OSIPP Discussion Paper : DP-2006-E-001

Dynamically Sabotage-Proof Tournaments
May 10, 2006
Junichiro Ishida

Associate Professor, Osaka School of International Public Policy (OSIPP)【キーワード】Sabotage; Tournament; Fast track; Late selection; Supermodularity;
Submodularity【要約】This paper examines a two-period tournament where agents may possibly
engage in destructive sabotage activities. Under plausible circumstances, sabotage
proves to be an effective tool for low-ability agents, especially when they are faced withhigh-ability agents.

The possibility of sabotage then gives rise to a dynamic concern,

similar to the Ratchet effect, because an agent runs a risk of becoming the target ofsabotage by signaling his high ability in early stages. In this dynamic setting, we firstestablish an impossibility result where the mere possibility of sabotage makes it
impossible to implement the first-best effort due to this dynamic concern. Given thisresult, we then offer two distinct incentive schemes, fast track and late selection, tocircumvent this problem. The fast-track scheme is likely to prevail when the production
process values diversity in inputs (submodular technologies) while the late-selection
scheme is to prevail when it values homogeneity (supermodular technologies). The
present model thus offers a mechanism through which both fast track and late selection
arise in a unified framework, which can explain the difference in managerial practices
between the US and Japan.

Correspondence to: Junichiro Ishida, OSIPP, Osaka University, 1-31 Machikaneyama
Toyonaka, Osaka 560-0043, Japan. E-mail: jishida@osipp.osaka-u.ac.jp

1

Introduction

Relative compensation schemes in general, or tournaments in particular, are a prevalent form ofevaluation adopted in many situations. Despite several virtues,1 however, it is also well known that
relative compensation schemes have some drawbacks that are inherent to their relative nature. One
particular form of these drawbacks, as suggested by Lazear (1989), is the fact that there may arise
an incentive to engage in destructive sabotage activities, instead of more productive counterparts,
because reducing the opponent’s output can be a close substitute for improving one’s own output
under relative compensation schemes. Since it is an essential task for any managers to build andsustain a cooperative, or at least non-hostile, work environment, it is of critical interest to see in whatways sabotage activities can be mitigated.
In this paper, we address issues that stem from the possibility that agents may resort to destructive
sabotage activities in a dynamic context. Consider a two-period tournament between two agents who
may differ in innate ability where each agent has three alternatives to choose from: exerting productive
effort, sabotage effort or no effort at all. To appreciate the dynamic inefficiency that arises in thissetup, we start the analysis by showing that sabotage tends to be an effective tool for low-ability
agents, especially when they are faced with high-ability opponents. That is, when the perceived
difference in innate ability is sufficiently large, it becomes the preferred option for the less able agent
to sabotage the opponent to fill this gap. This fact gives rise to a serious dynamic implication when
agents share some information about each other over time. Suppose that, at the end of each period,
each agent can observe the opponent’s productivity, which reflects both ability and effort. In such asituation, a high-ability agent essentially runs a risk of becoming the target of sabotage by signaling
his high ability in early stages. There then arises an incentive for high-ability agents to control theireffort in order to conceal their private information and appear less able.
1 There are several reasons why the principal uses relative compensation schemes, or tournaments in particular, as away to motivate agents. One reason is that with a relative compensation scheme, the principal can fix the total amountof prizes paid to the agents, which is especially important when the agents’ performances are not verifiable to a third
party. Also, relative compensation schemes become more effective when the agents are subject to common stochastic
shocks. Finally, it is often argued that relative performance evaluation is cheaper and easy to obtain in many regards
than absolute performance evaluation.

1

This dynamic concern yields a critical implication for the design of optimal incentive schemes.
Suppose that the cost of productive effort is so small that it is always optimal to induce productive
effort, whenever it is feasible to do so. Within this setup, we first show that any tournament that can
implement the first-best effort (always inducing productive effort) is not sabotage-proof under fairly
plausible conditions: in other words, there exists no contract that can implement the first-best effort inboth periods when sabotage is a viable option. The logic behind this result is fairly simple. When thefirst-best effort (everyone exerting productive effort) is implemented in the first period, any differencein the productivity must be attributed to the difference in innate ability. There then inevitably arises
a situation where the perceived difference in innate ability is so large that it is optimal for the less able
agent to resort to sabotage activities in the second period. This result thus indicates that although
the costs arising from sabotage activities are well recognized in the static setting,2 the possibility ofsabotage invites more serious problems in the dynamic setting than previously recognized.
Given this impossibility result, we then explore ways in which to mitigate sabotage in search of thesecond best, with a particular focus on how much weight to place on the ranking in each period. The
main issue here is how to device a tournament that can mitigate the incentive for low-ability agentsto exert sabotage effort in the second period. In general, there are two distinct ways to achieve thisgoal. More precisely, we show that the initial tournament that can implement the first-best effort in
the absence of sabotage can be made sabotage-proof by shifting the weight in either direction, i.e.,either towards the ranking in the first period or in the second. While these two schemes can equally
prevent sabotage from actually taking place, each comes at a cost with different implications.
First, consider a tournament which places more weight on the ranking in the first period: thatis, high-powered incentives are provided in the first period, followed by low-powered incentives in thesecond. The key aspect of this scheme is the pay compression to make sabotage less effective in thesecond period, as Lazear (1989) points out. Since less is at stake in the second period, this reduces the
incentive for low-ability agents to exert sabotage effort in that period. There is a cost associated with
2 An obvious cost of sabotage is that it substitutes for more productive effort. This also leads to another inefficiency,

as pointed out by Chen (2003), that the principal may fail to select the most deserving agent in the presence of sabotage
activities.

2

this scheme, however, because with low-powered incentives, it also fails to induce desirable productive
effort from low-ability agents. An illuminating property of this type of incentive scheme is that it
always induces productive effort from high-ability agents at a potential cost that low-ability agentsexert no effort in the second period. For expositional purposes, we refer to this as the fast-trackscheme since this scheme rewards the first-period winner more heavily.
Besides the fast-track scheme, there is another, totally opposite, way to mitigate sabotage in thissetting. Now consider a tournament which places more weight on the ranking in the second period:that is, low-powered incentives are provided in the first period, complemented with high-powered
incentives in the second. Since more is at stake in the second period, high-ability agents have an
incentive to conceal their private information in order not to get too much ahead of others because
they run a risk of becoming the target of sabotage in later stages by doing so. An illuminating propertyof this type of incentive scheme is that it always induces productive effort from low-ability agents at
a potential cost that high-ability agents exert no effort in the first period. For expositional purposes,
we refer to this as the late-selection scheme since this scheme rewards the second-period winner moreheavily.
The present model thus provides a potential explanation for both fast track and late selection ina unified framework, from a previously unexplored perspective. At the same time, the explanation is
also consistent with the difference in managerial practices between the US and Japan. It is argued
that the US firms often adopt the fast-track scheme by selecting promising candidates early on (alsoknown as the early selection of ‘stars’). This draws clear contrast to many Japanese firms which tend
to adopt the late-selection scheme where they do not differentiate workers for a substantial period oftime, roughly 10-15 years. We argue that the single most important factor in choosing between thetwo schemes is the nature of the production process. The fast-track scheme makes sure that highability agents always exert optimal effort. Since effort is positively related to ability, i.e., more able
agents exert more effort, effective inputs from each agent are highly diversified under this scheme.
This implies that the fast-track scheme becomes the optimal choice when the production process3values diversity, rather than homogeneity, in inputs. It is argued that this feature, often referred to as
submodularity, is more common in industries such as software, fashion and entertainment which value
new ideas and creativity, and are more prevalent in the US. In the late-selection approach, on theother hand, effort is negatively related to ability, which makes effective inputs from each agent fairly
homogenized. The late-selection scheme is then more likely to be efficient when the production process
values homogeneity in inputs. It is argued that this feature, often referred to as supermodularity, ismore common in industries such as automobile and consumer electronics which require careful andprecise implementation of tasks, and are more prevalent in Japan.3
While the argument thus far lays out an explanation for why fast track or late selection is more
often adopted given technological factors, we can also make an argument from the other way around.
For instance, many sociologists and historians alike point out that harmony (called ‘wa’ in Japanese)is one of the most important and salient concepts that prevail in the Japanese society: as the oldsaying goes, “a tall tree catches much wind,” which is believed to be one of the principals guiding andregulating people’s behavior in Japan. The cultural tendency that emphasizes harmony is believed tostem originally from Confucianism, which has had enormous influences on the Japanese culture in its
formative period. Many Japanese firms seem to inherit this cultural tendency where maintaining harmony or homogeneity within organizations once again appears to be the objective of utmost concern.
In this sense, compensation schemes which tend to minimize diversity (or preserve homogeneity) aresimply a natural fit for the Japanese society from the beginning. The main logic here can then be
regarded as an explanation for why industries with supermodular technologies have flourished andbeen so successful in Japan from a cultural perspective.
The paper is related to several strands of literature. First, there are many works that examine
the optimal timing of promotion or compensation with different approaches and focuses.4 Just to
name a few, one of the most influential approaches to explain delayed compensations is the incentive
3 Grossman and Maggi (2000) argue that this difference in the nature of technologies arises from the distributionof talent within each country. It is argued that submodular technologies are more prevalent in the US because the
distribution of talent is more diverse in the US.
4 For more extensive surveys, see Gibbons and Waldman (1999a) and Prendergast (1999).

4

approach by Lazear (1979), which posits that compensations should be delayed to maintain career
incentives. The signaling approach, most notably by Waldman (1984), states that promotions in earlystages tend to be inefficiently few because a promotion signals the worker’s ability, which in turnraises the retention wage.5 As for the difference in managerial practices between the US and Japan,
Prendergast (1992) argues that Japanese firms are able to delay promotion since workers are limited intheir mobility in the labor market. As an explanation for fast track, on the other hand, Meyer (1982)
constructs a two-period model and shows that it is optimal to bias the second-period tournament infavor of the first-period winner.6 Gibbons and Waldman (1999b, 2006) show that workers who receive
large wage increases early in their stay at one level of the hierarchy are promoted quickly to the nextlevel because workers who receive large wage increases are likely to be those with high ability.
The paper is also related to works that focus on negative aspects of relative compensation schemes.
A seminal paper on sabotage in a tournament is Lazear (1989). The paper is most closely related toChen (2003) who shows in a static tournament that able members are likely to be subject to sabotage
attacks, illustrating an inefficiency that the most able member might not have the best chance ofbeing promoted. The focus of the present paper differs from Chen (2003) as it is placed on dynamic
interactions between the agents and the ways in which to mitigate sabotage activities.7 The paper isalso related to a literature that deals with collusion under relative performance evaluation.8 Along thisline, Ishiguro (2004) shows that the principal can prevent collusion by offering asymmetric contracts,
even though agents are symmetric with respect to productive abilities. This paper is similar in spirit
as it seeks for sabotage-proof contracts, instead of collusion-proof, where agents have no incentive toexert sabotage effort.
The paper is organized as follows. The basic model is outlined in the next section and analyzed in
section 3. Section 4 characterizes the equilibrium without sabotage activities as a benchmark. Section
5 Owan (2004) applies this approach to explain the US-Japanese difference by complementing it with firm-specifichuman capital. Bernhardt (1995) extends this signaling approach to account for fast-track promotions as well as otherstylized facts of internal labor markets.
6 In this paper, we consider a situation where the principal is unable to bias the tournament in order to focus out
attention.
7 Chen (2003) discusses several schemes to mitigate sabotage activities, although not in a formal analysis.
8 Mookherjee (1984) is one of the first to point out that tournaments are vulnerable to collusion.

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5 then extends the analysis to incorporate potential sabotage activities and obtain the main results.
Section 6 discusses key properties of the model and section 7 makes some concluding remarks.

2

The model

2.1

Environment

Consider a two-period model in which a principal (female) hires two agents (male), denoted by i ∈{1, 2}, to produce output. Each agent differs in his innate ability ai ∈ {L, H}. We say that agent i
possesses high (low) ability if ai = H (ai = L). The prior distribution of the ability type is given byprob{ai = H} = θ ∈ (0, 1).
The ability type is the agent’s private information while the prior distribution is common knowledge.
Let µi,t ≡ prob{a−i = h | Ωi,t }, i 6= −i, denote agent i’s belief about the other agent’s ability type atthe beginning of period t, based on his information set Ωi,t . By construction, µi,1 = θ, i = 1, 2.

2.2

Production

In each period t, each agent must decide his effort levels (ei,t , di,t ) ∈ {0, 1}2 where ei,t indicates thelevel of productive effort while di,t indicates the level of (destructive) sabotage effort. We assume thateach effort level is a binary choice and moreover that each agent can exert at most one type of effort,
i.e., ei,t + di,t ≤ 1. The cost of productive effort is c while that of sabotage effort is (1 + λ)c.
The individual productivity of each agent, denoted by yi,t , depends on his ability and effort as wellas the other agent’s sabotage effort, if it is positive. We assume that the role of sabotage effort is tonegate the other agent’s productive effort. More precisely, the productivity of each agent is given by
yi,t = f (ai , ei,t − d−i,t ) wheref (H, 1) = h, f (H, 0) = f (L, 1) = m, f (H, −1) = f (L, 0) = l, f (L, −1) = 0.
We assume that the marginal value of productive effort is larger for high-ability agents.
Assumption 1 h − m > m − l > l > 0, i.e., productive effort is complementary to ability.
62.3

Information

The crux of the model is its information structure, i.e., who can observe what. In this model, we
consider a situation where the agents have access to more precise information regarding their productivity. More precisely, each agent i can observe the other agent’s productivity y−i,t in each period(although not the ability type nor the effort level). The principal, on the other hand, can only observe
the relative ranking of each agent ri,t ∈ {0, 1} where ri,t = 1 if agent i outperforms the other agent inperiod t. The relative ranking is an imperfect signal of the productivity and is given byprob{ri,t = 1 | yi,t , y−i,t } = G(yi,t − y−i,t ).
We assume that the distribution function G has the following properties: (i) it is strictly increasing in
its argument; (ii) G(0) = 0.5; (iii) limz→−∞ G(z) = 0 and limz→∞ G(z) = 1; and (iv) the corresponding density g is single-peaked and symmetric around zero, i.e., argmaxz g(z) = 0 and g(z) = g(−z) forall z.9 These properties imply that G(z) + G(−z) = 1 and G(x) − G(0) > G(x + z) − G(z) for any
x > 0 and z > 0, which we repeatedly use in the subsequent analysis.
If the relative cost of sabotage is too small, it is always optimal to exert sabotage effort; if it is toolarge, it is never to optimal to do so. To focus our attention to more interesting cases, therefore, we
make the following assumption regarding the relative size of these costs.
Assumption 2

2.4

G(h − m) − G(m − l)
> λ > 0.
G(m − l) − G(0)Contracts

Under the current information structure, the relative ranking is the only available measure of performances for the principal.10 Since the principal must rely on the relative ranking to motivate theagents, sabotage evidently becomes a serious issue for all parties involved in the transactions. Besides
9 All those properties hold when each agent’s productivity is subject to a shock drawn from the same distribution.
10 When agents are asymmetric with respect to the productivity, it is often optimal for the principal to set up a

tournament with a handicap. See Lazear and Rosen (1984). Meyer (1992) also shows that the possibility of biased
tournaments have important dynamic implications. In order to exclude this effect and focus our attention, we assume
that it is not feasible to bias the tournament one way or the other because the principal can only observe the relative
ranking.

7

this, we place two more restrictions on the class of contracts we consider in this paper. First, the agents
face a liquidity constraint so that the wages paid to the agents must be nonnegative. Second, the relative ranking is not verifiable to a third party so that the principal cannot offer asymmetric contracts
contingent on each agent’s identity. Under those restrictions, the loser always receives the minimum,
which is zero, and a feasible contract can generically be written as (w1 , w2 ) where wt is the wage paid
to the winner in period t. Alternatively, if this is a promotion tournament, (w1 ri,1 + w2 ri,2 )/(w1 + w2 )can be regarded as the promotion probability with the total prize w1 + w2 , and wt /(w1 + w2 ) as theweight given to each period t.

2.5

Payoffs

Since each agent has three alternatives, i.e., productive effort, sabotage effort, and no effort, there arepotentially many equilibria. To focus on more interesting cases, throughout the analysis, we restrict
our attention to sabotage-proof equilibria in pure strategies where the agents never exert destructive
effort by assuming that the principal’s payoff is prohibitively negative when sabotage activities evertake place. This implies that each agent’s equilibrium strategy profile is completely summarized by
L
H L
a
(eH
t (µi,t ), et (µi,t )), or (et , et ) for short, where et (µ) ∈ {0, 1} is the level of productive effort in periodaLt, conditional on the ability type a and the belief µ. We also define (pHt , pt ) where pt ∈ [0, 1] is theex ante probability with which an agent with ability a exerts productive effort in period t.11
For simplicity, we assume that the principal’s (gross) payoff is specified exclusively as a function
H L
Lof (pHt , pt ). Let π(pt , pt ) denote the principal’s payoff. The payoff is not necessarily the sum of yi,t

as it also captures the degree of externalities (or complementarities) between the agents. We assume
that productive effort is sufficiently valuable for the principal so that it is always optimal (and sociallyLefficient) to implement (pHt = 1, p1 = 1), t = 1, 2, at any finite cost whenever it is feasible.
11 The ex ante probability refers to the one at the beginning of period 1, that is, before any additional information(namely, yi,1 ) is observed.

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2.6

TimingNotice that the relative ranking observed by the principal reveals no relevant information about theactual realization of the ability type, and there is no incentive for the principal to restructure the initial
contract at the interim stage. For this reason, without loss of generality, we consider the following
timing of the model.
1. The principal offers a contract (w1 , w2 ), which may or may not be accepted by the agents. If thecontract is rejected, both parities receive zero and the game ends at this point.
2. In each period t = 1, 2, each agent determines the effort levels (ei,t , di,t ) and the outcome (yi,t , ri,t )is realized.
3. At the end of period 2, the principal pays the wage w1 ri,1 + w2 ri,t as specified by the contract.
Both the principal and the agents simply maximize the sum of the net payoffs with no discounting.
L
More precisely, the principal maximizes π(pHt , pt ) − wt over the two periods, subject to various incen-

tive compatibility constraints. Similarly, each agent maximizes the expected wage minus the cost ofeffort over the two periods.
In this model, fast track is regarded as a compensation scheme which emphasizes early performances and thus places more weight on the ranking in period 1 (w1 > w2 ). In the late selection approach, on the other hand, late performances become more important so that a compensation scheme
places roughly equal weight (w1 ≈ w2 ) or even more weight on the ranking in period 2 (w2 > w1 ). We
show that both situations can be optimal, depending largely on the nature of the production process.

3

Incentives for productive and sabotage effort

3.1

The second-period problem

To solve the model backward, we begin with the second-period problem. The agents’ incentives are
relatively straightforward in period 2 as it is virtually a static problem with no future concerns.
L
H
L
Let (eH−i,t (µ−i,t ), e−i,t (µ−i,t )), or (e−i,t , e−i,t ) for short, denote the opponent’s strategy profile

in period t, conditional on the ability type. Since each agent have three alternatives (productive
9

effort, sabotage effort and no effort), three constraints are sufficient to characterize the preferencesLamong them. First, given the opponent’s strategy (eH−i,2 , e−i,2 ) and the belief µi,2 , productive effort

is preferred to no effort if
L[µi,2 G(f (ai , 1) − f (H, eH−i,2 )) + (1 − µi,2 )G(f (ai , 1) − f (L, e−i,2 ))]w2 − cL≥ [µi,2 G(f (ai , 0) − f (H, eH−i,2 )) + (1 − µi,2 )G(f (ai , 0) − f (L, e−i,2 ))]w2 .

(1)Second, productive effort is preferred to sabotage effort if
L[µi,2 G(f (ai , 1) − f (H, eH−i,2 )) + (1 − µi,2 )G(f (ai , 1) − f (L, e−i,2 ))]w2 − cL≥ [µi,2 G(f (ai , 0) − f (H, eH−i,2 − 1)) + (1 − µi,2 )G(f (ai , 0) − f (L, e−i,2 − 1))]w2 − (1 + λ)c. (2)Finally, no effort is preferred to sabotage effort if
L[µi,2 G(f (ai , 0) − f (H, eH−i,2 )) + (1 − µi,2 )G(f (ai , 0) − f (L, e−i,2 ))]w2
L≥ [µi,2 G(f (ai , 0) − f (H, eH−i,2 − 1)) + (1 − µi,2 )G(f (ai , 0) − f (L, e−i,2 − 1))]w2 − (1 + λ)c. (3)To induce productive effort, a contract w2 must satisfy both (1) and (2). Moreover, any sabotage-proof
contract must satisfy at least one of either (2) or (3).
We first consider incentives for sabotage effort. Since productive effort is complementary to ability
and the role of sabotage effort is to negate productive effort, it is easy to imagine that the incentive
to engage in sabotage activities is stronger for low-ability agents. The following statement establishes
that when productive effort is less costly than sabotage effort, high-ability agents never exert sabotage
effort.
Proposition 1 Productive effort always dominates sabotage effort for high-ability agents.
Proof: See Appendix.

Even though productive effort is less costly, the same statement cannot be made for low-ability
agents because the return to productive effort is lower for them. For ai = L, (2) can be written asλc≥

H{µi,2 [G(l − f (H, eH−i,2 − 1)) − G(m − f (H, e−i,2 ))]

10L+ (1 − µi,2 )[G(l − f (L, eL−i,2 − 1)) − G(m − f (L, e−i,2 ))]}w2 .

(4)

Given this, two remarks about incentives for sabotage are in order. First, as Lazear (1989) points
out, one way to inhibit sabotage activities is to compress the pay scale. Second, notice that theleft-hand side is positive when eH−i,2 = 1 and µi,2 is sufficiently large. Since this is from the viewpointof low-ability agents, this implies that sabotage activities are more of a problem when the perceived
difference in ability is sufficiently large. This aspect of the model yields a critical dynamic implication.
In order to induce productive effort, the principal must satisfy both (1) and (4) simultaneously.
Examining these constraints closely, however, one can show that there is a range of the belief for which
this is not feasible. This yields the following statement.
Proposition 2 There exists some threshold µ̄ ∈ (0, 1) such that, given some belief µi,2 and eH−i,2 = 1,
low-ability agents never exert productive effort for µi,2 > µ̄.
Proof: See Appendix.

3.2

The first-period problem

The problem becomes more complicated in period 1 since the situation is now dynamic and what
each agent does in period 1 may influence the opponent’s belief, and the expected payoff, in period 2.
Define uai (ei,1 , di,1 ), ai = L, H, denote agent i’s expected payoff in period 2 as a function of his own
effort levels in period 1.
The incentive compatibility constraints need minor modifications in this dynamic setting. As
above, three constraints characterize the dominance relationship among the three alternatives. First,
Lgiven the opponent’s strategy (eH−i,1 , e−i,1 ), productive effort is preferred to no effort ifaiL[θG(f (ai , 1) − f (H, eH−i,1 )) + (1 − θ)G(f (ai , 1) − f (L, e−i,1 ))]w1 − c + u (1, 0)aiL≥ [θG(f (ai , 0) − f (H, eH−i,1 )) + (1 − θ)G(f (ai , 0) − f (L, e−i,1 ))]w1 + u (0, 0).

11

(5)

Second, productive effort is preferred to sabotage effort ifaiL[θG(f (ai , 1) − f (H, eH−i,1 )) + (1 − θ)G(f (ai , 1) − f (L, e−i,1 ))]w1 − c + u (1, 0)Lai≥ [θG(f (ai , 0) − f (H, eH−i,1 − 1)) + (1 − θ)G(f (ai , 0) − f (L, e−i,1 − 1))]w1 − (1 + λ)c + u (0, 1). (6)Finally, no effort is preferred to sabotage effort ifLai[θG(f (ai , 0) − f (H, eH−i,1 )) + (1 − θ)G(f (ai , 0) − f (L, e−i,1 ))]w1 + u (0, 0)Lai≥ [θG(f (ai , 0) − f (H, eH−i,1 − 1)) + (1 − θ)G(f (ai , 0) − f (L, e−i,1 − 1))]w1 − (1 + λ)c + u (0, 1). (7)The incentive compatibility constraints are now dynamic in that they include not only w1 but also
w2 .

4Equilibrium with no sabotage activities: a benchmark

We first consider as a benchmark a case where sabotage effort is not a viable option in order to single
out the impact of potential sabotage activities. We in particular seek for an equilibrium where theLHL
agents always exert productive effort in both periods, i.e., {(pH1 = 1, p1 = 1), (p2 = 1, p2 = 1)}.

Assuming that effort is sufficiently valuable, the optimal contract in this benchmark case is the onethat implements this effort profile, if such a contract exists.
With no sabotage activities, the agents’ behavior is completely characterized by a single constraint
in each period, (1) and (5). To induce productive effort with probability one, both of these constraints
must be satisfied for any possible contingency. The following statement presents a contract that canachieve this.
Proposition 3 Suppose that sabotage effort is not a viable option. The optimal contract is then givenby
w1 = w1∗ ≡cc, w2 = w2∗ ≡,θ[G(h − l) − G(h − m)] + (1 − θ)[G(m − l) − G(0)]G(h − l) − G(h − m)L
H
L
which implements {(pH1 = 1, p1 = 1), (p2 = 1, p2 = 1)}.

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Proof: See Appendix.

Notice that w2 > w1 , i.e., the principal must provide stronger incentives to induce productive
effort in period 2. This is because the agents possess more information about each other after a periodof production. Since the additional information sometimes discourages low-ability agents to exertproductive effort, stronger incentives are needed to overcome this.12

5Equilibrium with sabotage activities

We now introduce sabotage effort into the model and see how the mere possibility of sabotage activities
alters the optimal structure of incentives. What is especially important in this respect is proposition
2, which implies the following impossibility result.
Proposition 4 Suppose that sabotage effort is a viable option. Then, there exists no contract thatL
H
L
can implement {(pH1 = 1, p1 = 1), (p2 = 1, p2 = 1)}.

Proof :L
If (pH1 = 1, p1 = 1), the agents are able to identify the opponent’s ability type withprobability one and, hence, µi,2 ∈ {0, 1}. The situation described in proposition 2 is then bound toarise with some positive probability.
Q.E.D.
The proposition indicates that the principal is unable to implement the first-best effort and must
instead settle for the second-best. The main problem here is the possibility of sabotage activities inperiod 2. We now consider two distinct incentive schemes that can suppress sabotage activities.
One possible scheme is to provide low-powered incentives in period 2 and directly suppress sabotage
activities, as suggested by Lazear (1989). The well-known cost associated with this scheme is that with
low-powered incentives, the principal also fails to induce desirable productive effort. An illuminating
property of this type of incentive scheme is that it always induces productive effort from high-ability
12 To be more precise, the marginal value of productive effort (the marginal increase in the winning probability) for
low-ability agents decreases as the belief µi,2 increases, i.e., the opponent becomes more able.

13agents. We refer to this incentive scheme as the fast-track scheme since more weight is placed on early
performances.
H
The fast-track scheme (pH
1 = p2 = 1): Sabotage activities are circumvented by offering high-powered incentives in period 1, followed by low-powered incentives in period 2. The compensation
scheme places more weight on early achievements so that there are not sufficient returns for low-ability
agents to exert sabotage effort in period 2.
In this dynamic setting, there is another way to circumvent sabotage activities in period 2. Supposethat the principal provides low-powered incentives in period 1, complemented with high-powered
incentives in period 2. Since more is at stake in period 2, high-ability agents would rather choose toexert no effort in order to conceal their ability type. If µ̄ > θ, this can prevent sabotage activities
because the productivity reveals no relevant information and hence µi,2 = θ with probability one
Lwhen (eH
1 = 0, e1 = 1). An illuminating property of this type of incentive scheme is that it always

induces productive effort from low-ability agents, in contract to the fast-track scheme. We refer tothis incentive scheme as the late-selection scheme since more weight is placed on late performances.
L
The late-selection scheme (pL
1 = p2 = 1) Sabotage activities are circumvented by offering low-

powered incentives in period 1, complemented with high-powered incentives in period 2. The compensation scheme places less weight on early achievements so that more is at stake in period 2.

5.1

The optimal fast-track contract

L
In this equilibrium, the principal implements (pH1 = 1, p1 = 1) so that the ability type is perfectly

identifiable in period 2. To prevent any sabotage activities, incentives must be weak enough to satisfy(4). This obviously comes at a cost: satisfying (4) necessarily implies the violation of (1) for low-ability
agents with positive probability.
Proposition 5 The optimal fast-track contract is given by
w1 = w1FT ≡cc, w2 = w2FT ≡.
θ[G(h − l) − G(h − m)] + (1 − θ)[G(m − l) − G(0)]G(h − l) − G(m − l)
14

L
H
L
which implements {(pH1 = 1, p1 = 1), (p2 = 1, p2 = 0)}.

Proof: See Appendix.

Compared to the case without sabotage activities, the optimal first-track contract provides weaker
incentives in period 2, i.e., w2∗ > w2FT . This is because the principal must compress the pay structure,or minimize the difference between the winner and the loser, in order to mitigate potential sabotage
activities, at an expense that it fails to induce productive effort from low-ability agents. The incentive
scheme places more weight on early performances, i.e., the ranking in period 1 ri,1 , as a way toprotect high-ability workers from sabotage attacks from colleagues by designating the winner early
on: in other words, fast track is used as a safeguard for sabotage activities.

5.2

The optimal late-selection contract

Things are somewhat more complicated in the late-selection scheme. Under this scheme, the principal
L
implements (pH1 = 0, p1 = 1) so that the ability type is not identifiable in period 2. Although it is

not possible to induce productive effort only from low-ability agents in period 2, this may be possible
in period 1 because high-ability agents face different dynamic incentives.
Proposition 6 If θ is sufficiently small, the optimal late-selection contract exists and is given by
w1 = w1LS ≡cc, w2 = w2LS ≡,G(m − l) − G(0)
θ[G(h − l) − G(h − m)] + (1 − θ)[G(m − l) − G(0)]L
H
L
which implements {(pH1 = 0, p1 = 1), (p2 = 1, p2 = 1)}.

Proof: See Appendix.

In this late-selection approach, the principal starts with weaker incentives (w1∗ > w1LS ) and raises
the stake later on (w2LS > w2∗ ). Since more is at stake in period 2, high-ability agents have an incentive
to conceal their true type by exerting no effort even though it certainly lowers the probability ofwinning in period 1. As a result, agents are not initially differentiated for some time, maintaining
harmony within organizations.
156Discussion: diversity vs homogeneity

The previous section illustrates two distinct schemes that can suppress sabotage activities. The
question is then which scheme the principal should adopt. The principal adopts the fast-track scheme
over the late-selection scheme if
π(1, 0) − w1FT − w2FT ≥ π(0, 1) − w1LS − w2LS ,(8)which can be written as∆π ≡ π(1, 0) − π(0, 1) ≥cc−.
G(h − l) − G(m − l) G(m − l) − G(0)(9)

Among other factors, a critical determinant turns out to be the nature of the production function or,
more specifically, ∆π.
Our interpretation of ∆π is as follows. The first term π(1, 0) indicates the payoff when only
high-ability agents exert productive effort. Since effort is positively related to ability in this case,the distribution of effective inputs is more diversified. This term is then likely to be large when
the productivity depends more crucially on the best idea or the luckiest draw, disproportionatelyreflecting the input of a few highly talented individuals. It is argued that this feature, often referred
to as submodularity, is more common in industries such as software, fashion and entertainment which
value new ideas and creativity, and are more prevalent in the US.
The situation is totally opposite, on the other hand, when only low-ability agents exert productive
effort. Since effort is negatively related to ability in this case, the distribution of effective inputs ismore compressed and homogenized. The second term π(0, 1) is then likely to be large when there
are strong complementarities between the tasks and each task needs to be done equally well. It isargued that this feature, often referred to as supermodularity, is more common in industries such as
automobile and consumer electronics which require careful and precise implementation of tasks, andare more prevalent in Japan.
In light of this interpretation, the fast-track (late-selection) scheme is likely to prevail if the underlying production technology exhibits submodularity (supermodularity) and hence values diversity16

(homogeneity) in inputs. The present framework then provides an explanation for the difference inmanagerial practices between the US and Japan by focusing on the nature of production technologies.
It is often pointed out that Japanese firms typically do not differentiate workers for a substantial
period, roughly 10-15 years, which draws clear contrast to the early selection of ‘stars’ in the US.
Since Japanese firms are willing to evaluate workers over a long span of time, their workers are inno hurry to show off their talent and potential, and hence it is wise for those with high ability tokeep pace with others in early stages of their career. This pattern of behavior reflects a belief widelypervasive in the Japanese society that “a tall tree gets much wind,” which is believed to be one of
the most important principles guiding and regulating people’s behavior in Japan. We argue that thislate-selection scheme contributes in important ways to corporate culture that emphasizes harmonyand homogeneity in workplaces – one of the defining features of Japanese firms.

7

Conclusion

This paper constructs a two-period model of a tournament to illustrate dynamic inefficiencies thatarise from that possibility that agents may engage in sabotage activities. We first show that whensabotage is a viable option, it is impossible to implement the first-best effort under fairly plausible
circumstances. Given this result, we then show that the initial contract that can implement the firstbest effort without sabotage can be made sabotage-proof by shifting the weight in either direction, i.e.,either towards the first-period winner or towards the second-period winner. The critical determinantof the optimal scheme turns out to be the nature of the production process. The fast-track scheme
which rewards the first-period winner more heavily is optimal when the production process values
diversity in inputs; the late-selection scheme which rewards the second-period winner more heavily is
optimal when it values homogeneity. We argue that this result provides a mechanism through which
both fast track and late selection arise in a unified framework, which can explain the difference inmanagerial practices between the US and Japan.

17Appendix: the proofs
Proof of Proposition 1: For ai = H, (2) can be written asλc≥

H{µi,2 [G(m − f (H, eH−i,2 − 1)) − G(h − f (H, e−i,2 ))]L+ (1 − µi,2 )[G(m − f (L, eL−i,2 − 1)) − G(h − f (L, e−i,2 ))]}w2 .

(A.1)

Productive effort is then preferred to sabotage effort ifµ[G(m − f (H, eH − 1)) − G(h − f (H, eH ))] + (1 − µ)[G(m − f (L, eL − 1)) − G(h − f (L, eL ))] ≤ 0,

(A.2)

for any µ and (eH , eL ). Alternatively, we need to show thatG(m − f (H, eH − 1)) − G(h − f (H, eH )) ≤ 0,

(A.3)

G(m − f (L, eL − 1)) − G(h − f (L, eL )) ≤ 0,

(A.4)

both of which hold for any given (eH , eL ).
Q.E.D.
Proof of Proposition 2: Solving (1), we can show that productive effort is preferred to no effort if w2 ≥L
w(µi,2 ; eH−i,2 , e−i,2 ) wherew(µ; eH , eL ) ≡c.
µ[G(m − f (H, eH )) − G(l − f (H, eH ))] + (1 − µ)[G(m − f (L, eL )) − G(l − f (L, eL ))]L
Similarly, it follows from (4) that productive effort is preferred to sabotage effort if w̃(µi,2 ; eH−i,2 , e−i,2 ) ≥ w2

wherew̃(µ; eH , eL ) ≡λc.
µ[G(l − f (H, eH − 1)) − G(m − f (H, eH ))] + (1 − µ)[G(l − f (L, eL − 1)) − G(m − f (L, eL ))]Given that eH−i,2 = 1, there exists no contract that can induce productive effort from low-ability agents if
w(µ; 1, eL ) > w̃(µ; 1, eL ). The existence of the threshold µ̄ is guaranteed if w(1; 1, eL ) > w̃(1; 1, eL ), which canbe written as
G(h − m) − G(m − l)> λ.
G(h − l) − G(h − m)

18

(A.5)

Note that this condition holds becauseG(h − m) + G(m − l)G(h − m) − G(m − l)>> λ.
G(h − l) − G(h − m)G(m − l) − G(0)(A.6)

The first inequality holds since G(m − l) − G(0) > G(h − l) − G(h − m), which hold when the distribution issymmetric and single-peaked around zero. The second inequality holds by assumption 1.
Q.E.D.
Proof of Proposition 3: It follows from (1) that productive effort is preferred to no effort for high-ability
L
agents if w2 ≥ w(µi,2 ; eH−i,2 , e−i,2 ) wherew(µ; eH , eL ) ≡c.
µ[G(h − f (H, eH )) − G(m − f (H, eH ))] + (1 − µ)[G(h − f (L, eL )) − G(m − f (L, eL ))]Similarly, as can be seen in the proof of proposition 2, productive effort is preferred to no effort for low-ability
L
H
L
agents if w2 ≥ w(µi,2 ; eH−i,2 , e−i,2 ). Note that w > w for any µi,2 and (e−i,2 , e−i,2 ) so that w2 ≥ w holds if

w2 ≥ w. To induce productive effort with probability one, therefore, we need to focus on w.
LH
L
If (pH1 = 1, p1 = 1), then µi,2 ∈ {0, 1}. To implement (p2 = 1, p2 = 1), therefore, a second-period contract

must satisfy both w2 ≥ w(1; 1, 1) and w2 ≥ w(0; 1, 1). Note that w(1; 1, 1) > w(0; 1, 1). This implies that the
optimal contract in period 2 is given by
w2 = w(1; 1; 1) =c.
G(h − l) − G(h − m)

(A.7)

L
We now shift attention to the first-period problem. To implement (pH1 = 1, p1 = 1), the following condition

must be satisfied:
w1 ≥c + uai (0, 0) − uai (1, 0).
θ[G(f (ai , 1) − h) − G(f (ai , 0) − h)] + (1 − θ)[G(f (ai , 1) − m) − G(f (ai , 0) − m)](A.8)

The first-period constraint thus hinges critically on uai (0, 0) − uai (1, 0).
We first consider incentives for low-ability agents. For those agents, µ−i,2 = 0 no matter what they doin period 1. This implies that uai (0, 0) = uai (1, 0), and the problem is totally identical to that in period 2.
Low-ability agents exert productive effort if w2 ≥ w(θ; 1, 1).

19

Now consider incentives for high-ability agents. The equilibrium expected payoff for high-ability agents
under the candidate contract can be computed asuH (1, 0) = [θG(0) + (1 − θ)G(h − m)]w(1; 1; 1) − c.

(A.9)

Suppose that a high-ability agent unilaterally deviates and chooses ei,1 = 0. This signals to the opponent
that the agent is of the low-ability type. This does not influence the opponent’s behavior, though, because
the candidate contract always induce productive effort for any given belief. Again, the problem is reduced tothat in period 2. High-ability agents exert productive effort if w1 ≥ w(θ; 1, 1).
Since w(θ; 1, 1) > w(θ; 1, 1), the optimal contract in period 1 is given by
w1 = w(θ; 1, 1) =c.
θ[G(h − l) − G(h − m)] + (1 − θ)[G(m − l) − G(0)](A.10)
Q.E.D.

L
Proof of Proposition 5: Note that the ability type is not identifiable only when (pH1 = 0, p1 = 1); for any

other profile, the ability type is perfectly identifiable. Provided that effort is sufficiently valuable, it is optimal
LH
L
for the principal to implement (pH1 = 1, p1 = 1), if not (p1 = 0, p1 = 1).

Given this, we first obtain the following result.
L
L
Lemma 1 If (pH1 = 1, p1 = 1), then p0 = 0.

L
L
L
Proof: If (pH1 = 1, p1 = 1), then µi,2 takes either zero or one. We thus need to show that e2 (0) = e2 (1) = 0.
H
L
Suppose first that eL
2 (1) = 1. This implies that w2 ≥ w(1; e , e ), can be written as

w2 ≥

c
,
G(m − f (H, eH )) − G(l − f (H, eH ))(A.11)H
H
L
for some given eH∈ {0, 1}. To implement eL
2 (0) = e
2 (1) = 1, however, it is also necessary that w̃(1; e , e ) ≥w2 , which can be written as
w2 ≤

λc
.
G(l − f (H, eH − 1)) − G(m − f (H, eH )))(A.12)

Note that these two conditions do not hold simultaneously if w(1; eH , eL ) > w̃(1; eH , eL ). When eH = 1, thisbecomescλc>,
G(h − l) − G(h − m)G(h − m) − G(m − l)

20

(A.13)

which holds because
c
c
λc
>
>
.
G(h − l) − G(h − m)G(m − l) − G(0)G(h − m) − G(m − l)

(A.14)The second inequality holds by assumption 1. This implies that a necessary condition for eL
2 (1) = 1 iseH
2 (0) = 0 or, more precisely,
w2 < w(0; eH , 1) =c.
G(h − m) − G(0)(A.15)

Also, given that eH
2 (0) = 0, (A.11) becomes
w2 ≥

c
.
G(m − l) − G(0)(A.16)

This is a contradiction since (A.15) and (A.16) do not hold simultaneously. We can thus conclude thateL
2 (1) = 0.
H
Now suppose that eL
2 (0) = 1. Similarly as above, this implies that w2 ≥ w(0; e , 1), can be written as

w2 ≥

c
.
G(m − l) − G(0)(A.17)For this contract to be sabotage-proof, we must also have w̃(1; eH , 1) ≥ w2 . These two conditions do not holdsimultaneously ifcλc> w̃(1; eH , 1) =.
G(m − l) − G(0)G(l − f (H, eH − 1)) − G(m − f (H, eH ))(A.18)

Again, this condition holds only when eH = 0. We can then apply the same argument as above and show thateL
2 (0) = 0.
Q.E.D.
H
L
Given that pL
2 = 0, the best the principal can do is (p1 = 1, p1 = 0). We now construct a contract
L
which implements this. Note first that w(0; 1, 0) > w(1; 1, 0). To implement (pH2 = 1, p2 = 0), therefore, it isnecessary that w2 ≥ w(0; 1, 0). Besides this, any optimal contract must also satisfy (3) for low-ability agentsL(for any given µi,2 ∈ {0, 1}). Given (pH1 = 1, p1 = 0), the condition for this can be written as(1 + λ)c≥ w2 ,µ[G(h − l) − G(m − l)] + (1 − µ)[G(l) − G(0)]21(A.19)

for some given µi,2 = µ. It immediately follows from this that this condition is the most stringent when µ = 1.
Sabotage-proofness thus requires that
(1 + λ)cc> w(0; 1, 0) =,G(h − l) − G(m − l)G(h − l) − G(m − l)

(A.20)

which holds by assumption 1.
L
We now construct a contract that implements (pH1 = 1, p1 = 1). First, low-ability agents cannot influence

the opponent’s belief no matter what they do. This again implies that the problem for low-ability agentsin period 1 is identical to that in period 2, and the same constraints apply for this case. To implement
L(pH
1 = 1, p1 = 1), it is necessary thatw1 ≥ w(θ; 1, 1) =c.
θ[G(h − l) − G(h − m)] + (1 − θ)[G(m − l) − G(0)](A.21)

Now suppose that a high-ability agent deviates and chooses ei,1 = 0. This signals to the opponent that hepossesses low ability. The agent then exerts productive effort only if
θG(0) + (1 − θ)G(h − m) + uFT (1) − c ≥ θG(m − h) + (1 − θ)G(0) + uFT (0),(A.22)

which is reduced to
w1 ≥c + uFT (0) − uFT (1).
G(h − m) − G(0)(A.23)L
Given that the optimal second-period contract implements (pH2 = 1, p2 = 0), the deviation does not change

the opponent’s behavior: high-ability agents always exert productive effort while low-ability agents always
exert no effort, regardless of the belief. This implies that uFT (0) = uFT (1), and that the problem is again
identical to that in period 1. Since w(θ; 1, 1) > w(θ; 1, 1), the constraint for high-ability agents is not binding,and the optimal first-period contract is given by w1 = w(θ; 1, 1).
Q.E.D.
L
Proof of Proposition 6: If (pH1 = 0, p1 = 1), then the ability type is not identifiable, i.e., µi,2 = θ. To
L
implement (pH2 = 1, p2 = 1), it is necessary thatw2 ≥ w(θ; 1, 1) =c.
θ[G(h − l) − G(h − m)] + (1 − θ)[G(m − l) − G(0)]22(A.24)

For this to be sabotage-proof, we must also havew̃(θ; 1, 1) =λc
≥ w2 .
θ[G(h − m) − G(m − l)](A.25)The second-period contract must satisfy both of these conditions.
Given this, we now examine the first-period problem. We first consider incentives for low-ability agents.
If a low-ability agent deviates from the equilibrium and chooses ei,1 = 0, this signals to the opponent histrue type. Given the optimal contract in period 2, however, this does not change anyone’s behavior becausew(θ; 1, 1) > w(0; 1, 1) > w(0; 1, 1). This implies that the problem for low-ability agents in period 1 is identicalLto that in period 2, and the same constraints apply for this case. To implement (pH1 = 0, p1 = 1), it isnecessary thatw1 ≥ w(θ; 0, 1) =c.
G(m − l) − G(0)(A.26)The situation is more complicated for high-ability agents. If a high-ability agent deviates and chooses
ei,1 = 1, this signals to the opponent his true type. This may lead the opponent to exert sabotage effort inperiod 2 if he is of the low-ability type. Taking this dynamic aspect into account, the incentive compatibility
constraint needs some modification. It follows from (5) that productive effort is preferred to no effort for
high-ability agents if
w1 ≥H
c + uH
2 (0, 0) − u2 (1, 0).
θ[G(h − f (H, eH )) − G(m − f (H, eH ))] + (1 − θ)[G(h − f (L, eL )) − G(m − f (L, eL ))(A.27)

It is easy to obtain the expected payoff on the equilibrium path:uH
2 (0, 0) = [θG(0) + (1 − θ)G(h − l)]w2 − c.

(A.28)

It is, on the other hand, more complicated to see what happens off the equilibrium path. To this end, we firstestablish the following result.

Lemma 2 Suppose that a high-ability agent deviates and chooses ei,1 = 1, and moreover that the second-period
contract satisfies (A.24). Then, in period 2, (i) the opponent exerts sabotage effort if he is of the low-ability
type and productive effort if he is of the high-ability type; (ii) the deviating agent exerts productive effort.

23

Proof:

We examine what happens off the equilibrium path, when a high-ability agent deviates. Suppose

first that the opponent is of the low-ability type. The deviation then induces sabotage effort becausecλc> w̃(1; 1, 1) =,θ[G(h − l) − G(h − m)] + (1 − θ)[G(m − l) − G(0)]G(h − m) − G(m − l)

(A.29)

by assumption 1. Now suppose that the opponent is of the high-ability type. In this case, the opponent still
exerts productive effort because w(θ; 1, 1) > w(1; 1, 1).
Given this, the deviating agent still chooses productive effort if[θG(0) + (1 − θ)G(m − l)]w2 − c ≥ [θG(m − h) + (1 − θ)G(0)]w2 ,(A.30)[θG(0) + (1 − θ)G(m − l)]w2 − c ≥ [θG(0) + (1 − θ)G(l)]w2 − (1 + λ)c.

(A.31)

The first condition holds because
w2 ≥ w(θ; 1, 1) >c.
θ[G(h − m) − G(0)] + (1 − θ)[G(m − l) − G(0)](A.32)The second condition always holds for any nonnegative w2 .
Q.E.D.
This implies that the expected payoff is given byuH
2 (1, 0) = [θG(0) + (1 − θ)G(m − l)]w2 − c.

(A.33)The incentive compatibility constraint (A.27) can then be written as
w1 ≥c + (1 − θ)[G(h − l) − G(m − l)]w2.
θ[G(h − f (H, eH )) − G(m − f (H, eH ))] + (1 − θ)[G(h − f (L, eL )) − G(m − f (L, eL ))(A.34)

L
To implement (pH1 = 0, p1 = 1), therefore, the following condition must be satisfied:c + (1 − θ)[G(h − l) − G(m − l)]w2> w1 .
G(h − m) − G(0)(A.35)

L
To sum up, any contract that implements (pH1 = 0, p1 = 1) must satisfy (A.24), (A.25), (A.26), and (A.35).

These conditions are summarized asc
λc> w2 ≥.
θ[G(h − m) − G(m − l)]θ[G(h − l) − G(h − m)] + (1 − θ)[G(m − l) − G(0)]24(A.36)

c + (1 − θ)[G(h − l) − G(m − l)]w2
c> w1 ≥.
G(h − m) − G(0)G(m − l) − G(0)(A.37)

An obvious candidate is
w1 =

c
c
, w2 =.
G(m − l) − G(0)
θ[G(h − l) − G(h − m)] + (1 − θ)[G(m − l) − G(0)]This is optimal if it satisfies (A.37):
c + (1 − θ)[G(h − l) − G(m − l)]w2c>,
G(h − m) − G(0)G(m − l) − G(0)(A.38)G(h − m) − G(m − l)(1 − θ)[G(h − l) − G(m − l)]>.
θ[G(h − l) − G(h − m)] + (1 − θ)[G(m − l) − G(0)]G(m − l) − G(0)(A.39)

which can be written as

This condition holds when θ is sufficiently small.
Q.E.D.

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26