PokéBase - Pokémon Q&A
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The damage taken by the player's trainer is loosely communicated by a red bleed border effect and audio cues, but I'm curious about how the power of a move actually affects the amount of damage accumulated. Broadly, stronger Pokemon seem to deal more damage, but anecdotally, it doesn't seem entirely consistent. For instance, my own lv 100 Pokemon appear to consistently do minimal (but not zero) damage when their attacks hit my player character, despite having higher stats and moves with higher base power than most Pokemon in the game, including some of the "special" enemies like Alphas and Rogues, yet special fights (especially Zygarde and Hyperrogue Ange Floette) seem to do relatively intense levels of damage to the trainer even if they don't appear to be all that strong, as they don't damage my Pokemon that much. This leads me to wonder if damage is calculated differently than it is for Pokemon attacking Pokemon and/or whether there are special exceptions for your own Pokemon and for special fights. There also appears to be inconsistency in how long recovery seems to take.
I'm sure these observations probably aren't entirely accurate, but I am curious about how it really works under the hood since monitoring the damage on your trainer is a fairly relevant part of the new battle system.

I'm looking for an answer that describes what specific factors influence the damage taken by the trainer beyond just "stronger" = more damage. If possible, I'd appreciate a quantitative answer from the perspective of how the game calculates it that mentions the specific numerical point at which the trainer will black out (does the trainer have HP?), how the number of Green Canari Plushes owned affects this threshold, and how the rate of recovery works.

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https://www.serebii.net/legendsz-a/playerhealth.shtml
This explains most of what you're asking about.
thanks, if you want to post that with a quick summary I'd be satisfied with your answer. Through further gameplay I've noticed that it doesn't seem like my own Pokemon actually do damage to my trainer, the trainer just reacts as though they did.

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So let's take this bit by bit.

Does the trainer have HP?
Yes, the amount of HP you have depends on the level of your green Canari plush.
No Green Canari Plush = 100 HP
Level 1 Green Canari Plush = 150 HP
Level 2 Green Canari Plush = 200 HP
Level 3 Green Canari Plush = 250 HP

When the red border appears on the screen, you have taken a lot of damage and will black out soon.

When you are not being targeted by an opposing Pokemon you will regain 20 HP/second.

How do you take damage?
When hit by an opposing Pokemon's move, you will take damage according to 2 factors
1. The base power of the move used: this table was copied from the Serebii Page.

Move Base Power Damage Inflicted
25-49 30
50-74 40
75-99 50
100+ 60
Rogue Mega Evolution Special 70

2. The base stats of the opposing Pokemon and other factors (also called Total Power) This will create a multiplication factor of this damage from 0.5-1.5.
While Serebii does not include all the factors, this is likely why Alpha Pokemon do a boatload of damage to you.

Source: https://www.serebii.net/legendsz-a/playerhealth.shtml

I will update this as more information is available.

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