Game data mined and analyzed by 101Leafy, and tested against real gameplay by 101Leafy since 2015.
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Formulas were traced in the Yo-kai Watch, Yo-kai Watch 2, and Yo-kai Watch 3 game code; the game logos show which games a formula or value applies to. Each topic has its own page.
When a Yo-kai attacks
The Yo-kai with the lowest wait counter acts (Turn Order). Between turns you can spin the Yo-kai Wheel.
It picks a move by its Attitude (Attitudes). Your own Yo-kai do the same unless you tell them what to do.
The damage comes from the move’s power, the attacker’s Strength or Spirit, and the target’s Defense, with stat changes (Damage). Techniques are multiplied by the target’s element (Weak & Strong Elements).
A guarding target takes less (Guarding). Otherwise the damage varies a little at random.
The Soul gauge fills. A hit on a Yo-kai charging its Soultimate can interrupt it (Soultimate Charge).
After the battle you get EXP and money (EXP & Money), more if you pinned a Wisp.
Yo-kai Watch 4
Battles run in real time, with no wait counter and no Yo-kai Wheel. The known values (critical damage, elements) are on Damage, Critical Hits, and Weak & Strong Elements.