Am I a bad DM?

[SpyOne]

It seems to me you had a valuable learning experience. You have decided that didn’t go well, that it was at least partly your fault, and to try handling things differently in the future. All good.
So no, you do not appear to be a "bad DM".

Should I have:

• Not have had an assassin that the players would find it difficult to immediately catch?
• Rolled with my player's ideas of, I don't know, mathematically proving her destination?
• Retconned my own pre-laid plans for how well-equipped and well-prepared the assassin is, so the player was capable of catching her?
• Just told him flat-out that it was impossible for him to catch her from the start, so he never left the city in the first place?
• Something else?

My votes would be:
Maybe
No
Maybe
Maybe
Sure

That is, the one thing I would recommend against is supporting the player's notion that a high Intelligence can substitute for skills you don't have.
None of the other ideas are inherently good or bad.

Having the characters arrive in the immediate aftermath of the assassination (he went thatta way!) sets a certain mood, but it is easy to see that if the assassin had been long gone when they arrived, they wouldn't have given chase. It wasn't a wrong decision, but ... Investigating plane crashes gets into what they call an "event chain". Like, if it hadn't rained on Thursday the farmer would have plowed on Thursday, but instead he plowed on Friday and so was struck by debris falling from the sky.
The decision to plow tomorrow wasn't in any way wrong, but changing it means the farmer doesn't die.

Changing your plans based on the players' choices can be good but isn't mandatory. (A DM I know once ditched the design for the bad guy's home because the players expected him to have a greek style villa because his name sounded Greek. The DM had chosen the name on the fly, seeing "second level" in his notes he read "second" backwards and turned it into "Donassis".)

But it seems most likely that the player was expecting more immediate feedback about his ultimate failure. Not a railroading shutdown, but give him a minute, ask for a roll against something appropriate (perception, inspiration, whatever - the roll is mostly theater) and say something like "you've lost sight of her, you don't know how to track, she has a better horse than you and is an experienced rider and you aren't. This plan doesn't seem to be working"

[BRC]

Both myself and the other player did point out the flaw in his logic regarding catching up with a quarry moving faster than him a couple of times along the way, but he disagreed. Perhaps I should have sort of told him out of character, voice-of-god style?

This is a key lesson here.

The Player seemed to be under the impression that "And then they catch the assassin" was part of your intended plan for the session, and communicating the difference between "A thing you are supposed to be able to do", "A thing that you will probably fail at" and "A thing that is supposed to be totally off the table" is a tricky skill for a GM, especially because succeeding at Difficult and Dangerous things is kind of part of the whole Fantastic Adventure package.

Of course the GM is going to describe the dragon as terrible and dangerous, that's to make you feel good when you kill it. Nobody gets excited about fighting a dragon that the GM describes as "Pretty tough looking, but well within your abilities".

You DID try to communicate this, both with the situation itself (Trying to catch up with somebody who had a faster horse and a head start), and, it sounds like, with NPC's discouraging them, but that apparently didn't get this across. So, some voice of god "You can keep trying, but your chances of catching up are basically nil" might have served to calibrate their expectations. It sounds like at some point they hit a sunk-cost , where you had let them spend so much time on the chase that they assumed you must have been planning something to make it worthwhile.

[Jay R]

One fact worth considering. A horse cannot gallop for long. After one or two miles, the horse is fatigued and cannot run at all. it's true that a good rider can catch up, by using another gait.

If the assassin took off in a nice, easy canter, she'd be much harder to catch. She could be caught by a rider at a gallop, unless she is already too far ahead. Remember, the pursuing galloper would have to catch her in the first couple of miles.

The ideal, perfectly knowledgeable DM would have said, "Does your PC have the Ride skill?" If not, he will not catch up to the assassin, who presumably knows how to avoid capture. But no DM is perfectly knowledgeable on every relevant subject.

So the slightly more complex answer is that a well-trained rider might be able to catch up to an untrained rider. But if your assassin knows horses and the PC doesn't, he doesn't know how.

---

Now, get rid of the question "Am I a bad DM?" In fact, get rid of the phrase "bad DM". All DMs are imperfect. You will make mistakes. That doesn't mean you're a "bad DM". It means you're human.

DMing takes experience, just like anything else. You are a low-level DM, looking to gain experience points. And that's fine.

---

I'd like to offer some advice: one tool I've found useful as a DM. Once I've listened to the player's objections, and decided that there is no reason to change anything, and I can't get the player to move on, I will sum up the whole thing, including his points, as a conclusion.

"Look, the character with superior riding skill, on a superior horse that she chose for this getaway, and which she knows well, tried to escape from your character. She rolled the rolls she needed to roll, and succeeded at what she did. She knows more about horses than you do, and used that knowledge to successfully escape. Your character is on a strange horse that does not know you. It's a peasant's horse, not trained for the chase. You do not know what skills she has, and you did not see what happened after you lost sight of her. Yes, sometimes a better rider can catch a worse one by using a slower but steadier gait. That didn't happen here because she made the right choices and got the rolls she needed. I know where she is now, and you're not there.

"Nothing you have said suggests that I have applied the rules mechanics wrong, or that the rules mechanics shouldn't apply here. She has gotten away from you. That encounter is over. What does the party do next?"

[vasilidor]

you should never assume that the villain will get away. yes, it would make sense for them to have a getaway plan, but sometimes those get countered. in this scenario it is apparent that the villains plan bore fruit. you should never just hand wave "and the villain escapes" if you have a combat system for the game you are running, I would have everything go in combat turns using relative skill checks and actions. in example if the characters are in a chase scene on foot, have both the player and the villain do athletics checks. if one of them gets an advantage, such as getting to a vehicle or horse, then the other needs to get something comparable or simply fall behind. you also need to have a way to resolve if the person running gets away. the easiest way to do this is determine if the fleeing person got away is if the pursuer would reasonably lose track of their quarry, such as having lost line of sight and are unable to track.

[kyoryu]

you should never assume that the villain will get away. yes, it would make sense for them to have a getaway plan, but sometimes those get countered.

Never put a piece on the board you're not willing to lose.

[I3igAl]

There was nothing wrong with what you did ...

... however a good idea would have been to give the monk (and maybe the rest of the group) some partial success.

He could have found the assasin's clothes, a used up disguise kit and a dead horse in front of a medium sized city for example. Or a burned out "Disguise Self scroll". Or a burned out scroll, that turns him into a bird. Now he knows the dude will be basically impossible to recognize, but he has a lot some new clues he can bring back to the group for further investigation.
Some peasants could turn out to be the assasin's henchmen and either try to rough him up or try to a (honey)trap him and invite him to dinner. After handling said encounter, he could question them for some new clues.
You could have had more assasins especially since you fed one to an NPC. Leave one to be taken out by the players as well, while the rest flees.

This way the assasins get away, but your heroes still get to feel good.