players are asked to contribute the period’s stake level. The stake
level may vary across game periods. In each period, each player has to choose whether to
contribute zero or to contribute exactly tS (no other contribution amount is allowed).
We adopt a minimum-effort (also referred to as “weakest-link”) payoff structure. With a
minimum-effort payoff structure, the value of the project output for everyone is tSα (
1α > ) if all I players contribute tS , and zero otherwise. Therefore, for each player i (in
period t) player payoff is as follows:
54
=≠∃=−=
≠∀==−+=
DAtsijandCAifSE
DAifE
ijCAandCAifSE
Earnings
tjtit
ti
tjtit
it
,,
,
,,
..,
,)1(α
Each player does not know other players’ actions when he or she makes a contribution
decision. At the end of each period, however all players are told whether all members in
the group have contributed the pre-defined stake in that period.
The essential feature in this model is level-k thinking. Central to the model is the
assumption that level-0 players are non-strategic while level-1 players best respond to
level-0 players, level-2 players best respond to level-1 players and so on. Generally, level-k
players best respond to level-(k-1) players for any 1k ≥ (e.g., Costa-Gomes et al., 2001;
Costa-Gomes & Crawford, 2006; Costa-Gomes et al., 2009). A level-k player views all his
opponents as level-(k-1) players.58,59
We assume that a level-0 player has a constant “willingness-to-contribute”, the amount he
would have liked to contribute towards to the common pool had there not been a binary
constraint (set by the stake level for each player’s decision in each period).60 Qualitative
interviews conducted upon completion of the study reveal that some players do adopt such
a rule (e.g., some individuals indicated that they would contribute towards the common
pool “as long as the period stake is 8 points, or 10 points, 12 points or pre-determined
number of points in their mind, etc.”)
The definition of a level-0 player varies in the empirical literature: some studies assume
level-0 players randomize over available actions, while other studies assume that level-0
players play according to some constant. In an auction game, Crawford and Iriberri (2007)
allow the coexistence of these two types of level-0 players, namely, random level-0 players
58 For example, a level-1 player views all his opponents as level-0 players; A level-2 player views all his
opponents as level-1 players. 59 In cognitive hierarchy models (e.g., Camerer et al., 2004), level-2 players best respond not to level-1 players
alone but to a mixture of level-0 and level-1 players. The difference in defining what level-k players best
respond to will not affect our general theoretical predictions (Propositions 1-3). 60 The assumption that each player has a constant “willingness-to-contribute” is a natural extension to a case
where each player has a continuously defined “willingness-to-contribute”.
55
and truthful level-0 players, respectively.61, 62 Differences in the definition of what
constitutes a level-0 player do not influence our theoretical predictions.
From the standpoint of a level-0 player the decision to contribute is based on the following
rule: if his willingness-to-contribute is larger than or equal to the period stake (“high type”
contributor), then he intends to contribute; otherwise he does not contribute (“low type”
contributor). Note that because stakes change across periods, this particular definition of
“high” and “low” types only applies to a given period.63
Because a player could make a mistake based on his or her decision rule, we introduce
some noise to level-0 players’ actions.64 With this caveat, a “high” contributes with a
probability of (1 ε− ) and does not contribute with a probability of ε .65 Similarly, a “low”
type contributes with a probability of η and he does not contribute with a probability of (
1 η− ).66 Because the “high” type is more likely to contribute than the “low” type, we
assume 1 ε η− > . We assume that level-1 players know (or believe) the decision rule of
level-0 players. Similarly, we assume that level-2 players know that level-1 players know
the decision rule of level-0 players. In general, we assume that higher level types know the
strategic rule of lower-level types and that lower-level types know and assume the same for
even lower-level types.
In this setup, a level-1 player adopts a prior belief about each level-0 player’s “willingness-
to-contribute” level. Each level-1 player chooses his or her best response according to his
61 In addition, the empirical literature provides no consensus exists on whether level-0 players really exist or are
just a convenient assumption on how level-1 players view other players. 62 There are two other ways to define level-0 players in our game: they are either driven by a random (rather
than a constant one) but continuously defined “willingness-to-contribute” and they decide to contribute (or
not) according to the comparison of its realization and the current stake, or they randomize over the binary
actions per se (as defined in Ho & Su, 2013 for centipede games). Under these alternative definitions, because
of the weakest-link payoff structure, level-1 players will intend to contribute as long as the stake is lower than
a critical value which depends on her risk attitude and her belief about the possibility that level-0 players will
contribute. Thus level-1 players behave like level-0 players defined in our original model, level-2 players behave
like level-1 players defined in our original model, and so on. This will not change our main theoretical
predictions. Detailed proofs are available upon request. 63 A “high” type for a given stake may not necessarily be a “high” type for a different-level stake. 64 This setup is similar to Fudenburg et al. (2011) on the experimental play of repeated prisoners’ dilemma
when intended actions are implemented with exogenous noises. By introducing errors by level-0 players, we
allow level-1 players to play “leniently”. In other words, they may not necessarily retaliate for the first
defection of others. This does not seem to matter for our main experimental results, but can help explain a few
cases when a group can switch between coordination failure and success. 65 Because ε is a probability of an event, 0<ε<1. 66 Because η is a probability of an event, 0<η<1.
56
or her belief regarding each level-0 player’s “willingness-to-contribute” level. After
observing each period’s outcome, each level-1 player updates his belief about the
“willingness-to-contribute” levels of level-0 players. Analogously, higher-level players
update their beliefs according to the same rule.
All I players are risk averse. A level-1 player i’s belief about a level-0 player j’s
willingness-to-contribute isijB , which follows a cumulative distribution function , ()i
j tF ,
j i≠ .
θt(St) denotes the “reserved probability of successful group coordination.” During a
period t, a level-1 (or any level-k player for 1k ≥ ) player i will contribute if and only if his
belief, regarding whether each of his group members67 will contribute, exceeds θt(St).
Assuming each player is risk averse, we can prove the following lemma:
Lemma 1. With θt(St) denoting the reserved probability of successful group coordination
between 0 and 1, the higher the stake, the higher the reserved probability of success (i.e.,
0 ( ) 1i tSθ< < , and ( ) / 0i t tS Sθ∂ ∂ > ).
Proof: Let 1 0β α= − > . Ui(·) represents person i’s utility function and because we
assume each person is risk averse, we know ' ''( ) 0, ( ) 0i iU x U x> < . iθ denotes the
minimum probability of success, which induces person i to contribute. Therefore, we
obtain ( ) ( ) (1 ) ( )i i i t i i tU E U E S U E Sθ β θ= + + − − .
Rearranging for θt, we obtain:
67A level-1 player views all his opponents as level-0 players; similarly, level-2 player views all his opponents as
level-1 players.
( ) ( )
( ) ( )i i t
i
i t i t
U E U E S
U E S U E Sθ
β− −
=+ − −
57
Partially differentiating with respect to St, we obtain:
From ' ( ) 0 ( ) ( ) ( ) ( ) 0 0 1i i t i t i i t iU x U E S U E S U E U E Sβ θ> ⇒ + − − > − − > ⇒ < <
We know that B>0. Therefore, for us to prove that ( ) / 0i t tS Sθ∂ ∂ > , we just need to show
that A>0.
From the above expression for A, we have:
' ' '
2
( )[ ( ) ( )] [ ( ) ( )][ ( ) ( )]/
[ ( ) ( )]
i t i t i t i i t i t i ti t
i t i t
U E S U E S U E S U E U E S U E S U E SS
U E S U E S
A
B
β β βθβ
− + − − − − − + + −∂ ∂ =
+ − −
≡
' ' ' '
' '
' '
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) [ ( ) ( )] ( )[ ( ) ( )]
i t i t i t i t i i t i t i t
i i t i t i t
i t i t i i t i i t
A U E S U E S U E S U E S U E U E S U E S U E S
U E U E S U E S U E S
U E S U E S U E U E S U E U E S
β β β β β
β β β
= − + − − − − + + − +
− − + − −
= − + − − + − −
''
' ' ' '
( )( )( ) ( ) ( ) ( )0 (B.1)
( ) ( ) ( ) ( )
t
t
EE S
ii E Si t i i i t E
i t i t i t i t
U x dxU x dxU E S U E U E U E SA
U E S U E S U E S U E S
β
ββ β β β
+
−+ − − −> ⇐ > ⇐ >
+ − + −∫∫
58
Because (B.1) holds, A must be positive. Therefore, ( ) / 0i t tS Sθ∂ ∂ > .
Q.E.D.
Intuitively, the higher the pre-defined stake level for each game period, the more the
“reserved probability of success” that induces one to contribute jumps.
Assuming that player i’s beliefs about all his opponents’ types (i.e., “high or “low”) are
independently distributed, a level-1 player i, during period t, will contribute if and only if:
, ,[(1 ) (1 )] ( )i ij t j t i t
j i
s s Sε η θ≠
− + − ≥∏ (B.2)
Where , ,Pr ( ) 1 ( )i i ij t j t j t ts ob B S F S= ≥ = − is player i’s belief, during period t, that a
level-0 player j is a “high" type.
Proposition 1. The lower the 1S (i.e., the stake level at t=1), the higher the probability
that coordination at t=1 will succeed is.
'' ' '
' '
( ) 0 ( ) ( ) for all [ , ),
( ) ( ) for all ( , ].i i i t t
i i t t
U x U x U E S x E E S and
U x U E S x E S E
β β< ⇒ > + ∈ +< − ∈ −
' '
' '
''
' '
( ) ( )LHS of (B.1)
( ) ( )
( ) ( )RHS of (B.1)
( ) ( )
t
t
E S
i t t i tEt
i t i t
E
i tE S t i t
t
i t i t
U E S dx S U E SS
U E S U E S
U E S dx S U E SS
U E S U E S
ββ β β
β β β β
+
−
+ +> = =+ +
− −< = =− −
∫
∫
59
Proof:
Because ,1 1 ,1 1 1/ ( ) / 0i ij js S F S S∂ ∂ = −∂ ∂ ≤ , 1 0ε η− − > and , ,(1 ) (1 ) 0i i
j t j ts sε η− + − ≥ , we
obtain , , 1[(1 ) (1 )] / 0i ij t j t
j i
s s Sε η≠
∂ − + − ∂ ≤∏ .
However, 1 1( ) / 0i S Sθ∂ ∂ ≥ . According to (B.2) above, the lower the 1S is, the higher the
probability that the LHS of (B.2) will be larger than the RHS (i.e., the higher the
probability that a level-1 player i will contribute during period t=1.)
A given level-2 player k will contribute towards the common pool if and only if he believes
that the probability that all his opponents68 will contribute at least ( )k tSθ (i.e., his
“reserved probability of success.”)
A level-2 player knows the strategic rule that dictates the contribution rule for all level-1
players. Therefore, during period t=1, a level-2 player believes that the lower the 1S is, the
higher the probability that a level-1 player i will contribute towards the common pool.
Correspondingly, a level-k (k>2) player uses a similar rationale: the lower the 1S is, the
higher the probability that he will contribute during period t=1.
The probability that a given level-0 player j will contribute is
(1 )1( ) 1( )j t j tB S B Sε η− ≥ + < , where l(●) equals one if the argument in the parenthesis is
true, and zero otherwise. jB is his willingness-to-contribute. Because we assume that
(1 )ε η− > , the lower 1S is, the higher the probability that a level-0 player j will
contribute during period t=1 is.
As we show above, the lower the stake 1S at t=1 is, the higher the probability that
any level-0, level-1 and level-k (k ≥2) player will contribute is (in other words, the higher
the probability that coordination at t=1 will succeed.)
68 A given level-2 player views all his opponents as level-1 players.
60
Q.E.D.
Proposition 1 suggests that successful coordination is more likely to occur the lower
the stake levels are.
Upon observing the outcome of group coordination at the end of each period, a level-1
player i updates his beliefs regarding cooperative likelihood of his group members.69,70,71
Using the Bayesian rule72, he updates his posterior beliefs about the probability that his
opponent, player j , is a “high” type according to:
, , , , ,Pr ( | ) / [ (1 )(1 )]i i i i ij t j t j t j t j t j th ob B S A D s s sε ε η= ≥ = = + − −
During period t+1, if the stake is still tS , then player i believes that player j will
contribute towards the common pool that the probability , ,(1 ) (1 )i ij t j th hε η− + − .
If player i observes that player j contributes during period t, then player i’ posterior belief
that player j is a “high” type is:
, , , , ,Pr ( | ) (1 ) / [ (1 ) (1 ) ]i i i i ij t j t j t j t j t j tk ob B S A C s s sε ε η= ≥ = = − − + −
During period t+1, if the stake is still tS , then player i believes that player j will
contribute towards the common pool with probability , ,(1 ) (1 )i ij t j tk kε η− + − .
69 In this study, player i may not necessarily observe the actions of j after each period but gets to observe the
outcome. 70 In this study, player i may not necessarily observe the actions of j after each period but gets to observe the
outcome. 71 Player i views a given opponent j is a level-0 player. 72 If we rule out the possibility of players making mistakes (i.e., let ε=η=0), the Bayesian updating process
will degenerate. Once a level-1 player i observes that his opponent j does (not) contribute, then he believes
that player j is a “high” (“low”) type with a probability one.
61
We can show that , ,i ij t j tk h≥ and , , , ,(1 ) (1 ) (1 ) (1 )i i i i
j t j t j t j tk k h hε η ε η− + − ≥ − + − provided
that 1 ε η− > .
Intuitively, given that the probability the a “high” type player will contribute is higher
than the probability that a “low” type player will contribute, then the posterior
probability that player j is a “high” type goes up by more if we observe that player j
contributes than if we observe that player j does not contribute.73
Assuming that, at period t+1, player j’s actions are independently distributed,
player i will believe all players j will contribute at period t+1 when the stake is 1t tS S+ =
with probability:
, , , ,{[ 1( ) 1( )](1 ) }i ij t j t j t j t
j i
h A D k A C ε η η≠
= + = − − +∏
l(●) equals one if the argument in the parenthesis is true, and zero otherwise. We assume
that player i observes m (0 1m I≤ ≤ − ) number of contributing opponents74 during period
t. , ,i ij t j tk h≥ and 1 ε η− > . If the number of players m is larger, then player i will believe
that all opponents will contribute at period t+1 (when 1t tS S+ = ) with a higher
probability. As a result, the probability that player i will contribute at during period t+1
is also higher.
A level-2 player k knows what rules dictate the decision of level-1 players to contribute and
how level-1 players update their beliefs. Therefore, if the number of contributing players
m is larger, then player k believes that all players i will contribute during period t+1
(when 1t tS S+ = ) with a higher probability. As a result, the probability that player k will
contribute at t+1 is also higher. Analogously, a level-3 player knows what strategic rule
level-2 players use when level-2 players decide whether to contribute or not. Therefore, if
the number of contributing players m is larger, then the probability that a level-3 player
73 This also applies to the probability that a player j will contribute during the next period when he faces the
same stake level. 74 Player i views all opponents as level-0 players.
62
will contribute during period t+1 is higher. This process continues analogously and
iteratively for players of higher level.
A level-0 player j will always contribute with a probability (1 )1( ) 1( )j t j tB S B Sε η− ≥ + < .
This probability does not depend on m.
When players get to interact during period t within a group of (m) number of
contributors, all players facing a stake level tS , then the following statement holds true:
If the number of contributing players m is larger, then all players will contribute during
period t+1 (if 1t tS S+ = ) with a higher probability. As a result, the probability that
coordination will succeed during period t+1 is higher if 1t tS S+ = . We can formalize the
last statement in the proposition below.
Proposition 2. When players are informed that they will get to interact with (m)
number of contributing players during period t (facing a stake level tS ) then the following
statement holds: the larger the number of contributing players (m) is, then coordination
during period t+1 will succeed with a higher the probability if 1t tS S+ = .
Proof: See above.
Because of the information structure in our experiment, if a group successfully coordinates,
then all contributors know that all group members have contributed. If at least one
member fails to contribute, then contributors only know that not all group members have
contributed. However, they do not get to know the exact number of other contributors.75
Therefore, following Proposition 2, a successful group is more likely to maintain its success
in the next period than a previously failed group if the stake level does not change. If
75 When all members in a group do not contribute, then everyone only knows that their group has failed
cooperating. They do not know how many or who among the other group members have (or have not)
contributed. In this case, a rational player cannot update his beliefs about his opponents. Therefore, his actions
will not change in the next period if he faces same stake level. As a result, a group that has no contributors
will tend to fail cooperating during the next period.
63
players do not make mistakes to their decision rule on whether to contribute or not (i.e.,
0ε η= = ), then once a rational player observes that (not) all his opponents contribute
when faced with a stake level tS , he will (never) contribute in the next period when the
stake level is 1t tS S+ = . As a result, once a group succeeds (fails), it will succeed (fail)
during the next period with the same stake level. Barring a few exceptions, the results of
this study depict this general pattern. If we introduce the possibility that players make
mistakes to their decision rule on whether to contribute or not (as previously described in
this model), then the model will explain not only the general patterns, but also the
exceptions we note.
With the proposition below, we show that, in regards to coordination, a gradual increase in
the stake level can never be worse than a sudden increase in the stake levels.
Proposition 3. No matter whether a group succeeds or fails during period t when the
group faces a stake level tS , when the stake level 1( )t tS S+ ≥ is lower, the probability that
the group will succeed coordinating during period t+1 is (weakly) higher.
Proof: See the proof of Proposition 1 and replace t=1 with t = t +1.
In summary, Proposition 3 simply suggests that a gradual increase in the stake level can
never be worse, in reference to group coordination, than a sudden increase in the stake
levels.
64
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