Consider Battle Hunter for PS1
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It's a board-game type dungeon exploration video game compatible with Multi-tap and up to 4 players.
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The dungeons are randomly generated on a grid, and your characters are all given cards with which to use during different situations.
There are four types of cards, indicated by color, and they are Attack cards, Defense cards, Movement cards and Trap cards, which you can lay for the other players to land on which has a variety of effects.
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Battles take place in a 1v1 format where players each 'roll' a number, matching up their stats to determine how much damage is taken per combat turn.
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All in all, it is a great party game and has a lot of interesting mechanics to discover.
It's easy to learn and fun for up to 4 players.