Fact :
(Supercell) generated earnings before taxes and other items of $1 billion, up from $897 million a year earlier. That’s an enormous profit for a company with just 213 employees and only four games on the market.
First, a bit of background reading:
Compulsion loop (in game design)
Why Mobile Games Make You Their Slave and You Pay for the Privilege
Five tips on beating video game addiction
...And of course the famous Candy Crush Saga: The Science Behind Our Addiction.
Oldskool video games were only a bit addictive. Taking online FPS as an example, leaving during a game is rude but games are usually short, so you have an opportunity to leave after every game, and you can come back whenever you want. In this case, you keep playing because it's fun, and the biggest problem of these games is that they are too much fun.
Free to play (ie, pay to win) mobile games work on a completely different paradigm. They are designed to create addiction in order to turn the player into a cash cow. Letting the player have fun is only one of the tools in the toolbox to achieve this goal. Candy Crush is applied science (in this case, existing knowledge on gambling addiction).
A survey by Ask Your Target Market polled 1,000 players and found that 32% of them ignored friends or family to play the game, 28% played during work, 10% got into arguments with significant others over how long they played, and 30% said they were “addicted.”
These games create addiction by:
Giving you some fun, then taking it away. The waiting part is essential. You have to wait while your troops train or your buildings upgrade, or your energy recharges. Waiting creates craving and anxiety, it builds up the tension, until you're allowed to have your fun again (or you pay).
Creating a rhythm: for example your energy bar fills up in two hours, and if you don't use it then it stops filling. This is designed to make you open the game every 1-2 hours, to reinforce the compulsion. Click on your gold mines! They're full! You're losing gold! The more evil the developer, the shorter the delay.
Random events with notifications: Oh noes! Someone is attacking your village! Open the game now!
Random loot: this is also essential (see: RPGs). If casino jackpot machines always returned 90% of the coins you put in, no-one would play. It's the randomness that makes it exciting. The machine will return some part of what you put in (on average), but it'll do it only when it wants. So you have to keep playing in the hopes of winning next time. Maybe the next loot box contains a rare item?
No end: Most games (except sandboxes) have an ending. In single player, it's the end of story, in multiplayer it's the end of the match. Endless games (like Clash of Clans) are specifically built for addiction. There is no reward at the end, and also you'll never get there as you either need to play every day for 5 years or spend $10k.
Social pressure: this one is the worst, as demonstrated by WoW zombies. If you can't be there for that 5 hour raid, your friends will hold a grudge!
Now to answer your question:
What's a good plan to make her at least reduce the amount of hours spent on the game without compromise our relationship?
Reducing the hours won't really work, the game is designed to be addictive, so it will creep back in. Complete cut off is required.
Now a possible solution:
1) Working on her defenses
Read up on the stuff, and watch her play, take note of the compulsive loops in the game, understand this game and figure out what makes it addictive, with an aim to find the weaknesses in the game in order to dissuade her. Then...
Ask if she's actually having fun while playing the game. Most likely not. These games hand out equal amounts of fun and frustration, and you have to pay to alter the balance in your favor.
After doing your research, explain the addiction mechanisms built into that game (see above, that's why I wrote this) and show her how it's designed to sucker her into wasting her time and money.
Explain how she's being manipulated. Your goal is to teach her how to recognize these mechanisms herself, so she will think for herself and come to the realization on her own that she's wasting her time.
Remind her the game has no end, therefore it's all pointless. She will get bored eventually, and will then switch to another game. The point above is also meant to prevent this.
Try to make her realize that her addiction is a problem. There may be evidence in the house, like piles of unwashed dishes, big messes everywhere, piles of unanswered job applications, her being dressed in pajamas all day, etc. If she stays home all day playing she might also get health problems. When you come home at night, and ask her how her day went, and she only talks about the game, you can ask "yeah, but what did you actually do?"
Make gentle fun of her (Are you there? Do I still have a girlfriend?)
2) Social engineering
Invite people (her friends) for dinner, board games, etc. Make sure it's something she likes and would love to participate in.
If she has fun without her phone, then she's on the right track.
If she spends the evening on her phone, someone will inevitably say something. Make sure this someone is not you. You will have more success in making her realize that her addiction is a problem if several people independently confirm, plus you don't have to be the annoying one who says it to her, it can be someone else. If this makes her think and she asks you "do you really think it's a problem" then yes, you should be sincere.
3) Don't be too harsh
If she gets into the game to escape from something (like stress) then you don't want to add more stress, which would cause her to retreat more into the game. Also, when stressed, an addict will turn to the drug (that's why it's called a fix), so avoid confrontation, be (or at least sound) understanding, etc.
Example: if you talk about cancer to a smoker until they freak out, they will have a cigarette to calm themselves down. So this doesn't work.
Perhaps her reason for escaping into a game is stress from not having a job, or your upcoming marriage, or boredom, unhappiness, lack of self-esteem, or any other reason. You should try to find out. If she tries to escape from unhappiness, giving her an ultimatum is pretty much guaranteed to backfire.
This also means no ultimatums, or blackmail, etc.
If this gentle approach fails, escalate to phase two:
Organize a week-long hike in an area without cell coverage. Or a boat cruise, whatever, anything you want. It has to be something she would enjoy. If you don't find an area without coverage, then make it a condition that she doesn't bring her cell.
Thus you give her a huge reward: the trip she's always been dreaming about! (she's probably been talking to you about something like that...) and you pay for everything! (if you plan to marry her, surely you can spare a couple thousand on curing her addiction)... in exchange for a small compromise on her part: her cellphone stays home. This is the enhanced version of the cosy social evening with friends.
If she says yes and you spend 2 weeks together in a tent and you don't murder each other, then you will end up a lot closer together...
If this fails (ie, she would rather stay with her phone than go on a cruise with you) then let's be honest, the outlook is bleak, and you're gonna have to escalate to stage 3: Dread, aka Cheating Simulator.
The point of this is not to cheat on her but rather to remind her that leaving her is a definite possibility if she keeps dating her phone more than you. It's quite simple to do: spend a nice evening with some friends or colleagues, leave her home with her phone, and mention in passing that you think Stacy has a crush on you.
This is a bit hardcore, so use only as a last resort. Not as bad as an ultimatum though.
Now some loose ends:
she plays this game from the morning to late night continuously, she doesn't work so she plays all the day.
...Is she at least looking for work?...
Even when we hang out together for some romantic time in restaurants and so on she always has the smartphone with her, with the game on.
That's a total lack of respect. I would walk out, even if I said to avoid confrontation before. The reason is simple: you can't help her out of this if she doesn't respect you, and she can't respect you if you don't respect yourself, and if you respect yourself, you're walking out of the restaurant.
We had the plan to marry in the next few months
You're using the past here, so you sound like it's canceled, at least in your mind. You can always reconsider if she comes out of it and wins you back.
I thought to give her an ultimatum of the kind "stop with this game or I quit", but knowing her she wouldn't accept this and she'd only get upset and probably leave.
Well if she values the game more than you, then you're in trouble...
I tried to get involved in the situation to help her make the game easier, I setup for her a bot to automate some extremely time spending actions, but it didn't help...
You're validating her addiction. When someone attempts to quit smoking, would you offer to buy them cigarettes? No! It's much better to offer alternatives (ie, social life, for example).
EDIT: Rat Park - excellent comic on addiction from LinuxBlanket.