Research costs money, whether it be for buying equipment, compensating drug test subjects or simply to pay the salaries of researchers. Much of the money that funds academic research comes in the form of competitive research grants from a variety of national and supra-national research councils (e.g. EPSRC in the UK, the European Research Council, or the National Science Foudation in the USA), or private foundations (e.g. the Wellcome Trust).

Funding from these bodies is allocated by a competitive process where researchers submit grant applications to the relevant body, describing the project they wish to carry out and justifying the cost. Panels of experts then decide which grants to fund. As well as deciding which research gets done, this also has a profound impact on the academic's career progression as universities depend on the money from these grants to maintain their operations.

Because of the high importance of these grant awards, many academics in research positions will spend a great deal of time and effort on preparing applications. Universities may support them by allocating them dedicated time for preparation, or by giving them smaller amounts of money to perform preliminary studies which make the full application stronger. Each applicant knows that competing researchers from other universities will be working hard on their applications too, so they must go the extra mile to succeed.

I wondered about the 'deadweight costs' of this process. Time and money spent on applications is time and money that cannot be spent on the research itself. Ultimately we want a system that produces as much excellent research as we can get. Competition may spur academics to do better research, but it is also costing resources that could be devoted to research alone.

We can model this mathematically. Imagine there are two research teams vying to obtain a grant of size

*G*(a typical grant may be a few hundred thousand pounds). Each team can increase its probability of winning the grant by devoting more initial resources to preparation, either in time, salaries, experiments etc. Call the investment of team 1,*A*and the investment of team 2*B*.
A simple model might suggest that the probability of winning the grant is proportional to the initial investment. In that case, for team 1 the probability to win is:

*P = A / (A + B)*

The expected

*reward, R,*for team 1 is the probability of winning, multiplied by the grant amount,*minus*the amount invested*R = GP - A*

*R = GA / (A + B) - A*

Now, in order to work out how much team 1 will invest we need to introduce two ideas. The first is a Nash equilibrium. This is a situation where both teams have decided on investments

*A*and*B*and neither wants to change, i.e. neither can increase their reward by changing. This implies that*dR/dA = 0*

which implies that

*0 = G/(A+B) - GA/(A+B)*

^{2}- 1*0 = GB - (A+B)*

^{2}
the second idea is symmetry. Since in this simple model both teams are identical, they should come to the same conclusion about how much to spend, so when both teams are 'happy' (at the Nash equilibrium)

*A = B = X*so:

*0 = GX - 4X*

^{2}*0 = G - 4X*

*X = G/4*

so eventually each team will invest one quarter of the grant value into its preparation. One team will be lucky and end up

*3G/4*better off, while the other will be*G/4*poorer than before.
With two identical teams then the deadweight cost is

*G/2.*This much must be invested by both teams together to decide who gets the final grant. Nothing is produced from this investment, and each team still ends up with a 50% chance of winning the grant.
With more teams one can perform a similar analysis to find that when there are N teams, each will invest

*G(N-1)/N^2*in the process. So as*N*becomes large (as it is in most cases), the total deadweight cost will become*G(N-1)/N ~ G*. In other words, the deadweight costs reach the value of the grant itself. For every £1m the government or private trusts puts on the table to be fought over, another £1m will be wasted in application preparation.
But what if this model isn't correct. Maybe the team that puts in the best application always wins. Maybe it really is worth spending another week, another month of research time on preparation? Well in this case the situation is even worse. The resulting incentives look a lot like a famous hypothetical game called a 'Dollar Auction' (https://en.wikipedia.org/wiki/Dollar_auction). In this game players bid against each other to win a single dollar, with the caveat that everyone must pay their highest bid (just like everyone must pay the cost of their application). Initially bids are low, since bidding a few cents for a dollar seems like a good deal. But once the bids grow over a dollar something strange happens. Players with bids lower than the highest still want to increase their bid, even if it is now more than a dollar, because if they don't win they will lose even more. In such a game the only way to win is not to play!

These are very simple models and reality will be a lot more complex. Applications can be improved and resubmitted. Reviewing applications filters out poorly thought out ideas. But when we consider the effects of competition in science, we should not only see the positive incentive to produce better research, but also the damaging waste that competition over a fixed pool of resources can produce.

These are very simple models and reality will be a lot more complex. Applications can be improved and resubmitted. Reviewing applications filters out poorly thought out ideas. But when we consider the effects of competition in science, we should not only see the positive incentive to produce better research, but also the damaging waste that competition over a fixed pool of resources can produce.