I had gathered a bunch of semi-obsolete data through a Spy Master play style from before the 1.1.1.1 beta patch, most of which looked like this...

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...including a chart in which I periodically checked the chance of success of every action if it was above 15%...

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...and notable events.

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If you want to look at that save game for whatever reason, here it is:
I did a handful of additional tests in the same galaxy dozens of years later into it after letting it run on auto for a while, just to be sure that even larger, more powerful empires didn't change anything up.
Unfortunately, it only shows what we already know concerning the topic. After some 50 lines of entries, I realized it wouldn't help, but I kept checking the numbers anyway in case anything telling happened (nothing really). However, I did figure out a few things while testing this. The big one is that the chance of success is inversely proportional to the target empire's population. Revealed by game editing, I deleted the capital of the #2 empire and checked how that affected success estimations, then reloaded and, subsequently, their largest population world otherwise. That removed their leader, all spies, and their spy facility for the capital. The chance of success was bumped up about the same in both cases.
I suspect, but have not confirmed, that the spy's host empire population contributes, meaning that it could be
proportional population is the significant factor. Not essential to verify, IMO, but worth noting. Other possible contributors could be the number of colonies in the target empire, income, the total amount of corruption $ lost, etc... in any case, all those possibilities correlate to population. And, of course, this doesn't apply to pirates. Other factors like total technology may matter, but that was beyond the scope of my tests.
Since Aurora, I've only played the Haakonish, whose spies have been ... utterly useless ... they haven't succeeded in a single mission against an empire worth anything, including territory map stealing. Since it is on very difficult settings, this isn't a surprise. Anyway, the beta patch made low-success-chance missions significantly more likely to succeed, such that despite having
the worst spies I remember having, they still have close to the same chances of success as the Zenox empire I tested before the 1.1.1.1 beta when it comes to those 'high-value' missions.
Now, despite not getting much out of the data (sigh), spending all that time with the Spy Master run did force me to reflect on espionage, made further evident in the subsequent Haakonish run with some hindsight.
- Spies are inherently free, and their actions are too.
- Their value comes from scarcity and the potential for successful actions.
- As the chance of success is inversely proportional to population, the value of spies against high-population empires is lower due to the much smaller potential for success not keeping up with the desirability of success.
- The cost of getting caught is only a tiny reputation hit and some diplomatic ire from the target empire. Concealment seemingly makes getting caught less likely. Getting caught leads to capture far less frequently as of 1.1.1.1.
- The cost of a spy getting captured is that the spy can no longer take action; in other words, they lose their value until freed. There is also a significant chance they'll develop a very bad trait.
- The cost of freeing a skilled spy is almost always worth it, as their value far exceeds the price. Bribing a skilled spy is undoubtedly worth it unless they are patriotic.
- The fewer you have, the more likely you'll acquire a new spy. It seems the bigger population you have also affects how likely you'll get another.
- A captured spy seems to count toward your total spy number, making the appearance of further spies less likely. However... with automated Character Locations, your AI will periodically dismiss bad spies that are captured. You can't do that. This is probably not intended. Perhaps it's a bug, though I've been able to duplicate it easily.
- Successful technology and map stealing directly benefit the empire. The value of these missions is clear.
- Counter-espionage effectively slightly hurts the reputation of the offending party and has a chance to net you a captured spy, and as aforementioned, is likely worth bribing or selling back.
- Other actions (sans Deep Cover) damage other empires, the value of which can't be easily measured. The value of these missions is largely dependent on the user.
- The value of numerous successful direct-value missions outweighs those of others- an espionage spy is far more valuable than a saboteur/assassin/psy-op. This is amplified by Ambassadors' aid to espionage actions, which doesn't contribute to other missions. They can have a counter-espionage bonus, but I can't tell how impactful that is or how it works on an ambassador.
- Although higher technology is more difficult to steal, they are worth multiple times as much research as earlier tech (never mind the weird values coming from diplomatic trade right now), so they do not gain or lose value but are more likely to cost in failure. They are still worth a net gain until the population becomes a prohibitive factor.
- Counter-espionage already had significant diminishing returns for multiple agents performing it. It is even more so now, so a few spies good at counter-espionage similarly reduce other spies' value at it.
- Excess spies only have significant value in the potential of successful missions in assassination, psy-ops, and sabotage.
Yeah, that's a lot of points, but they're all relevant. Something very, very important about all that?
Spies aren't as precious as they feel.
They are all expendable. The fewer you have, the more you get. The bigger you are, the faster they come. You can get them back diplomatically with location information, pirate contacts, or just throw money at the problem. Dismiss them without a second thought if they don't have a good trait or desirable skills. Hope they get killed instead of being captured when they don't have great skills. Don't feel bad about that. They know what they're doing is a top-tier dangerous activity.
There are a lot of hold-ups because espionage gives so much direct value; the other options seem to be a waste by comparison. Yes, espionage is great, but it tapers off, and it's only good if you can get good spies. That's not always a viable option. Meanwhile, the other missions don't need much to get a lot out of them. Just takes a bit of luck and some persistence.
All you need is 1-3 counterespionage buffs and a few (or no) espionage experts, then use the rest on more risky missions. If you lose an espionage expert because they've gone traitor or got selected as Boskara dinner, train a new one, and don't sweat the loss. With that, I think it's worth continuing into the value of sabotage, psy-ops, and assassination.
Sabotaging destroys whatever is being built or a station. Pretty straightforward. In the save above, I saw an agent with a mere 5% sabotage skill from a trait alone target the #2 empire with a 22% chance to sabotage construction and a 37% to get a research base. During a war, if you claim their capital, you regain your captured spies if you don't glass the planet and aren't unlucky in the invasion. So
why not send your spies that don't have value otherwise to try to blow up some ships or stations? Just don't throw out your espionage or counterintelligence experts at this.
Psy-ops: Inciting a rebellion just applies a negative modifier to approval for a single colony. The chance of success shoots up the closer the target colony is to rebellion. This is useful when fighting an enemy that refuses to give up in a long war. Instead of committing to an invasion of a world, just blockade it, then after a while, send in your PsyOps agent(s) to incite a little chaos. It might just be a 30% chance to work, and you might make it happen without a spy's help,
and it might just be a -4 to -12 (or higher) bump to approval. If it works, though, they will join at 100% assimilation. That will save you
a lot. Very worthwhile.
The revolution? If you have five spies with a 10% chance to incite a revolution against a warring Military Dictatorship, that's a good deal. A coup d'etat eats up an admiral/general, maybe even 2 or 3, and imposes a -18 approval. That's not even taking into account whatever bonuses the leader had. Your spies are worth far less than that.
Assassination seems to be the least impactful of these. You can't see enemy character traits, just their skills. It also seems to be the least likely to succeed overall, so I have to wonder if there's a trick to using assassins like there is for inciting a rebellion. I have no idea. Nonetheless, a spy with the right trait and skill in assassination could be more useful doing just that instead of the other missions.
A tip. A spy with the Inspiring Presence trait instantly makes them the best waifu writer for your spies, and you should never let them leave home. Being a Demoralizing Spy makes them worse than a mutant in the Imperium of Man, which is grounds for instant execution. Both these continue to affect spies so long as they aren't captured.