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You Cut I Choose
Keywords: Theory
You-Cut-I-Choose is a general principle for fair division of just about anything between two parties. One side divides the stuff into two lots, and the other side selects which party gets which lot. In Go, it is relevant to a wide variety of topics, e.g.
etc. There's an interesting extension of You-Cut-I-Choose when more than 2 players are involved. Suppose you have a cake that you want to divide 3 ways. Player A cuts a piece that he estimates to be 1/3 of the cake, that he would be willing to accept. Player B can either reject the piece, or cut another piece off. Player C then can either reject the piece, or cut another piece off. The last player to cut the piece gets that piece. The remaining 2 players play the 2-person You-Cut-I-Choose for the rest of the cake. This method is fair, because each player is guaranteed what they estimate to be 1/3[1]. This method generalizes to a large number of players. [1] I hope I don't understand it, because if I do, it doesn't work. A is not a perfect estimator and say he cuts off a piece that is bigger or smaller then 1/3. Say B and C are better estimators than A. How can they both get a piece that they estimate to be 1/3 ? In the two player game I don't see this complication. A cuts the cake into two pieces he is willing to accept. They can be 1/3 and 2/3 of the cake, it does not matter. B chooses from those pieces. They may not get what they think to be half the cake, but both do get pieces they are willing to accept. -- mAsterdam The real generalisation of "You cut I choose" is as follows: Player A cuts Player B can take the piece cut by A or pass Player C same... ... If all players players pass A gets the piece he cut Repeat until all have pieces lavalyn: Close. You are assuming that there would be no alliances between players. Player A and B scheme to get an advantage, decides to cut 4/5 of the cake as chunk. B naturally accepts. C gets screwed. Rather, player A cuts "the smallest" piece he is willing to accept, and all players that would accept the piece must play the game on the little piece, slowly cutting tiny little slices off until only one player is willing to accept. And that player gets it. This of course assumes that cake is infinitely divisible[2] and that the tiny slices that would result are Riemann summable :) Henry: I don't think that an infinite process qualifies as a solution to this problem. The possibilities for collusion can be removed by randomising who out of B and C gets first dibs on A's cut. Obviously in this case A can't promise both B and C a larger share to get them to collude, since any collusion will increase A's share. B's and C's "expected" share is to get exactly half the remainder each, so it's not in either of their interests to collude with A before the toss. And after the toss it is too late for A to change his cut. lavalyn: I didn't say it was an infinite process. I claimed that the little pieces of subcake will have to be summed up. The process at worst is O(n^3) time, since at least one person (the person doing the cutting) will drop out each cut. (I have corrected an omission above which addresses this.) And as for randomizing - that doesn't work either. Player A cuts exactly half. Regardless of what either B or C does, A wins. Henry: Oh, I see your method. I don't understand what you mean by Riemann summation, though. Obviously we are only talking about measurable subsets of the cake, or you could decompose it into four pieces adding to two copies of the original cake... (And in any case I dislike the notion that the last chooser is going to be left with a nasty, even if finite, collection of crumbs.) If A cuts half and B and C get to choose randomly who picks first, then A is going to get no more than a quarter of the cake, a definite loss.
[2] The moving knife procedure can take care of that. One randomly chosen person moves the knife over the (lineair) cake until one of the participants says "stop", accepting the piece that is cut off there. (This is not just theory - big auctions, for instance This is a copy of the living page "You Cut I Choose" at Sensei's Library. ![]() |