I think 3 player games are fundamentally different from other games because you end up balancing the game all the time. I talked about this in length in my book. In short you have to balance between not being the weakest player and not being the strongest player. I.e. you want to end up in the middle. If you become the weakest player, the chances are high that other players will gang up on you and kick you out. If you become the strongest player, the chances are that other players will target you more than normal in fear of you winning the game.
So in effect, the game is about gathering enough momentum so that you can eliminate one and damage another enough so that the remaining won’t be able to bite back.
I used to play many 3 player games a lot, and eventually we discovered that if players are very experienced, the game may never finish. Because at each stage of the game, two players get together against a third until the strongest player become weak. Then players switch sides and the cycle start all over again. You worry about another player being too strong, or you yourself become the strongest player and start to play defensively against others.
To see how you can win, look at other games when you start with 5 or 6 players and then get to 3. It seems that in these games we never have a problem and the cycle never happens. The reason is momentum. Since the beginning of the game, there has been lots of historical development, agreements, revenge, enemies and allies and so on which when you get to the 3 player stage will not stop you from finishing off what you started. In other words, player miss the chance to balance the game so that would go into a cycle, because they are so focused on their target.
So to win in a 3 player game, create momentum. That means you need to become powerful without showing it. You need to establish links with another player so when the right time comes you can exploit it. Create a history in the game.
For example, since your game is played over days, you can negotiate with other players during this time and over the back of the other. Of course you also have to anticipate against others doing this to you. Play a fair game, but with a twist. Make sure everyone has fun (I am not suggesting to backstab them, just negotiate with them), and try to win by being deceptive. The less they know about your power the better. Then unleash it with max effect.
Let us know how it plays out and what your friends end up cooking for you.