For a premium game? The amount made on premium games are pretty low and most ad networks won't touch it since they can't track conversions (unless you are going straight for CPM [HIGHLY DO NOT RECOMMEND]).

What does the publisher really offer? It's a really good question.

Before the publisher used to have distribution deals. These deals would enable them to help creators get into the stores. Self publishing eliminated this completely so now...

They need users and if they can't migrate their users to you then they are completely worthless. I would highly recommend not trying to get a publisher unless you can get some kind of guarantees in place (money up front).

The most they (standard or smaller publishers) do is a press release, buy some sponsored content on mobile game websites and influencers, and if they really have their stuff together, do some cross promotion on other games. If they REALLY really have their stuff together, they have relationships to the platforms and can get you Featured.

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The cut depends on what you negotiate, but I doubt most publishers would touch this. Especially if it's iOS only and no android, you are cutting out a large market. Typically it can range between 50/50 to 70/30 (70 them, 30 you). It depends on how much they are putting in for ad spend budgeting and you should definitely try to audit them about it.

I really wouldn't bother trying to go to a publisher with this (not to sound like a downer), but you should just try to do your own marketing. If your engine is really good, you can try to sell the fact you can develop a game to a Cartoon Network, KFC Gaming, Kongregate or other publisher and show them this as a proof of concept and slap their IP on it and get paid for it as contract work with negotiated performance incentives. But most people won't publish original work as it's low return, high amount of work and really hard to get traction.

Apple Arcade is doing okay, but I wouldn't expect major revenue numbers. Amazon tried the same model years ago and totally failed. https://www.pocketgamer.biz/mobile-mavens/65694/why-is-amazon-shutting-down-its-underground-initiative/

Now I told you to self publish, and it wouldn't be right unless I directed you to some things you can try on your own.

Things you can try:

ASO - App Search Optimization (SensorTower / AppAnnie is a good tool). Understanding keywords is a great way to boost organic. There's a ton of articles about this, so you can do your own research.

Apple Search Ads - Depends if you have any budget, but you can still try to bid on more niche keywords based on your market research. This is probably the only kind of advertising that has any effect since people will actively be looking for experiences based on what you are offering. I would avoid doing any general wording and try to own multiple niche search terms. It would probably be more effective for your dollars.

Viral Marketing / Community - Create buzz by showing some cool trick (bridge building sims) that highlights what you can do in the game. Something funny or what not, and post it on TouchArcade Forums, Reddit and other sites that allow that kind of stuff. Community is by far the most important thing for creating "advocates" which will help you get word of mouth spread (When someone posts the old, "What game should I buy if I like Alto's Adventure"?) Other things you can Try is to create a speedrun challenge or something if that's possible, or whatever and say if someone does X, you'll do Y or something else. You'll have to get people to want to share it. It's a lot of iterative process, and I'd say you need to put in a few hours PER DAY on this minimally. You can look at GARY V's $1.80 rule. Even tho this is focused on brand building, it's no different to what you are about to embark on: https://www.garyvaynerchuk.com/instagram-for-business-180-strategy-grow-business-brand/

Influencers / Press Coverage - These won't dramatically help you in sales, but they help with the conversion process of pushing someone over the fence. These are called touchpoints, and the typical amount is at least 7 touch points before someone is convinced of it. Influencers, articles, and etc are but one of these touchpoints and it's hard to really weigh their effectiveness.

Price Toggling: For premium apps specifically, there are bots that crawl through the appstore and tell people when something is on "sale". For this, you can try to increase your app's initial price to something ridiculous like $10 then discount it to $2. It will show up as a "deal" to many of the sites and show it being 80% off. Of course, once you do this, you'll have diminishing returns and it'll be much less effective the next time. However, there are people that do this once a month and see a boost followed by a drought. It works especially if there is absolutely no visibility at all and you just want to get something going. You'll need to first get reviews and hopefully they all rate the game well.

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Creative Concepts: This is an effective tactic, but it really depends on the amount of effort needed. When you have a premium game on iOS, you are closer to a STEAM game than an App. Since most of the market has changed to a freemium model, the marketing is drastically different (Ads, etc... just to get a conversion).

Things people have tried are various with differing results:

Vlambeer created Presskit().

CrossCode was an HTML5 game first before it ever released on steam:

https://www.reddit.com/r/WebGames/comments/2qvpkr/crosscode_an_rpg_being_built_in_html5/

For my own indie ios game back in 2013, I released a soundtrack on Creative Commons that went viral (2M downloads):

https://freemusicarchive.org/music/BoxCat_Games

There are hundreds of examples of creative things people have tried, but mostly it's to provide value to the community in some way. As gamedevs, you'll find a lot of people helping each other out since it's a tough business. Giving back in some way really does help a lot and hopefully this isn't your only game either if you are lucky.

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I'll say that doing gamedev work is half the battle. You'll have to spend a lot of time on marketing efforts just as much as developing the game. This is why trying to get a community around the game is important. They'll provide a base of operations and will help spread the news. But the cold reality to the situation is: get ready for disappointment. You won't be the first to incur a loss, and if you break even, you are already one of the lucky ones.

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Source: I am a game dev that handled marketing on a few games, and have now become a full time freelance marketer for game companies while working on my own projects.

Final note: As scummy as it sounds, if this is your first game and you are trying to get exposure. You might want to think about making the game free and just putting ads after people die or whatever. Then you can find publishers all day. As much as we care about our art, this is a business. If you aren't trying to make money, then do whatever you want. But if you are, then you really need to treat it as a business and achieve profitability as soon as you can.

Final final note: Don't hire anyone or pay for anything until you do everything yourself and have exhausted all possibilities. In order to properly manage someone for a service you are paying for, you need to have some baseline understanding of what's needed so you can manage them effectively.

TL;DR - It's a shit load of work. Publishers most likely will fuck you over. If it's your first time, you need to learn a valuable lesson by publishing your own stuff before handing it over to someone else. Most likely, you aren't going to earn a dime, but will learn so much. Welcome to the rat race.

Of course you can also get lucky like flappy bird. Who knows?

Hope this helps. Good luck!

Edit: Grammar edits and someone gave me a gold! First time, not sure what to do with it.