It took a while before I decided that "Game" was probably the best category for this Idea. Other possibilities were "Election", "Voting", and "Politics", for example, simply because if you have a bunch of people arguing about what Decision to make, those possibilities are often relevant.

But sometimes
people make a game of it, such as "Rock, Paper, Scissors" and flipping a coin. So, here we go....

The *typical* coin flip "game" involves one person flipping the coin and the other calling "heads" or "tails". But what if three or more people *all* want *different* things (example, what piece of music to play)?

So, we need a Fair and Generic Deciding Game.

Everyone flips a coin, two people at a time. This lets others referee the coin flips. Obviously one of the things to check, first of all, is to make sure every coin used has a "heads" side and a "tails" side (grin). If the total number of involved people is odd, the last coin-flip is done by three people at the same time.

Let the process of "every participant flipping a coin" be called a "round". The subdivisions of each round can be called "flips". If N is the total number of participants in a round, then the number of flips in that round will be N/2 (when N is odd ignore the fraction, because the last flip involves 3 people.)

For two people flipping the coins, each person calls "heads" or "tails" on the *other* person's coin flip. The calls are made while the coins are flipping in the air. So, if you happen to be an expert coin-flipper, who can greatly enhance the odds that a particular side of the coin will be exposed, it won't do you any good. Once you've flipped it, the other person will have a 50/50 chance of guessing the result.

When three people flip the coins, they stand facing each other in a triangle, and each calls the flip made by the person on the right (or on the left; decide this in advance!).

In general, those who guess wrong will be "out". They will not be involved in the next "round". The remaining lucky guessers try again, in another round of coin-flipping. It may sometimes happen that *everyone* in the round guesses wrong (relatively likely when the number of participants in a round has shrunk to two or three people). In this case the result is ignored, and everyone in that round plays another round.

Eventually this Game will have a fairly-selected winner, who thereby gets to make the Decision that was being contested.

Soooo.., everyone splits into pairs or pairs and a group of three if there's an odd number of people? Then one person from each pair or 'three' goes through to the next round? This means that you're at a massive disadvantage if you get put into the 'three' in the first round.

I think there's a better way, needing only one coin-toss per round, and at most ceiling(log(2)n) rounds[1].

1) Divide the people into two groups of roughly equal size[2]. Declare one of the groups as 'heads' and the other 'tails'.
2) Toss the coin
3) People on the losing side are out.
4) Repeat from step (1) until one person remains.
5) The remaining person makes the decision.

This is better than both Vernon and Ling's procedures because only one coin-toss is needed for a round, so less checking needs to be done; it's therefore faster. It is also guaranteed to complete in a certain number of rounds.

[1] For as-even-as-possible splits, i.e. splitting even numbers into two groups of n, and odd numbers into a group of n and a group of n+1.

[2] For large groups it may be more efficient to divide people into groups of roughly the same size (eg. by dividing line), rather than exact splits. This won't generally increase the number of rounds needed very much.

[Loris], your version does not address the fundamental problem that everyone might want to be that *one* person doing all the coin flips. And it doesn't address the problem that the actual coin-flipper may be skilled enough to cheat.

How do you cheat at coin-flipping? Well, if you always hold your flipping-hand the same height above the ground, and always have the coin in the same orientation in your hand before flipping it, and if you always apply the same amount of force to the flip, then you will greatly improve the odds that the result will be skewed toward heads or tails.

Remember, there are jugglers out there who are expert enough to do it totally blindfolded. People *can* apply consistent forces to objects!

//[Loris], your version does not address the fundamental problem that everyone might want to be that *one* person doing all the coin flips. And it doesn't address the problem that the actual coin-flipper may be skilled enough to cheat.//

I assumed I didn't have to go into that, since in the absence of a referee the solution is trivial, widely practiced in the generic 2-player version, and indeed the same as you go to some lengths to describe in your article. A person on one side flips, a person on the other side calls.
To sum up, that's a non-issue.