The current blog post is inspired by a grant rejection. With a grant proposal, one invests a lot of time with a relatively small chance of success. In this sense, submitting a grant proposal feels like buying a lottery ticket. A very expensive lottery ticket. One that costs months of work, and has the unlikely scenario of the realisation of a project idea, and a red carpet towards a professorship as the potential gain. We know that buying a lottery ticket is irrational: the long-term expected value, the gain relative to the investment, is negative: if it wasn't negative for the consumer, there would be no gain for the organisers and therefore no motivation to run it.

If we take just the base rate of grant success rates (e.g., for the ERC), we will find that most grants have a higher success rate than lottery tickets (e.g., the Australian National Monday Lottery Ticket). However, the investment into an ERC grant is also much higher. A single lottery ticket costs AU$2.42, an ERC proposal takes months of work. This made me wonder if it's not worth, in the long run, to buy lottery tickets instead of writing proposals. This suggestion did not come off well with my boss.

So I decided to do the numbers to see if this is indeed the case. In the following I compare the expected value of buying an Australian National Monday Lottery Ticket compared to submitting a proposal for an ERC Starting Grant. How to calculate an expected gain for the lottery is explained here, and with a bit of researching I learned a bit about how the lottery works and managed to find or estimate the necessary values. As a simplification, we consider only the possibility of winning the jackpot (AU$1,000,000). The odds of getting a jackpot, according to the lottery's website, are 1 in 8,145,060. Thus, we have a success rate (

*p*), the potential gain (

*V*) and the cost of a ticket (

*C*). To calculate the expected gain, we still need to estimate the number of people who buy this type of lottery ticket. I could not find this number on the official website, but I found the overall number of winners of a recent draw on a different page. Given that 58695 people won the last division, and that the probability of success of this division is 1 in 144, then, if I understand how the lottery works, we can estimate that around 8,452,224 lottery tickets are bought. Plugging these numbers in the formula below:

This means that, as we predicted, the expected value is negative: if we play the lottery, we expect, in the long term, to lose money.

Now, let's do the same thing for the ERC grant. Here, the success rate already takes into account the number of applicants, so we can use the simpler formula:

Here, we need to somehow estimate the cost of submitting an ERC proposal. Assuming two months of focussing only on writing the proposal, and my current salary, the cost comes to approximately 5,000 Euros. The immediate financial gain of the proposal is 1,500,000 Euros; for now, we don't consider the additional gain of 5 years' occupational security and high chance of a permanent professorship position afterwards. The probability of success, according to the ERC, is 12.7%. This gives us:

*E*= 0.127 * 1,500,000 - 0.8 * 5,000 = 186,500.

So, the good news is: The expected value of submitting an ERC proposal is not only positive, but with all the simplifying assumptions that we made, and assuming I made no mistakes in the calculations, it's also quite large! I don't fully trust my calculations. But even if my lottery calculations are wrong, there is the reasonable assumption that the expected value is negative. Furthermore, the expected value for the ERC Starting Grant is conservative, in the sense that it considers only the immediate financial gain: in reality, a success comes with additional gains. The estimation may be too optimistic, if the time spent writing the grant is underestimated in my calculations. However, keeping the other parameters constant, one would need to work for >7 years on the proposal for the expected value to turn negative.

There are some additional factors which may decrease the expected value. Perhaps the psychological loss associated with getting rejected again and again and again adds to the financial loss of spending time on the grant. It is up to each individual to decide how much rejection affects them, and whether it will tip expected value towards a negative one. Furthermore, the current calculations show that it's more rational to submit ERC Starting Grants than to buy lottery tickets for the rest of your life. However, that's a really low bar to set: the gains and losses associated with finding a stable, well-paying non-academic job could far outweigh the gains associated with applying for grants.

Of course, there is one important difference between lottery tickets and grants, at least in theory. A lottery is controlled by completely random processes: I'm no more or less likely to win the lottery than any person sitting next to me in the bus on my way to work (provided we both buy the same kind of ticket). Grants, at least in theory, are awarded based on merit, not based on a random number generator. Whether or not this is actually the case is a matter of debate. Still, me applying for a grant this year and me applying for a grant next year are not independent events: my chance of success depends on my grant writing skills, connection to the reviewers, how impressive my track record looks, and so on.

Is it rational to submit proposals with the hope to stay in science? Well, maybe. For me, at least, the thought that I have a greater chance of success in academia than if I were to buy lottery tickets and hope for the best is an encouraging one.