Right, ZK's post was wonderful as expected, I've had many ideas in the past that could lend themselves to this system perfectly if people were willing to take the time to create scenarios. I've used very basic examples in the past and will add links and such in this post to give you all a clearer understanding of how I see the mission system being utilised in part. I have a few different versions of similar ideas, each focused around different elements of RP but with the same base foundations. Hopefully I do a good enough job of getting across what I want.
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Puzzle Rooms
My first idea would be an extension to something I tried
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that I used not too long ago. So here all of my posts were detailed descriptions of the environment, I left many doors open on how to proceed with the training and had a different plan set for every decision my student could have made, everything was pre-thought out and had a direct link to the next stage of the training but would cause a different reaction in the story. I set-up scenes reminiscent of those found in the 'Tomb Raider' game series, you have a clear end goal and a clear starting point but everything in between is fairly free for you to decide hat to do, the way you choose to RP the situation effects the environment in different ways and brings forth new challenges that you must overcome before you reach your end goal. Now, with access to all the techniques most of us know then obviously a lot of these situations would be too easily accomplished, but then it's down to the 'narrator' or 'gamemaker' as I'd prefer to put it to think of new and inventive ways to limit the participating RP'er.
This type of mission could be taken up by the kage of one village who sends one representative to take on the RP. If they succeed then their village gains whatever benefit or prize the RP'er collected upon completing the puzzle. They fail, then it gets offered out to the next village and so on and so forth.
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Contests.
This is taken from the Training Centre portion of the build-up to any Hunger Games in that series, or from the pokemon games where different moves/abilities have different moods (Cute, Cool, Smart? etc.). We invite 1 or 2 RP'ers from each village to a central location where their is a contest taking place. Depending on the type of contest, Kage may choose to send different members of their villages depending on whom they think will handle the 'style' of contest better. This is good because it still grants the RP'ers a competitive environment, but one in which we use words instead of power.
We have a panel of judges which all score each participant on a scale of 1-10, highest overall score wins, numerous rounds can be used to show off different skills... Maybe villages can send a specialist for each of the rounds and tag in when needed.
Potential contests could include things such as 'Cute Appeal' contest. So participants will be put inside a room of our design and given any number of items and abilities which they are allowed/not allowed to use. They have a single post in order to score the highest cute score they can, they are able to use whichever means they have access too within the set rules and can be as creative as possible.
Same thing with other contest styles. 'Cool', 'Power', 'Terror', etc.
I see this being done with PM's to a head judge in order to prevent any cheating. Each participant has 3 days to message their performance to the head judge who will post them all once he/she has received every participants entry, or the time runs out.
From there the judges vote for round one. Before round two starts and the process starts again.
I also have some nice ideas for prizes on this as each village is taking part. It'd be a village benefits system in which the winner/kage can choose between a defensive or offensive one time use boost for their village in the NW. Defensive boosts would be things such as a 'Help from the Locals' boost which can be placed within a landmark within their own village. The kage picks a landmark, which they tell to a moderator so they cannot change their mind. Then the next time one of their members bios is killed inside that landmark, they are able to bring their second bio into that event in order to keep on fighting. This is a single use item and is only applicable within the selected landmark. I like this idea as it gives some tactical experience to wars that is currently lacking, you can try and bait an opponent into fighting inside your own boosted zone and reap the benefits of that zone should you hit the criteria. Opponents could see you're forcing them into a certain zone and try to counterplay that with different routes.
Offensive boosts could be things such as giving a 'checkpoint' within another villages landmarks. This could grant your village a chakra refill or an extra fight for any members who manage to enter that zone in the event that you choose to activate it. Obviously this would need limitation on the number of times its used etc. But again, it would add some tactics to wars.
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Betting
This isn't an idea so much as a potential risk/reward idea for potential 2+ player RP adventures. Both participants put something up for grabs, be it a CJ or CE w/e. Winner takes all.
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My last idea would be a longer version of the first part of this post and based on those fightting fantasy gamebooks. My father and I used to use them all the time when I was little, he'd sit down with my brother and I and we'd create our characters, rolling die to create our strength, luck, intelligence scores and such before setting off on an adventure which twisted and turned based on the decisions we made, come across a troll in an alley? more luck than strength? You attempt a sneak! It doesn't work! Troll Smash.
This would be a lot of work on the mods part as these adventures are complitcated to make as every decision needs to lead to a new already thought out piece of RP, it needs to be difficult enough but completable. Only thing that puts this into near impossibleness is incorporating it into the Ninja World, with our bios that can perform miracles.
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Final Note
If I'm ever allowed to run a Hunger Games event on this RP, I promise it would be brilliant. I have it all in place, all the balancing is done and the pre-planning has been planned. It would rely on the participants being active and would involve fighting at times, but as I said, I have it all down on paper and ready to go.

#HedaforGamemaker.
Final Final Note
If any mod of NWA reading this would like examples of any of this, most of it can be found in basic form in my Hunger Games RP group. I had the 'contest' thing running in the Training Centre thread as well as other things.