I don't think anyone would/could provide a detailed brief to Pirelli, or any other manufacturer for this matter, for a variety of reasons.
Firstly, the one giving a detailed specification would need to assume responsibility if it proved insufficient, think about a Indianapolis/Michelin scenario, who would want to assume responsibility for something like this? Pirelli turning around and saying, well, we are sorry, but that's what you have ask for. I don't nobody in their right mind, neither FIA nor Eccelstoane/FOM would be crazy/stupid enough, to do that.
Secondly, even if the FIA would hire someone, with the necessary technical skills, to spec such a tyre(s), that does not mean Pirelli (or anyone else) would be able to produce them, that way.
Not because, they are incompetent, but because every manufacturer goes about "making tyres" in their own way. Michelin has their own way/philosophy as have Bridgestone, Pirelli and others. Some maybe closer together in their way then others, but still everyone has it's own philosophy about their product.
Therefore any manufacturer, keeping in mind, it is his name (Pirelli in this case) on the line, would take it too kindly, if you were trying to tell them how to do their job.
There are some other things to consider such as patents etc. etc., Pirelli (or someone else) may know how to do something, but can't do so, because someone else holds a patent on a specific application.
To make a long story short, at the end of the day, the buck stops at Pirelli's desk. They signed up for the job, now they have to do it.
Having said all this, I think it is possible, that they were ask, to have a look at some aspects of how they approach the job.
This race in Canada in 2010 is said to have been given as an "example" - Like "could we have more of these"?
This, I think is a possibility, and Pirelli may have agreed to this, but then again, it is/was their decision to do so, I doubt anyone could "force" such a approach onto them. (against their will).
Keep in mind, that at the end of 2010, their were not that many companies overly enthusiastic about F1, Bridgestone just walked out, saying thanks and giving a s... about F1, and not many others putting up their hand. Could/would have Avon (one of the others candidates in the running AFAIR) have done a "better" job? Maybe, who knows, but who is Avon?
A small(ish) company doing mainly clubsport and track day tyres, being quite popular in things like Rallycross, Hillclimbs and Historic Motorsports (Boss series). They have done F3000 (GP2) in the past and make tyres for the BOSS series (old(er)/ex F1 and Indycars run by privateers) AFAIK. ( put I maybe not completely up to date with the latest developments).
Would they apart from the engineering expertise have the financial, logistic and marketing muscle to do such a project? Maybe, but maybe not.
Who else was "seriously" interested at the time?
So Pirelli may have seen this as an opportunity and jumped on it, fair game IMHO.
But let's have a look from a technical perspective, to what happened in Canada, basically Bridgestone "f.... u..", less dramatic then Michelin in Indy, but doing a mistake, "guessing wrong" nonetheless.
I don't blame them, not do I glee them for it, these things happen (shit happens) and sometimes it happens to the best, no shame in it.
They brought the "wrong" tyre for the conditions at the weekend, people struggled to get them into the window, mainly having cold graining issue (not reaching Tg), struggling to "switch the tyre on", and therefore experiencing higher then normal wear rates. (the compound being to "brittle" due to the low temperatures).
It made for an unpredictable and entertaining (for same) race, because everyone was "on their back foot" trying to adjust "on the go" to the situation at hand.
Now, it may be possible at some places with low temperatures and smooth tarmac, to do something like this, but I'm reasonable sure, that you will "struggle" to do that on a regular basis, at most of the F1 tracks. Bringing a tyre that is "too hard". Technically it's "safe", you are unlikely to see any structural problems,you just have a slow(ish) tyre, that wears a lot and is difficult to drive, but unlikely to fail otherwise.
The second point is, that it worked well in Canada, because nobody was expecting/anticipating this. People (teams/engineers) in F1 are normally quite "sharp" (quick learners), you bring such a tyre three weekends in a row, they will raise to the challenge and (try to) adapt to it. The "surprise moment" will be lost, and the "trick" will become less effective over time.
Pirelli choose the other way, bringing tyres that are "too soft", because you can almost always do so, the other side effect is, that such a tyre is still "quick(er)" at least for a short while. But technically it's a bit more challenging, to find the balance, because if you get it "wrong" you can get into hot water, and seeing structural failures. Something nobody like to see, no PR will paper over this.
No tyre manufacturer in their right mind, wants to see a tyre explode or disintegrate in front of a TV audience of millions of people. It's a no-no.
Performance and wear issues are one thing, that's not the "end of the world" in a controlled tyre formula, with enough PR you can explain that away, and sooner or later people will believe it, but safety is paramount. There always will be some tyre failures, in any form of racing, if people see two cars touching and one ends up with a flat/damaged tyre, most people can rationalize this, no problem.
If they see, someone running over a piece of debris or going way off track and having a puncture, they can relate to it, just what happens to you and me on the road from time to time.
Tyre failures "out of the blue" are a different story, and in this respect, I'm reasonable sure that Pirelli was not "happy" with what happened in Bahrain, whichever way they may paint this in the press.
To be fair, technically their is the possibility, that they just got unlucky, and that all failures were caused by debris, that could happen.
But while debris is a valid reason, it also can be abused as a default excuse, nearly everything can be explained with "debris strike" when it comes to catastrophic tyre failures, and it's hard to prove the opposite. So let's see what the next races bring.
All in all, I think, they have done a reasonable job, maybe not the best possible but reasonable, and most of the time, with a margin of safety, which would not be a concern to me (but maybe to others).
Spa when RBR hat their blistering issues, has shown, at least to me, that they approaching a limit condition. It's not the end of the world, but at this moment, this tyre on this car, was on the limit.
It's something you are not seeing very often in a "controlled tyre" formula, because most manufacturers don't want to cut it that close, and in a controlled formula, their is no need to do it, but you see it regularly in a "tyre war" as people will/are forced to push the limits. Bahrain has raised some flags at least for me, and the next races especially Spa, Monza and Suzuka will show if their is reason for concern or if they have a "handle" on it.
IMHO (which is that this), they have painted themselves into a corner, they chose to go "on the aggressive" side of the curve, and I have no issue with this whatsoever, but as people adapt to the situation, and they try to keep them guessing, they are pushed more and more down this road, and their is a end, as in how far you can go.
As long as they keep a handle on safety/structural side of things, I don't mind too much what they are up to. It's what they do, and the teams will have to deal with it. Sure some will not be happy, nobody is, when things are made more difficult for them, but I don't mind that to be honest.
If someone claims to be a top team, top engineer or top driver, he should prove it, again and again, and raise to the challenge, not relay on things he has may done in the past.
Horner had to say this after Bahrain:
As long as this is the case, I see not much wrong with it. Nobody said it was meant to be easy. As long it is possible to get it right, and getting it wrong does not land someone in the hospital or worse, I can life with it. The onces aspiring to be "the best" or "top", should raise to the challenge and do so weekend after weekend - INHO.
To be fair so far, RBR has done just that, sure they make a lot of noise and try to sway the balance in their favor, but that is just part of F1, call it politics, gamesmanship or what you like, but despite all the complaining, they also react on the technical side, and try to get on top of things, and I for one respect this. If they would only complain, and not doing anything about it technically, it would be a different story.
So after a lot of ramblings, for me the buck stops with Pirelli, but I don't think, that on balance, they have done such a bad job overall.
But they may find themselves between a "rock and a hard place" in the future. Unless they come up with some "wonder material" I don't think, they have much more scope trying to be more and more aggressive without risking structural/catastrophic tyre failures too, and I doubt, that they take this light hearted.
So, they may need to come back a bit, or at least can't keep pushing the compounds down a notch year after year. In both cases, that would mean, things become "predictable" again, and their is nothing any engineer likes more, because it makes life a lot easier. :-) Better the devil you know, and all this.
But for F1 it means, that things will settle down a bit, and some sort of pecking order will emerge again (may different from the one before), with little room for "surprises".
If this is considered "bad", then maybe it's time for some "full course yellows" or the like.
Well, it's a long rant anyway, so at the end, maybe a short excursus into racing history, not even F1.
There was a time when some of Bridgestone's best (rain)tyres were willingly traded in for a set of Pirelli's and/or Michelins.
At the time, that was quite big news, and met with a lot of laughter/sarcasm. So, as you see, everyone started somewhere, and Rome was not build in a day.
Not even for Bridgestone - all in good fun & spirit.
Makes sense - exemplary write up. Thanks, TC3000.