In a normal game of mafia/werewolf, there are a few high-level strategic elements:

• Night Play: During night phases, the mafia has to choose who to try to kill and the players with night actions (cops, doctors, and in some cases trackers, watchers, vigilantes, etc.) have to choose who to target.
• Day Play: During day phases, the townies try to share information without exposing too much information about the identities of their power roles (cops, doctors, etc.) while the mafia tries to provide misleading information. Everyone is trying to affect (1) future night actions and (2) the daily execution vote.

The goal of the prediction market mafia game is to replace the voting mechanism with a set of markets, one for each living player on whether they’re a member of the mafia. At the end of each day phase, the player with the highest market price is executed. 

If you replace the election with a market that the players participate in, the game will break because townies have an incentive to bet their own markets down to 0% and to pressure mafia players to do the same and drain their capital (I didn’t think about this that hard; maybe a few adjustments actually suffice to make this setup work, but I don’t think so). 

A simple way to make it work is to let the spectators bet on the prediction markets. So now there are multiple parties to the game: 

• the town players, trying to kill all of the mafia players;
• the mafia players, trying to kill the town players; and
• the speculators, trying to maximize their profits.

I used the term “spectators” above, but the speculators need to have everyone’s night actions hidden from them the same way players do.

Tweaks/considerations:

• The end of the day phase could be decided by some kind of random process that avoids the problem of rapid trading activity right before the phase ends.
• There can be other speculative markets that aren’t execution-relevant, such as on the identity of particular players or roles, the number of phases the game will take, the eventual winning side, and so on, but this might lead to weird outcomes where speculators move the execution-relevant markets with the intent of manipulative other markets, such as by trying to take a loss by getting a town player executed because this will increase profits on other markets.
• In most setups, most dead players won’t be able to join the speculators, because they’ll possess game-breaking information.