Are there multiplayer equivalents to trading places after the first move?

In many two-player games a first move advantage can be mitigated by giving the second player the option to trade places if the first move appears desirably strong, which encourages the first player to make a mediocre first move. This is referred to as the pie rule, named after a fair way to divide a pie between two people (one person cuts, the other chooses).

Can this be generalized to more than two players in a fair manner? If so, which games would this mechanic work well with? Which is it entirely inappropriate for (such as games with secret information)?

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In which games have you ever seen this? – Joe Z. Apr 5 '13 at 19:00
@JoeZeng I edited the question with a wikipedia link, which includes mention of Go, Hex, and Twixt. – Sparr Apr 5 '13 at 21:31
Thanks for the link. – Joe Z. Apr 6 '13 at 1:19
My first thought is this would be absolutely horrible for Magic the Gathering. But then I started thinking about it, and it might actually be a helluvalotta fun to swap decks! – corsiKa Apr 8 '13 at 19:14

Mathematically the Pie rule can be generalized. This is actually a well known problem.

One of the solutions is to let people take it in turns to add what they want from the common lot to a set of goods until one person calls that he wants that set. For example, in the 40 thieves problem, the goal is to share the bounty in 40 equal sets, which seems difficult, since there are jewels, gold, weapons etc. So the leader prepares a set and each thief adds one thing at a time until one thief claims he wants that set. Then the thief leaves with his share and the problem is reduced to n-1 people.

I never heard of a board game using exactly this principle, however in some board games the players bid point to determine the order of the players for the turn. For example, in the expert-variant of the game Tikal, each turn you draw as many tiles as there are players, and they bid to play first.

The strict application of the principle to a board game would be to have a first turn where you set up some initial position on the board. And afterwards, the players declare: "I add X money to the white meeple", "I add X pieces of wood to the white meeple", "I add X card to the blue meeple", "OK STOP, I take the blue meeple".

Actually, there are probably some games that use this kind of mechanic, I just don't recall one now.

[Edit:] You probably wonder what to do when several players wants to take a share. In that case, you just subtract something until nobody or (only one person) wants it any more, and then you continue adding.

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Nice answer. Coloretto and its spin-offs would be examples of games that use the principle of everyone adding items to buckets, or taking a bucket and being removed from the process. – Johno Apr 8 '13 at 8:15
True, Coloretto and Zooloretto uses this kind of mechanic. This is used at each turn. There are several games which use this mechanic at each turn, however, I don't recall any game that uses the 40 thieves sharing process to determine a starting position (and then the game continues with another mechanic). – Djaian Apr 8 '13 at 8:18
Is there another name for the 40-thieves sharing process? I'd like to read up more about it, but I can't find anything other than this question. The terms are too generic :/ – Bobson May 26 '15 at 4:51