It’s Now or Never:
Deadlines and cooperation
Kaz Miyagiwa
Emory University
Abstract: Cooperation among opportunistic agents often breaks down when agents cannot
observe one another’s actions. The standard remedy for such a problem is a two-mode
approach pioneered by Green and Porter (1984), where agents switch back and forth
between cooperation and punishment mode. Here, I consider use of a deadline as an
alternative. I find that, under certain conditions, imposing a deadline can induce
cooperation with unobservable actions, and that the optimal deadline can payoffdominates the optimal two-mode strategy.
Correspondence: Kaz Miyagiwa, Department of Economics, Emory University, Atlanta,
GA 30322, U.S.A.; E-mail: kmiyagi@emory.edu; Telephone: 404-727-6363; Fax: 404727-4639
Acknowledgments: I have benefited from comments by seminar participants at Emory,
Florida International, Kobe, and Osaka Universities. Errors are my responsibility.
1. Introduction
Today most of us academic economists do research with someone else. Having a
co-author makes research more enjoyable than working alone, and improves the chance
of solving tough problems, such as proving theorems and identifying next research
agendas. However, it can also give rise to the temptation to free-ride on one’s co-author’s
effort. Such temptations are greater especially when co-authors are in separate locations
and cannot observe each other’s actions. If both researchers succumb to such temptations,
of course, co-authorship breaks down.
How to induce cooperation among opportunistic agents has long been a subject of
intensive research among social scientists, possibly dating back to Rousseau’s problem
over stag hunting.1 Game theory has taught us that, if agents interact repeatedly over
time, cooperation can be induced with threats of punishment.2 However, in stochastic
environments where bad outcomes occur due to bad luck as well as due to lack of effort,
it is impossible to mete out punishment if agents cannot observe one another’s actions.
The standard remedy for this type of problem is a two-mode strategy pioneered by Green
and Porter (1984), in which agents switch back and forth between cooperation and
punishment mode over time. Although all agents are aware in equilibrium that no one has
ever shirked, occasional self-flagellation is necessary in the wake of poor outcomes so as
to make agents toe the line in cooperation mode.
In this paper I consider use of a deadline as an alternative solution to this type of
1
As retold in Fudenburg and Tirole (1991, p. 3), each hunter of a group on a stag-hunting expedition has a
hard time resisting the temptation to catch a rabbit that happens cross his path and to call it a day instead of
hunting a stag that requires cooperation and coordination.
2
A contract-theoretic approach has recently been applied to study how a principal can induce cooperation
among agents. The present paper concerns the desirability to cooperate outside principal-agent
relationships.
2
problem. The setting I consider is as follows. I suppose that agents are engaged in some
‘search activity’ and exert efforts until there is success. The probability of success
depends on total efforts of agents. Agents however cannot observe each other’s action, so
it is impossible to disentangle failures due to shirking from those due to bad luck.
Therefore, if shirking goes undetected an agent who shirks faces the same continuation
payoff as an agent who exerts effort.
Suppose that in such circumstances all agents succumb to the temptation to shirk
so cooperation cannot be maintained. Now consider the effect of a deadline, which
specifies the number of periods during which agents agree to exert efforts. When the
deadline arrives, agents know that it is the last chance to cooperate. Since there is no
future cooperation, agents know that failure leads to a smaller continuation payoff than
when there is no deadline. This decrease in continuation payoff motivates agents to exert
efforts to succeed. Once agents have the incentive to exert efforts at the deadline without
observing each other’s action, I can move back to check their incentive to shirk in earlier
periods. My analysis shows that if the deadline is set too much in the future agents cannot
cooperate. In fact, I find that there is the unique optimal deadline, yielding the maximum
expected payoff to each agent. I then compare the expected payoff from the optimal
deadline with that from the two-mode strategy a la Green-Porter (1984) and show that the
former can payoff-dominate the latter.
Now that I have outlined my analysis it is time to turn to the relevant literature.
As already mentioned, research on cooperation with unobserved actions in stochastic
environment was pioneered by Green and Porter (1984) Focusing on collusion
enforcement in oligopoly under stochastic demand, these authors proposed a two-mode
3
strategy in the class of trigger strategies, whereby firms start out in cooperation mode but
switch to punishment mode when the price falls below a certain level. Abreu, Milgrom
and Pearce (1991) advanced this approach and applied it in a situation similar to the one
considered here. However, their paper differs considerably from the present work. Their
main interest lies with the folk theorem, i.e., the effect of changing discount factors on
the stability of cooperation under the two-mode strategy, whereas here focus is on the
cooperation-inducing effect of a deadline for a given discount factor.
The effect of a deadline has also been examined in the recent bargaining
literature.3 The principal result in this literature is that, when there is a deadline, agents
stall out, making no attempt to reach an agreement until the deadline arrives. Thus, in
bargaining situations a deadline causes inefficient delays in reaching an agreement. In
contrast, in the present paper a deadline has a positive outlook.
The remainder of the paper is organized in six sections. Section 2 describes the
basic environment and presents the benchmark case, in which cooperation fails without
observable actions. Section 3, the main section, introduces a deadline and examines its
property. Section 4 compares the expected payoffs between the optimal deadline and the
optimal two-mode strategy a la Green-Porter (1984) and shows that the former may
dominate the latter. Section 5 extends the analysis to the case of asymmetric agents in
success probability, while Section 6 considers another extension, in which agents never
leave the game when there is success.
See. e.g., Hendricks, Weiss and Wilson (1988), Fershtman and Seidmann (1993), Ponsati (1995), and
Damiano et al. (2010)
3
4
Section 7 concludes the discussion and suggests possible applications of the
analysis to three specific cases; joint research projects in industrial economics, law
enforcement efforts as regards statute of limitations and commitment in military
campaigns.
2. Setup
2.A. Going it alone
Consider a single agent in an infinite time horizon. Time is discrete and all actions
take place at t = 1, 2,…., which are called ‘dates.’ At each t, conditional on not having
succeeded to date, an agent decides whether to exert effort or not. Exerting effort costs
the disutility e, but leads to success at t + 1 with (conditional) probability (1 – p) > 0,
where p > 0 is constant over time. Success entitles an agent to the perpetual flow of
benefits at the rate b, suming up to π = b/(1 - δ), where δ ∈(0, 1) is the discount factor.
If there is failure, a probability p event, an agent faces exactly the same
continuation payoff as at date t due to the stationary environment. Therefore, v, the
expected payoff satisfies this recursive structure:
v = - e + (1 – p)δπ + pδv.
Collecting terms yields
v=
(1 − p)δ b − e
.
1−δ p
5
On the other hand, if an agent chooses not to exert effort, there is possibility of success,
yielding the normalized utility of zero to an agent. I assume v ≥ 0 so that an agent exerts
effort.4
2.B. Cooperation with observable actions
Now suppose that m (≥ 2) identical agents agree to independently and
simultaneously exert efforts and to share the reward yielded by success. If probability of
failure per agent remains constant at p, cooperation reduces the team’s probability of
failure to pm < p. Thus, cooperation functions as insurance here.
Cooperation can also change the rewards. Let B denote the benefit per-period to
each agent and let Π = B/(1 - δ). B need not be equal to b.
In the remainder of this subsection, I assume actions are observable and seek the
condition for cooperation under the standard trigger strategy: “At t = 1, agree to
cooperate and exert effort. At any t ≥ 2, exert effort as long as all agents exerted efforts at
all dates up to t – 1; otherwise go it alone.”
If each agent plays the above strategy, by an argument similar to the one given for
an individual agent, the expected payoff V to each agent must satisfy this recursive
equation:
(1)
V = – e + (1 − pm)δΠ + δpmV
Collecting terms yields
4
This is just for convenience’s sake. If the inequality is reversed, an agent’s reservation utility is till zero,
and the rest of the analysis is unaffected.
6
(1 − p m )δΠ − e
V=
.
1 − δ pm
I assume that cooperation is desirable; i. e., V > v.
The above trigger strategy is an equilibrium strategy if no agent has the unilateral
incentive to shirk. Shirking saves the disutility e, but increases the probability of failure
from pm to pm-1. It also triggers punishment mode, reducing the continuation payoff from
V to v. Thus, an agent who shirks would get the expected payoff
Vd = (1 – pm-1)δΠ + δpm-1v
Therefore, no one shirks if and only if V ≥ Vd, which I assume. This condition can be
expressed, after substitution and rearrangement, as
pm-1(1 − p)δ(Π - V) + pm-1δ(V - v) ≥ e.
Assumption 1: When actions are observable, agents can cooperate as a team in the sense
that V ≥ Vd.
2.C. Unobservable actions
Suppose next that agents cannot observe one another’s actions. If all agents exert
efforts, each agent faces the expected payoff V defined above. However, shirking goes
undetected. If an agent can shirk with impunity, he faces the continuation payoff V insead
of v. Since shirking also increases the probability of failure to pm-1, an agent sho shirks
expects the payoff
(2)
Wd = (1 - pm-1)δΠ + pm-1δV.
7
I assume that
V – Wd = pm-1(1 − p)δ(Π - V) – e < 0.
Assumption 2: With unobserved actions, cooperation is impossible, i. e.,V < Wd.
3. Deadlines
In this section I focus my attention on the cases in which both assumptions 1 and
2 hold; i.e., Wd > V ≥ Vd. In other words, agents prefer to cooperate but cannot because
they cannot observe each other’s actions. Using the defining expressions, these
conditions can be expressed, after manipulation, as
pm-1(1 − p)δ(Π - V) + pm-1δ(V - v) ≥ e > pm-1(1 − p)δ(Π - V).
Since V > v, there is a range of e satisfying the above conditions.
The standard remedy to this problem is a two-mode strategy a la Green-Porter
(1984). Translated into the present context, the Green-Porter (1984) approach has all
agents exert efforts at t = 1. If there is failure, agents go solo for a given number of
periods, after which they return to cooperation mode, and so forth. By temporarily halting
cooperation, this strategy reduces the continuation payoff and induces cooperation. The
objective of the present section is to present a deadline as an alternative strategy for
inducing efforts.
Let me being by explaining what I mean by deadlines. First, define the sequence
of strategies Cn; n = 1, 2,… , which says “At dates t (where 1 ≤ t ≤ n) cooperate and exert
effort. At t > n, go it alone.” That is, agents cooperate and exert efforts only up to date t =
8
n and go it alone from t = n + 1 on. In this case, I say that there is a deadline at t = n, and
call Cn the deadline-n strategy.
Let U(n) denote the expected payoff to each agent at t = 1, given that all agents
play Cn, and let Ud(n) the corresponding expected payoff to an agent who shirks at t = 1.
With C1, these are expressed as
U(1) = – e (1 – pm)δΠ + pmδv
Ud(1) = (1 – pm-1)δΠ + pm-1δv.
No agent shirks if and only if
U(1) – Ud(1) = pm-1(1 - p)δ(Π – v) – e ≥ 0.
Since V > v, there is a range of e satisfying:
(3)
pm-1(1 - p)δ(Π – v) ≥ e ≥ pm-1(1 − p)δ(Π - V).
The first inequality ensures that U(1) > Ud(1) while the second equality implies V < Wd.
It is easy to check that an e satisfying (3) also satisfies assumption 1. We have proved
Proposition 1: If an e satisfies (3), it satisfies assumptions 1 and 2. For such an e, all
agents exert effort at date 1 under strategy C1 without observing one another’s action.
Proposition 1 has the following intuitive explanation. Shirking always increases the
probability of failure from pm to pm-1. Further, while the continuation is V without the
deadline, it is now v < V with the deadline. Thus, the deadline reduces the continuation
payoff, prompting agents to exert efforts, as does the Green-Porter strategy.
9
If condition (3) holds so that agents cooperate for at least one period, then I
consider C2. If all agents play C2, the expected payoff at t = 1 is
U(2) = – e + (1 – pm)δΠ + pmδU(1),
whereas shirking yields
Ud(2) = (1 – pm-1)δΠ + pm-1δU(1).
There is no shirking if
U(2) – Ud(2) = pm-1(1 – p)δ(Π - U(1)) – e ≥ 0.
If this inequality holds, agents cooperate at least for the first two periods, in which case I
move to C3 and so on.
This process of extending the deadline eventually comes to an end because the
deadline extended indefinitely is no deadline at all so that by assumption 2 agents shirk.
To show this formally, assume that Cn-1 induces cooperation (for the first n – 1 periods).
Then, the expected payoff under the strategy Cn is:
(4)
U(n) = – e + (1 – pm)δΠ + pmδU(n – 1).
This is a first-order difference equation, with the solution
U(n) = (v – V) (pmδ)n + V.
Since v < V, U(n) increases monotonically; that is, the longer agents can cooperate, the
larger the expected payoff.
Shirking yields the expected payoff:
(5)
Ud(n) = pm-1(1 – q)δΠ + pm-1δU(n - 1)
10
so there is no incentive to shirk under Cn if
U(n) – Ud(n) = – e + pm-1(1 – p)δ(Π - U(n - 1)) ≥ 0.
(6)
Because U(n) is monotone increasing, (6) implies that U(n) – Ud(n) is monotone
decreasing.
I now show that there is an n* ≥ 2 such that U(n) – Ud(n) < 0 for all n > n*.
Substituting from (1), I rewrite (4) as
U(n) = V – pmδ(V – U(n – 1)).
Similarly, substituting from (2) into (5) yields
Ud(n) = Wd – pm-1δ(V – U(n - 1)).
Therefore,
U(n) – Ud(n) = V – Wd + δpm-1(1 – p)(V – U(n - 1)).
Now, as n → ∞, U(n – 1) → V since a deadline set at infinity is no deadline at all,
and hence U(n) – Ud(n) → (V – Wd), which is negative by assumption 2.
I have proved
Proposition 2: If U(1) – Ud(1) ≥ 0, there exists the unique integer n* ≥ 1 such that
U(n* + 1) – Ud(n* + 1) < 0 ≤ U(n*) – Ud(n*).
The uniqueness of n* comes from the monotonicity of U(n) proved above. Proposition 2
has the following explanation. When the deadline t = n is reached, an agent faces the
11
continuation payoff v because there will no more be cooperation. Moving back one
period, agents at t = n – 1 faces the continuation payoff U(1) because there is one more
period of cooperation left even if there is failure. Since U(1) > v, each agent has a greater
incentive to shirk at t = n – 1. The incentive to shirk is even greater at t = n – 2, when the
continuation payoff is U(2) > U(1). Since the continuation payoff U(n) is increasing,
moving back in time, agents eventually reach the period n* so that U(n* + 1) < Ud(n* +
1).
If the deadline cannot be extended beyond n* without destorying cooperation, the
monotonicity of U(n) implies that the deadline n* yields the maximal welfare to each
agent; i.e., n* is the optimal deadline. The next proposition gives further characterizations
of the optimal deadline n* (see Appendix A for the proof).
Proposition 3.
(A) The greater the reward Π of cooperation, the greater n* tends to be.
(B) Given Π, the greater the number of cooperating agents m, the smaller n* tends to be.
Note that to derive part B of proposition 3 I assumed constancy of Π with respect to the
number of agents. In applications, Π may fall with the number of agents involved,
especially if success yields the fixed reward to be divided among agents. In such cases,
12
depending the rate at which Π changes, part B may not hold locally, although it must at n
sufficiently large.5
4. A comparison with the two-mode strategy a la Green-Porter (1984)
Although the original work of Green and Porter (1984) analyzed a situation
drastically different from the one considered here, their methodology is fully applicable
to the present model. Therefore, it would be of considerable interest to compare the two
approaches.
Translated into our context, the two-mode strategy a la Green-Porter (henceforth
the GP strategy) has agents exerting efforts in the first period, only to switch to
punishment mode if there is failure. Once in punishment mode, agents go it alone for a
fixed number of periods, after which they return to cooperation mode. In equilibrium this
pattern repeats itself over time. Inclusion of punishment mode in the strategy lowers the
continuation payoff and makes shirking less appealing to agents. The main difference is
that the two-mode strategy spreads out cooperation periods over the entire horizon
whereas the deadline frontloads them.
Now I derive the optimal two-mode strategy in the present context. Begin by
letting Gc and Gp denote, respectively, the cooperation-model and the punishment-mode
expected payoffs. Since failure triggers punishment mode, Gc can be expressed as
Gc = – e + (1 − pm)δΠ + δpmGp.
5
With n approaching infinity, cheating by one agent will have no effect on success probability but a cheater
saves the disutility e and hence deviates.
13
Substituting from the definition of V in (1), I can rewrite this as
(7)
Gc = (1 - δpm)V + δpmGp.
Similarly, if agents stay in punishment model for τ – 1 periods, Gp can be expressed, after
some manipulation, as
(8)
Gp = [1 – (δp)τ]v + (δp)τGc.
Solving (7) and (8) simultaneously yields
(9)
[1 − (δ p)τ ]δ p m v + (1 − δ p m )V
Gc =
[1 − (δ p)τ ]δ p
Gp =
[1 − (δ p)τ ]v + (δ p)τ (1 − δ p m )V
.
[1 − (δ p)τ ]δ p
Now, let Gd denote the payoff to a cheater when agents are in cooperation mode,
i. e.,
Gd = (1 – pm- 1)δΠ + pm-1δGp.
There would not be cheating if
(10)
Gc – Gd = – e + pm-1(1 − p)δ(Π - Gp) ≥ 0.
All these payoffs depend on τ so write Gc(τ), Gp(τ) and Gd(τ).
It is easy to check that Gp(τ) is decreasing, because the longer agents stay in
punishment mode, the smaller the expected payoff. Thus, the optimal GP strategy
minimizes Gp(τ) with respect to τ subject to the no-deviation constraint (10) and the
integer constraint. Let τ* be the optimizer of this program.
14
Since the above is a well-structured optimization problem, one may be
tempted to conclude that the optimal GP strategy payoff-dominates the optimal
deadline strategy. However, the next proposition shows that the converse can be
true.
Proposition 4: If Gp(τ*) ≤ U(τ*), then the optimal deadline strategy payoff-dominates
the GP strategy; that is, U(n*) > Gc(τ*).
Proof: U(n) is increasing and Gp(τ) is decreasing. Further, U(0) = Gp(∞) = v and U(∞) =
Gp(0) = V. Thus, there is a unique positive integer, say, τo, so that U(τ) ≥ Gp(τ) for all τ
≥ τo. Now, consider:
(11)
0 ≤ – e + pm-1(1 − p)δ(Π - Gp(τ*))
≤ – e + pm-1(1 − p)δ(Π - U(τ*)) = U(τ* + 1) – Ud(τ* + 1).
The first inequality holds because τ* is the integer satisfying constraint (10). The second
is due to the assumption of proposition 4. The equality follows because of (6).
(11) imply that the deadline strategy Cτ* + 1 is incentive-compatible, meaning that
the optimal deadline strategy yields the expected payoff at least as large as U(τ* + 1).
The proof is now complete because
U(τ* + 1) = – e + (1 − pm)δΠ + δpmU(τ*)
15
≥ – e + (1 − pm)δΠ + δpmGp(τ*) = Gc(τ*),
where the first equality follows from the recursive definition of U(n), while the second
equality follows from the definition of Gc(τ) given in (9).

6. Asymmetric agents
In this section I relax the assumption that agents are symmetric, and study the
effect from the differences in probability of success among agents. To lighten notation, I
consider the case with two agents, whom I call Senior (s) and Junior (j). Senior has a
better chance of success, or a lower probability of failure, than Junior; i.e., ps < pj. They
are identical in all other respects agents. In particular, they have the same discount factor,
incur the same disutility from effort and enjoy the same benefits b and B from individual
and team success, respectively. The analysis follows closely that of the preceding section,
so details are omitted.
Going it alone yields the expected payoff vi to agent i = s, j, where:
(12)
vi = [(1 – pi)δπ – e]/(1 – δpi).
The difference in failure probability implies that vj < vs; Junior’s expected payoff is
smaller because he is more likely to fail than Senior. Assume vj ≥ 0 so both agents find
individual search worthwhile undertaking.
If agents agree to cooperate and exert effort, the probability of failure is pspj, so
the expected payoff V to each agent from cooperation satisfies
V = – e + (1 - pjps)δΠ + δpjpsV,
16
and hence is given by:
V = [(1 - pjps)δΠ – e]/(1 – δpjps).
If agents cannot observe each other’s actions, the expected payoff from shirking is
Vid = (1 - pk)δΠ + δpkV;
i, k = j, s (i ≠ k).
Therefore, there is no cooperation if
(13)
V – Vid = – e + pk(1 − pi)δ(Π - V) < 0.
Since Π > V and
ps(1 − pj) – pj(1 − ps) = ps – pj < 0,
(13) implies that V – Vjd < V – Vsd; i.e., Senior has more of an incentive to cooperate
than Junior. Intuitively, the team’s success depends more on Senior’s effort. Put
differently, Junior’s shirking does not decrease the team’s success probability as much as
Senior’s and yet both agents face the same disutility of effort and receive the same
reward. Therefore, Junior has a greater temptation to free-ride on Senior’s effort. Thus,
this asymmetry in incentives may be corrected with a side payment from Senior to Junior,
provided that V – Vjd < 0 < V – Vsd. However, it can be checked that a side payment
reduces V – Vsd more than it increases V – Vsd, so such an arrangement may not always
be possible.6
by p (1 – p )(1 - δ)ΔΠ/(1 - δp p ) and V – V by p (1 – p )(1 sd
j
s
j s
sd
s
j
δ)ΔΠ/(1 - δp p ). Since p > p , the latter is smaller (in absolute value) than the former.
j s
j
s
6
Transfer of ΔΠ decreases V – V
17
For the rest of the analysis I assume that cooperation is impossible to maintain,
and consider the effect of deadlines. With the deadline n = 1, the expected payoff to agent
i is
Ui(1) = – e + (1 - pjps)δΠ + δpjpsvi.
The expected payoffs from cheating are
Uid(1) = (1 - pk)δΠ + δpkvi.
Hence, there is no cheating if
Uj(1) - Ujd(1) = – e + ps(1 − pj)δ(Π - vj) ≥ 0.
(14)
for Junior and
(15)
Us(1) – Usd(1) = – e + pj(1 − ps)δ(Π - vs) ≥ 0.
for Senior. The second terms on the right of (14) and (15) cannot be ranked a priori
because ps(1 − pj) < pj(1 − ps) = ps – pj < 0 whereas Π – vj > Π – vs. However,
substituting from (12) shows, after manipulation, that if
(16)
e + (1 – δpj)(1 – δps)Π < (1 – pjps)δπ,
then Us(1) – Usd(1) < Uj(1) – Ujd(1), implying that Senior has less of the incentive to
cooperate than Junior. Thus, imposing a deadline can result in an incentive reversal.
An incentive reversal occurs because with the deadline Junior has a lower
continuation payoff than Senior (vj < vs) and is harmed more from failure at the deadline.
Thus, while Junior’s shirking does not impact the probability of success as much as
Senior’s, success now means more to Junior than to Senior. When (16) holds, Junior is
more concerned with success than saving the disutility, and has a greater incentive to
18
cooperate than Senior. In that case, if Us(1) – Usd(1) < 0 < Uj(1) – Ujd(1), Junior may
offer a side payment to Senior to ensure cooperation. Such a payment increases Us(1) –
Usd(1) more than it decreases Uj(1) – Ujd(1), so that such a side payment can always be
arranged provided that Us(1) + Uj(1) > Usd(1) + Ujd(1). The next proposition summarizes
the findings.
Proposition 5: With unobserved actions, Junior has a greater incentive to shirk without a
deadline. If(16) holds, the deadline n = 1 results in an incentive reversal; Senior now has
a greater incentive to shirk.
If both agents have the incentive to cooperate with the deadline n = 1, the rest of
the analysis is similarly to the preceding analysis. The following result, proved in the
appendix, is analogue to proposition 4.
Proposition 6. If (14) and (15) are satisfied, a unique optimal deadline exists.
7. Cooperation with repeated search
In the preceding analyses, agents always exit the game when there is success. This
section, motivated by Rouseau’s parable concerning stag hunters (see footnote 1),
considers the case in which the rewards are transitory so that agents must return to search
when the rewards run out. To keep the analysis simple, I assume that the rewards last just
one period, although the qualitative results obtained here are valid for any finite number.
19
When an agent is going it alone, the expected payoff, v, now satisfies this
recursive structure
v = - e + (1 – p)(δb + δ2v) + pδv.
which differs only in the reward from success from the previous analysis. While success
brings b in perpetuity in the preceding analysis, here an agent enjoys the benefit b for one
period (at date t + 1) and returns to search the next date (t + 2), at which he faces the
same expected payoff v. That is, the total reward from success is b + δv instead of π.
Collecting terms yields
v=
(1 − p)δ b − e
.
1 − δ p − (1 − p)δ 2
I again assume that search is worthwhile for each agent, i. e., v ≥ 0.
Turning to cooperation, assume there are just two agents to ease exposition. Given
the reward B per agent, the expected payoff V satisfies
V = - e + (1 – p2)(δB + δ2V) + p2δV
and hence
(1 − p 2 )δ B − e
V=
.
1 − δ p 2 − (1 − p 2 )δ 2
Assume that V > v or cooperation is desirable.
If agents cannot observe each other’s action so that shirking goes unpunished, the
expected payoff to an agent who shirks is
Wd = (1 – p)(δB + δ2V) + pδV.
I assume
20
(17)
V – Wd = - e + p(1 – p)δB – p(1 – p)(1– δ)δV < 0
so that cooperation is impossible to maintain.
Assuming that (15) holds, I now consider the effect of a deadline. As before, let
U(n) denote the expected payoff at t = 1 if both agents play the strategy Cn, and let Ud(n)
be the corresponding expected payoff from shirking at t = 1. For C1 these are written as:
U(1) = - e + (1 – p2)(δB + δ2v) + p2δv
and
Ud(1) = (1 – p)(δB + δ2v) + pδv.
There is no shirking if and only if
(18)
U(1) – Ud(1) = - e + p(1 – p)δ[B – (1 – δ)v].
Substituting from(17) and (18) yields
(V – Wd) – (U(1) – Ud(1)) = pδ(1 – p)(1 – δ)(v – V) < 0.
This indicates that there is a range of e such that imposition of a deadline induces agents
to cooperate for at least one period, while they cannot without a deadline.
The remainder of the analysis is similar to the one in section 4. I show, in
Appendix C, that U(n) is monotone increasing and that U(n) – Ud(n) becomes negative
for n sufficiently large. This leads to the next conclusion (see Appendix C for the proof).
Proposition 7: Suppose that U(1) – Ud(1) > 0. Then there is the optimal deadline n* ≥ 1,
which yields the maximum welfare U(n*) per agent.
21
8. Summary and suggestions for applications and extensions
In this paper I examine stochastic environments in which cooperation among
opportunistic agents is impossible to maintain when they cannot observe one another’s
actions. The main result has been that in such circumstances imposing a deadline can
induce agents to cooperate if the deadline is not set too far in time. In fact, there exists the
unique optimal deadline, which allows cooperation for the maximum duration, yielding
the greatest payoff to each agent.
Since the standard remedy for effort inducement in such circumstance is a twomode approach a la Green and Porter (1984), I then proceed to compare the two
approaches, finding that neither approach can strictly payoff-dominate the other. In the
text, I furnish a sufficient condition in which the optimal deadline strategy yields a
greater expected payoff than the optimal two-mode strategy.
I also consider two extensions to the basic model. In one I examine the effect of
asymmetry in probability of success between two agents. Here I find that a deadline can
result in an incentive reversal; without a deadline an agent having a greater chance of
success has less of an incentive to shirk, but without an deadline he may more likely shirk
than his partner. In the second extension, I examined the case in which agents do not exit
the game with success. In both cases I find that a deadline can induce cooperation when
agents cannot cooperate without one.
Besides co-authorship, the present theoretical framework may illuminate the role
of a deadline for inducing cooperation in a number of diverse situations. I make three
suggestions. A first is in industrial organization. Specifically, for pharmaceutical and biomed firms trying jointly to discover a cure for diseases, joint research projects often
22
depend crucially on government and private grants. Practically all such grants come with
time limits. Our analysis suggests that research grants with time limits can help firms
exert effort. In the cases of more general research joint ventures, the durations are usually
predetermined and often overseen by government agencies wary of antitrust actions.7 For
example, the Advanced Technology Program (ATP), a program within the U.S.
Department of Commerce, requires firms to specify the duration of an RJV on its
application form submitted to secure exemptions from antitrust laws. In Europe, firms
participating Eureka projects must commit to the duration of joint ventures at the time of
applications for research grants. Although these requirements are intended to keep track
of RJV activities, our analysis shows that they may also have the effect of facilitating
cooperation among firms.
Our analysis can also be applied to an area in law and economics. In most
countries including the U.S., crimes that are considered exceptionally heinous by society
have no statute of limitations. In 2006, following the footsteps of other industrial
countries, Japan also finally abolished its statute of limitations for murders. However,
according to the present analysis, statute of limitations may serve as a deadline, inducing
cooperating among law enforcement officers or agencies. The implication is that
abolishment of statute of limitations may have a negative impact on arrest records in
murder cases, which can be empirically investigated. The Japanese case may present a
natural experiment to test the deadline effect.
As a third application, the present analysis can throw a positive spin in
commitment in military strategies. In particular, President Obama drew heavy criticisms
7
Miyagiwa (2009) examines the relationship between research joint ventures and product market collusion.
23
when he announced his commitment to withdraw American troops from Iran by certain
dates. The criticism was based on the fear that the enemies would then just wait out until
withdrawals are complete and then occupy the vacated lands unopposed. However, the
present analysis furnishes a more positive note to the President’s exit plan. Our analysis
suggests that the commitment to withdraw troops by a certain date can serve as a
deadline, inducing the military to exert more efforts in a coordinated manner among all
its branches, making a final victory over enemies more likely before troop withdrawals.
All these applications involve empirical investigation of the deadline effect,
requiring separate treatments, as the present framework must be re-specified to fit the
separate cases. I look forward to working together with empirical economists in these
investigations in the near future, if they agree to a deadline.
24
Appendices
Appendix A: Proof of propositions 3 and 4
To prove proposition 3, differentiate both sides of (10) with respect to Π, noting that
sgn {d(U(n) – Ud(n))/dΠ} = sgn {d(Π - U(n - 1))/dΠ > 0,
The inequality holds since U(n – 1) contains Π only with positive probability. Thus, an
increase in benefit intensifies the incentive to cooperate. Thus, if U(n*+ 1) – Ud(n* + 1) >
0 in (10), then cooperation can be maintained at least for the first n* + 1 periods instead
of the first n* periods. This establishes proposition 3.
Turning to proposition 4, I show that an increase in m increases the incentive to
deviate, given n. The first things to show is that U(n) increases with m. To do so, write
U(n) = U(n; m). Suppose U(n; m) is positive. Then I can show that U(n; m) < U(n; m + 1)
by induction on n. It is easy to check using (3) that the inequality holds for n = 1.
Suppose that U(n; m) < U(n; m + 1) for n ≥ 1. Then by (7)
U(n + 1; m + 1) - U(n + 1; m)
= (pm – pm+1)δΠ + pm+1δU(n; m + 1) - pmδU(n; m)
= δpm{(Π – U(n; m)) –p(Π – U(n; m + 1))} > 0.
However, the incentive to deviate increases in m even faster. To see it, let Ud(n) = Ud(n;
m) and observe that by (8)
[U(n; m + 1) – Ud(n; m + 1)] – [U(n; m) – Ud(n; m)]
= pm(1 – p)δ(Π - U(n – 1; m + 1)) – pm-1(1 – p)δ(Π - U(n – 1; m))
= δ(1 – p)pm{p[Π – U(n – 1; m + 1)] – [Π – U(n – 1; m)]} < 0.
25
The inequality follows because U(n – 1; m + 1) > U(n – 1; m).
Appendix B: proof of proposition 6. For the deadline n ≥ 2, assuming that both exert
effort between t = 1 and t = n, I can write Ui(n) to agent i as
Ui(n) = – e + (1 - pjps)δΠ + δpjpsUi(n – 1).
Ui(n) is monotone increasing for each agent. There will be no incentive for agent i to
cheat if
Ui(n) - Uid(n) = – e + pk(1 − pi)δ(Π - Ui(n-1)) ≥ 0; i, k = j, s (i ≠ k).
As before, it can be shown that the right-hand side of this equation converges to V – Vid
< 0 as n tends to infinity. Thus, there is ni* satisfying that
Ui(n) - Uid(n) ≥ 0 > Ui(n + 1) - Uid(n + 1).
Let n* = min {ns*, nj*}.
Appendix C: Proof of Proposition 7. I first prove that U(n) is monotone increasing by
induction. Define ΔU(n) = U(n) – U(n – 1), with ΔU(1) = U(1) – v > 0. Given that ΔU(1)
> 0, a calculation shows that
(1b)
ΔU(2) = p2δΔU(1) > 0
Now assume that ΔU(n – 2) and ΔU(n – 1) are both positive. Then
(2b)
ΔU(n) = (1 – p2)δ2ΔU(n – 2) + p2δΔU(n – 1)
which is positive under the assumption. Thus, U(n + 1) > U(n), for all n ≥ 1.
26
Next I prove the existence of the optimal deadline n*, given that U(1) – Ud(1) > 0.
Note that U(n) approaches V as n tends to infinity. The expected payoff from cheating is
given by
Ud(n) = (1 – pm-1)(δB + δ2U(n – 2)) + pm-1δU(n – 1)
= Wd + pmδ(U(n – 2) – V)
which approaches Wd as n tends to infinity. Thus, as n tends to infinity, U(n) – Ud(n)
approaches V – Wd < 0. Thus the deadline cannot be extended infinitely while
maintaining cooperation.
27
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