It should be warned, that this Persona 5 trophy can be a slight pain in the arse to get thanks to the fact that it is heavily RNG based. The Competent Negotiator trophy itself is rewarded to you upon successfully rescuing one of your characters once they have been taken hostage by an enemy unit. Negotiating your unit free is exceedingly easy.
If you are connected to the internet and the Thieves’ Guild, then you can merely call upon them to instantly save your unit. If you are not connected, then complying with the enemies’ demands will save your unit and earn you the trophy.
The hard part is getting the enemy to actually take one of your units hostage. They can only take your units hostage after they have landed a critical hit or a weakness blow and left you at low HP. However, the AI has 2 options at this juncture: 1) Land another free hit. 2) Take your unit hostage. The fact that I am writing this guide proves that they usually choose the former option.
Strategies
There are some strategies, however, that can help you out. One option is to go to earlier dungeons, thus allowing you to withstand several attacks while at low health. You can also put the game on a harder difficulty level, which increases enemy critical hit rate. You can even choose to play the game extremely risky by choosing to heal infrequently.
Abyssal King Strategy
There is also one, more specific, strategy that can work, courtesy of Timio95x. It does contain minor spoilers for the final dungeon. Continue at your own risk.
First, change the difficulty to safe. This will allow you to revive your whole party when they die (and they will die with this strategy).
Second, you will want access to the Revolution skill, which increases enemy and allied crit rate for 3 turns. This isn’t technically essential, but it does help speed up the battle quite a bit.
Now you will need to be in the last dungeon. On the second floor is an enemy named the Abyssal King of Avarice (Lv74). You will want to kill all the other enemies, leaving him for last. Once in combat you will want to use Revolution to raise his crit chance. His physical, AoE attacks should now crit quite often.
Since he is immune to physical attacks you can use Rush attacks to speed the fight up. You merely need to stop to activate Revolution after every 3rd turn. When your team dies, choose to have your team revived (hence why Safe mode is important).
Merely rinse and repeat this until the Abyssal King takes one of your units hostage and then give him what he wants (probably avarice).
There you have it. Several different strategies to get the process for acquiring this trophy started.
