Individual scientists do science because it is fun and pays the bills. Why would society finance this? I think it is done to achieve prediction and control. Control over whatever can be controlled (physical things, people, abstract concepts) and prediction of what cannot be controlled (weather, earthquakes, evolution of the universe).
Control is something evolution has made people desire (control is commonly used to change the environment to improve people’s survival and reproduction). Prediction allows people to adjust themselves to aspects of the environment that they cannot change to suit themselves. Prediction is also part of learning to control: predict a response, take an action, observe if the response to the action is the one predicted.
If prediction and control are the goals humanity sets science, then scientists and fields of study should be evaluated based on their contribution to these goals. For example, studying history is useful if it helps predict or control future history.
A small contribution of science may be entertainment (popular science shows and news), which may justify financing interesting and entertaining science even if it does not contribute much in other respects.
When evaluating an artwork, the guilt of a suspect or the quality of theoretical research, the usual procedure is to gather the opinions of a number of people and take some weighted average of these. There is no objective measure of the truth or the quality of the work. What weights should be assigned to different people’s opinions? Who should be counted an expert or knowledgeable witness?
A circular problem appears: the accurate witnesses are those who are close to the truth, and the truth is close to the average claim of the accurate witnesses. This can be modelled as a set of signals with unknown precision. Suppose the signals are normally distributed with mean equal to the truth (witnesses unbiased, just have poor memories). If the precisions were known, then these could be used as weights in the weighted average of the witness opinions, which would be an unbiased estimate of the truth with minimal variance. If the truth were known, then the distance of the opinion of a witness from it would measure the accuracy of that witness. But both precisions and the truth are unknown.
Simultaneously determining the precisions of the signals and the estimate of the truth may have many solutions. If there are two witnesses with different claims, we could assign the first witness infinite precision and the second finite, and estimate the truth to equal the opinion of the first witness. The truth is derived from the witnesses and the precisions are derived from the truth, so this is consistent. The same applies with witnesses switched.
A better solution takes a broader view and simultaneously estimates witness precisions and the truth. These form a vector of random variables. Put a prior probability distribution on this vector and use Bayes’ rule to update this distribution in response to the signals (the witness opinions).
The solution of course depends on the chosen prior. If one witness is assumed infinitely precise and the others finitely, then the updating rule keeps the infinite and finite precisions and estimates the truth to equal the opinion of the infinitely precise witness. The assumption of the prior seems unavoidable. At least it makes clear why the multiple solutions arise.
If the environment changes, then there is a fitness benefit to being an adaptable organism. On the other hand, adaptability is costly (bigger brain for adjusting behaviour, various backup systems in the body like the camel’s hump need to be carried around). So adaptability gives a net benefit if the environment changes sufficiently rapidly.
If periods of change alternate with periods of constant environment, then it would be useful to have the ability to switch adaptability off for a while. This is adaptable adaptability. The ability to switch adaptability off is in turn costly. It is useful to have if periods of environmental change alternate with periods of stability sufficiently rapidly. It would be good to have an ability to switch off the ability to switch adaptability off if changes and stability alternate with different frequency over time. Even more complex patterns may necessitate the ability to switch the ability to switch the ability to switch adaptability, etc. Hierarchies of abilities controlling abilities arise.
Perhaps after an infinite hierarchy, there is some meta-ability that can switch all lower order abilities. Self-awareness or something similar.
Ronald Fisher analyzed signalling in biology through traits that do not confer direct fitness advantage (higher survival or fecundity), but are desired by the opposite sex. This attraction is an equilibrium in a coordination game – if a potential mate has traits desired by the opposite sex, then the offspring with that mate are likely to have these traits as well and succeed in attracting the opposite sex. The traits confer a mating advantage, which is part of a fitness advantage, justifying the desirability of the traits.
It is a coordination game, because in a different equilibrium, traits without a direct fitness advantage are not desired. Then these traits do not give a mating advantage to the offspring and therefore do not have an indirect fitness advantage either. Then it is not fitness-enhancing to desire them. In summary, if a trait is expected to be desirable in the future, then it is desirable now, and if a trait is expected to be neutral or undesirable, then it is neutral or undesirable now.
Fiat money is inherently worthless, but in one equilibrium of the money game, has positive value in terms of other goods. If everyone expects that others will accept money in return for goods in the future, then it is useful to obtain money now. So everyone is happy to deliver goods in return for (a sufficient sum of) money now. The money game is a coordination game, because if everyone expects money not to be accepted in the future, then they do not give goods for money now. If money is expected to be worthless, then it is worthless, and if money is expected to be valuable, then it is valuable.
An overview of signalling in biology is at http://en.wikipedia.org/wiki/Signalling_theory and Fisher’s theory at http://en.wikipedia.org/wiki/Fisherian_runaway
The coordination game of money is studied by Kiyotaki and Wright (1989, 1993): http://www.jstor.org/stable/1832197 http://www.jstor.org/stable/2117496 and more simply explained in van der Lecq “Money, coordination and prices” https://books.google.com.au/books?id=r1r40SB0Wn8C&pg=PA29&lpg=PA29&dq=fiat+money+coordination&source=bl&ots=iI0M96m-qz&sig=lRHBAWIXYZs2V5S-iNFeH-2yar8&hl=et&sa=X&ei=d3lqVeneJ8TvmAX4vIGgDg&ved=0CEoQ6AEwBw#v=onepage&q=fiat%20money%20coordination&f=false
If only those who fail are allowed to retake an exam and it is not reported whether a grade comes from the first exam or a retake, then the failers get an advantage. They get a grade that is the maximum of two attempts, while others only get one attempt.
A simple example has two types of exam takers: H and L, with equal proportions in the population. The type may reflect talent or preparation for exam. There are three grades: A, B, C. The probabilities for each type to receive a certain grade from any given attempt of the exam are for H, Pr(A|H)=0.3, Pr(B|H)=0.6, Pr(C|H)=0.1 and for L, Pr(A|L)=0.2, Pr(B|L)=0.1, Pr(C|L)=0.7. The H type is more likely to get better grades, but there is noise in the grade.
After the retake of the exam, the probabilities for H to end up with each grade are Pr*(A|H)=0.33, Pr*(B|H)=0.66 and Pr*(C|H)=0.01. For L, Pr*(A|L)=0.34, Pr*(B|L)=0.17 and Pr*(C|L)=0.49. So the L type ends up with an A grade more frequently than H, due to retaking exams 70% of the time as opposed to H’s 10%.
If the observers of the grades are rational, they will infer by Bayes’ rule Pr(H|A)=33/67, Pr(H|B)=66/83 and Pr(H|C)=1/50.
It is probably to counter the advantage of retakers that some universities in the UK discount grades obtained from retaking exams (http://www.telegraph.co.uk/education/universityeducation/10236397/University-bias-against-A-level-resit-pupils.html). In the University of Queensland, those who fail a course can take a supplementary exam, but the grade is distinguished on the transcript from the grade obtained on first try. Also, the maximum grade possible from taking a supplementary exam is one step above failure – the three highest grades cannot be obtained.