Discussion Paper No. 826

SEEKING HARMONY AMIDST DIVERSITY:CONSENSUS BUILDINGWITH NETWORK EXTERNALITIES

Chia-Hui Chen
Junichiro Ishida

December 2011

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Seeking Harmony Amidst Diversity: Consensus
Building with Network Externalities
Chia-Hui Chen∗and Junichiro Ishida†
December 16, 2011

Abstract
A group of individuals, with a potential conflict of interest, face a choice amongalternatives. There is a network externality such that the chosen alternative yields value
only if sufficiently many individuals get on board. Their preferences for each alternative
and the benefit derived from a successfully formed network are known only privately andmight vary between the players who determine whether to make their choices early orlate. We characterize the equilibrium timing of adoption as well as the efficient timing
which maximizes the total expected payoff. We also show that the efficient timing ofadoption can be implemented by a simple fee scheme. The analysis gives an insight into
why consensus is often hard-won in some societies and suggests a potential role of social
norms in improving the efficiency.

∗†

Institute of Economics, Academia Sinica. Email: chchen@econ.sinica.edu.tw
Institute of Social and Economic Research, Osaka University. Email: jishida@iser.osaka-u.ac.jp1

1

Introduction

A group of individuals, with a potential conflict of interest, face a choice among alternatives.
There is a network externality such that the chosen alternative yields value only if sufficiently
many individuals get on board. The group members thus need to cooperate, to some extent, toreach consensus among them to realize greater value. Consensus is often hard-won, however,
because there are no coordination devices or central planners to sort out differences in opinions
in many cases; as a result, consensus often needs to be built in a highly decentralized manner.
With this decentralized nature, coordination failures become a real concern to be reckoned
with. One potential problem is that too many individuals may take a stance early on, inorder to sway others and have things in their own favor. On the flip side, it could equally beproblematic if too many of them abstain and wait for others to take a stance. In either case,when the right person does not step up at the right timing, we may fail to reach consensus
swiftly enough to fully reap the benefit of a network.
This type of problem abounds in many facets of our life. For instance, firms often need tochoose a technology or a platform among possible alternatives. In many cases, in the presenceof network externalities, only one of them will survive in the end as the standard technology:
a classic example of this case is the competition of the videotape format between Sony’s
Betamax and JVC’s VHS; more recent examples include the choice among the operating
systems, wireless communication protocols, electric payment technologies, and many others.
Consumers also face a similar coordination problem when they need to purchase network
goods, such as telecommunication devices, personal computers (operating systems), social
networking services, and video game consoles, whose value depends on and increases with
the number of users. In a broader perspective, the problem can also arise in almost any
circumstances in which individuals hold different opinions to various extents, but benefit fromchoosing the same side as others, e.g., when politicians or countries hold different politicalstances but want to join a larger coalition (political parties, international treaties, free trade
agreements, etc) to achieve greater benefit,1 or when committee members rank candidates
differently but would like to vote for the winner for some reasons.
The main purpose of the paper is to provide a synthetic illustration of the nature ofinefficiency that could arise in this type of environment and a potential remedy for that. The
1

Free trade agreements are often initiated by a small group of countries which set the tone for the basic
framework and, if successful, later followed by other countries. For instance, the current European Union
was initiated by six countries (Belgium, France, Italy, Luxembourg, the Netherlands and West Germany) as
the European Coal and Steel Community, which have subsequently expanded and developed into its current
form through several expansion phases. As a more recent example, the Trans-Pacific Partnership (TPP) was
initiated by four countries (Chile, New Zealand, Singapore and Brunei) but has now expanded to include the
US and other countries.

2

tradeoff of our focus is simple: by taking a stance and making his intention clearer earlyon, one may attract more people to take his side, but loses chance to learn of what others
think. To illuminate this tradeoff in the simplest manner, we consider a two-period problem
faced by two individuals who must choose either one of the two alternatives, which we call“projects” for expositional clarity. In each period, the two individuals simultaneously choose
whether to adopt either project or to wait and see the other’s decision. A decision to takea side entails some fixed cost and is hence irreversible once it is made. They can reap the
benefits of a network when both choose to work on the same project, but all the potential
benefits are lost when they fail to coordinate their decisions.
The problem is compounded by the differences in their preferences. We assume thateach individual’s preferences, which are only privately known, could differ in two dimensions.
In one dimension, individuals are heterogenous with respect to their intrinsic tastes for the
projects: some individuals have a clear-cut stand on what they like and dislike (the typewith strong preferences), while others are more accommodating and indifferent between the
projects (the type with weak preferences). In the other, individuals also differ in the extent
to which they can benefit from a successfully formed network: some derive a large marginal
benefit from a network (the dependent type), while others do not derive much from it (theindependent type). We assume that the type in each dimension is independent of that in theother, so that there are four different types of individual to be considered. The differencesin preferences amount to the potential conflict of interest which makes the model highly
strategic in nature.
We first show that, perhaps quite intuitively, the types with strong preferences adopt
earlier than those with weak preferences in equilibrium. This is because a player with strong
preferences can induce the other player to follow his decision by letting his intention known
and thereby increasing the probability that his preferred project will be chosen. Between the
two types with strong preferences, the dependent type is less inclined to adopt early because
he derives a larger benefit from reaching a consensus, and so acts more cautiously to ensurethat the cooperation will eventually emerge. This equilibrium allocation is, however, often
not efficient, in that it does not maximize the expected total payoff, because each player
considers only his own payoff and ignores the fact that his timing of adoption will also affectthe other player’s payoff.
To explore welfare implications of the model, we proceed to characterize the efficient
allocation, which turns out to be far more complicated than the equilibrium allocation evenin a simple setup like this. To characterize the efficient allocation, there are two distinct
cases we need to consider, depending on the difference in the values of a network between the3dependent and independent types. When the difference is small, the important dimension iswhich project to work on as a group. It is then efficient to let the types with strong preferences
adopt first, followed by those with weak preferences. While this ordering could emerge inequilibrium, the equilibrium allocation can still be inefficient because it does not realize theright allocation of types between the periods, i.e., the types with strong preferences adopt
in the first period either too much or too little. The inefficiency is exacerbated even morewhen the players are more heterogenous in productivity. In this case, the efficient allocation
should assure to have the dependent types in the network, regardless of the strength ofpreferences. The way to achieve this is either to let the dependent types adopt first and thenthe independent types follow, or vice versa. As stated, though, this ordering never occurs inequilibrium because the types with strong preferences always adopt earlier.
Given that the efficient allocation cannot often be sustained in equilibrium, we thenexplore ways in which to restore the efficient allocation. We obtain two results on this.
First, we show that the (constrained) efficient allocation can be achieved by imposing fees on
individuals adopting in different periods, without resorting to more complicated mechanisms.
The appropriately designed fee scheme, which reallocates the costs and benefits among thetypes, corrects not only which type of individual will step forward to be the leader but also
the probability that each type will step forward. We argue that this simple property is criticalgiven the spirit of the problem that consensus is often built in a decentralized manner: evenwhen an almighty central planner can achieve a better allocation by designing an incentive
compatible mechanism and implementing it in a more complex and dictatorial manner, suchan option is rarely available or highly costly in the class of problems under consideration.2
In contrast, our solution suggests that the efficiency can be restored simply by placing moreor less burden on those who speak up first, which is supposedly much easier to implement.
As one possibility, social norms may play an important role in reallocating the costs between
the leader and the follower.
Second, the current analysis also clarifies when it is more desirable to place an additional
cost on the leader. We find that it is welfare-improving to impose a positive fee on the leader
either when the players are relatively homogeneous in productivity or when they are diverse
in both productivity and preferences. This result suggests that the norm which shifts moreburden to the leader works well in a wide range of circumstances, thereby lending support
to a prevalent social norm that leaders should assume more responsibility and accept the2

Our proposed fee scheme achieves a constrained optimum as it only allocates the right proportion oftypes to each period but does not resolve the coordination problem. In this sense, the central planner can dobetter by designing an appropriate direct mechanism and assigns the players to each period in a way to avoid
coordination failures.

4

burden of accountability. Our analysis reveals, however, that this type of social norm does
not always improve welfare. When the players are diverse only in productivity but have morehomogeneous preferences, they are reluctant to speak up and tend to abstain too often in
equilibrium. In this case, it is desirable to give an extra reward to the leader to encourage
more people to step up to lead the way.
Our analysis is closely related to studies on voting where the timing of expressing one’s
opinion could be decisive. In the literature, considerable attention has been devoted to
studying how simultaneous voting and sequential voting affect electoral outcomes (see e.g.
Dekel and Piccione (2000), Battaglini (2005), Battaglini, Morton, and Palfrey (2007)). In
particular, Callander (2007, 2008) considers the situation in which voting is sequential andvoters prefer to vote for a winner.3 The environment is similar to that under consideration
here, but in traditional sequential voting models, voters cast their votes in an exogenously
fixed order, whereas in ours, voters strategically choose when to cast their votes.4
The timing of expressing one’s opinion is also critical in the legislative branch of government. In the US Congress, for instance, legislators need to determine not only how to vote,
but also when to announce their position on the issue during the weeks or months before
the final vote. This type of strategic interaction is obviously not a unique phenomenon in
the US, as it can be observed virtually in any legislative process. Legislators are generally
concerned with the opinions of constituents, interest groups, and party leaders, and thus
need to weigh the costs and benefits when they decide when to make their opinions clear.
Box-Steffensmier, Arnold, and Zorn (1997) and Caldeira and Zorn (2004) provide evidence
showing that legislators whose constituencies, interest group influences, and partisanship arein alignment announce their positions earlier, and those with conflicting or ambiguous signals
delay their announcements. Provided that the former case of aligned interests corresponds
to the types with strong preferences, we argue that this observation is in line with our main
results.
This paper is also related to the literature on the diffusion of technologies with network
externalities. Farrell and Saloner (1986) and Farrell and Shapiro (1993) consider the timingthat different groups of people choose to switch from an old technology to a new technology.
Farrell and Shapiro (1993) further study the situation in which different groups initially use
incompatible technologies and analyze which group switches first in equilibrium. This paper
3

Callander shows that a desire to conform with the majority induces momentum and bandwagons, two
phenomena commonly observed in real sequential elections.
4
Ottaviani and Sorensen (2001) consider decision making in a committee consisting of members who wish
to appear well informed and analyze the optimal order of speech (who to speak up first). In this work, however,
the order is also exogenously imposed form outside.

5

goes one step further by examining how users with different characteristics choose between
different new technologies and the timing to adopt them.
The remainder of this paper is organized as follows. Section 2 describes the model. Section
3 characterizes the equilibrium timing of adoption. Section 4 characterizes the efficient timingof adoption and makes a point that the equilibrium timing is often inefficient due to network
externalities. Section 5 shows that one can restore the efficient outcome by a simple fee
scheme. Finally, Section 6 concludes.

2

Model

Environment: Two risk-neutral players, denoted by i = 1, 2, are asked to make a choice
between two projects, a and b, over two periods. In each period, each player chooses his
stance si,t ∈ St where S1 = {a, b, A} and S2 = {a, b}. For expositional purposes, we say thata player “adopts” when either si,t = a or si,t = b, while he “abstains” when si,t = A.5 The
decision is irreversible once a player adopts in the first period, one way or the other, i.e.,si,1 = si,2 if si,1 = a or si,1 = b. A network is formed successfully and the chosen projectyields value only if s1,2 = s2,2 , i.e., the two players agree to take the same side.
Payoffs: The payoff for player i is given by
ui = [vi − c(s̃i , si,2 )]x,
where x = 1 if a network is formed successfully and x = 0 otherwise. Each player’s preferences are characterized by (vi , s̃i ), which captures two-dimensional preference heterogeneity
in this context. We assume that both vi and s̃i are the player’s private information and notobservable to the other player.
First, the value of a network may vary between the players, where some derive a larger
marginal benefit from a network while others derive a smaller benefit. For expositional purposes, we refer to the former as the “dependent” type and to the latter as the “independent”
type. The difference in the values captures one source of heterogeneity between the players
which yields at least two distinct (and possibly conflicting) ways to interpret, depending on
the underlying context: in one interpretation, the dependent type can be regarded as the one
who needs to rely more on the network, hence the label dependent, whereas the independent
type is the one who is “large” enough by himself and cares less about what others do; in the5

We assume that the option of abstention is available only in the first period. This assumption is madepurely to simplify the exposition and absolutely inessential because each player possesses veto power in thissetup. To be more precise, a player would abstain in the second period only when there is a leader who chosethe project he dislikes in the first period. In this case, however, the player can reject the proposal by simply
choosing the other project.

6other, the dependent type can also be the more productive type who can exploit more fromthe existing network. The probability that a player is dependent is p, i.e.,Prob(vi = vH ) = p, Prob(vi = vL ) = 1 − p.
Second, the players may also differ in the strength of preferences with respect to theproject, where each player’s preferred project is denoted by s̃i ∈ {a, b, n}. A player has a firm
stand on what he likes and dislikes when s̃i = a or s̃i = b (the type with strong preferences),
whereas he is fairly neutral and indifferent between the positions when s̃i = n (the type withweak preferences). The strength of preferences determines the cost of working on the project,wherec(a, a) = c(b, b) = 0, c(n, a) = c(n, b) =d, c(a, b) = c(b, a) = d > 0.
2

That is, a player with strong preferences incurs a disutility of d (zero) when he works on the
project he dislikes (likes); a player with weak preferences incurs a disutility of d2 no matter
which project is chosen.6 We also assume that vL > d2 to ensure that for each type, there
exists a project on which he can cooperate with the other player and derives a positive payoff.
Finally, the probability distribution of each type is given byq
Prob(s̃i = a) = Prob(s̃i = b) = , Prob(s̃i = n) = 1 − q.
2
Each player’s strength of preferences is independent of the benefit derived from a network
vi . There are hence four types of player: the dependent type with strong preferences (DS), thedependent type with weak preferences (DW), the independent type with strong preferences(IS), and the independent type with weak preferences (IW). The distribution of player typeis summarized in the following table.

Dependentp
Independent
1−pStrong preferencesq

Weak preferences
1−qpq(1 − q) p

q (1 − p)(1 − q) (1 − p)Interpretation: The set of parameters (vH , vL , d, p, q) totally characterizes the nature of
the underlying environment. Among them, our primary focus is on two parameters, which6
We specify that c(n, a) = c(n, b) = d2 in order to isolate the unintended “productivity effect”. In a fully
symmetric environment where the project is chosen randomly, the ex ante expected cost becomes independentof the strength of preferences. This property is important when we characterize the efficient allocation.

7

we call the productivity dispersion ∆v := vH − vL and the preference (ideological) dispersion
q. First, the productivity dispersion measures the degree of heterogeneity among the playerswith respect to the productivity,7 which seems to play a critical role in the case of technology
adoption or any other cases in which the players differ greatly in the extent to which they
can benefit from a network.8 On the other hand, the preference dispersion measures how
likely a given player has a certain stance about the direction of the project, one way or theother, and hence the severity of the potential conflict of opinions. The preference dispersion
plays a critical role in political situations, but is likely to be small when the players sharesome goal, such as profit maximization, and care more about the outcome (what is achieved)
rather than the process (how it is achieved). As we will see below, these two parameters
have a profound impact on the socially efficient timing of adoption, thereby yielding critical
welfare and policy implications.
Timing: To sum up, the timing structure of the game is as follows.
1. Nature randomly draws vi and s̃i for i = 1, 2.
2. In the first period, each player simultaneously makes his choice si,1 . The choice isobservable to the other player.
3. In the second period, a player who has abstained in the first period, i.e., si,1 = A makes
his choice si,2 once again.
4. The payoffs are realized.
We seek for symmetric equilibria of this game in which the two players adopt the same
strategy.

3

The Equilibrium Timing of Adoption

This section analyzes the equilibrium timing of adoption. The strategies of this game consistof: (i) the type-contingent probability distribution over the first-period choice; (ii) that overthe second-period choice given the other player’s first period choice. Instead of describing theequilibrium timing in terms of these strategies, which can be quite cumbersome, we employ an
7As mentioned above, however, it is not clear without any concrete context which one (vi = vH or vi = vL )is the more productive type, and the interpretation of this parameter should hence be made with caution.
8
In contrast, we do not place much emphasis on p. Aside from the fact that this parameter does noteasily admit an economically relevant interpretation, its role is qualitatively similar to that of ∆v , though
not completely identical: the productivity dispersion becomes less of a factor as ∆v decreases, which roughly
corresponds to the case where p is close to either zero or one.

8

alternative way of characterizing the timing: any equilibrium of this game can be thoroughly
described by the order of adoption across the types and (r1 , r2 ), where• r1 : the unconditional probability that a player adopts in the first period;• r2 : the unconditional probability that a player adopts and follows the choice made by
the other player in the second period.
For clarity, we refer to the set of the order and the probabilities (r1 , r2 ) as the timingof adoption. The key to this representation is that if one type is indifferent between thetwo choices, then all the other types listed higher (lower) in the order have a strict incentive
to adopt (abstain). Exploiting this feature, the order and the unconditional probabilities(r1 , r2 ) are all we need to characterize the equilibrium allocation which can straightforwardly
be computed from the primitives of the model (see Appendix B for more detail).
We start with the second-period problem, which is meaningful only when at least one
agent chooses to abstain in the first period. There are two cases we need to consider. First,
if both abstain, i.e., s1,1 = s2,1 = A, no additional information is revealed. One can theneasily check that the following constitutes a Bayesian Nash equilibrium of this static game:(i) the types with strong preferences choose si,2 = s̃i ; (ii) the types with weak preferences
randomize between a and b, each with equal probability.9 If only one player adopts in thefirst period, the choice for the other player in the second period is either to follow the choiceor to deviate from it. Player i follows the choice made by the other player if
vi − c(s̃i , s−i,1 ) ≥ 0, i 6= −i.
Given this, we now proceed to the first-period problem, where each player must decide
whether to adopt or to abstain and hopefully wait for the other player to lead the way.
Note first that there is clearly no incentive for a player with strong preferences to choose
the project he dislikes, so that if he adopts in the first period, he always chooses the project
he likes.10 We also suppose that whenever a player with weak preferences chooses to adopt
in the first period, he randomizes and chooses each project with equal probability. It thenfollows from these that the other player is expected to choose each project with probability
r1 /2. If vL − d ≥ 0, which implies that even the independent type finds it profitable to work
9
Since this is effectively a coordination game, there is also a pure-strategy equilibrium where all the indifferent players coordinate on one side. Given the spirit of decentralized consensus building, however, we
assume that such coordination is either impossible or prohibitively costly, so that if a player who is indifferent
between the two projects chooses to take a stance, he has no choice but to randomize and choose each withprobability one half.
10
If a network is formed on the project a player likes, the payoff is vi ; if on the project he dislikes, it isvi − d. Choosing the project he dislikes is hence weakly dominated.

9

on the project he dislikes, then a player waiting until the second period will certainly choose
whichever the project chosen by the other player in the first period, if any, and so r2 = 1 − r1 .
Otherwise, a player waiting until the second period chooses a different project if the project
chosen by the other player in the first period is unfavorable, and hence r2 < 1 − r1 .
Given this and (r1 , r2 ), if a player chooses to adopt in the first period, the probability
that a network forms successfully is r21 + r2 . If a player chooses to abstain, the other player
adopts in the first period with probability r1 and in the second period with the remaining
probability. In each case, the probability that the other player chooses the project player ilikes is one half.11 It then follows that a player with strong preferences chooses to adopt in
the first period only if
r

1

2



+ r2 vi ≥

r1 1 − r1
+
2
2



vi +

r1
max (vi − d, 0) .
2

Similarly, a player with weak preferences chooses to adopt in the first period only if
 



r
r1 1 − r1 r1
d
d
1
+ r2
+
+
≥
vi −
.
vi −
2
2
2
2
2
2

(1)

(2)

Notice that both of the types with weak preferences face the same incentives which depend
only on (r1 , r2 ).
We denote the equilibrium values of (r1 , r2 ) by (r1∗ , r2∗ ). The following two lemmas proveseveral properties of (r1∗ , r2∗ ) which will be useful for the subsequent analysis. Lemma 1 showsthat if a player chooses to abstain, then he will always follow the choice made by the otheragent in the first period, if any, so r1∗ + r2∗ = 1. It is thus implied that if working on the
project he dislikes yields a negative payoff to a player, then in equilibrium, he never abstains
in the first period.
Lemma 1 In equilibrium, r1∗ + r2∗ = 1.
Proof: If r1∗ + r2∗ < 1, it is implied that among those who abstain in the first period, there
exists some type that always chooses the project he likes no matter which project the otherplayer chooses in the first period. This type cannot be the types with weak preferences
because it is always optimal for these types to agree with the choice made in the first period
when vL ≥ d2 . Moreover, we can show that it cannot be the types with strong preferences
either. To see this, suppose that, for a type with vi and with strong preferences‘, it is optimal
11

Note that among the types with a strong preference, one half of them prefers project a and the otherhalf prefers b. Also, by assumption, the types with a weak preference randomize and choose each projectwith equal probability. Therefore, for any type distribution, the probability that the other player chooses the
project player i likes is one half.

10

to wait and choose the project he likes, regardless of the choice made in the first period. Thismeans that vi < d and

 ∗

r1
r∗
1
1
∗
+ r2 vi ≤ vi ⇔ 1 + r2∗ ≤ .
2
2
2
2

r1∗
1
∗
2 + r2 < 2 , all the other types are strictly better off by abstaining in the first period.
r∗Since there must be some type who adopts in the first period in equilibrium, 21 + r2∗ = 21 .

If

Now notice that: (i) the unconditional probability that a player adopts in the first period
is r1∗ ; (ii) the unconditional probability that a player abstains and chooses the project her∗likes in the second period is 2(1 − r1∗ − r2∗ ).12 Since 21 + r2∗ = 21 , r1∗ + 2 (1 − r1∗ − r2∗ ) = 1,
meaning that only the types with strong preferences exists, a contradiction.
This equilibrium property substantially simplifies the analysis, as (1) and (2) can now be
written as(1 − r1 )vi ≥ r1 max (vi − d, 0) ,(3)

1 ≥ 2r1 ,(4)

respectively. Define
Ri :=

vi
,
vi + max(vi − d, 0)

as the threshold probability, with which (3) is reduced to Ri ≥ r1 . Note that RL ≥ RH > 0.5.
Next, Lemma 2 shows that the probability that a player adopts in the first period is atleast as large as the probability that he abstains and waits until the second period.
Lemma 2 In equilibrium, r1∗ ≥ r2∗ , i.e., r1∗ ≥ 21 .
Proof: Suppose, on the contrary, that r1∗ < r2∗ .

By Lemma 1, r2∗ = 1 − r1∗ , so aplayer with weak preferences obtains a higher payoff by adopting in the first period:
 ∗





∗
r1
d
∗
∗ + r2vi − d2 . The same also holds for a player with strong
+
r
>
r
v
−
i
2
1
2
2
2


 ∗
 ∗
r∗
r∗
r
r
preferences: 21 + r2∗ vi > 21 + 22 vi + 21 max (vi − d, 0). So in equilibrium, r2∗ = 0, acontradiction.

With the help of Lemmas 1 and 2, we can characterize the equilibrium timing of adoption.
If we list type i ahead of type j, it means either (a) that type i adopts in the first period withprobability one and type j adopts with probability α ∈ [0, 1], or (b) that type i adopts in
12
1 − r1∗ − r2∗ is the probability that a player abstains and disagrees with the choice made by the other player.
Since there is equal probability that the other player chooses the project he likes, the probability is multipliedby two to obtain 2(1 − r1∗ − r2∗ ).

11

the first period with probability α ∈ [0, 1) and type j adopts with probability zero. In either
case, therefore, type i adopts no later than type j even in the ex post sense.
Proposition 1 The equilibrium order of adoption is (i) IS (ii) DS (iii) IW/DW, where

1


2




q



r1∗ = RH

(1 − p)q





RL

h
i
if q ∈ 0, 12iif q ∈ 12 , RHiRHif q ∈ RH , 1−p
i

RH RL
, 1−p
if q ∈ 1−p
iRLif q ∈ 1−p,1 ,Proof: The equilibrium order of adoption is evident from (3) and (4). See Figure 1 inAppendix B for the computation of r1∗ .
The order characterized in Proposition 1 is fairly intuitive. As we have already seen, thetypes with weak preferences face exactly the same incentives and are ranked in the sameorder. Moreover, these types feel indifferent between the two projects, and the expected
benefit from adopting first and inducing the other to follow is not as much. Between thetypes with strong preferences, the dependent type derives more benefit from a network, so
he acts more cautiously to make sure that a network will eventually emerge, one way or theother, which can be seen from the fact that RL ≥ RH . By adopting in the first period, aplayer expects that the probability of the two players agreeing on the same project is r21 + r2 ;by adopting in the second period, a player expects the probability to be r1 + r22 . Since r1 ≥ 12 ,the probability is larger if a player adopts in the second period, meaning that the incentive toabstain in the first period is stronger for the dependent type than for the independent type.

4

The Efficient Timing of Adoption

In this section, we characterize the efficient timing of adoption under the restriction thatdecisions are made in a totally decentralized manner. More precisely, the social planner here
can only dictate the probability that each type adopts in each period, but is not equippedwith a tool to resolve coordination failures. This notion of constrained efficiency yields
implications of more practical importance, because it could be implemented by a simple, andmore realistic, institution as we will discuss in the next section.13
13
While the fully efficient allocation can be implemented by an incentive-compatible direct mechanism, it
is in practice not easy to find a vehicle though which to achieve this. In contrast, as we will see in the next
section, the constrained efficiency can be implemented by simply imposing fees contingent on the timing ofadoption.

12

As it turns out, the characterization of the efficient timing is much more complicatedthan that of the equilibrium timing. To obtain clear-cut results, we make two assumptions
to restrict our attention to cases that are more relevant and capture the spirit of the current
analysis.
Assumption 1 Any of the four probabilities, pq, q (1 − p), p (1 − q), and (1 − p) (1 − q),does not exceed the sum of any other two probabilities.
Assumption 2 vL ≥ d.
Assumption 1 indicates that the types are reasonably evenly distributed, and none of themis hence negligible. With some computation, one can show that a sufficient condition for √ √ 
1 2
the assumption is p, q ∈ 3−2 5 , 5−1,,
whereasanecessarycondition
is
p,
q
∈
2
3 3 . With
Assumption 2, we restrict our attention to the case where the value of a network is so largethat if only one player adopts in the first period, the other player will always follow the firstplayer’s decision, i.e., r1 + r2 = 1.
In this problem, the social planner assigns the probability for each type to adopt in eachperiod. This can be done in the same manner as we characterize the equilibrium timing: thesocial planner’s problem is effectively to determine the order of adoption across the four types
and r1 , i.e., the unconditional probability that a player adopts in the first period.14 The way
we characterize the efficient timing is as follows. First, we fix r1 and find the efficient order of
adoption for each given r1 . We then later shift r1 to find the efficient level, which we denoteby r1∗∗ , which maximizes the expected total payoff. Note that, given the order and r1∗∗ , theefficient allocation can be derived straightforwardly from the type distribution characterized
by p and q, as in the case of the equilibrium allocation.
dLemma 3 Define δ := 4∆. The efficient order of adoption isv
A: (i) DS (ii) DW (iii) IS (iv) IW for r1 ∈ 0, 21 − δ ,
B: (i) DS (ii) IS (iii) DW (iv) IW for r1 ∈ 12 − δ, 12 ,
C: (i) IS (ii) DS (iii) IW (iv) DW for r1 ∈ 12 , 12 + δ ,
D: (i) IS (ii) IW (iii) DS (iv) DW for r1 ∈ 21 + δ, 1 .

Proof: See Appendix A.
The lemma indicates that there are four distinct candidates for the efficient order, labeled
as A, B, C, and D, in this setup. In fact, as we will see shortly, any of the four orders can be
14

Since vL ≥ d and r1 + r2 = 1 always holds, r1 alone is a sufficient statistic.

13

efficient under some conditions. This fact suggests that the welfare properties of the model
are much more delicate and complicated compared to the equilibrium properties that arequite simple.
With this lemma, we can now characterize the efficient timing of adoption. We startwith the case where the productivity dispersion is relatively small (a small ∆v ), so that only
regions B and C in Lemma 3 are relevant.
Proposition 2 Suppose ∆v < d2 . Then,
1. if q > 21 , r1∗∗ ∈ 12 , q , and the efficient order is (i) IS (ii) DS (iii) IW (iv) DW;
2. if q < 21 , r1∗∗ ∈ q, 12 , and the efficient order is (i) DS (ii) IS (iii) DW (iv) IW;
3. if q = 21 , r1∗∗ = 12 , and the efficient order is (i) DS/IS (ii) DW/IW.

Proof: See Appendix A.
The proposition characterizes the efficient timing of adoption when δ > 21 . Given that the
productivity dispersion is small, the important dimension necessarily becomes the preference
dispersion, and the focus of attention is on how to choose the right project between the two.
It is hence efficient to let the types with strong preferences adopt first, followed by the types
with weak preferences, in any event. Given this, the next priority is to have the dependent
types in a successful network, which depends largely on the likelihood of a player having
strong preferences. When q > 12 and the preference dispersion is large, it is efficient to let
more people express their preferences early on (r ∗∗ > 12 ). Since there is more likely to be aleader to show which way to go, the dependent type with strong preferences should take aback seat to the corresponding independent type, so that he can follow the leader’s choice
to be in a successful network. When q < 12 , on the other hand, it is efficient to let lesspeople express their preferences (r ∗∗ < 21 ). Now that there is more likely to be a follower, thedependent type with strong preferences should step up first to allow the player to follow his
choice.
We now turn to the case where the productivity dispersion is relatively large.15 In thiscase, the focus of attention shifts to having the dependent types in a successful network.
¯ v , q̄, and q such that for ∆v > ∆¯ v,
Proposition 3 Given p ∈ 31 , 32 ,16 q, and d, there exist ∆
1. if q ∈ q̄, 23 , r1∗∗ ∈ 0, 21 − δ and the efficient order is (i) DS (ii) DW (iii) IS (iv) IW;
2. if q ∈ 13 , q , r1∗∗ ∈ 12 + δ, 1 and the efficient order is (i) IS (ii) IW (iii) DS (iv) DW.

15
Aside from the fact that ∆v is large enough, this also requires q to be sufficiently close to one half, because
the effect of ∆vwould be diminished when p is close to either zero or one.
16
p, q ∈ 31 , 23 is a necessary condition for Assumption 1.

14

Proof: See Appendix A.
When the productivity dispersion is large, the first priority is to assure to have thedependent types in a successful network. The efficient order to achieve this is either that thedependent types adopt before the independent types or vice versa, regardless of the strengthof preferences. The dependent types should adopt early when there is a greater possibilityof attracting a follower, i.e., a small r1∗∗ (region A in lemma 3). By the same token, thedependent types should adopt late when there is a greater probability of having a leader, i.e.,a large r1∗∗ (region D).
The deciding factor between the two cases is therefore which one, a small r1∗∗ (case 1)
or a large r1∗∗ (case 2), is less likely to have the types with strong preferences work on the
project they dislike. Rough intuition for the proposition goes as follows. When the preference
dispersion is large, i.e., q ∈ (q̄, 32 ), more of the types with strong preferences are inevitably
allocated to the second period. If the likelihood of having a leader, i.e., r1∗∗ , is large, that
also increases the likelihood that a follower must work on the project he dislikes. As such,therefore, it is relatively costly to have a larger r1∗∗ and allocate more people to the first
period when the preference dispersion is relatively large. A similar argument can be applied
to show that case 2 emerges as the socially efficient allocation when the preference dispersion
is relatively small.
The equilibrium timing of adoption derived in Section 3 is often different from the efficienttiming characterized in Section 4, because each player does not take into account the impactof his move on the other player’s payoff when he decides when to adopt. To see this, notethat the timing choice in this context is a strategic substitute, where the value of adopting in
the first period is decreasing in r1 . When a player moves from the second period to the first,
therefore, it lowers the expected payoff for the types who adopt in the first period and raisesthat for those who adopt in the second. These costs and benefits do not necessarily coincide
at the equilibrium level r1∗ .
Proposition 4 If ∆v < d2 , r1∗∗ ≤ r1∗ . Moreover, there exists q̄ ∈r1∗∗ = r1∗ if and only if q ∈ 21 , q̄ .

vH
1
2 , 2vH −disuch that

Proof: See Appendix A.

Proposition 5 Given p ∈¯ v,∆v > ∆

1 2
3, 3



¯ v , q̄ ≥ 1 , and q ≤ 1 such that given
and d, there exist ∆
2
2

15


1. if q ∈ q̄, 23 , r1∗∗ < r ∗ ;
2. if q ∈ 13 , q , r1∗∗ > r ∗ .
Proof: See Appendix A.
These propositions indicate that the equilibrium allocation is largely inefficient, though
not always, and moreover that r1∗ > r1∗∗ holds for a wide range of parameter values, meaningthat too many people tend to adopt in the first period. This is mainly due to the presence ofthe types with strong preferences: for those players, the benefit of adopting in the first period
is excessively strong because they can increase the probability of attracting other people tofollow and having things in their favor by doing so. As a result, we tend to have too many
leaders, with too few followers, generating excessive congestion in the first period.

5

Implementation of the Efficient Timing of Adoption

The results presented thus far illuminate inherent difficulties associated with consensus building in the presence of network externalities. The equilibrium timing of adoption can beinefficient on two distinct levels. First, r1∗ is either too high or too low, thereby resulting
in excessive congestion in either period. Second, the order of adoption can also be wrong,
especially when the productivity dispersion is sufficiently large. These results provide an
insight into why we often end up with a seemingly inefficient consensus or standard when it
is built up in a totally decentralized manner: for instance, it is well known that the standard
technologies are often not the most technically advanced (Arthur, 1994; Lee et al., 1995). In
many cases, we cannot expect a right person to step up at the right timing in the presenceof network externalities.
Given that decentralized consensus building is often inefficient, a natural question is thenwhether there is any feasible way to restore the efficient outcome. To this end, we consider
the following simple fee scheme: a player must incur a cost c1 if adopting in the first period,and c2 if adopting in the second.17 The fees are collected by a financial institution, which
reimburses each player half of its expected revenue from fee collection in advance. It is
important to note that what is referred to as a fee here should be interpreted broadly: while
it certainly includes monetary transfers, it can also refer to psychological costs or benefits ofbecoming a leader in a specific context. Without loss of generality, we assume that at least
one of c1 and c2 is 0: let c1 = c and c2 = 0 for normalization. As we will show next, despite
the fact that the conflict of interests arises from two-dimensional heterogeneity in preferences,17

We assume that the players do not have any budget constraints.

16this simple fee scheme can implement the efficient timing of adoption when the fee c is set at
an appropriate level.
Proposition 6 For any given set of parameters (vL , vH , d, p, q), the efficient timing can be
implemented by imposing a fee c on players adopting in the first period. Moreover, c > 0 if
r1∗ > r1∗∗ and c < 0 if r1∗ < r1∗∗ .
The result suggests that the ex post reallocation of costs and rewards through the fee
scheme could influence the way consensus emerges in a group. A positive fee should beplaced on early adopters when r1∗ is too high, i.e., r1∗ > r1∗∗ whereas a negative fee should beplaced when r1∗ is too low. By changing the expected share of costs and rewards, one candesign a scheme which alleviates the coordination problem, and realizes the efficient outcomein the absence of proper coordination devices.
In reality, when a group of people engage in a project, the costs and benefits are notnecessarily allocated evenly among them; moreover, the allocation often depends on thetiming of adoption, even without explicit ex ante agreements. Among many possibilities, we
focus on social norms which often provide a way to reallocate the costs and benefits withina group. For instance, there is a strong belief that a leader must assume more responsibility
and accept the burden of accountability: the leader is expected to take an aggressive role inrecruiting and motivating fellow members in initiating a project, and to be the one to takethe blame whenever things go wrong. “The Buck Stops Here,” the well-known sign on thedesk of Harry S. Truman, exemplifies this spirit. The leadership literature also emphasizes
that one of the most important qualities to be held by a leader is self-sacrifice (Conger andKanungo, 1998). This type of social norms or expectations certainly affects one’s incentive
to step forward to lead the way, and when the incentive to step forward is excessively strong,that could correct inefficiencies inherent in decentralized consensus building.
The allocation of costs and benefits through social norms might be harder in the context
of technology adoption (either by firms or consumers), but there may still be a mechanism
to allocate the costs and benefits between leaders and followers. In developing a new technology standard, pioneering firms typically need to invest more and take more risk in theprocess. Since not all of the costs can be legally protected, followers can easily freeride on
the leader’s efforts. While this freerider problem might be problematic from the viewpoint ofprotecting the investment incentives, the current analysis provides an alternative view thatthe asymmetric share of costs could improve the way the economy reaches consensus.
Of course, as can be seen from Proposition 5, it is not always welfare-improving to have apositive c. The norm which shifts more burden to the leader might work well either when the17

productivity dispersion is small or when both the productivity dispersion and the preference
dispersion are large, e.g., in situations that are highly political in nature. In contrast, when
the productivity dispersion is large but the preference dispersion is small, not enough people
step up in equilibrium and the norm which imposes more cost to the leader aggravates thesituation even worse. If the initial burden is too large, only the dependent types, who expect
to derive large benefits, are willing to step up, including the type with weak preferences.
Since the type with weak preferences have no strict preference over the alternatives, they justrandomly pick one alternative and wait for others to join. The independent types follow thelead, but they include the type with strong preferences who has a strong stand as to which way
to go and suffers from joining the alternative they dislike. As such, if the allocation of costs
and benefits is not appropriate, the group cannot maximize the probability of successfully
forming a network, thereby failing to pick an alternative favored by the majority.

6

Conclusion

Two players, with a potential conflict of interest, are asked to make a choice between two
projects. They derive benefit only if they agree on the project to implement. In this context,
we characterize the equilibrium timing of adoption as well as the efficient timing and show thatthe equilibrium timing is often different from the efficient timing in the presence of network
externalities. Given this inefficiency result, we then show that the efficient timing can be
implemented by a simple fee scheme which imposes fees on players contingent on the timing ofadoption. The analysis gives an insight into why we often end up with inefficient consensus or
standard, and suggests a potential role of social norms in resolving the coordination problem.
As a final note, the model is deliberately kept simple to illuminate the basic tradeoff. For
this reason, there are several avenues to extend the current analysis. An obvious extension
is to increase the number of players and time periods. The most general form along thisline is perhaps a T period model with n players in which at least m, n ≥ m ≥ 2, players
need to take the same side to yield any value. While the extension apparently makes theanalysis more complicated and its underlying logic less transparent, we expect that the same
principle applies for this more general setting as well because one can gain from expressing
one’s opinion early to lure other players to his side at a potential cost that he loses his chance
to learn others’ preferences. It is nonetheless of some interest to analyze this general modelin future.

18

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Arbor: University of Michigan.
[2] Marco Battaglini. (2005) “Sequential Voting with Abstention.” Games and Economic
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409-424.
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Appendix A: Proofs
Proof of Lemma 3: Let Utj (r1 ), j = DW, DS, IW, IS, denote the expected payoff when atype j player adopts in period t for a given r1 , and let ∆U j (r1 ) := U1j (r1 ) − U2j (r1 ) denotethe payoff difference for type j. The efficient order is obtained by ordering the types in terms
0of the payoff difference, where type j is listed higher than type j 0 if ∆U j (r1 ) > ∆U j (r1 ).
It is straightforward to compute the payoff difference for each type. It follows from (1)and (2) as well as the fact that r1 + r2 = 1 for any efficient allocation thatd
1 − 2r1 vH −,∆U DW (r1 ) =2
2∆U DS (r1 ) =1 − 2r1
r1
vH + d,2
2∆U IW (r1 ) =1 − 2r1 
d
vL −
,
2
2

∆U IS (r1 ) =r1
1 − 2r1
vL + d.
2
220

Simple algebra then establishes the lemma.
Proof of Proposition 2: Since ∆v < d2 , we can focus on the orders labeled as B and C in
Lemmas 3. Given the orders, we need to choose the efficient value of r1 which maximizes thetotal profit.
First consider the case where q > 21 . When r1 = 21 and the efficient order is C in which
the independent type with strong preferences and part of the dependent type with strong
preferences adopt in the first period. The expected total payoff is
π (r1 ) =

r1 
r1  iq (1 − p) 1 −vL + (r1 − q (1 − p)) 1 −
vH
2
2


1
r1
+ (q − r1 )vH + (vH − d)
2
2






1 r1
d
d
1 r1
+
+
vL −
+ (1 − q) pvH −.
+ (1 − q) (1 − p)
2
2
2
2
2
2
h

It follows from this that
π 0 (r1 ) =1
1[(1 − r1 ) vH − r1 (vH − d)] − [r1 v̄1 (r1 ) − (1 − r1 ) v̄2 (r1 )] ,
2
2

(5)

where v̄1 (r1 ) = r11 (q (1 − p) vL + (r1 − q (1 − p)) vH ) is the expected value of those who adopt
in the first period while assuming that the types with strong preferences work on the project
1
they like, and v̄2 (r1 ) = 1−r[(q − r1 ) (vH − d) + (1 − q) (1 − p) vL − d2 + (1 − q) p vH − d2 ]1is the expected value of those who adopt in the second period while assuming that thetypes with strong preferences work on the project they dislike. Note that when r1 = 21 ,(1 − r1 ) vH − r1 (vH − d) = d2 and r1 v̄1 − (1 − r1 ) v̄2 < d2 , so π 0 (r1 ) > 0. Furthermore, for
r1 ∈ 12 , q , π 0 (r1 ) decreases in r1 . For r1 > q, some proportion of the types with weak

preferences also adopt in the first period, and we can show that π 0 (r1 ) < 0 and π 0 (r1 )decreases in r1 . Therefore, we know that for r1 ≥ 21 , the total payoff is maximized at some
r1 ∈ 21 , p . For r1 ∈ 1 − q, 12 , the efficient order is B, in which the dependent type withstrong preferences and part of the independent type with strong preferences adopt in the first
period. In this case,
π 0 (r1 ) =1
1[(1 − r1 ) vL − r1 (vL − d)] − [p1 v̄1 (r1 ) − (1 − r1 ) v̄2 (r1 )] ,
2
2

where

1
(pqvH + (r1 − pq) vL ) ,
r1




1
d
d
v̄2 (r1 ) =[(q − r1 ) (vL − d) + (1 − q) p vH −+ (1 − q) (1 − p) vL −].
1 − r122v̄1 (r1 ) =

21

(6)

When r1 = 21 , (1 − r1 ) vL − r1 (vL − d) = d2 , and






d
∆
+ (2q − 1) p vH − vL −r1 v̄1 (r1 ) − (1 − r1 ) v̄2 (r1 ) = (1 − q) p vH − vH −
2
2



d
+ ((1 − q) − pq) vL − vL −
2


d
1
− (1 − q) (vL − (vL − d)) < ,+2
2
by Assumption 1. We can further show that π 0 (r1 ) decreases in r1 for r1 < 21 . Therefore,r1∗∗ ∈ 12 , q .
Applying similar arguments, we can also prove that r1∗∗ ∈ q, 21 if q < 12 , and r1∗∗ = 12 if
q = 21 .

Proof of Proposition 3: Suppose q > 12 . When r1 = 21 and the efficient order is C,∂π 0 ( 12 ;vH )= 21 (1 − p) (2q − 1) > 0, so π 0 12 > 0 when vH is large enough. Furby (5),∂vH

thermore, π 00 (r1 ) ≥ − (2vH − d) for r1 > 21 , and we can find u such that for vH > u,dvH −vL + d2
π 0 12 − (2vH − d) 2(vH2−vL ) > 0, so for vH > u, π 0 (r1 ) > 0 for r1 ∈ 21 , 2(v, andH −vL )

vH −vL + d2
r1∗∗ does not fall in 12 , 2(v. Similarly, we can show that when vH is large enough,H −vL )
vH −vL − d2 1vH −vL − d2 10∗∗
π (r1 ) < 0 for r1 ∈ 2(vH −vL ) , 2 , so r1 does not fall in 2(vH −vL ) , 2 . Similar arguments vH −vL − d2vH −vL + ∆2,
1
when q 6= 12 .
∪can be applied to the case when q < 21 , so r1∗∗ ∈ 0, 2(v2(vH −vL )H −vL )

When q = 21 , for r1 ∈ 12 , 12 + (1 − p) (1 − pH ) , the types with strong preferences andpart of the independent type with weak preferences adopt in the first period. When vHvH −vL + d22(v
H −vL )< 21 + (1 − q) (1 − p), and applying the same logic as before,



∆
d
1 vH −vL + 2
1 vH −vL + 20∗∗
π (r1 ) < 0 for r1 ∈ 2 , 2(vH −vL ) , so r1 cannot fall in 2 , 2(vH −vL ) . Then we show that
vH −vL + d2
when vH is large, choosing r1 ∈ 2(vH −vL ) , 1 yields a higher total payoff than choosing




1
vH −vL + ∆1
1 ∂π ( 2 ;vH )32
r1 = 2 . Given q = 2 ,= 4 p. With r1 ∈ max 2(vH −vL ) , 1 − p , 1 − (1 − q) p ,∂vHis large enough,the independent types and some of the dependent type with strong preferences adopt in the
1 ;vH )
first period, and ∂π(r= p 12 + r21 > 43 p. Therefore, there exists v such that if vH > v,
∂vH



vH −vL + d2
1
π (r1 ) > π 2 , where r1 ∈ 2(vH −vL ) , 1 . Similarly, we can show that when vH is large,
vH −vL − d2vH −vL − d2 1choosing r1 ∈ 0, 2(v,yieldsahighertotal
payoffthanchoosingr∈
1
2(vH −vL ) 2 .
H −vL )
 vH −vL − d2vH −vL + d2
So, r1∗∗ ∈ 0, 2(v, 1 when q = 21 .
∪ 2(vH −vL )
H −vL )

22

Next we compare the payoffs when r1 ∈vH −vL − d2
0, 2(v
H −vL )

and when r1 ∈vH −vL + d2
2(vH −vL ) , 1.

By deriving
π 0 (r1 )as in the proof of Proposition 2, we can showthat asvH increases, ifvH −vL − d2vH −vL + d2
1+p
1 5
∗∗
∗∗
∗∗
r1 ∈ 0, 2(vH −vL ) , r1 converges to 4 ∈ 3 , 12 ; if r1 ∈ 2(vH −vL ) , 1 , r1∗∗ converges

7 2
∈ 12
, 3 . Therefore, given a small number  > 0, there exists v 0 > v suchto 1 − 1+p
4




1+p
3−p
3−p
−
,
+

or
in
−
,
+

. Suppose thatthat if vH > v 0 , r1∗∗ falls either in 1+p
4
4
4
4


1+p
∗∗
0
r1∗∗ ∈ 1+p
4 − , 4 +  . By choosing r1 = 1 − r1 , the expected total payoff increases by(q (1 − p) + max {pq − r1∗∗ , 0}) r1∗∗ − max {r1∗∗ − p (1 − q) , 0} r10 d






1+p
1+p
3−p
1+p
,0
− max
− p (1 − q) , 0'
q (1 − p) + max pq −
d.(7)
4
4
4
4


18 so choosing r 0 is better than choosing r ∗∗ . Therefore,
When q is small, (7) is positive,
11


when q is small enough, r1 ∈vH −vL − d2enough, r1 ∈ 0, 2(vH −vL ) .

vH −vL + d2
2(vH −vL ) , 1

Proof of Proposition 4: When q <1
2,. Similarly, we can show that when q is large

r1∗ =

1
2

by Proposition 1, and r1∗∗ <

1
2

by

Proposition 2, so r1∗∗ < r1∗ . For q ≥ 12 , given π 0 (·) defined in (5),19 let r̄ (q) be such that
2v − 3 d2(1−p)(v −v )− dH
L
42
> 12 . Furthermore, r̄ 0 (q) =, and
π 0 (r̄ (q)) = 0. For q = 21 , r̄ (q) = 4vHH −2d4vH −2dr̄ (q) ∈ 21 , 2vvHH−d for q > 21 . By Proposition 2, for q ≥ 12 , r1∗∗ = r̄ (q) if r̄ (q) ≤ q, and r1∗∗ = q
iif r̄ (q) > q. By Proposition 1, r1∗ = q for q ∈ 12 , 2vvHH−d . Therefore, when q = 21 , r1∗ = r1∗∗ ,and when q = 2vvHH−d , r1∗ < r1∗∗ . Since r1∗ increases in q linearly, and r̄ (q) increases ori
decreases in q linearly, there exists q̄ ∈ 12 , 2vvHH−d such that r1∗∗ = r1∗ if and only if q ∈ 12 , q̄ .

Proof of Proposition 5: By comparing the results of Propositions 1 and 3, it is obviousthat when q ∈ q̄, 23 , r1∗∗ < r ∗ , and when q ∈ 13 , q , r1∗∗ > r ∗ .
Proof of Proposition 6: When deriving the efficient order given r1 , we compare theexpected payoffs when a type chooses to adopt in the first period and in the second period.
A type with a larger payoff difference should be placed higher in the order. This is consistent
with the players’ incentive compatibility constraints. When a type is expected to get a higher
payoff by adopting in the first period, he has a stronger incentive to adopt in the first period.
18

For q such that pq − 1+p< 0,4

q(1−p)
max{ 1+p−p(1−q),0}
4> 0, 1+p− p (1 − q) > 0, andsuch that pq − 1+p4
4

decreases in q, so (7) is positive when q is small. For qq(1−p)+pq− 1+p4
1+p−p(1−q)4

increases in q, but it can be shown that evenwhen q = 23 , (7) is negative. Therefore, there exists q̄ such that when q < q̄, (7) is positive, and when q > q̄,(7) is negative.
19
Note that π 00 (q) < 0.

23

Therefore, to implement the efficient order given r1 , we only need to find out the type thatis allocated to both periods, which we call the cutoff type for expositional purposes,20 andfind out c that makes his payoffs in the two periods equal. Given c, the other types behave
according to the efficient order.
To see how c changes with r1 , let r1 increase from a to a + , where  > 0 is some smallv −v − dv −v + d

H
L 2
H
L 2
, 12 , 2(v∈/ (a, a + ), then r1 stays in the same region (A, B, C , or
number. If 2(vH −vL )
H −vL )

D in Lemma 3). If the cutoff type does not change, since r1 increases, the type’s first-period
payoff decreases and the second-period payoff increases. Therefore, to make the two payoffs
equal, c must increase. If the cutoff type changes to the next type in the order, then weknow that the difference between the two payoffs
than that of the old of this type is less 
cutoff type, so c must increase. Finally, if b ∈vH −vL − d2 1 vH −vL + d2
2(vH −vL ) , 2 , 2(vH −vL )and b ∈ (a, a + ),

when r1 passes through b, the order changes. We already know that when r1 increases froma to b or from b to a + , c must increase as discussed just above. We thus need to see how
c should change when r1 passes through b. If the cutoff type does not change, then nothing
affects c. But if the cutoff type changes, it is implied that at r1 = b, the differences between
the payoffs in the two periods of the new and the old cutoff types are the same, so c is notaffected either. This completes the proof that c must increase as r1 increases from a to a + .

Appendix B: Computation of equilibrium
In this appendix, we describe how we obtain the equilibrium allocation (the type-contingent
probabilities of adoption) from the equilibrium order and (r1 , r2 ) as indicated in Proposition
1. First, by Lemma 1, r2 = 1 − r1 , so r1 is a sufficient statistic. Suppose that a player expects
the other to adopt in the first period with probability r. For each given r, we can then check
whether it is optimal to adopt or abstain in the first period for each type: (i) the dependent
type with strong preferences adopt in the first period only if RH ≥ r; (ii) the independent
type with strong preferences adopt only if RL ≥ r; (iii) the types with weak preferences
adopt only if 0.5 ≥ r. From these incentive conditions, we can define a correspondence F
which maps this unconditional probability r to the probability that a player adopts in thefirst period. The equilibrium r1 is then obtained as a fixed point of F , i.e., r1 ∈ F (r1 ). See
Figure 1 for a typical correspondence and an equilibrium. In this particular example, r1 = q
20

For example, suppose that the efficient order is B. If pq < r1 < p, then the independent type with strong
r1 −pqof the type chooses to adopt in the firstpreferences is allocated to both periods. In this case, a fraction q(1−p)
period, while the remaining fraction chooses to adopt in the second period.

24where the types with strong preferences adopt in the first period and the types with weak
preferences adopt in the second.

q(1 − p)q

r1∗

RH

RL

Figure 1: A typical correspondence and an equilibrium

25