Setting the Budget for Targeted Research Projects

Proof of proposition 1

First, the agency must set R ≥ π_H if it wants to induce participation of agents in state (AB); in fact, a budget cap greater than π_H is suboptimal as it would generate an (extra) rent for agents in state (A) and (AB) without effects on their participation. When R = π_H, agents in state (AB) participate and submit a price bid t = π_H. In the rest of the proof, we focus on the price-bidding behavior of agents in state (A) and we restrict attention to symmetric equilibria where each agent in state (A) follows the same bidding strategy.

Agents in (A) will never submit bids below k, for if they win with such a bid, they will not be willing to implement the project. In addition, they will not submit bids above π_H or their bids will be discarded. Furthermore, there is no equilibrium in which all agents submit t = k as they would get no surplus whereas an agent could secure a payoff (1−PA)n−1πH−ϵ−k>0 by submitting a bid arbitrarily close to π_H. They will not submit bids equal to π_H, either, as they would also face the competition of agents in (AB), which they could avoid by undercutting them even slightly by bidding t = π_H − ϵ, with ϵ > 0.

We now argue that there is no symmetric equilibrium in which all agents in (A) submit the same bid t̃∈(k,πH). With a bid equal to t̃ agent i in state (A) expects to get:

An agent in (A) could profitably deviate by submitting t̃−ϵ, with t̃−k>ϵ>0, as he would obtain t̃−ϵ−k>0 with probability one. We now claim that there cannot be point masses in the equilibrium bid strategy. Suppose that an agent bids t ∈ (k, π_H) with some positive probability. Akin to the argument put forward above, another agent could then gain by placing zero weight on t and positive weight on t − ϵ. Moreover, the agents randomize over a connected support. Let f(t) be the probability that an agent in (A) submits a bid t. It cannot be that f(t̂)=0 for t̂∈(t1,t2), with t₂ > t₁ and f(t₁) > 0, f(t₂) > 0. In each instance in which a bid t₁ wins, also t̂ wins but yields a strictly higher payoff. Therefore, t₁ would not be part of the equilibrium strategy and f(t₁) = 0. Hence, the equilibrium in mixed strategy is described by the cumulative distribution function of price bids F(t). To pin down this c.d.f. we need the expected payoff of an agent i who is in state (A) when other agents adhere to the same strategy. The probability that exactly n − 1 − j agents are in state (A) is:

In this instance, i’s expected payoff if he bids t ∈ (k, π_H) is given by t − k times the probability that each of the n − 1 − j competitors submit more than t, i.e. [1−F(t)]n−1−j(t−k). Agent i’s expected payoff from submitting t is

Agent i must obtain the same expected payoff from each bid in which f(t) > 0, or else it would be better off submitting those bids associated with a higher expected payoff. To find the equilibrium bids, we take the derivative of the above expected payoff with respect to t and set it equal to zero:

This is a differential equation whose solution yields the expression reported in the statement of the proposition. To see this, rewrite the above expression as:

Adopting this change of variables, y = F(t) and dy = f(t)dt, and integrating:

Because at t₀ = π_H, it holds that y₀ = 1,

from which it is immediate to recover the expression reported in the proposition once variable y is transformed back into F(t) and the solution is unique. QED

Proof of corollary 1

Take the derivative of the right hand side of condition (6):

where

The above expression is zero for P_AB = 0 and it is increasing in P_AB, as:

Therefore the derivative of the right hand side of Condition (6) is nonnegative, which implies that the condition in Proposition 2 is more difficult to satisfy the greater is n. QED

Proof of proposition 3

We begin by recovering F^c(t). The agency can set R ≥ k + c such that an agent in (A) wins if and only he is the only one in state (A) and gets no rent:

From this we recover the budget cap and the endpoint of the equilibrium bids. The cumulative distribution function of the bids by agents in (A) is obtained following the same procedure as in Proposition 1. It is easy to see that no agent has incentive to deviate. To obtain the expected winning bid, that we call t_A′, consider that the sum of the expected payoff of all agents in (A) is:

where 1−(1−PA)n is the probability that at least one agent is in (A). Therefore,

The agency’s utility is:

Suppose that the agency induces agents in state (AB) to participate. We have already shown that R=πH+cPBN−1. The other endpoint of the interval over which agents in state (AB) bid is obtained from this equation:

That is, by bidding x=πH+c(1−PA)n−1, an agent in state (AB) would undercut all the other agents in (AB) but none in (A). To determine FABc(t) we follow the same approach as in the Proof of Proposition 1. See that an agent in (AB) who submits a bid t∈πH+c(1−PA)n−1,πH+cPBn−1 gets t − π_H if he wins, which occurs if his competitors are either in state (B) or in (AB) but submit a bid higher than t. Thus, agent i’s expected payoff from submitting such t is:

To find the equilibrium bids, we take the derivative of the above expected payoff with respect to t and set it equal to zero:

The above equation can be rewritten as:

Adopting this change of variables: y=FABc(t) and dy = f_AB(t)dt, and integrating

Because y₀ = 1 at t0=πH+cPBn−1, and P_B = 1 − P_A − P_AB, we have:

We then transform y back into FABc(t).

Consider now the bidding behavior of agents in state (A). By bidding t=πH+c(1−PA)n−1 an agent in (A) expects to get:

The left endpoint of the interval, y, is a bid which wins with probability 1 and gives an expected payoff equal to (1−PA)n−1(πH−k); it therefore solves:

The cumulative distribution function of the bids over the interval is obtained following the same steps as in Proposition 1.

To check whether these are equilibrium bidding strategies, let us see whether an agent would have an incentive to deviate. If an agent in state (AB) bids below πH+c(1−PA)n−1, that is, say, he submits some bid

then he wins with probability p(t′) and he obtains an expected payoff equal to:

To see why this is negative, notice that p(t′)(t′−k)−c=(1−PA)n−1(πH−k) is the expected payoff of an agent in state (A) who submits bid t′ and that p(t′)>(1−PA)n−1, where the latter is the probability of winning if an agent bids πH+c(1−PA)n−1. Similarly, let us show that an agent in (A) does not want to bid above πH+c(1−PA)n−1. Specifically, suppose that such an agent submits a bid t″∈πH+c(1−PA)n−1,πH+cPBn−1, and let p(t″) denote the associated probability of winning. The agent in state (A) would get:

To understand why, consider that p(t″)(t″ − π_H) − c = 0 because agents in state (AB) get no rent and p(t″)<(1−PA)n−1 because the latter is the probability of winning with a bid πH+c(1−PA)n−1<t″. It is also straightforward to show that an agent would not bid above the budget cap or otherwise his bid would be discarded, nor below k+c+(1−PA)n−1(πH−k) as it would decrease the payoff conditional on winning without affecting the probability of submitting the lowest bid.

The expected utility of the agency is:

Notice that the agency’s new utility functions, (A1) and (A2), differ from the ones in the baseline model (1), and (5), only by the expected participation costs. As n(1 − P_B)c(1 + λ) > nP_Ac(1 + λ), the scope for setting a high budget cap decreases with c. QED