Contest design for a multi-tier organization: Leveraging informational environment to motivate downstream agents

2.1.1 Setup of the contest

Consider a contest organized by the owner or headquarters of a company with two identical subordinate branches (e.g., individual dealerships), i = 1, 2. Each branch consists of one manager and one agent, both assumed to be risk-neutral. As shown in Figure 1, the owner offers a winning prize, W , for these two branches to compete, with the aim of eliciting higher performance from agents (e.g., greater sales, higher quality of service, or improved customer satisfaction). Then, each branch manager decides on the value of a reward, B ( ≤ W ), for her own agent. Note that the values of B could be different across branches since each branch manager independently chooses to incentivize her agent. If a branch wins the contest, manager of that branch pays her agent the promised reward, B, and keeps the rest, W − B , as her own payment. Otherwise, she will not receive any prize from the owner nor pay reward to her agent.

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Figure 1.

Sequence of moves in the contest.

Once an agent is informed of the reward decision by her manager, she will determine the effort level exerted in the contest. Agents are assumed to have an identical individual production function. When agent i exerts an effort level e i , her output y i is calculated as y i = e i + ε i . The stochastic term ε_i captures the uncertainty faced by an agent, which is assumed to be uniformly distributed over the interval [ − q , q ] and independent across agents. Agent i will win the contest for her branch if her output is higher than that of her competing agent (who is working for the other branch), that is, y i > y − i , where y − i represents her opponent's output. ⁵

In this contest, the objective of the owner is to elicit a high level of effort from downstream agents, which entails branch managers offering large rewards to their own agents. For branch managers, however, they must carefully make tradeoffs to maximize their own payoffs. Specifically, by promising a larger reward to their agent, managers could expect a higher level of effort, which would lead to a greater chance of achieving a larger output than the competing branch. However, it also means that they must pay their agent more if their agent indeed outperforms the opponent and wins the contest, leaving managers with lower payoffs. For agents, they face another type of tradeoff when making their effort decision: a higher level of effort means a greater chance of producing a larger output than the competing agent, signifying a higher probability of securing the reward promised by their manager; but a higher level of effort also incurs a higher cost, which must be subtracted from their payment regardless of the outcome of the contest. Thus, there exist two pairs of vertically arranged principal-agent relationships in this contest (i.e., owner-manager and manager-agent), which is absent in a typical contest examined by previous research.

To solve for the optimal level of reward offered by branch managers and equilibrium effort from agents, we begin with the analysis of agents’ best response given managers’ reward decisions.

We assume that the two agents are risk neutral and have homogeneous utility functions that are separable in the reward received and the cost of effort exerted. According to the setup of the contest, agent i's expected utility is given by the following equation:

E U i = B i * P r o b ( y i > y − i ) − c ( e i )

(1)

e 1 = 2 B 1 q 8 k q 2 + B 1 − B 2

(2)

Similarly, Agent 2's best response is e 2 = 2 B 2 q 8 k q 2 + B 1 − B 2 .

Note that the best response of agents implies that their effort decisions are directly influenced by the rewards offered by the branch managers, but not the winning prize W.

In this contest, both branch managers are facing the same decision problem: maximizing the expected payoff by choosing an optimal level of reward for their agent. For the manager of branch i, her expected payment π_i is

E π i = P r o b ( y i > y − i ) * ( W − B i )

(3)

The optimal level of reward for manager i is the solution to the following first-order condition: ∂ E π i ∂ B i = ∂ P r o b ( y i > y j ) ∂ B i * ( W − B i ) − P r o b ( y i > y − i ) = 0 . Next, by plugging the best response of Agents 1 and 2 into the probability of winning the contest, that is, P r o b ( y i > y − i ) , and then solving the first-order conditions for both managers simultaneously, we obtain the symmetric optimal value of B 1 and B 2 as

B 1 * = B 2 * = W − 4 k q 2

(4)

Given these symmetric reward decisions by both managers, from agents’ best response (equation (2)), we find that both agents will exert the symmetric equilibrium effort at ⁷

e 1 * = e 2 * = W 4 k q − q

(5)

Note that in the best response equation of agents (equation (2)), the opposing agent's reward amount enters the function. While this information is not available to agents, it is reasonable to assume that they hold the belief that both branch managers, who are symmetric in this contest, should choose a symmetric level of reward, that is, B 1 * = B 2 * . Therefore, the best response of each agent in this contest becomes simply e i = B i 4 k q . Plugging this best response into the probability of winning the contest and solving the first-order conditions simultaneously, we find that both branch managers will still choose the symmetric optimal reward at the level of W − 4 k q 2 , which is identical to that presented in equation (4). As a result, the symmetric equilibrium effort for the agents remains at the level of W 4 k q − q (equation (5)).

The theoretical analysis above produces the following benchmark prediction on managers’ optimal reward decisions and agents’ equilibrium effort in this contest.

The core intuition behind Proposition 1 is that when agents choose effort to maximize their pecuniary payoffs from the contest, their calculations rely solely on directly relevant information—namely, the rewards set by their branch manager. As a result, the winning prize offered by the owner to branch managers affects agent effort only indirectly, through its impact on managers’ reward decisions. Moreover, since all branches are symmetric, agents can reasonably infer that their opponent receives the same reward when such information is absent and base their effort decisions on this belief. Branch managers’ optimal reward decisions thus remain unaffected by the additional contest information agents possess, and agents’ equilibrium effort levels are likewise unaffected by the additional information revealed to them.

2.2.1 Formulation of the benchmark

Following the studies on fairness concerns (e.g., Cui et al., 2009; Cui and Mallucci, 2012), we propose that agents assess the fairness of their reward B i by comparing it against the benchmark B ¯ . A reward is deemed fair if B i − B ¯ ≥ 0 ; otherwise, it would be considered unfair.

The central feature of our approach to capture fairness concerns lies in the formulation of the benchmark B ¯ , which is flexible in many ways. For example, when the owner discloses the size of winning prize W, agents naturally form an expectation of the “fair” share B ¯ = γ * W with 0 < γ < 1 , and compare their own reward B i against this benchmark to assess fairness. This constitutes a vertical comparison, in which agents compare their prospective payoff with that of their manager who supervises them and allocates the prize. Alternatively, if agents know the reward offered to their competing agent, the benchmark becomes B ¯ = B − i , where B − i denotes the opponent's reward. In this case, the perceived fairness arises from a horizontal comparison against the other agent (Ho and Su, 2009). Last, when agents are informed both W and B − i prior to making effort decision, both vertical and horizontal fairness concerns could be simultaneously activated. Thus, the type of fairness concerns activated in the contest is primarily determined by the information disclosed to agents. Because B ¯ depends on the information possessed by agents, the types of fairness concerns activated in the contest could be multidimensional. This dependency provides the owner with leverage to enhance the contest's effectiveness: by strategically selecting the information disclosed to agents, she can elicit the most conducive type of fairness concern, thereby better aligning managers’ decisions with her primary objective of maximizing agents’ effort in the contest. A key question we address in this study is which informational setting—vertical, horizontal, or both—is more effective. Together, these elements distinguish our study from prior research on fairness concerns.

2.2.2 Modified agent utility function

Building on existing behavioral economics models that incorporate the effects of social preferences (Bolton and Ockenfels, 2000; Charness and Rabin, 2002), we modify the utility function of an agent by accounting for the effect of fairness considerations into equation (1). Specifically, we let

E U i = { B i + Δ ( f a i r n e s s ) } * P r o b ( y i > y − i ) − c ( e i )

(6)

Δ ( f a i r n e s s ) = α * m a x { B i − B ¯ , 0 } − β * m a x { B ¯ − B i , 0 }

(7)

Based on the owner's decision about information disclosure, there are a total of four possible informational settings. We define and label each of them with two letters. The first letter represents whether the size of the winning prize offered to their managers is made known to agents, where “W” means yes and “N” means no. If the owner chooses to facilitate all agents knowing the reward offered to their opponent agent, the second letter is “O”; otherwise (i.e., “N”), agents will not have this knowledge. For instance, “NN” indicates the informational setting wherein agents are only aware of their own reward in the contest. ¹⁰

2.2.4 NN setting

We begin our discussion with the NN setting, which serves as the baseline when we compare the effects of agents’ fairness concerns across different information settings. In the absence of information regarding the winning prize W and opponent's reward B − i , agents in the NN setting do not have a viable way to evaluate the fairness of their reward, vertically or horizontally. As a result, Δ ( f a i r n e s s ) = 0 . Therefore, the analysis reverts to that of the standard economic model—the optimal reward decision for managers is B 1 = B 2 = W − 4 k q 2 and both agents would exert effort at the level of e 1 * = e 2 * = W 4 k q − q .

That intuition behind the predictions of the NN setting is: when additional contest information is withheld, agents cannot establish the benchmark B ¯ and thus are not able to judge the fairness of their reward, even if they value fairness. Agents therefore choose effort solely to maximize monetary payoffs (as presented in equation (1)). Aware of this, branch managers would not adjust their behavior—they make reward decisions as if their agents were purely motivated by monetary incentives.

2.2.5 NO setting

When agents are informed of the reward offered to their opponent agent, they gain necessary insight to draw comparisons across branches with B ¯ = B − i , thereby the horizontal fairness concern is activated. Recall that we assume B 1 ≥ B 2 when solving the standard economic model, so we have

Δ ( f a i r n e s s ) = α H * ( B 1 − B 2 )

(8)

Δ ( f a i r n e s s ) = − β H * ( B 1 − B 2 )

(9)

Plugging these two specifications into equation (6) for both agents, respectively, we have the best response for each agent as e 1 = 2 [ B 1 + α H ( B 1 − B 2 ) ] q 8 k q 2 + ( 1 + α H + β H ) ( B 1 − B 2 ) and e 2 = 2 [ B 2 − β H ( B 1 − B 2 ) ] q 8 k q 2 + ( 1 + α H + β H ) ( B 1 − B 2 ) .

Branch managers would then choose their reward at

B 1 * = B 2 * = W − 4 k q 2 1 + α H + β H

(10)

Since α H and β H are both assumed to be positive, this equilibrium reward is larger than that in the NN setting where the optimal reward is W − 4 k q 2 . Given the equilibrium reward offered by branch managers, agents’ equilibrium effort in the NO setting also increases from that in the NN setting to

e 1 * = e 2 * = W 4 k q − q 1 + α H + β H

(11)

Note that the increase in managers’ reward decisions leads to higher effort from agents, which better serves the objectives of the owner. However, it also implies that the expected payoffs for managers from the contest decrease since they have to share more of the prize with their agents.

When agents observe the value of the winning prize rather than their opponent's reward, vertical fairness concerns can arise. In this case, agents assess the fairness of their own reward relative to the benchmark B ¯ = γ * W , with 0 < γ < 1 . Under this informational setting, we derive managers’ optimal reward decisions based on the levels of γ .

(a) When γ is sufficiently small, managers have the incentive to set the reward above the threshold γ * W . As suggested by equation (7), such a strategy enables them to activate and leverage agents’ positive psychological utility (captured by α V ≥ 0 ), thereby inducing higher effort to improve the probability of winning the contest. Under this scenario, the fairness utility for agents is given by the following equation:

Δ ( f a i r n e s s ) = α V * ( B i − γ * W )

(12)

Following a procedure similar to that in the NO setting, we have the best response for each agent as e 1 = 2 [ B 1 ( 1 + α V ) − α V γ W ] q 8 k q 2 + ( 1 + α V ) ( B 1 − B 2 ) and e 2 = 2 [ B 2 ( 1 + α V ) − α V γ W ] q 8 k q 2 + ( 1 + α V ) ( B 1 − B 2 ) . Then, the optimal reward for branch managers is

B 1 * = B 2 * = W − 4 k q 2 1 + α V

(13)

e 1 * = e 2 * = W [ 1 + α V ( 1 − γ ) ] 4 k q − q

(14)

(b) When γ is sufficiently large, agents’ expectations for what is considered “fair” is too large for managers to satisfy in a profitable manner. According to equation (7), agents will then incur a utility loss characterized by β V ≥ 0 , and the psychological utility component becomes

Δ ( f a i r n e s s ) = − β V * ( γ * W − B i )

(15)

Under this condition, managers’ primary objective is to adjust rewards in order to mitigate the negative impact of vertical fairness concerns on agents’ effort. Specifically, anticipating the best response of each agent e 1 = 2 [ B 1 ( 1 + β V ) − β V γ W ] q 8 k q 2 + ( 1 + β V ) ( B 1 − B 2 ) and e 2 = 2 [ B 2 ( 1 + β V ) − β V γ W ] q 8 k q 2 + ( 1 + β V ) ( B 1 − B 2 ) , branch managers optimally increase the reward to the level of

B 1 * = B 2 * = W − 4 k q 2 1 + β V

(16)

which is again higher than in the NN setting. This optimal reward implies that when γ ∈ [ 1 − 4 k q 2 W ( 1 + β V ) , 1 ) , agents derive utility loss due to the fact that their expectation of reward is not met. The corresponding effort is

e 1 * = e 2 * = W [ 1 + β V ( 1 − γ ) ] 4 k q − q

(17)

(c) When γ is at the intermediate level, managers’ optimal reward depends on the relative magnitude of α V and β V . From the analyses in (a) and (b), we defined two benchmark intervals of γ : ( 0 , 1 − 4 k q 2 W ( 1 + α V ) ] and [ 1 − 4 k q 2 W ( 1 + β V ) , 1 ) , respectively. When β V > α V , we have 1 − 4 k q 2 W ( 1 + β V ) > 1 − 4 k q 2 W ( 1 + α V ) , which implies the existence of an intermediate range of γ , ( 1 − 4 k q 2 W ( 1 + α V ) , 1 − 4 k q 2 W ( 1 + β V ) ) . Within this range, managers can set rewards at the threshold γ * W , preventing the negative impact of vertical fairness concerns on agents’ effort. At the same time, any further increase in rewards is not profitable. Thus, the optimal rewards are B 1 * = B 2 * = γ * W . Because the threshold is exactly met, neither α V or β V will be activated, and the equilibrium effort for γ within this range becomes e 1 * = e 2 * = γ W 4 k q .

In contrast, when α V ≥ β V , we have 1 − 4 k q 2 W ( 1 + α V ) ≥ 1 − 4 k q 2 W ( 1 + β V ) , which implies an overlap between the intervals ( 0 , 1 − 4 k q 2 W ( 1 + α V ) ] and [ 1 − 4 k q 2 W ( 1 + β V ) , 1 ) . In this case, no distinct intermediate region arises, and managers’ optimal reward follows the characterizations established above in (a) and (b). Specifically, for γ ∈ ( 0 , 1 − 4 k q 2 W ( 1 + α V ) ] , the optimal reward is given by equation (13), whereas for γ ∈ ( 1 − 4 k q 2 W ( 1 + α V ) , 1 ) , managers choose reward level characterized by equation (16).

Thus, like in the NO setting, agents possessing additional information of the contest in the WN setting (i.e., the size of the winning prize) results in branch managers sharing more of the potential winning prize, and as a result, agents respond with larger effort, compared to the NN setting. This means that despite the branch managers incurring lower expected payoffs, the owner should expect a more favorable outcome when the information of winning prize is disclosed to agents.

In this informational setting, agents have complete information about the contest before making effort decisions, which means that they possess sufficient information to make comparisons horizontally and vertically. As a result, the specification of Δ ( f a i r n e s s ) should reflect the effects from both types of fairness concerns. One viable and intuitive way is to include both types of utilities discussed in the previous two settings (i.e., NO and WN) and assign each with a non-negative weight in the behavioral model. The analysis of managers’ and agents’ behavior in the WO setting follows the procedure analogous to that in the WN setting.

(a) When γ is sufficiently small, it is profitable for managers to activate α V . Therefore, the psychological utility component for Agent 1 is

Δ ( f a i r n e s s ) = m * α H * ( B 1 − B 2 ) + n * [ α V * ( B 1 − γ * W ) ]

(18)

Δ ( f a i r n e s s ) = m * ( − β H ) * ( B 1 − B 2 ) + n * [ α V * ( B 2 − γ * W ) ]

(19)

In the above specifications, m ≥ 0 and n ≥ 0 represent the weight of psychological utility derived from the horizontal and vertical fairness concerns, respectively. We opt for this way of capturing and presenting the psychological utilities mainly to keep the behavioral model parsimonious while its predictions tractable. ¹¹

In this case, Agent 1's best response function is

e 1 = 2 q [ B 1 ( 1 + m α H + n α V ) − m α H B 2 − n α V γ W ] 8 k q 2 + ( B 1 − B 2 ) [ 1 + m ( α H + β H ) + n α V ]

e 2 = 2 q [ B 2 ( 1 + m β H + n α V ) − m β H B 1 − n α V γ W ] 8 k q 2 + ( B 1 − B 2 ) [ 1 + m ( α H + β H ) + n α V ]

Based on these best responses, we can solve the optimal reward decision for both branch managers as follows:

B 1 * = B 2 * = W − 4 k q 2 1 + m ( α H + β H ) + n α V

(20)

This optimal reward implies that when γ ∈ ( 0 , 1 − 4 k q 2 W [ 1 + m ( α H + β H ) + n α V ] ] , agents should experience utility specified by equations (18) and (19). Facing this level of reward, each agent would then decide their equilibrium effort at

e 1 * = e 2 * = W 4 k q + n α V ( 1 − γ ) W 4 k q − q ( 1 + n α V ) 1 + m ( α H + β H ) + n α V

(21)

(b) When γ is sufficiently large, it becomes not profitable for managers to satisfy agents’ expectations, and therefore β V is activated. The psychological utility function for Agent 1 becomes

Δ ( f a i r n e s s ) = m * α H * ( B 1 − B 2 ) + n * [ β V * ( γ * W − B 1 ) ]

(22)

Δ ( f a i r n e s s ) = m * ( − β H ) * ( B 1 − B 2 ) + n * [ β V * ( γ * W − B 2 ) ]

(23)

We first solve for agents’ best response as

e 1 = 2 q [ B 1 ( 1 + m α H + n β V ) − m α H B 2 − n β V γ W ] 8 k q 2 + ( B 1 − B 2 ) [ 1 + m ( α H + β H ) + n β V ]

e 2 = 2 q [ B 2 ( 1 + m β H + n β V ) − m β H B 1 − n β V γ W ] 8 k q 2 + ( B 1 − B 2 ) [ 1 + m ( α H + β H ) + n β V ]

B 1 * = B 2 * = W − 4 k q 2 1 + m ( α H + β H ) + n β V

(24)

Comparing this reward decision with γ * W , we find that when γ ∈ [ 1 − 4 k q 2 W [ 1 + m ( α H + β H ) + n β V ] , 1 ) , the effect of vertical fairness concerns on agents’ effort decisions is captured by β V . Accordingly, both agents will exert effort at

e 1 * = e 2 * = W 4 k q + n β V ( 1 − γ ) W 4 k q − q ( 1 + n β V ) 1 + m ( α H + β H ) + n β V

(25)

(c) Similar to the WN setting, for the case that β V > α V , we have

1 − 4 k q 2 W [ 1 + m ( α H + β H ) + n β V ] > 1 − 4 k q 2 W [ 1 + m ( α H + β H ) + n α V ]

Therefore, there exists an intermediate level of γ , which is

( 1 − 4 k q 2 W [ 1 + m ( α H + β H ) + n α V ] , 1 − 4 k q 2 W [ 1 + m ( α H + β H ) + n β V ] )

Within this range, managers’ optimal reward becomes B 1 * = B 2 * = γ * W and the equilibrium effort is e 1 * = e 2 * = γ W 4 k q .

In contrast, when α V ≥ β V , we have

1 − 4 k q 2 W [ 1 + m ( α H + β H ) + n α V ] ≥ 1 − 4 k q 2 W [ 1 + m ( α H + β H ) + n β V ]

Thus, no distinct intermediate region arises. In this case, managers’ optimal reward follows the characterizations established above in (a) and (b). Specifically, for γ ∈ ( 0 , 1 − 4 k q 2 W [ 1 + m ( α H + β H ) + n α V ] ] , the optimal reward is given by equation (20), whereas for γ ∈ ( 1 − 4 k q 2 W [ 1 + m ( α H + β H ) + n α V ] , 1 ) , managers choose a reward level characterized by equation (24).

Since the psychological utility parameters and the weights of utility arising from the vertical and horizontal fairness concerns are all assumed to be non-negative, the optimal reward and equilibrium effort in the WO setting are also higher than that in the NN environment. Similar to the NO and WN settings, the owner will benefit from agents knowing more about the contest at the expense of lowered expected payoffs of managers.

Table 1 summarizes the theoretical analyses derived from our behavioral model. These results offer an alternative prediction to Proposition 1.

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Table 1.

Summary of behavioral predictions.

Informational setting	Reward	Effort
NN	W − 4 k q 2	W 4 k q − q
NO	W − 4 k q 2 1 + α H + β H	W 4 k q − q 1 + α H + β H
when β V > α V	W − 4 k q 2 1 + α V , for γ ∈ ( 0 , 1 − 4 k q 2 W ( 1 + α V ) ] γ W , for γ ∈ ( 1 − 4 k q 2 W ( 1 + α V ) , 1 − 4 k q 2 W ( 1 + β V ) ) W − 4 k q 2 1 + β V , for γ ∈ [ 1 − 4 k q 2 W ( 1 + β V ) , 1 )	W [ 1 + α V ( 1 − γ ) ] 4 k q − q γ W 4 k q W [ 1 + β V ( 1 − γ ) ] 4 k q − q
WN
when α V ≥ β V	W − 4 k q 2 1 + α V , for γ ∈ ( 0 , 1 − 4 k q 2 W ( 1 + α V ) ] W − 4 k q 2 1 + β V , for γ ∈ [ 1 − 4 k q 2 W ( 1 + α V ) , 1 )	W [ 1 + α V ( 1 − γ ) ] 4 k q − q W [ 1 + β V ( 1 − γ ) ] 4 k q − q
when β V > α V	W − 4 k q 2 1 + m ( α H + β H ) + n α V , for γ ∈ ( 0 , 1 − 4 k q 2 W [ 1 + m ( α H + β H ) + n α V ] ] γ W , for γ ∈ ( 1 − 4 k q 2 W [ 1 + m ( α H + β H ) + n α V ] , 1 − 4 k q 2 W [ 1 + m ( α H + β H ) + n β V ] ) W − 4 k q 2 1 + m ( α H + β H ) + n β V , for γ ∈ [ 1 − 4 k q 2 W [ 1 + m ( α H + β H ) + n β V ] , 1 )	W 4 k q + n α V ( 1 − γ ) W 4 k q − q ( 1 + n α V ) 1 + m ( α H + β H ) + n α V γ W 4 k q W 4 k q + n β V ( 1 − γ ) W 4 k q − q ( 1 + n β V ) 1 + m ( α H + β H ) + n β V
WO
when α V ≥ β V	W − 4 k q 2 1 + m ( α H + β H ) + n α V , for γ ∈ ( 0 , 1 − 4 k q 2 W [ 1 + m ( α H + β H ) + n α V ] ] W − 4 k q 2 1 + m ( α H + β H ) + n β V , for γ ∈ [ 1 − 4 k q 2 W [ 1 + m ( α H + β H ) + n α V ] , 1 )	W 4 k q + n α V ( 1 − γ ) W 4 k q − q ( 1 + n α V ) 1 + m ( α H + β H ) + n α V W 4 k q + n β V ( 1 − γ ) W 4 k q − q ( 1 + n β V ) 1 + m ( α H + β H ) + n β V

Behavioral prediction. Compared to the NN settings, agents’ knowledge of the additional contest information (i.e., in the NO, WN, and WO settings) will drive branch managers to strategically increase rewards; as a result, agents will respond with a higher level of equilibrium effort. ¹²

The key intuition behind this behavioral prediction is as follows. When agents who value both the size and fairness of their reward receive additional contest information, branch managers would voluntarily raise the rewards in order to keep their branch competitive in the contest. ¹³ Higher rewards, in turn, elicit greater effort from agents. This interaction between agents’ fairness concerns and managers’ reward choices provides the owner with the actionable insights and guidance for designing the optimal informational structure that better aligns managerial decisions with her overarching objective, maximizing organizational performance. For example, when γ is sufficiently small, managers have the incentives to offer rewards larger than agents’ expectations. Despite this, if the positive effect of vertical fairness concerns on effort is insufficient—the parameter α V , which captures how much agents are positively impacted by the vertical fairness concern, is small such that α V < 4 k q 2 ( α H + β H ) ( 1 − γ ) ( 1 + α H + β H ) W —then revealing the opponent's reward (activating horizontal fairness concerns) would induce higher effort than disclosing the winning prize alone. Otherwise, the owner can boost higher effort by activating vertical fairness concerns.

3.1.1 Experimental design

The primary goal of our experimental study is to delve into the influence of the informational environment, as shaped by the owner, on the decision-making processes of branch managers and the effort responses of agents. To achieve this, we exogenously varied the informational settings (i.e., contest-related information disclosed to agents) that are up to the owner to alter. We then focus on the interactions within dyads—comprising branch managers and agents—given the chosen informational environment by the owner. This approach enables clear observation and measurement of which informational setting would best serve the objective of the owner.

We implemented a 2 × 2 between-subject design in the laboratory by varying the additional information agents are aware of before they make effort decisions. As shown in Table 2, consistent with our theoretical analysis, there were a total of four treatments: NN, NO, WN, and WO. Each treatment consisted of four experimental sessions, and the duration of each session was around one and a half hours. We recruited a total of 256 participants from a large public research university in the United States. Each participant partook in only one of the four treatments. The experiment was implemented using the z-tree program (Fischbacher, 2007). Besides partial course credit, each participant received a sign-up bonus and earned cash payments based on their decisions in the lab experiment (on average, $18 in total).

The parameter values chosen for the lab experiment are q = 30, k = .008, and W = 65 or 85. ¹⁴ Based on these parameters, the standard economic theory predicts that both managers would choose B = 36.2 for their agents when W = 65, and B = 56.2 when W = 85. Correspondingly, agents’ equilibrium effort would be 37.7 and 58.5, respectively.

The experimental procedure closely follows the sequence of moves presented in Figure 1. To clearly explain the experimental procedure across different treatments, we describe the NN treatment in detail first and then contrast the other three treatments against it.

NN Treatment. In each session, participants were informed that they would participate in an experiment by completing the identical decision task for 12 rounds, and their monetary payoffs from the experiment depended on their own decisions and the decisions of others.

Role assignment and formation of group and contest: In each session, half of the participants were randomly and anonymously designated as Type 1 players (i.e., Branch Managers) while the rest were assigned as Type 2 players (i.e., Agents) throughout the entire session. Each round, one Type 1 player would be randomly and anonymously paired with one Type 2 player to form a Group.

NO, WN, and WO Treatments. The other three treatments share a very similar procedure to that in the NN treatment, except for two major differences. First, before branch managers made their decisions on the reward for their agent, they were reminded of the specific additional information their agent will receive according to the experimental design: in the NO treatment, the value of the reward for the competing agent; in the WN treatment, the value of W; and in the WO treatment, both pieces of information. Second, for agents, in addition to knowing the reward B chosen by their own manager, they also received additional treatment-specific information corresponding to the experimental design (i.e., the reward for their competing agent, the winning prize, or both) prior to their decisions on effort.

3.2.1 Managers offer higher average rewards when agents possess additional information

We begin the analysis with branch managers’ reward decisions. The left panel of Table 3 presents the average reward (B) offered by branch managers to their agent in each treatment, while Figure EC-1 (in the E-Companion) visually shows the average reward in each of the 12 decision rounds for W = 65 and W = 85. For both levels of W, consistent with the prediction of our behavioral model, the average rewards in the NN treatment are consistently lower than that in the other three treatments, where agents have the knowledge of additional contest information. ¹⁵ In fact, the largest average reward appears in the WN treatment, where agents are aware of only the winning prize offered to branch managers, followed by the NO and WO treatments. This is persistent across most of the 12 decision rounds.

To gain more insights on how branch managers’ behavior systematically differed across different treatments, we take the entire managers’ reward decisions and run a regression on the presence of different additional contest information made available to agents, controlling for the two levels of the winning prizes (65 and 85). From the left panel of Table 4, we observe that branch managers on average offer a larger reward when agents are aware of more information. Specifically, the coefficients on indicator variables “W revealed to agents” and “B revealed to competing agent” (8.7 and 5.6, respectively) are both significantly greater than zero. These increases in reward suggest that managers anticipate agents will be influenced by the perceived fairness of reward when making effort decisions. Consistent with our behavioral predictions on rewards in Table 1, the adjustments managers made in response to the change in informational setting indicate that they believe both ( α V + β V ) and ( α H + β H ) to be positive.

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Table 4.

Regression of reward and effort decisions on informational settings.

	Manager's reward decision	Agent's effort decision
Constant	33.9***	41.9***
	(1.28)	(2.08)
W revealed to agents	8.7***	0.8
	(2.09)	(2.96)
B revealed to competing agent	5.6***	6.0*
	(2.08)	(3.07)
Both W and B revealed	−10.5***	−1.4
	(2.97)	(4.34)
I (W = 85)	14.3***	10.7***
	(.64)	(.98)
Number of observations	1536	1536
R 2	.337	.093
Adjusted R2	.335	.090

* p < .1; ** p < .05; *** p < .01.

Robust standard errors clustered at the subject level are reported in parentheses.

It is noteworthy that branch managers’ reward decision is not monotonically increasing in the additional information available to agents. The coefficient on the indicator variable “Both W and B revealed” (–10.5) is significantly below zero and less than the sum of the other two positive coefficients (“W revealed to agents” and “B revealed to competing agents”). ¹⁶ This finding is particularly noteworthy as it suggests that while branch managers tend to offer higher rewards in the WO treatment compared to the NN scenario, they opt for lesser rewards when agents have access to more than one piece of additional information about the contest. ¹⁷ This implies that the owner providing more information may not necessarily prompt branch managers to further increase the size of rewards.

The average effort ( e i ) exerted by agents in each treatment is reported in the right panel of Table 3. This is complemented by Figure EC-2, which visually presents the average effort across the 12 decision rounds for W = 65 and W = 85, respectively. To systematically distinguish agents’ behavior among the four treatments, we conduct a regression analysis of agents’ effort decisions on the presence of various additional contest information and report the results in the right panel of Table 4.

We first notice that the coefficient for the variable “W revealed to agents” (0.8) does not significantly differ from zero, consistent with what we observed in Figure EC-2. We will revisit this observation later. Next, the coefficient for “B revealed to competing agent” (6.0) is significantly positive at the 10% level. This indicates that when agents are aware of their competing agent's reward—in scenarios where we’ve seen managers choose rewards averaging 5.6 points higher—there is an increase in average effort in the NO treatment compared to the NN treatment. Lastly, the coefficient for “Both W and B revealed” (–1.4) displays a negative direction but lacks statistical significance. However, the Wald test, comparing the WO treatment with the NN treatment, suggests a higher average effort in the former. ¹⁸ This implies that when agents possess both pieces of information, there is a discernible increase in effort compared to an environment devoid of such information.

In summary, the contest-related information disclosed to agents plays a critical role in shaping contest outcomes. We find that average effort is higher in the NO and WO treatments than in the NN setting, consistent with our behavioral predictions. However, a puzzling pattern arises in the WN treatment: despite substantially higher average rewards, agents’ effort does not significantly differ from that in the NN condition. From the owner's perspective, only disclosing the size of the winning prize prompts managers to allocate larger rewards, but this does not necessarily translate into greater agent effort. Instead, our findings indicate that revealing the opponent's reward—either alone or alongside the winning prize—proves more effective in boosting average agent effort than disclosing the winning prize in isolation.

3.2.3 Fairness concerns affect individual agents in an asymmetric way

While the above analysis is informative, examining aggregate agent effort alone may not fully capture the impacts of fairness concerns. In the contest, agents choose effort only after their manager sets the reward. Their effort is therefore shaped mainly by two distinct forces: (a) the monetary incentive generated by the specific reward they received (which may vary across agents even within the same treatment), and (b) the informational environment that moderates the influence of fairness concerns on effort provision. As a result, a simple cross-treatment comparison of raw effort levels (as presented in Table 4, Column 2) would confound monetary incentives with fairness concerns, making it difficult to isolate the effects of latter. ¹⁹

To address this issue, we refine our approach in three ways. First, we distinguish between agents’ theoretical best response and their observed effort by calculating the former from the standard economic model and subtracting it from the latter, yielding the “Deviation from the Best Response” (DBR), which captures the non-monetary influences. Second, since agents perceive fairness before choosing effort, we classify rewards as “fair” or “unfair” in each treatment, allowing more precise inference about the utility parameters, α and β , and the possibility of asymmetric activation of fairness concerns. ²⁰ Finally, recognizing that factors borne from experimental setup would likely operate uniformly across treatments, our analyses compare the three treatments with additional contest information (i.e., the NO, WN, and WO settings) against the benchmark “no information” NN treatment. Thus, DBR and baseline NN treatment play complementary roles: DBR removes the rational, incentive-driven component associated with endogenous rewards while NN anchors the comparison to ensure that identified effects can be attributed specifically to the disclosure manipulation.

Deviation From the Best Response. We begin by visually examining the relationship between DBR and the predicted best response in Figure 2, adding smoothing lines with 95% confidence intervals to illustrate trends. ²¹ If effort were driven purely by monetary incentives, DBR would align with the horizontal line at zero (black solid line). Instead, the smoothing lines for the NO and WO treatments remain consistently above zero, suggesting agents exert more effort than predicted—a surplus unexplained by incentives alone. In the WN treatment, the line starts near zero, but dips negatively as predicted effort rises. Even in the NN baseline, where agents know only their own reward, DBR is nontrivial: the smoothing line departs from zero across a wide range of predicted effort and, similar to WN, declines as predicted effort increases.

(a) What is driving the higher average effort in the NO treatment?

By answering this question, we aim to explore the possibility that the horizontal fairness concern ( α H and β H ), stemming from equations (8) and (9), have impacted agents in the NO treatment. We first divide agents in a contest into two categories: those who receive a reward higher than their competing agent and those who receive a lower one. It should be noted that agents in the NN treatment cannot determine whether they belong to the “higher” or “lower” category, as they are not informed about their competing agent's reward. Despite this, we also categorize those in the NN treatment into two groups. The purpose of this is to assess whether there are any systematic differences in agent behavior between the two treatments that can be attributed to the knowledge that facilitates the horizontal comparison. Next, using the indicator variable of I(Higher) = 1 for agents who receive a higher reward and 0 otherwise, along with the treatment indicator variable NO = 1 for the NO treatment and 0 for the NN treatment, we run a regression of DBR on the specification presented in the first panel of Table 5 (I. NO vs. NN). For this regression, we choose agents in the NN treatment who receive a lower reward than their opponent as the baseline.

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Figure 2.

Deviation from the Best Response (DBR) against the best response.

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Table 5.

Comparison of DBR against the benchmark.

I. NO versus NN		III. WO versus NN
Base (NN × Lower Reward)	8.8***	Base (NN × Lower Reward × Unfair Split)	12.3***
	(2.61)		(2.98)
NO	5.0	I (Higher Reward) × I (Unfair Split)	2.9
	(3.36)		(2.46)
I (Higher Reward)	−5.9***	I (Lower Reward) × I (Fair Split)	−9.3**
	(1.92)		(4.48)
NO × I (Higher Reward)	7.6***	I (Higher Reward) × I (Fair Split)	−11.5***
	(2.00)		(3.42)
I (W = 85)	−3.1**	I (W = 85)	−3.1**
	(1.23)		(1.36)
Observations	768	WO	−3.2
R 2	0.082		(4.07)
Adjusted R2	0.077	WO × I (Higher Reward) × I (Unfair Split)	−5.9
II. WN versus NN			(6.20)
Base (NN × Unfair Split)	12.9***	WO × I (Lower Reward) × I (Fair Split)	14.6**
	(2.87)		(5.65)
WN	−11.0**	WO × I (Higher Reward) × I (Fair Split)	20.6***
	(5.39)		(4.74)
I (Fair Split)	−11.2***	Observations	816
	(3.53)	R 2	0.146
WN × I (Fair Split)	6.9	Adjusted R2	0.137
	(5.75)
I (W = 85)	−3.4***
				(1.27)
Observations	768
R 2	0.124
Adjusted R2	0.119

* p < .1; ** p < .05; *** p < .01.

Robust standard errors clustered at the subject level are reported in parentheses.

First, we inspect the behavior of agents who received a lower reward than their competitors. The regression results show that, in the NN treatment, agents who receive a lower reward than their opponent exert greater effort than the predicted best response by 8.8 (t = 3.40, p = .001). At the same time, the coefficient on the indicator variable NO is 5.0 (t = 1.47, p = .140), indicating that agents in the NO treatment display a similar pattern of deviation from the best response. This suggests that β H is not significantly different from zero. ²²

Next, for agents who received a higher reward than their competitor in the NN condition, as indicated by the coefficient on I(Higher Reward), their DBR is significantly lower than the baseline (NN × Lower Reward) by 5.9 (t = −2.95, p = .004). This result is consistent with the downward-sloping trend observed in Figure 2. More importantly, the average DBR for their counterparts in the NO treatment is significantly larger (Wald test: 5.0 + 7.6 vs. 0, χ² = 13.99, p = .000). This suggests that when agents are clearly aware that their reward is higher than their opponent's, they exert effort greater than the best response to a greater extent than those who are not aware of it. Thus, our findings indicate that α H is positive. The scatterplots in Figure EC-3A corroborate this observation.

(b) Why do higher rewards not lead to higher effort in the WN treatment?

Recall that in the WN treatment, agents are informed about the winning prize offered by the owner to branch managers, so if any fairness concerns were to be activated, it would involve the vertical fairness concerns (i.e., α V and β V ). To examine how agents’ perceived fairness of the reward (relative to the winning prize) may have impacted their effort decisions, we again divide agents into two categories: those who received a reward that is considered a fair split of the winning prize and those who did not. We achieve this by taking 50% of the winning prize (W) as the cutoff point (i.e., γ = 0.5 ), and those rewards that are more than 50% of W are considered a fair split and an unfair one otherwise. ²³ Similar to what was done in the comparison of DBR across the NO and NN treatments, to compare the WN treatment against the benchmark NN treatment, we also divide agents in the NN treatment into two categories based on the same criteria. Next, we construct an indicator variable of I(Fair Split) = 1 for the fair split and 0 otherwise, along with the treatment indicator variable WN. We then run the regression of DBR on the specification presented in the bottom left panel (II. WN vs. NN) of Table 5. This time, we choose agents in the NN treatment who receive an unfair split as the baseline.

We first examine agents who received an unfair split of the winning prize by inspecting the first two coefficients. Notice that agents in the NN treatment who receive an unfair split exert significantly higher effort than the best response by 12.9 (t = 4.49, p = .000). By contrast, the average DBR for their counterparts in the WN treatment is 11.0 lower (t = −2.04, p = .042). This pattern indicates that when agents are able to vertically compare their reward against the winning prize, awareness of receiving an unfair split (i.e., lower than 50% of the winning prize) has a clear negative impact on effort compared to those who are not aware of this unfairness. This finding suggests that β V is significantly different from zero in our modified utility function.

Next, when inspecting agents who receive a fair split of the winning prize, we find that the average DBR in the WN treatment does not differ significantly from that in the NN treatment (Wald test: −11.0 + 6.9 vs. 0, χ² = 1.26, p = .262). This suggests that when agents in the WN treatment know precisely how much they and their manager could potentially earn and the division of the winning prize is perceived as “fair,” their effort deviation from the best response is similar to that of agents in the NN treatment. Accordingly, α V is not significantly different from zero. ²⁴

In summary, our findings reveal that the perceived unfair split stemming from vertical comparisons may have prevented agents in the WN treatment from exerting sufficient effort. Meanwhile, we do not find evidence of the activation of a positive effect derived from vertical fairness concerns. This not only explains why the overall average agent effort remains close between the WN and NN treatments but also accounts for the lower average DBR in the WN treatment compared to the benchmark NN treatment. The visual evidence presented in Figure EC-3B provides support for this finding.

(c) How do different types of fairness concerns influence effort provision in the WO treatment?

In contrast to the NO and WN treatments, the WO treatment is anticipated to activate both types of fairness concerns. To this end, we estimate a three-way interaction regression (as shown in the right panel of Table 5), using NN × Lower Reward × Unfair Split as the baseline.

The regression results show that when agents in the WO treatment learn that their reward is < 50% of the winning prize (i.e., an unfair split), those who get a reward lower than their competitor behave in a similar pattern to agents in the NN treatment (–3.2 vs. 0, t = –.80, p = .427). Moreover, for agents who receive a higher reward than their competitors, the difference in the average DBR between the WO and NN treatments remains statistically non-significant (Wald test: – 3.2–5.9 vs. 2.9, χ² = 2.35, p = .125).

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Table 6.

Summary of key findings and inferences of behavioral parameters.

Role	Decision	NO	WN	WO
Manager	Reward (vs. NN)	Higher than NN	Higher than NN	Higher than NN, but lower than NO or WN
	Perceived behavioral parameters	α H + β H > 0	α V + β V > 0	There is a diminishing effect when both types of fairness concerns present (i.e., m + n < 1 )
Agent	DBR (vs. NN)	Fair rewards increase effort, while unfair ones have no effect	Unfair rewards decrease effort, while fair ones have no effect	Vertically fair (unfair) rewards increase (decrease) effort, regardless of the horizontal fairness
	Inferred effects of behavioral parameters	α H > 0 β H = 0	α V = 0 β V > 0	α V > 0 and β V > 0 ; the effect of vertical fairness concern dominates that of horizontal fairness concern

Notes: Managers’ rewards in the NO, WO, and WN settings are all higher than that in the NN setting (Table 4), consistent with our behavioral prediction. Agents’ effort, as predicted by the behavioral model, is higher in the NO and WO settings, but not in the WN setting (Table 4). However, since simple cross-treatment comparisons of raw effort levels confound monetary incentives with fairness concerns, DBR analysis was used to isolate the effects of behavioral parameters. The table therefore reports the key results from the DBR analyses.

In contrast, when agents receive a fair split of the winning prize from their manager, but the reward is lower than their competitor's, those in the WO treatment exhibit a significantly larger increase in effort from the best response relative to their counterparts in the NN treatment (Wald test: – 3.2 + 14.6 vs. – 9.3, χ² = 6.37, p = .012). The difference in the average DBR becomes even more pronounced for agents whose reward exceeds that of their competitor (Wald test: – 3.2 + 20.6 vs. – 11.5, χ² = 26.99, p = .000). These findings indicate that agents in the WO treatments demonstrate much larger increase in effort than the baseline NN treatment whenever the manager allocates the prize fairly, regardless of whether their reward is higher or lower than that of their competitor.

Taken together, while both winning prize size and opponents’ reward affect agents’ effort, fairness in splitting the prize with managers dominates horizontal fairness concerns. ²⁵ Average effort is higher in the WO treatment than the NN treatment, largely because agents receiving more than 50% of the prize (representing 74% of cases) exert extra effort (over the best response). This large effort effectively offsets the lower effort from the rest of the agents facing unfair splits. Figure EC-3C clearly demonstrates this conclusion with the top two graphs displaying the most frequently occurring observations (i.e., the fair split) with the divergence in behavioral pattern between the NN and WO treatments. ²⁶

Overall, our analysis of DBR offers deeper insights into how behavioral parameters shape agents’ effort across informational settings. Coupled with our findings on managers’ perceptions of these behavior parameters (Section 3.2.1), these results yield valuable findings, which are summarized in Table 6. ²⁷