It is very difficult for a group to make a decision that
involve a member to vote against his own best
for the benefit of the whole group

Today I have to examples of decision making that, in the moment, were very difficult but could have been solved but science.

Split or steal?
The rules are very simple:
You have a jackpot you play for. In the following example it's more than £100.000.
You have to choose split or steal and you opponent does the same without knowing each others choice.
If you both chose split you will split the jackpot and go home with £50.000 each.
If you both chose steal you will go home with nothing.
If one chose split and the other chose steal the person ho chose steal will take home all £100.000.

Let's watch this clip:




This problem is called the prisoner's dilemma. Let's remove the feelings involved in the decision making and take a purely mathematical point of view.

There are two scenarios:
1) You choose split: If you opponent chooses split you win 50% pot. If he chooses steal you get nothing.
2) You choose steal: If you opponent chooses split you win 100% pot. If he chooses steal you get nothing.

Now, from your perspective the only reasonable option is to choose steal. You can't consider split because your opponent will also know that the only reasonable option is steal. This way all split or steal should end up as steal vs. steal and the game show wins. That's not sad or anything - that's simply the way the game show works.

Choosing seats
Sune and I are working at an office with eight other guys at the moment. At the end of this month we are relocating to a new location and now we have to find a (fair) system to get everyone to agree on a seating plan. As one of the guys said: “Position is everything!”. We decided to do this as adults and argue until we ended up in a deadlock. Then we decided the tiebreaks with the luck of a die.

There were three seats that people in general didn't like. Good for me because I thought they were the three best seats. After endless discussion four people were going for the same group of seats. They ended up being assigned to seats by random. How could we have solved this problem in a mathematical way?

I suggested that the last four guys made a prioritized list of the remaining six possible seats, without knowing each others chooses. Then they would hand them to me and I would minimize their overall discomfort. This field is called combinatorial optimization.

The idea is to try every possible combination of seatings. A given seating have a discomfort score that are the sum of the priorities each person got. Now we simply choose the seating with the lowest   discomfort score, meaning that the group in total benefit the most. If there are more best discomfort scores we choose the one where the unluckiest person are screwed over the least.

In reality nobody wanted to hear about this solution and before I could explain it to them the dice were in the air.