The core idea behind the Games of Ni comes from classic auction theory and mechanism design, and is backed by research of a large number of Nobel Prize economists thinking on the subject (William Vickrey and Roger Myerson to name just a few). But the origins of the idea can be traced to earlier times.
Goethe and the first second-price auction ever
You are Johann Wolfgang von Goethe, who is about to finish an epic poem Hermann and Dorothea. You are going to have to deal with a publisher to negotiate royalties for the poem. Selling at a higher price is desirable, but there is a price, which you conider reasonable. Everything below it is much less reasonable, and you would rather keep the poem and look for a different publisher.
Any publisher, being himself a person and all, has his valuation of the poem as well. As befit Germany's greatest poet and a humanist, you do not want to haggle over the price of poetry. Instead, you invent the second price auction two centuries before economists will rediscover it:
I am inclined to offer Mr. Vieweg from Berlin an epic poem, Hermann and Dorothea, which will have approximately 2000 hexameters.... Concerning the royalty we will proceed as follows: I will hand over to Mr. Counsel Böttiger a sealed note which contains my demand, and I wait for what Mr. Vieweg will suggest to offer for my work. If his offer is lower than my demand, then I take my note back, unopened, and the negotiation is broken. If, however, his offer is higher, then I will not ask for more than what is written in the note to be opened by Mr. Böttiger.Why complicate the matter in such a way? Had Goethe just stated his price, he would never know the publisher's offer. Through his mechanism, Goethe discovers the publisher's offer for the poem, and publiher has no incentive to lie. Goethe devised what present-day economist would call a dominant-strategy fully revealing mechanism. This is the same idea that governs second-price auctions and ultimately lies at the core of the Games of Ni.
The mechanism employed by the Games of Ni app is a modified Vickrey-Clarke-Groves mechanism. This is hands-down the one auction theory idea that had the biggest impact outside academia, beginning with the spectrum auctions of 1994 (or, again, 18 century German literature, because YMMV).
Games of Ni attempts to bring this great concept to even wider use. What follows is slightly more technical description of what makes VCG so great (and what does not). For less economics and more slack, go directly to "How-To" page instead or add the app for your team. 
Vickrey-Clarke-Groves mechanism
The Vickrey-Clarke-Groves (or VCG) mechanism is a sealed-bid auction that charges each individual the harm they cause to other bidders. It is a generalization of a second-price auction, in which the highest bidder gets the item, but pays the second-highest bid.
The primary benefit of VCG is "truth-reporting". It implies that nobody has to overthink his decisions and can simply bid his valuation (roughly what one considers the highest fair price for the item).The idea of "truth-reporting" lies in game theoretic concept of a dominant strategy. Imagine yourself in a horrifying world where the Games of Ni have not been invented. Among the chaos and distopian scenery, you (along with your comrades) attempt to make a decision with any other mechanism. Bidding your true value does not seem like a very sophisticated choice. Being a smart distopian citizen you would think strategically - you would try to predict what other people would bid and use that to make a decision. For example, in a first-price sealed bid auction, you would try to bid just a little higher than the highest bidder.
Second price auction spares you this. Strategizing does not help and can not help - the amount you pay does not depend on your bid if you win (only on the second-highest bid). So the only thing you have to consider is - how much you actually value whatever it is you are bidding for.
The game theory class usually explains it in the following way:
Say you want to bid for something, something worth v dollars to you. If you believe us, you should bid exactly b=v. But if you are being strategic you could:
a) Underbid b<v
In this situation you are hoping to pay less for the item. But note that you only pay the second price if you win - the price does not depend on the margin by which you win. Therefore by underbidding you only decrease your chances of winning and get nothing in return.
b) Overbid b>v
In this situation you are hoping to increase your chance of winning. But note that you don't want to win if you do - your bid is higher than what you think the thing is worth in the first place.
Though trivial in appearance, second-price auction is a powerful concept. VCG mechanism is a generalization of this and allows for much more complicated things - public goods, combinations of items and more.