Assume that field strength is equivalent. Is your expected value in a tournament the same, whether the field is 100 or 1000, if the buy in is the same? My instincts tell me that, just like running it twice, your EV is the same but your variance changes.

Similarly, assuming the same buy-in and structure, if my ROI is the same for both a smaller tournament that can be completed in one long day vs a three day tournament, is my hourly rate going to be the same for both tournaments?

I'm mainly a cash game player who is dabbling more in tournaments and I am trying to figure out how much time I can afford to spend on tournaments when I am likely to be in Vegas during June.

Its likely that your $ per hour is exponentially higher when you cash the larger tournament. So cashing a three day mega tournament will earn more per hour than 3 single day tournaments.

Or, cashing a 7 hour MTT should pay more per hour than 7 individual 1-hour MTT.

However, if you need to cash often to have money to live on, then the smaller the tournament the more often you cash.

Somewhere in the middle is what is best for you to grind.

Why is this likely?

Well, a positive ROI has to come from the negative ROI of your opponents. Gathering more such opponents together in a larger tournament provides you with larger cashes but less frequent cashes. Think of yourself as the fortunate few that actually have a positive ROI, hopefully, and it makes more sense.

A tournament that takes 3 days will have a player pool much larger than 3x the players of a 1 day tournament, also.

I should repeat that this is “when you cash” your $ per hour is better. Obviously, when you bust you earn zero per hour in either format.

When you go to Vegas, your concern should be to put in as many hours as possible since your overhead cost is the same either way. You need some cashes to recover your travel cost. If you think it is possible that you are exactly average or nearly average, then you should play as many smaller tourneys as possible. If you are a crusher, then you want to monetize your ROI with the largest tournaments possible.

Well, I have a huge win rate in pots that I win, ignoring pots that I lose.

I want a math-based, theoretical explanation of how field size affects long-term results.

Ferguson put on a ROI clinic last year, so his method is guaranteed the most mathematical theory based approach. I would do as he did. Good luck in Vegas!

Not a tourney player or ICM expert by any stretch of the imagination, but what seems most relevant here is the skill-edge differences. I'd imagine you have more control over your ROI in smaller fields rather than big ones for two reasons:

1) Since there's less variance, your results will more quickly approximate your winrate. This is especially true of live tournaments where the opportunity cost of playing a tournament is massive, both in terms of buy-in and the hours upon hours upon days of single-tabling.

2) In tournaments with massive fields, your EV is in a wide open system that far exceeds the bounds of what's happening at your table, much less what's happening in any given pot you're a part of. cEV is so unimportant for so long that there is little you can do to improve your odds of winning, beyond just blinding yourself out for most the day. This is why, in fact, the big name pros don't even bother showing up in time for the tournament, because they exert their skill advantage by blinding out while all the fish win phyrric victories over each other while donating 50% of their buy-in to the rest of the field. If they showed up, they'd barely improve their skill advantage by occasionally hitting the lotto ticket that is a hand/situation with such fat cEV that the value of those chips exceeds the risk of putting a large portion of their stack at risk. Since most players there made a long trip and paid a lot of money to be in the tournament, their default play will probably more closely approximate the correct play than under different circumstances.

In the other extreme, at a single-table-tournament you at least in some small way have a say in how every hand in the tournament plays out. The opportunities to leverage that are obviously limited, but still there's no such thing as an unfavorable table dynamic. If your table is tight-passive, you can easily tread water in terms of cEV until the blinds become worth fighting for; if your table is wild, then you'll quickly be ITM.

I'm struggling with this assumption. Could you be more specific? Do you mean something like "Your skill level places you in the nth percentile of the field", or something like that?
It's problematic to do this if you compare payout structures of different field sizes.
e.g. Someone that somehow always beats the worst 50% of HU players and loses to the top 50% would break even playing random HUSNGs (assuming no rake), since he'd cash in 50% of his games, but someone that always outlasted 50% of MTT players would not cash 50% of the time (only 15% of so get paid), nor would he win the big prizes very often. I'm not sure how this "variance" impacts ROI, but in the real world it's clear that MTT players have a higher potential ROI (due to the sheer number of weaker players that are donating buyins to them) than HU hyper SNG players (who often have an ROI of <5%), but the hourly earn of the latter is arguably higher, and certainly steadier. (Or at least it was when they were getting rakeback).
I think the problem is kind of unsolvable, or at least not without making assumptions that aren't practicable in the real world. :/ Good luck in Vegas!

I just want to avoid any scenario where the relative "softness" of the field is cited as a reason for a reason to play one tournament over another.

I'm a cash game player, not a tournament player, so I think about what poker I want to play in terms of hourly rate.

Let's say that one can make enough to be a professional tournament player by playing a $500 one-day tournament every day. (And I have no idea how accurate that number is.) If you could instead play a series of $500 three-day tournaments (with field sizes appropriate for a three-day tournament), being able to start a new one the next day after you bust from one so that you spend about the same hours playing over the course of a year, would you be making more money, less money, or about the same, on average?

Just for fun, here's a very naive model for winner-take-all tournaments with 10, 100 or 500 players. We assume that all the other players are of equal strength with one another, and we expect to win 51% of the time against a single opponent. We also assume there's no rake, no friction and no air resistance. We then calculate our expected value from the tournament for every field size.

EV for 10 players: -1 BI + 0.51/(0.51+9*0.49)*10 BI = 0.037 BI
EV for 100 players: -1 BI + 0.51/(0.51+99*0.49)*100 BI = 0.040 BI
EV for 500 players: -1 BI + 0.51/(0.51+499*0.49)*500 BI = 0.041 BI

Same calculations if we expect to win 55% of the time against a single opponent:

EV for 10 players: -1 BI + 0.55/(0.55+9*0.45)*10 BI = 0.197 BI
EV for 100 players: -1 BI + 0.55/(0.55+99*0.45)*100 BI = 0.220 BI
EV for 500 players: -1 BI + 0.55/(0.55+499*0.45)*500 BI = 0.222 BI

So at least under these circumstances, it would be more profitable over the long term to play two 10-player tournaments than one 500-player tournament over the same time frame. Of course, it's a separate question whether the above assumptions have any relevance to reality

What about introducing a random element to the original assumption that includes the possibility of bad play?

It seems that shorter tournaments are both more profitable and less variance at the same ROI

If I play a 1 day $500 with 50% ROI i expect to make $250 on average per day. If I can play 10 on these in 1 day I expect to make $2500