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What metrics could be used to know what certain people think about the game and how much they liked it? I would prefer parameters that don't need players to rate by themselves. Would hours of playing in a single seat be a good parameter about how much players wanted to spend time on game?

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This is a very tricky concept to measure for many reasons. DMGregory suggests very common metrics in modern games. I would suggest that these metrics, again very common in the industry, do not accurately measure "player engagement" as much as they do "player use". To put that more accurately, these metrics measure both player engagement AND unengaged addiction or unengaged habit. Mobile games, specifically, have become global leaders in using psychology, addictiveness, and gamification to encourage obsessive, addictive, or habitual play for corporate gain.

If you want to measure "fun" or "engagement", I would suggest looking at (some of) those metrics with many, many others to gain a fuller picture of play style.

Metrics that would be useful for this process include:

  • Biometrics: Expressions, microexpressions, pulse, brain waves, verbal feedback, nonverbal audible feedback, nonverbal body language, etc. These can require different types of equipment which can be cost prohibitive, or different types of software to utilize facial recognition which could also be cost prohibitive.
  • Survey/polling: This can be polling for opinions after the play session, after multiple play sessions, during the play session, or even integrated into the play session as a play mechanic either overtly (the game systems ask for ratings of different levels, areas, enemies) or covertly (the player is asked to choose which level to repeat or which enemy they would rather face again)
  • Typical play time metrics: Whether measuring attrition (how many people stop playing after the first jump, the first enemy, the first level, the second level, the third level) or direct play time (how long do players play before quitting for the day? For the week? Forever?), this can give an idea of player engagement, but can also be misleading if the game is designed for an addictive or habitual play style
  • Less typical play time metrics: If there are multiple challenges or goals presented in parallel, which are chosen more often? Which less often? Those that are chosen less often, are they never chosen in the first place (meaning the problem lies in the presentation) or are chosen but never chosen again (meaning the problem lies in the execution)
  • Attempting to measure frustration or boredom: If player engagement is how much "fun" or "invested" a player is in a game, then the opposite would be how "frustrated" or "bored" the player is. If engagement is "flow", attempt to measure the lack of flow. Is the player dying or repeating a certain segment repeatedly before quitting for the day/week/forever? Is the player breezing through a rather simple segment of the game before quitting for the day/week/forever? Is there a point when the player starts mashing buttons on the controller or mouse or tapping frantically on the phone? Is there a point when frantic mashing or tapping happens right before a loss? Does the player lose "a little" and come back or does the player seem to lose with frantic mashing followed by the session ending? The former suggests normal play while the latter is something everyone has experienced and can easily recognized as "not fun" or "not engaging".
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    \$\begingroup\$ One thing I'd always make sure to do is include a "congratulations, you ran the game" achievement (especially for Stream released games) so you can get an idea of how many people have actually played on Steam, as Steam does not release those numbers. \$\endgroup\$ – Draco18s Oct 4 '17 at 3:52
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There are a lot of metrics or KPIs (key performance indicators) in common use in mobile games, some which might touch on what you're looking for:

  • DAU (Daily Active Users) - how many distinct users play in a single day?
  • MAU (Monthly Active Users) - how many distinct users play at least once over a month?
  • Stickiness (DAU ÷ MAU) - what fraction of your players are coming back daily?
  • Churn - what fraction of players who were active last month don't return this month?
  • Duration (1 ÷ Churn) - how long does the average player keep playing the game?

For games with micro-transactions, we can also get a proxy of engagement via willingness to invest money into the game:

  • Conversion Rate - what fraction of your players eventually buy at least once?
  • ARPU/ARPPU (Average Revenue per (Paying) User) - how much do players spend?
  • ARPDAU (Average Revenue per Daily Active User)...

...etc.

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Simply tracking user per day and related metrics isn't the only way of measuring user engagement. Another approach involves tracking events, such as: how many users make it through the first level, how many went on the first sidequest, did any users find that secret shortcut you made, etc. Here is an article that should hopefully get you started with this sort of analytics.

https://www.gamasutra.com/blogs/ChristinaChen/20170215/291552/How_to_start_using_game_analytics_in_2017.php

However if you are interested in the data science behind basic game analytics and commonly used metrics (churn, KPI, etc.) here is an article on that:

https://www.gamasutra.com/blogs/AdamFletcher/20170817/303993/Applying_Data_Science_to_Understand_Your_Players.php

If you are looking for specific implementation for whatever engine you are using probably check the docs for your engine, and then ask another question here if necessary.

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