The hard tyre advantage relies on a number of if's, a few but's and a lot of finger crossing however it is all about timing and picking the right phase to get the maximum benefit from the extra performance afforded by the softer tyre.
None-the-less, starting on a harder tyre in theory, gives you a longer first stint where the pack is closer and there is less clear air and more too and fro shuffling trying to gain a passing advantage as drivers are not only attacking but defending their desired lines and positions in the condensed pack. It is very easy to burn a softer tyre out in this phase by pushing to hard to fast, especially on used soft's from qualifying when you are carrying a heavy, full fuel load on the start.
Of course if you make Q3, you are at a competitive disadvantage as to grid position to those running the prime that will be 0.5s quicker but you need to factor this in and we have seen a few runners like Mercedes run the harder tyre in Q3 and accept a grid penalty for the longer first stint where you think your car has better consistency over the quick for one lap but slow over a race distance cars. These are typically aero sensitive cars that do well in clear air, but not so good in a pack running in dirty, turbulent air.
So the harder tyre can allow you to run longer at a more consistent lap time over a greater period while the pack is bunched up and there is less opportunity to "stretch" a lead unless you are the first runner and can build a lead and back off as Vettlel did in Valencia to preserve the prime tyre.
Over this initial period, you may only be a very small fraction slower than those running the prime, however you can hold those behind you who burn out their softs trying to pass and need to stop while you can preserve the tyre and push on when they pit with good tyre performance left (while those that are either stopped or back out and are trying to warm either the prime of the harder compound) and look for the undercut during the pit stop interchange.
Also the softs overheat much easier and the performance advantage is either a small one over a longer period where you attempt to prolong their life or a greater one over a shorter period where you push to hard and over heat them and they either drop off a cliff or never fully come back and are slower than the harder tyre. Either way in the first stint you are most likely to push to hard to early or be stuck behind another car and use up the softs before you get a good chance to take advantage of their extra performance.
During the middle stint (regardless if you one, two or three stop) most runners will use the harder compound if the difference provides a noticeable disparity in longevity and performance.
Where a driver who started on the harder tyre and ran a longer first stint, managed to get the undercut at the first change while maintaining a good pace in the middle stint when the pack typically spreads out,they can get good clear air and most drivers will be running on the same harder tyre.
So you have the advantage of the other drivers needing to stop earlier for the last change to the harder tyre (as the soft may not last a longer last phase) which gives you another opportunity for clear air running and another undercut as well as the extra performance afforded by the prime in your shorter final phase where the cars should be more evenly spaced and this allows you to use the extra grip for better braking performance, mid corner speed and drive off the corners like Alonso did in Valencia when he drove around the outside of the other cars running the harder tyres.
Its not a sure thing, but you generally get more time in clear air as your stopped and timed differently to the lead drivers who are all timed relatively the same, as well as the grip advantage in the final phase where you competitors are running worn, harder compound tyres that have usually been pushed too hard or flat spotted due to fatigue/positioning etc.
That's how I have always run when I have an option tyre to run in my racing. When we could start on a harder compound we did, as long as we could get a short last phase without sacrificing the early to mid race pace. It affords more clear air running out of the pack, you are out of phase in the pits for stops and have a "green" options tyre to run in the last phase when the other runners are on worn harder compound tyres.
Its certainly not a perfect system as there are many factors that can throw the phasing out. But in general it works, sort of....