Friedrich Nietzsche claims that in the foundation of everything we will find the will to power. Even if we do not completely agree with this claim, most people have it in mind when it comes to politics. To avoid local bias of the story that follows, I will place it in Judea, in the days of Monty Python’s Brian (who, just like the Three Christs of Ypsilanti, is not the Messiah). If you recognise something from your own local political scene in this story, don’t be surprised. We’ve all seen it.
Let me introduce the Uniter. He is a man of vision, planning to make an All-People’s Federation of Fronts of Judea in which the four competing factions of Judea (People’s Front of Judea, Judean People’s Front, People’s Judean Front and Front of People of Judea) would be united. The Uniter decided to form a Federal Council with seven delegates, proportional to the number of members the competing fronts have: three delegates from PFJ, two from JPF, one from PJF and one from FPJ. This was completely unacceptable for the representatives of PJF and FPJ, so they went to see the Uniter.
PJF: Oh Great Uniter, what is this?
FPJ: Who gave you the right to give JPF double the power we, the FPJ have?
PJF: Yeah, what he said!
Uniter: Gentlemen… Mathematics is an exact science.
PJF: Sure it is, and two is greater than one.
FPJ: Even for very large values of one.
Uniter: It seems that I’ll have to tell you a story about three people: Banzhaf, Shapley and Shubik.
PJF: Oh, are those the delegates from PFJ?
Uniter: No, they are mathematicians.
FPJ: Is that a new front? The Maccabees?
Uniter: No, these guys don’t fight.
PJF: So, they’re deserters?
Uniter: ENOUGH! I’m telling you a story.
Power index is a measure introduced by scientists to numerically state the amount of power a faction in a council like ours has based on the number of seats they have. These indices can be defined in various ways, and the most famous ones are Banzhaf’s and Shapley-Shubik’s.
Banzhaf’s index is based on counting all the critical votes a faction can make. Here we consider a vote critical if it would overturn a decision if it weren’t made. Now, in order to make things easier, let’s denote I = PFJ, II = JPF, III = PJF, IV = FPJ.
PJF: Why are we using Roman numerals? Aren’t we trying to get rid of them?
Uniter: The strategic planning department suggested that we’re going to have some problems with Arabs in due course. Stick to the Roman numerals.
Uniter: For the council to accept something by voting, four votes are needed. The number of votes is four, and four shalt thou count. (That’s not from this film, PJF complained) These four votes can be obtained in the following scenarios:
I + II, I + III, I + IV, I + II + III, I + II + IV, I + II + III + IV, I + III + IV, II + III + IV.
Underlined votes are critical. Total underlined votes: 12. I was underlined 6 times, so the power of I is 6/12=50%. II was underlined twice, so the power of II is 2/12=16.67%. III and IV are also underlined two times each, so their power is also 16.67%.
PJF: That’s impossible! It’s impossible that we and the JPF have the same power! They have two delegates!
FPJ: Magic, deceit, fire, water, snow…
Uniter: Enough! If you don’t trust Banzhaf, here’s what Shapley and Shubik say. They count the pivot voting factions. You know, in the council meeting, there’s an order in which the factions vote and we make it usually random. First faction to ensure the motion to pass (i.e. make the count of votes larger than three) is said to be casting the pivot vote. There are 24 possible voting orders, and the pivot faction is underlined in each one of the cases.
I II III IV, I II IV III, I III II IV, I III IV II, I IV II III, I IV III II, II I III IV, II I IV III, II III I IV, II III IV I, II IV I III, II IV III I, III I II IV, III I IV II, III II I IV, III II IV I, III IV I II, III IV II I, IV I II III, IV I III II, IV II I III, IV II III I, IV III I II, IV III II I.
I is underlined 12 times, so the power of I is 12/24=50%. II is underlined 4 times, so the power of II is 4/24=16.67%. III and IV are again underlined 4 times, so their power is also 16.67%.
PJF: OK, we believe you now. We have the same power as JPF.
Uniter: I’m surprised you didn’t ask how does PFJ has 50% power without having 50% of seats in the council. If you all unite against them and make “PFJ 33 AD” you’d have 100% power in your hands.
The next antifeuilleton is again all about power and its name is “Is it me, or is “Veto” starting to sound really good”.