# Abstract

We consider a funding competition for targeted projects. Potential participants have stochastic opportunity costs, and do not know the number of competitors. The funding agency sets a budget cap indicating the maximum funding that participants may request. We show that raising the budget cap helps to attract more participants but causes an increase in the requested funds. A higher budget cap is optimal when the preferences of researchers and the funding agency are more congruent, competition is lower, targeted projects have larger social value, the cost of public funds is smaller, or bidding preparation costs are lower.

**Funding source: **Italian Ministry of Education

**Award Identifier / Grant number: **PRIN 2017 Y5PJ43-001

# Acknowledgements

For useful comments, we wish to thank Gianni De Fraja, Ester Manna, Markus Reisinger, and Emanuele Tarantino. An earlier version of this paper was circulated under the title: “How to set budget caps for competitive grants”. We gratefully acknowledge financial support from the Italian Ministry of Higher Education, and Research, PRIN 2017, Grant n. 2017*Y*5*PJ*43_001.

## 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 *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 *i* in state (*A*) expects to get:

An agent in (*A*) could profitably deviate by submitting *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 *t*_{2} > *t*_{1} and *f*(*t*_{1}) > 0, *f*(*t*_{2}) > 0. In each instance in which a bid *t*_{1} wins, also *t*_{1} would not be part of the equilibrium strategy and *f*(*t*_{1}) = 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. *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 d*y* = *f*(*t*)d*t*, and integrating:

Because at *t*_{0} = *π*_{H}, it holds that *y*_{0} = 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 *A*). Therefore,

The agency’s utility is:

Suppose that the agency induces agents in state (*AB*) to participate. We have already shown that *AB*) bid is obtained from this equation:

That is, by bidding *AB*) would undercut all the other agents in (*AB*) but none in (*A*). To determine *AB*) who submits a bid *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* = *f*_{AB}(*t*)d*t*, and integrating

Because *y*_{0} = 1 at *P*_{B} = 1 − *P*_{A} − *P*_{AB}, we have:

We then transform *y* back into

Consider now the bidding behavior of agents in state (*A*). By bidding *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

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

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

To see why this is negative, notice that *A*) who submits bid *t*′ and that *A*) does not want to bid above *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

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*_{A}*c*(1 + *λ*), the scope for setting a high budget cap decreases with *c*. QED

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**Received:**2020-10-10

**Revised:**2021-02-22

**Accepted:**2021-05-20

**Published Online:**2021-06-07

© 2021 Walter de Gruyter GmbH, Berlin/Boston