Rivalry among agents seeking large budgets

Abstract

An agent competing for resources from a principal may benefit from having the principal believe that the agent shares his preferences, whereas the principal may prefer that agents reveal their types, inducing a separating equilibrium. Such incentives are explored in a model with a principal who sets a budget in two separate periods, and two different agents allocate that budget among services. In the second period, the principal allocates a larger budget to the agent that he believes is more likely to share his preferences. In the first period, each agent may behave strategically, spending more on the service the principal prefers, thereby hiding the agent’s type; this benefits the principal in the current period, but hurts him in the future because he does not know which agent would spend in the way he prefers. The principal may induce separation by giving the agents a large budget in the initial period, or by hiding his preferences from them.

Keywords

Budget bureaucrats delegation hidden information reputation

1. Introduction

A principal often wants an agent to behave in a particular way, but also wants an agent to reveal his type. The problem significantly appears when a principal delegates to agents spending decisions, with an agent’s preferences differing from the principal’s. Here, we can think of Congress allocating a budget to the Federal Aviation Administration, with the Administration deciding where to allocate air traffic personnel, how many hours each facility should be open, and so on. A state legislature may give a budget to a state university, with the university deciding how many faculty to hire in the humanities, how many in the social sciences, and so on. Or, a central government may transfer funds to a local government, with the local government deciding how to spend the budget.

The problem is most interesting when an agent prefers a large budget; this assumption is consistent with much work, beginning with Niskanen (1971), which assumes bureaucrats want to maximize their budgets. We shall consider two agents, with different preferences about the services they can provide. The principal also has preferences over the services. Each agent’s utility increases with his budget, because it allows him to spend more on all services. In period 2, the principal allocates whatever resources that remain after period 1 to the agent who more likely shares his preferences. If the principal is unsure of an agent’s type, an agent’s expected utility in period 2 increases with his reputation for sharing the principal’s preferences. The agent may strategically allocate his budget in a way that encourages the principal to give a larger budget in the next period; thus, the principal may not learn an agent’s type, giving a larger budget to the agent who would spend on services the agent rather than the principal wants.

The analysis first shows the allocation of the total budget by the principal when the principal knows the agents’ preferences. In each period he gives the funds to the agents who share his preferences. We then consider a principal who does not know the agents’ preferences, and we consider the agents’ incentives in period 1 when they know the principal’s preferences. An agent who reveals that his preferences differ from the principal’s, by spending on the service the agent prefers, risks getting a small budget in period 2. Whether the benefit of sincere behavior is greater than its cost depends on how much remains to be allocated in period 2. By setting the aggregate budget left for period 2, the principal can affect the agents’ incentives to pool on his preferences. An increase in the aggregate budget allocated in period 1 weakens the agents’ incentives to pool with the principal’s type. Thus, the principal must spend more to induce a separating equilibrium in period 1. When the budget required for a separating equilibrium is sufficiently large, the principal may forgo inducing such separation. Separation may occur when the principal may want to hide his preferences from the agents, because agents can pool on his preferred policy only if they know his preferences. If the principal and agents do not know each other’s preferences, the principal can induce a separating equilibrium with a small budget.

The principal–agent problem we examine differs from the standard principal–agent model which has the principal give the agent incentive payments. In the standard model, the payments do not affect what the agent can do, only what the agent will choose to do. In our model, the payment made by the principal affects an agent’s resources, and so affects what he can do, not only what he will want to do. Though our analysis will speak of monetary budget allocations, a scarce resource the principal controls can take other forms. For example, heads of different agencies may seek face time with the President.

Evidence is consistent with our assumption that a political principal prefers to fund jurisdictions that share the principal’s preferences. Larcinese et al. (2006) show that a state in the US received more money from the federal government if it heavily supported the incumbent president in past presidential elections.¹ Hodler and Raschky (2014) found that subnational regions have more intense nighttime light when they are the birth region of the current political leader. Competitions among agents for funding are common and important. In the US, federal allocations to states based on agency decisions rather than on legislative formulas averaged US$677 per person, ranging from US$401 in Florida to US$2400 in Alaska.² Individual states also use competitive grants to local governments. For example, in 2015, New York state held a competition to fund microgrids, with only 83 municipalities of 130 that had applied receiving grants.³

Rivalry among governmental agencies, including competition for budgets, is especially well-documented for the military. The US Department of Defense was established to coordinate the various branches of the US armed services. Infighting was seen as hurting military effectiveness during World War II, the Korean War, the Vietnam War, and the Cold War. Even before that, the Navy and the Army fought over who would control military aviation (Wildenberg, 2013). A typical example is an evaluation written during the Cold War:

The values to be found in interservice rivalry, then, are the values of competition… But, one may ask, doesn’t competition with the Soviet Union provide enough stimulus? The answer is that the Soviet challenge often seems too distant and hypothetical and uncertain, whereas the possibility of losing part of the budget to a sister Service is a clear and present danger. (Enthoven and Rowen, 1961, p. 370)

A study of interservice rivalry during the Vietnam War concludes that

[I]nterservice rivalry over issues connected with the military application of airpower may be especially acute because of the enormous resources at stake in the competition between the armed forces over budgets and responsibilities. (Horwood, 2006, p. 2)

In our terms, such rivalry represents a separating equilibrium.⁴ But the fear of rivalry and of budgetary losses could also lead an agency to avoid developing its favored policy; for example, the US Navy limited development of the Trident II missile because it feared spurring competition from the Air Force. In our terms, this behavior reflects a pooling equilibrium (Coate, 1996).

2. Literature

The following discussion considers how an agent’s actions affect the principal’s beliefs about the agent’s type. A large body of literature examines behavior intended to affect reputation, with the principal often viewed as voters, and the agent as an elected official. Reputational concerns may lead a politician to end a policy that he, unlike the voters, knows has failed (Beniers and Dur, 2007). And reputational concerns can induce political correctness: an advisor who wishes to avoid a reputation for bias may hide his information (Morris, 2001). A career-concerns model, where the incumbent attempts to signal ability, is analyzed by Canes-Wrone et al. (2001). Fox (2007) shows that an agent who cares about his reputation may adopt policies commonly associated with a high-quality agent, though the state of nature would call for a different policy. He further shows that if an agent can hide his actions from the public, this distortion can be reduced.

Relatedly, a career-driven agent whose action is observed has an incentive to conform (Prat, 2005). The principal suffers from such behavior, and may want to commit to keep the agent’s action secret. The concern about how an agent’s actions in one period affects the principal’s beliefs in future periods builds on the career-concerns model of Holmstrom (1999). The idea has been applied to politics (e.g. Persson and Tabellini, 2000) to consider incumbent policy makers who have implicit incentives to perform well to appear talented to voters, and where the incentives are limited to a retain-or-fire decision. Carpenter (2004) uses a career-concerns model to argue that the US Food and Drug Administration may delay approving some drugs because it wants to safeguard its reputation for protecting the public’s health.

The following analysis builds on Terai and Glazer (2015) in considering how reputation affects budgetary allocations made by a principal. Unlike that paper, however, the concern here is on the principal’s actions rather than of the agents, including how the principal will allocate a fixed budget over two periods, and on whether the principal benefits from hiding his preferences. Our consideration of a principal allocating money among agents relates to work on the Good Samaritan Dilemma, where an altruistic donor gives more money to poor recipients (Buchanan, 1977).

The strategic behavior of agents relates to the ratchet effect, which considers a worker who may exert little effort today: he anticipates that the employer may infer that high effort signals a low cost of effort, inducing the employer to offer a lower wage in the future. For example, in Lazear (1986) and Gibbons (1987) the worker has private information about the firm (such as the job’s difficulty), which he is reluctant to reveal. In Aron (1987) and in Kanemoto and MacLeod (1992) the worker has private information about a worker-specific attribute, such as ability.

An agent’s preferences can differ from the principal’s because the agent is corrupt, or influenced by special interest groups. The differences can also appear when the agent is intrinsically motivated, caring about policy or outcomes, rather than only about the income he earns. Studies of public administration provide evidence of intrinsic motivation among public-sector employees (Crewson, 1997; Guyot, 1962). Other work studies whether individuals with greater intrinsic motivation more often work in the public sector. For example, Gregg et al. (2011) use British survey data to investigate whether prosocial behavior (as measured by the probability of working extra, unpaid, hours) is more prevalent in the nonprofit sector than in the for-profit sector. These authors find that individuals in the nonprofit sector are more likely to work such extra hours. Survey data studied by Georgellis et al. (2011) also support the hypothesis that individuals are attracted to the public sector more by intrinsic than by extrinsic rewards.⁵

Related work applied to politics considers how a candidate may gain from concealing information about himself. Shepsle (1972) shows that ambiguity pays when voters are risk-loving. Glazer (1990) shows that if each candidate is uncertain about the median voter’s preferred policy (and, therefore, faces the risk of stating an unpopular position), then in equilibrium both candidates may adopt ambiguous positions. The benefits of ambiguity rise further if the position announced by one candidate allows the other candidate to estimate more accurately the voters’ preferences. Similarly, Alesina and Cukierman (1990) show that a party can increase its popularity by concealing its preferences from voters.

An important analysis of competition among government agencies is provided by Hirsch and Shotts (2015), who consider agents differing in their preferences, and exerting effort to make policy proposals. They cite experience under President Franklin Roosevelt, where a bureaucrat who wanted the President to adopt a policy that is close to the bureaucrat’s ideology rather than competing alternatives had to produce a well-crafted policy that would also achieve the president’s policy goals. The Agriculture and Interior departments, for example, engaged in such competition. We too consider competition between agencies with different preferences. Ting (2003) develops a model in which a principal chooses the number of agents, and shows that multiple agents might be less effective than a single agent. But our analysis, unlike the analysis in those papers, considers a principal who is unsure about the agents’ preferences, and who can allocate budgets to induce a separating equilibrium. In our model, multiple agents benefit the principal because after discovering agents’ types, the principal can allocate the budget to ones who share his preferences. Strayhorn et al. (2016) model a principal who budgets his time spent on monitoring agents, focusing on the tension between the benefits of monitoring in the current period and the threat of monitoring in future periods.

3. Assumptions

The principal has a fixed budget, R > 0. He allocates the budget to agents A and B in the two periods. In each of the two periods, each agent allocates his budget between two services. An agent has either H-type or L-type preferences. Let $x_{t 1}$ and $x_{t 2}$ be provisions of services 1 and 2 in period t. The utility in period t of a type-j agent is:

v_{t}^{j} = α^{j} x_{t 1} + (1 - α^{j}) x_{t 2}, α^{L} = 0, α^{H} = 1

(1)

Thus, a type-H agent values service 1 more than a type-L agent does. An agent’s name is indicated by superscript $k = A, B$ . The prior probability that k’s type is H is called $π^{k}$ . Assume more specifically that $π^{A} = π^{B} = π$ ( $0 < π < 1$ ), independently from each other.

The budget the principal gives a type-j agent in period t is called $R_{t}^{j}$ . The agent spends it either on service 1 or on service 2. Let $y_{t}^{j}$ be the proportion of $R_{t}^{j}$ that a type-j agent spends on service 1. Then the utility in period t of a type-j agent is:

v_{t}^{j} = R_{t}^{j} (α^{j} y_{t}^{j} + (1 - α^{j}) (1 - y_{t}^{j})) = R_{t}^{j} f^{j} (y_{t}^{j})

(2)

An agent’s utility over two periods is

V^{j} = v_{1}^{j} + v_{2}^{j}

(3)

Without loss of generality, let the principal’s type be H. The important difference between the principal and the agents is that the principal benefits from the services both agents provide, whereas each agent benefits only from the services he himself provides. The principal’s utility in each period is $x_{t 1}^{A} + x_{t 1}^{B}$ , or

v_{t}^{P} = R_{t} f^{H} (μ_{t} y_{t}^{A} + (1 - μ_{t}) y_{t}^{B})

(4)

where superscript P represents the principal, $R_{t}$ is the budget in period t that the principal allocates between agents A and B as $R_{t}^{A}$ and $R_{t}^{B}$ , and $μ_{t} \in [0, 1]$ is agent A’s share of the budget in period t; that is, $μ_{t} = R_{t}^{A} / R_{t}$ . The principal’s utility over two periods is

V^{P} = v_{1}^{P} + v_{2}^{P}

(5)

4. Budget allocations and agents’ actions

4.1. Perfect information about agents’ preferences

Consider, first, behavior under perfect information: the principal and the agents all know everyone’s preferences. The game proceeds as follows. In period 1:

Nature determines the types of the principal and of the agents.

The principal allocates the budget $R_{1}$ among agents A and B as $R_{1}^{A}$ and $R_{1}^{B}$ .

Each agent k simultaneously and independently allocates his budget $R_{1}^{k}$ for the two services.

In period 2:

The principal allocates the budget $R_{2}$ among agents A and B as $R_{2}^{A}$ and $R_{2}^{B}$ .

Each agent k simultaneously and independently allocates the budget $R_{2}^{k}$ for the two services.

Under perfect information, we can determine a subgame perfect equilibrium by examining the game backward. In the final stage in each period t, a type-j agent allocates his fixed budget $R_{t}^{j}$ between service 1 and service 2 to maximize his utility in equation (2). We denote the proportion of the budget spent on service 1 that maximizes a type-j agent’s utility in each period, in the absence of signaling considerations, by $y^{j}$ ; it is straightforwardly derived that $y^{H} = 1$ and $y^{L} = 0$ .

In each period t, realizing how an agent will spend a budget he gets, the principal will favor an agent who shares his preferences. If either agent is an H-type, the principal gives the entire budget to such an agent, giving nothing to an L-type agent. Otherwise, the principal allocates the budget to the two type-L agents. If the agents have the same preferences – that is, both are of type H or of type L– the principal gets the same utility from any allocation $μ_{t} \in [0, 1]$ of $R_{t}$ among agents with the same preferences.

Concerning the intertemporal allocation of the total budget R, the first-order partial derivative of the principal’s utility function with respect to $R_{1}$ is

f^{H} (μ_{1} y_{1}^{A} + (1 - μ_{1}) y_{1}^{B}) - f^{H} (μ_{2} y_{2}^{A} + (1 - μ_{2}) y_{2}^{B})

(6)

Thus, a principal is indifferent about the intertemporal allocation.

4.2. Imperfect information about agents’ preferences

Let each agent’s type or preferences be private information. Each agent knows that the principal’s type is H, but the principal is unsure about the agents’ types; moreover, each agent is uncertain about the other agent’s type.⁶ Events in this model unfold as in Section 4.1, except that at the beginning of period 2, the principal updates his beliefs about each agent’s type before he allocates the budget $R_{2}$ to agents A and B.

We will explore a perfect Bayesian equilibrium. In doing so, we are interested in how an agent’s strategic behavior in period 1 affects the principal’s beliefs, and in how the principal can discover the agents’ preferences. An agent will be said to act sincerely if he ignores signaling, choosing to provide the services in period 1 that maximize his utility in period 1. A principal who knows that an agent acts sincerely learns the agent’s type. An agent is said to act strategically in period 1 when he cares about signaling, even if that does not maximize the agent’s utility in period 1. If an agent acts strategically, and the principal knows that he does, the principal does not learn the agent’s type. Thus, sincere behavior by a type-L agent results in a separating equilibrium; strategic behavior results in a pooling equilibrium.

In period 2, each agent acts sincerely; a type-H agent spends his budget on service 1; a type-L agent spends his budget on service 2. The principal allocates the budget $R_{2}$ to an agent whose preferences are more likely to be strictly closer to his compared to another agent, if any. Otherwise, we assume that the principal gives half of the budget $R_{2}$ to each agent; the principal is indifferent about the two agents, and discriminating between agents having the same characteristics may not be justified by a social norm. The posterior probability that agent k’s type is H is ${\tilde{π}}^{k}$ . Then the principal allocates the fraction $μ_{2}$ of the budget to agent A in period 2 as follows:

{\begin{matrix} μ_{2} = 1, & if & {\tilde{π}}^{A} > {\tilde{π}}^{B} \\ μ_{2} = 1 / 2, & if & {\tilde{π}}^{A} = {\tilde{π}}^{B} \\ μ_{2} = 0, & if & {\tilde{π}}^{A} < {\tilde{π}}^{B} \end{matrix}

(7)

Consider an agent’s behavior in period 1. Call the equilibrium proportion of spending on service 1 in period t by an agent of type j as $y_{t}^{je}$ . Assume, as is plausible, that the principal’s beliefs at the decision nodes in the information set off the equilibrium path satisfy the posterior beliefs that if an agent allocates his spending in any way other than some (to be determined) $y_{1}^{He}$ in period 1, then the principal believes that agent is an L-type. Assume that in equilibrium agents of the same type take the same actions; thus, we can say that the equilibrium is symmetric. Under this assumption of symmetric behavior, it is natural to let the principal allocate his budget $R_{1}$ in period 1 equally between the agents. In Section 5 we will examine the outcomes without imposing equal allocation in period 1.

The following relation holds between an agent’s incentive to act sincerely and his budget in period 1:

Lemma 1. Let a type-j agent’s budget in period 1 be $0 < R_{1}^{j} \leq R$ . Then a type-j agent’s utility in period 1 by choosing $y^{j}$ is greater than his utility in period 1 by a different choice $y_{1}^{j} \neq y^{j}$ . The difference increases with the budget $R_{1}^{j}$ .

Proof of Lemma 1. See Appendix 1.

This lemma assures that an agent’s benefit from acting sincerely, thereby revealing his type, increases with his budget. The intuition is that if the agents get a large budget in period 1, they cannot get large budgets in period 2. So, an agent prefers to spend his large budget in period 1 on the service he prefers; in period 2, he may suffer a reduced budget because he reveals that his type differs from the principal’s, but his budget would then be small anyhow.

We can define the critical budgets inducing a pooling and a separating equilibrium as follows:

Lemma 2. Define ${\bar{R}}_{1}^{s}$ such that in a separating equilibrium the aggregate budget the principal allocates in period 1 is not less than ${\bar{R}}_{1}^{s}$ . Also define ${\bar{R}}_{1}^{o}$ such that in a pooling equilibrium the aggregate budget the principal allocates in period 1 is not more than ${\bar{R}}_{1}^{o}$ . If type-H agents take the same action in a separating and a pooling equilibria, $R / 2 \leq {\bar{R}}_{1}^{s} = {\bar{R}}_{1}^{o} < R$ (the critical budgets inducing separation and pooling are the same).

Proof of Lemma 2. See Appendix 2.

Now examine the principal’s choice in period 1. In contrast to the results under perfect information, the principal prefers unequal allocation of his budget R over two periods.

Lemma 3. Let $R_{1}^{s}$ be the budget in period 1 that maximizes the principal’s utility in a separating equilibrium; the budget in period 1 that maximizes the principal’s utility in a pooling equilibrium is $R_{1}^{o}$ . Let a type-H agent allocate his budget in period 1 according to his ideal preferences, $y^{H} = 1$ . Then $R_{1}^{s} = 0$ and $R_{1}^{o} = R$ . That is, in a separating equilibrium, the principal wants to spend more in period 2 than in period 1; in a pooling equilibrium, the principal wants to spend more in period 1 than in period 2.

Proof of Lemma 3. See Appendix 3.

According to Lemmas 2 and 3, the budget in period 1 to induce a separating equilibrium should be large, although the principal wants to leave a large budget for period 2; he would then allocate the budget in period 2 to the agents sharing his preferences. Also, the budget in period 1 that induces type-L agents to act strategically (hiding their types) should be small, leaving a large budget for period 2, although the principal wanted to spend a large budget in period 1 – then, an agent would spend more on the service the principal prefers.

The following lemma compares the principal’s utility with the solutions $R_{1}^{s}$ and $R_{1}^{o}$ :

Lemma 4. If in period 1 a type-H agent allocates his budget in accord with his ideal preferences, $y^{H}$ , and if the conditions for an L-type agent to behave sincerely or strategically do not bind, the principal has higher utility under pooling than under separation, or $V^{Po} (R_{1}^{o}) > V^{Ps} (R_{1}^{s})$ .

Proof of Lemma 4. See Appendix 4.

Intuitively, the gain from a pooling equilibrium lies with the agents spending more on the service the principal prefers in period 1. The gain from a separating equilibrium is that in period 2 he need fund only a type-H agent, if any; however, the possibility remains that he has to fund type-L agents, who will not spend on the service the principal prefers.

Based on Lemmas 2, 3, and 4, we can now determine the principal’s decision in period 1:

Proposition 1. Let the principal be unsure about the agents’ types. Then there exists a pooling equilibrium, and the equilibrium is unique. The budget the principal chooses in period 1 is $R / 2$ ; a type-H agent allocates his budget in accord with his ideal preferences; a type-L agent acts strategically, allocating his budget the same way a type-H agent would. ⁷

Proof of Proposition 1. See Appendix 5.

In a pooling equilibrium, the agents spend more on the service that the principal prefers. The effect resembles, or offers another explanation for, herd behavior. In particular, Prendergast (1993) shows that when advisors want the principal to think highly of themselves, they have an incentive to conform to the principal’s opinion, behaving as ‘yes men.’ The mechanism we discuss can yield similar outcomes.

4.3. Imperfect information about the preferences of the principal and agents

Consider, next, a principal who can hide his preferences. The principal’s type is either H or L. While the principal knows that his type is H, the two agents only know the prior probability $π^{P}$ ( $0 < π^{P} < 1$ ) that the principal’s type is H. The game proceeds as in Section 4.2. Looking at the game backwards, we can apply the logic used in Section 4.2 to the agents’ decisions on provisions of two services in period 1. An agent who wants to get a large budget in period 2 may behave as if his preferences match the principal’s. The agents, however, are uncertain about the principal’s type.

In period 1 the principal gives each agent a budget $R_{1} / 2$ , as discussed in Section 4.2. As the following lemma shows, if the principal and agents are equally likely to have each type, to induce a separating equilibrium, the principal may give a smaller budget in period 1 when he hides his type than when the agents know his type.

Lemma 5. Consider imperfect information about the principal’s and agents’ preferences. Let the principal be an H-type with probability 1/2, and let the same hold for an agent. Then for any $R_{1}$ a type-L agent reveals his type. Then, in the separating equilibrium, if any, the budget the principal chooses in period 1 is 0. In both periods, a type-H and a type-L agent each chooses his most preferred way of spending; that is, $y^{H} = 1$ and $y^{L} = 0$ .

Proof of Lemma 5. See Appendix 7.

Intuitively, an agent gets a larger budget in period 2 if the principal regards the agent as having the same type as his but another agent as having the different type; the agent gets no budget if the principal regards the agent as having the different type from his but another agent as having the same type as his. If the principal regards both agents as having the identical type, he equally splits the budget in that period between them. If $π^{k} = π = π^{P} = 1 / 2$ , with the agent revealing or hiding his preferences, these events occur with the same probabilities so that the agent’s expected utilities in period 2 are the same. This induces the agent, given any budget in period 1, to spend it in his most preferred manner, thus revealing his type. Also, we can infer that if $π^{k}$ and $π^{P}$ are diverged from 1/2, different outcomes will occur.

The principal, by hiding his type, cannot induce agents to spend as he prefers with a larger budget in period 1.

Lemma 6. Under imperfect information about the principal’s and agents’ preferences, no budget in period 1 greater than ${\bar{R}}_{1}^{o}$ has a type-L agent pretend to have type H ( ${\bar{R}}_{1}^{o}$ is calculated under the assumption that the principal’s type was known to agents). Any pooling equilibrium never has $R_{1} > {\bar{R}}_{1}^{o}$ .

Proof of Lemma 6. See Appendix 8.

Based on Lemmas 5 and 6, we can determine the principal’s decision in period 1.

Proposition 2. Let the principal be uncertain about the agents’ types, and let agents be uncertain about the principal’s type. Let the probability that the principal is an H-type be 1/2, and let the same hold for an agent. Then there exists a separating equilibrium, and the equilibrium is unique. The budget the principal chooses in period 1 is arbitrarily close to 0; each agent allocates his budget in period 1 according to his ideal preferences.

Proof of Proposition 2. See Appendix 9.

Comparing Proposition 2 with Proposition 1 suggests that under imperfect information about the principal’s and the agents’ preferences, the principal may induce a separating equilibrium by giving a small budget in period 1 and a large budget in period 2, which makes the principal obtain the same utility as under imperfect information only about the agents’ preferences.

5. Extension. Asymmetric allocation in period 1

So far, we have assumed a symmetric allocation in period 1. Indeed, the principal may be constrained to treat the agents the same in period 1. For example, it may be considered unfair or even unconstitutional for a central government to arbitrarily discriminate among jurisdictions that appear very similar. After the agents provide services, the agents may be seen as different; so allocating more to one agent than to another can be justified.

If the principal is not so constrained, then in period 1 he may want to give unequal budgets to the agents. Assume that the agents know the principal’s type, and suppose that the principal wants to ensure that his favored service is provided in sufficient quantity in both periods. Then the principal may benefit from an asymmetric allocation. In period 1 he gives a large budget to one agent (say, agent A). That induces the agent to reveal his type. Also, the principal gives a small budget to the other agent (say, agent B) in period 1. That induces the agent in period 1 to spend on the service the principal prefers, because the agent loses little by doing that. In equilibrium, the posterior probability that agent B is an H-type is ${\tilde{π}}^{B} = π$ . If agent A is an H-type and reveals his type, the principal gives all the budget in period 2 to agent A. If agent A is an L-type, the principal gives all the budget in period 2 to agent B.

In period 1, the principal faces a trade-off when choosing $R_{1}^{A}$ and $R_{1}^{B}$ . Strategic behavior by agent B in period 1 benefits the principal. Giving a large budget in period 1 to agent B ( $R_{1}^{B}$ ), instead of to agent A ( $R_{1}^{A}$ ), increases the principal’s utility. With a smaller $R_{1}^{A}$ , however, agent A may not be induced to reveal his type. Indeed, from Lemma 1, the principal can induce a separating equilibrium with a large $R_{1}^{A}$ . Then $R_{1}^{B}$ will be small, and, hence, the principal’s benefit when a type-L agent acts strategically in period 1 will be small.

We derive the solutions $R_{1}^{Ae} = \frac{2}{5 - π} R$ , $R_{1}^{Be} = \frac{1 - π}{5 - π} R$ , and $R_{2}^{e} = \frac{2}{5 - π} R$ , inducing type-H agents A and B to allocate their budgets in accord with their ideal preferences; type-L agent A behaves according to his ideal preferences, while type-L agent B behaves as if his preferences are the same as the principal’s (see Appendix 10). Clearly, $\frac{\partial R_{1}^{Ae}}{\partial π} > 0$ , $\frac{\partial R_{1}^{Be}}{\partial π} < 0$ , and $\frac{\partial R_{2}^{e}}{\partial π} > 0$ . If $π$ is large – that is, agent A’s preferences are known to be more likely aligned to the principal’s, and they are expected to be revealed – the principal needs a large budget to be left for period 2, to induce type-L agent B to behave as the principal would like in period 1. The principal may benefit from this large budget in period 2 because then agent A’s type is revealed.

Figure 1 shows the principal’s utility under asymmetric allocation, symmetric allocation inducing separation, and symmetric allocation inducing pooling. A symmetric allocation inducing pooling assures the principal of the highest expected utility; asymmetric allocation is ranked at intermediate. The difference in the principal’s utility, however, is smaller, with higher probabilities that an agent has the same preferences as the principal’s.

Figure 1.

Principal’s utility under asymmetric and symmetric allocations.

6. Conclusion

We had considered three effects of an increased budget given to an agent in the initial period. First, an increased budget in the initial period increases the level of services agents can provide in that period, to the principal’s benefit. Second, and relatedly, the larger the budget in the initial period, the smaller the budget the principal could give in future periods, thereby reducing spending on the service the principal favors. These effects are standard. Third, an increased budget in the initial period increases an agent’s incentive to spend on the services he prefers, thereby revealing to the principal his type. Such revelation, in turn, allows the principal to allocate budgets in a later period to an agent who would spend the way the principal prefers. This consideration, which is neglected in standard models examining the allocation of limited resources among agencies, can be important when the principal faces multiple agents. Furthermore, we showed that a principal may hide his preferences, thereby inducing agents to reveal their types while giving them small budgets in the initial period.

Some leaders have recognized the benefits of competition among agencies, coupled, perhaps, with some ambiguity about preferences. The historian Burns (1970) writes of President Franklin Roosevelt that ‘whatever Roosevelt’s impatience with public brawling, he essentially did not mind—he even welcomed—competition. “A little rivalry is stimulating, you know.”’ And when Roosevelt’s advisors and editorial writers called for a centralized agency to manage mobilization, he refused, ‘driving his jostling horses with a loose bit and a nervous but easy rein’ (p. 342).

The analysis has implications for how the principal’s performance changes over the two periods. If agents act sincerely, then in the first period the principal learns each agent’s type, and so can better allocate resources in the second period, thereby better achieving his goals. If, instead, agents act strategically in period 1, providing the services the principal prefers, then the principal’s performance is better in the first period than in the second; in the second period, each agent pursues his own objectives, with the principal not knowing which agent would better use the resources he gave them. Contrast these outcomes to those if the principal knew the agents’ preferences, but not their ability. The strategic behavior we considered would then not appear; the principal would learn in the first period the agents’ abilities, and in the second period could improve his performance by allocating resources to agents who were revealed to have high ability. Poor performance in a president’s second term would then be consistent with several implications of our model.

Our model can be interpreted in additional ways. The principal may believe that an agent is either honest or dishonest (say, by mis-spending some of the budget he receives), but may not initially know each agent’s type. An agent who is dishonest may, nevertheless, behave honestly in period 1, aiming to attain a large budget in period 2, thereby allowing him to mis-spend even more. An honest agent, in contrast, would not mis-spend in any period. Another interpretation would have the agents differ not in the preferences, but in their costs of providing each service. An agent may then provide the service in period 1 that he believes the principal prefers, but in period 2 the agent would provide the service which requires him to exert less effort. These interpretations indicate that the problem we address can apply not only to government, but also to private firms which contract with other firms for services, with the suppliers recognizing that their actions in period 1 can affect the contracts they get in period 2.

The analysis can apply to the idea of managed competition in the governmental provision of services. Managed competition is usually viewed as forcing potential providers – private businesses and/or public agencies – to compete against each other for contracts, based on their performance and cost (Osborne, 2007). We extend the idea to consider what is provided, rather than only how efficiently a specified service is provided.

Footnotes

Appendix 1. Proof of Lemma 1

The difference between a type-j agent’s utility in period 1 by choosing $y^{j}$ and his utility in period 1 by choosing $y_{1}^{j} \neq y^{j}$ , for $0 < R_{1}^{j} \leq R$ , is

(8)

R_{1}^{j} (f^{j} (y^{j}) - f^{j} (y_{1}^{j})) > 0

and the term on the left-hand side strictly increases with $R_{1}^{j}$ .

Appendix 2. Proof of Lemma 2

Under the utility function (2),

(9)

y^{H} = 1; y^{L} = 0; f^{H} (y) = y; f^{L} (y) = 1 - y

The necessary conditions for a symmetric separating equilibrium are

(10)

(R_{1} / 2) y_{1}^{He} + π (R_{2} / 2) + (1 - π) R_{2} \geq (R_{1} / 2) + π 0 + (1 - π) (R_{2} / 2)

(11)

(R_{1} / 2) (1 - y_{1}^{He}) + π (R_{2} / 2) + (1 - π) R_{2} \leq (R_{1} / 2) + π 0 + (1 - π) (R_{2} / 2)

The second term on each side of the inequality is an agent’s utility when another agent is an H-type; the third term represents an agent’s utility when another agent is an L-type. The sum of the terms on the right-hand side represents the maximum payoff each type-j agent can obtain after deviating from $y_{1}^{He}$ to $y^{j}$ and given ${\tilde{π}}^{k} = 0$ instead of ${\tilde{π}}^{k} = 1$ . Note that relation (11) is rearranged as

(12)

(R_{1} / 2) (- y_{1}^{He}) + (R - R_{1}) / 2 \leq 0

From Lemma 1, for $R_{1} \in (0, R]$ , the first term on the left-hand side of relation (12) decreases with $R_{1}$ ; also, the second term there decreases with $R_{1}$ . Thus, we obtain ${\bar{R}}_{1}^{s}$ , which is given by

(13)

{\bar{R}}_{1}^{s} = \frac{1}{1 + y_{1}^{He}} R

Similarly, the necessary conditions for the existence of a symmetric pooling equilibrium are

(14)

(R_{1} / 2) y_{1}^{He} + R_{2} / 2 \geq R_{1} / 2 + 0

(15)

(R_{1} / 2) (1 - y_{1}^{He}) + R_{2} / 2 \geq R_{1} / 2 + 0

The sum of the terms on the right-hand side represents the maximum payoff a type-j agent can obtain after deviating from $y_{1}^{He}$ to $y^{j}$ and given ${\tilde{π}}^{k} = 0$ instead of ${\tilde{π}}^{k} = π$ . We can rearrange relation (15) as

(16)

(R_{1} / 2) (- y_{1}^{He}) + (R - R_{1}) / 2 \geq 0

The same logic enables us to set

(17)

{\bar{R}}_{1}^{o} = \frac{1}{1 + y_{1}^{He}} R

Comparing equations (13) with (17) with the same value of $y_{1}^{He}$ , we have $R / 2 \leq {\bar{R}}_{1}^{o} = {\bar{R}}_{1}^{s} < R$ .

Appendix 3. Proof of Lemma 3

In a separating equilibrium where agents act sincerely, the principal should set $R_{1}$ to maximize his expected utility:

(18)

V^{Ps} (R_{1}) = R_{1} (π^{2} y_{1}^{He} + 2 π (1 - π) (y_{1}^{He} / 2)) + (R - R_{1}) (2 π - π^{2})

The assumption that $y_{1}^{He} = y^{H}$ implies that equation (18) is proportionally decreasing in $R_{1}$ , meaning that $R_{1}^{s} = 0$ .

In a pooling equilibrium the principal chooses $R_{1}$ to maximize

(19)

V^{Po} (R_{1}) = R_{1} y_{1}^{He} + (R - R_{1}) (π^{2} + 2 π (1 - π) (1 / 2))

The assumption that $y_{1}^{He} = y^{H}$ implies that equation (19) is proportionally increasing in $R_{1}$ , meaning that $R_{1}^{o} = R$ .

Appendix 4. Proof of Lemma 4

Comparing equations (18) to (19), with $y_{1}^{He} = y^{H}$ , shows that if $R_{1}$ in equation (18) equals $R - R_{1}$ in equation (19), then the principal’s utility in period 1 in equation (18) takes the same value as his utility in period 2 in equation (19). Therefore,

(20)

V^{Ps} (R_{1}^{s}) < V^{Po} (R - R_{1}^{s}) \leq V^{Po} (R_{1}^{o})

Appendix 5. Proof of Proposition 1

By anticipating $y_{1}^{He}$ , the principal calculates ${\bar{R}}_{1}^{s}$ or ${\bar{R}}_{1}^{o}$ which incentivizes type-L agents. This means that in equilibrium the principal’s utility is maximized with the appropriate choice of ${\bar{R}}_{1}^{s}$ or ${\bar{R}}_{1}^{o}$ based on his anticipation of the agents’ behavior. The equilibrium choice of the budget in period 1 by the principal is solved by substituting ${\bar{R}}_{1}^{s} = \frac{1}{1 + y_{1}^{He}} R$ and ${\bar{R}}_{1}^{o} = \frac{1}{1 + y_{1}^{He}} R$ to $R_{1}$ in equations (18) and (19), and comparing the principal’s utility under separation and under pooling. It is straightforwardly derived that $y_{1}^{He} = y^{H} = 1$ , leading to ${\bar{R}}_{1}^{s} = {\bar{R}}_{1}^{o} = R / 2$ , maximizes both of equations (18) and (19). Also, this satisfies the incentive-compatibility constraints for a type-H agent in a separating and a pooling equilibria (relations (10) and (14)); that is, $R_{1} \leq \frac{R}{2 - y_{1}^{He}}$ . Thus, we derive a pooling equilibrium in which

(21)

R_{1}^{e} = R / 2, y_{1}^{He} = 1, y_{1}^{Le} = 1

and the equilibrium is unique.

Appendix 6. Partial pooling

We will show that the outcome by partial pooling, in which a type-H agent plays a pure strategy and a type-L agent plays a mixed strategy, is inferior to the pooling equilibrium outcome, for the principal.

Denote by q the probability that an agent (say, agent A) having type L hides his type in period 1. If the principal observes that agent A chooses $y_{1}^{He}$ , then the posterior probability that he is an H-type is ${\tilde{π}}^{A} = \frac{π}{π + (1 - π) q}$ . If the principal observes that agent A chooses $y^{L}$ , then the posterior probability is ${\tilde{π}}^{A} = 0$ .

If another agent (say, agent B) having type L chooses hiding his type, the posterior probability the principal attaches on agent B who chooses $y_{1}^{He}$ is ${\tilde{π}}^{B} = \frac{π}{π + (1 - π) q}$ . When agent A plays $q \in (0, 1)$ , agent B’s expected utility by hiding his type is

(22)

(R_{1} / 2) (1 - y_{1}^{He}) + (π + (1 - π) q) (R_{2} / 2) + (1 - π) (1 - q) R_{2}

If agent B having type L reveals his type, the posterior probability that he is an H-type is ${\tilde{π}}^{B} = 0$ . Then, agent B’s expected utility by revealing his type is

(23)

(R_{1} / 2) + (π + (1 - π) q) 0 + (1 - π) (1 - q) (R_{2} / 2)

Agent B must be indifferent between hiding and revealing his type, which gives us

(24)

\frac{R_{1}}{R - R_{1}} = \frac{1}{y_{1}^{He}}

The principal should maximize his expected utility as follows:

(25)

\begin{matrix} \frac{1}{1 + y_{1}^{He}} R ((ξ_{1} + 2 ξ_{2} + ξ_{4}) y_{1}^{He} + (2 ξ_{3} + 2 ξ_{5}) (y_{1}^{He} / 2) + ξ_{6} 0) \\ + \frac{y_{1}^{He}}{1 + y_{1}^{He}} R ((ξ_{1} + 2 ξ_{3}) + (2 ξ_{2}) (1 / 2) + (ξ_{4} + 2 ξ_{5} + ξ_{6}) 0) \end{matrix}

where

Also, expression (25) implies that $y_{1}^{He} = y^{H} = 1$ is optimal for the principal. Moreover, the principal’s expected utility in expression (25) increases with q, meaning that a pooling equilibrium ( $q = 1$ ) assures the principal of the highest expected utility. Thus, the outcome by partial pooling is inferior to the pooling equilibrium outcome.

Appendix 7. Proof of Lemma 5

The necessary conditions for the existence of a separating equilibrium are

(26)

\begin{matrix} (R_{1} / 2) y_{1}^{He} + π^{P} π (R_{2} / 2) + π^{P} (1 - π) R_{2} + (1 - π^{P}) π (R_{2} / 2) + (1 - π^{P}) (1 - π) 0 \\ \geq R_{1} / 2 + π^{P} π 0 + π^{P} (1 - π) (R_{2} / 2) + (1 - π^{P}) π R_{2} + (1 - π^{P}) (1 - π) (R_{2} / 2) \end{matrix}

(27)

\begin{matrix} (R_{1} / 2) (1 - y_{1}^{H e}) + π^{P} π (R_{2} / 2) + π^{P} (1 - π) R_{2} + (1 - π^{P}) π (R_{2} / 2) + (1 - π^{P}) (1 - π) 0 \\ \leq R_{1} / 2 + π^{P} π 0 + π^{P} (1 - π) (R_{2} / 2) + (1 - π^{P}) π R_{2} + (1 - π^{P}) (1 - π) (R_{2} / 2) \end{matrix}

The second, third, fourth, and fifth terms on each side of the inequality, respectively, are associated with an agent’s utility when the principal is an H-type and another agent is an H-type; when the principal is an H-type and another agent is an L-type; when the principal is an L-type and another agent is an H-type; and when the principal is an L-type and another agent is an L-type. Substituting $π = π^{P} = 1 / 2$ in these relations shows that these hold only with $y_{1}^{He} = y^{H} = 1$ . It also shows that the sum of the second, third, fourth, and fifth terms takes the same value on each side of the inequality in relations (26) and (27).

Appendix 8. Proof of Lemma 6

The necessary conditions for a pooling equilibrium are

(28)

(R_{1} / 2) y_{1}^{He} + π^{P} (R_{2} / 2) + (1 - π^{P}) (R_{2} / 2) \geq (R_{1} / 2) + π^{P} 0 + (1 - π^{P}) R_{2}

(29)

(R_{1} / 2) (1 - y_{1}^{He}) + π^{P} (R_{2} / 2) + (1 - π^{P}) (R_{2} / 2) \geq (R_{1} / 2) + π^{P} 0 + (1 - π^{P}) R_{2}

The second term on each side of the inequality is associated with the principal having type H; the third term is associated with him having type L. Under the assumption that agents know that the principal’s type is H, relation (15) in Appendix 2 never holds with $R_{1} > {\bar{R}}_{1}^{o}$ . Also, relation (15) corresponds to relation (29) with $π^{P} = 1$ . This means that for $0 < π^{P} < 1$ , no value of $R_{1} > {\bar{R}}_{1}^{o}$ satisfies relation (29).

Appendix 9. Proof of Proposition 2

From Lemma 5, the principal, giving $R_{1}^{s} = 0$ and inducing separation, obtains the expected utility $(3 / 4) R$ , while the principal, giving ${\bar{R}}_{1}^{o} = 0$ and inducing pooling, obtains the expected utility $(1 / 2) R$ , from Lemma 6. Thus, separation benefits the principal.

Appendix 10. Asymmetric allocation in period 1

For agent A, the necessary conditions for revealing his type are

(30)

R_{1}^{A} y_{1}^{AHe} + R_{2} \geq R_{1}^{A}

(31)

R_{1}^{A} (1 - y_{1}^{AHe}) + R_{2} \leq R_{1}^{A}

For agent B, the necessary conditions for hiding his type are

(32)

R_{1}^{B} y_{1}^{BHe} + (1 - π) R_{2} \geq R_{1}^{B} + (1 - π) R_{2} / 2

(33)

R_{1}^{B} (1 - y_{1}^{BHe}) + (1 - π) R_{2} \geq R_{1}^{B} + (1 - π) R_{2} / 2

From relations (31) and (33) we derive

(34)

R_{1}^{A} \geq \frac{1}{1 + y_{1}^{AHe}} (R - R_{1}^{B}); R_{1}^{B} \leq \frac{1 - π}{1 - π + 2 y_{1}^{BHe}} (R - R_{1}^{A})

Note that $R_{1}^{A}$ and $R_{1}^{B}$ with which relations in (34) hold with equality are the principal’s best choice, anticipating $y_{1}^{AHe}$ and $y_{1}^{BHe}$ . Thus, we obtain

(35)

\begin{matrix} R_{1}^{A} = \frac{2 y_{1}^{BHe}}{(1 - π) y_{1}^{AHe} + 2 y_{1}^{AHe} y_{1}^{BHe} + 2 y_{1}^{BHe}} R \\ R_{1}^{B} = \frac{(1 - π) y_{1}^{AHe}}{(1 - π) y_{1}^{AHe} + 2 y_{1}^{AHe} y_{1}^{BHe} + 2 y_{1}^{BHe}} R \\ R_{2} = \frac{2 y_{1}^{AHe} y_{1}^{BHe}}{(1 - π) y_{1}^{AHe} + 2 y_{1}^{AHe} y_{1}^{BHe} + 2 y_{1}^{BHe}} R \end{matrix}

The principal’s expected utility is

(36)

π y_{1}^{AHe} R_{1}^{A} + y_{1}^{BHe} R_{1}^{B} + (π + (1 - π) π) R_{2}

Substituting expressions (35) to (36) shows that inducing type-H agents A and B to choose $y_{1}^{AHe} = y_{1}^{BHe} = y^{H} = 1$ is optimal for the principal. This satisfies the necessary conditions for type-H agents A and B (relations (30) and (32)). Then we obtain

(37)

R_{1}^{Ae} = \frac{2}{5 - π} R; R_{1}^{Be} = \frac{1 - π}{5 - π} R; R_{2} = \frac{2}{5 - π} R

Appendix

$F^{j} (\cdot, \cdot)$ Utility function of a type-j agent defined on consumption of service 1 and service 2

$x_{ti}$ Quantity of service i provided in period t

$y_{t}^{j}$ Proportion of spending on service 1 by a type-j agent in period t

$y^{j}$ Proportion of spending on service 1 by a type-j agent in the absence of signaling considerations

$y_{t}^{je}$ Equilibrium choice of proportion of spending on service 1 in period t by an agent of type j

$v_{t}^{j}$ A type-j agent’s utility in period t

$V^{j}$ Utility of an agent of type j over two periods

$v_{t}^{P}$ Principal’s utility in period t

$V^{P}$ Principal’s utility over two periods

R Total budget given to the agents

$R_{t}$ The aggregate budget given to the agents in period t

$μ_{t}$ Agent A’s share of the budget in period t

$R_{t}^{j}$ Budget given to a type-j agent in period t

$R_{1}^{s}$ The aggregate budget in period 1 that maximizes the principal’s utility in a separating equilibrium

$R_{1}^{o}$ The aggregate budget in period 1 that maximizes the principal’s utility in a pooling equilibrium

${\bar{R}}_{1}^{s}$ Critical value of the aggregate budget in period 1 that induces a separating equilibrium

${\bar{R}}_{1}^{o}$ Critical value of the aggregate budget in period 1 that induces a pooling equilibrium

$π^{k}$ Prior probability that agent k’s type is H

${\tilde{π}}^{k}$ Posterior probability that agent k’s type is H

$π^{P}$ Prior probability that the principal’s type is H

Acknowledgements

The authors wish to thank two anonymous referees for their helpful comments. We also greatly value the comments and discussions by Toshihiro Ihori, Yukihiro Nishimura, Kimiyoshi Kamada, and other participants at the 2016 Annual Meeting of the European Public Choice Society and the 2016 Spring Meeting of the Japanese Economic Association.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Financial support from the Grants-in-Aid for Scientific Research (A) (24243042), the Grants-in-Aid for Scientific Research (B) (26285059, 26285065), and the Grants-in-Aid for Scientific Research (C) (26380370) from the Ministry of Education, Culture, Sports, Science, and Technology, Japan are gratefully acknowledged.

Notes

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