I am actually against this idea, and before *I* am the one being executed, let me explain why.
In-game mechanics should not be used to punish out-of-game transgressions.
If a person bought gold/wyr from a some child churning out change in China, and his character goes up for public execution.... what is the crime that will be read out before the character is executed?
On top of that a public execution is effectively an event, and some drama queen will break the rules just to have this event centred around their character and attend on their new account.
The stocks in Dragonrealms are there to punish the character who break provincial laws and are subsequently caught (doesn't always happen, and could take a while). Some of the less tolerant communities, such as the barbaric clans will chop off a thief's hands, and other punishments are also present in various nooks of the game. None of these are used to counter violations of policy (out-of-game laws). These are handled by GMs with things such as lock-outs, bannings, policy rooms, etc. all behind the scenes.
The deputy idea is flawed for very much the same reason. Players should not be responsible for policy enforcement, much like amusement park visitors don't get to vote on one of them to have a go at the ferris wheel controls.
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Now all that being said, I think the public execution and even the deputy thing are great ideas. I would love to see convicted murderers be publically executed.
The deputy thing could be awesome if, for example, the Arcanum got to elect 7 Spellsworn, whose function is to enforce the will of the Arcanum, and they would be free to engage in CvC against anyone at all times. A character could issue a ritual challenge to a Spellsworn for his/her position, by having a quorum of 21+ members of the Arcanum witness the battle. If the challenger wins, the assembled members of the Arcanum would get to vote on whether the defeated is to relinquish his position to the challenger.
All this would flesh out the world, providing for immersion, fun and wonderful depth.
Just my opinion anyway.