There is much talk these days about Israel's particular method of allocating mandates based on election returns. The relevant law is known as Bader-Ofer (after the two MKs who proposed it). So here comes a little tutorial on the much misunderstood Bader-Ofer Law.

First of all, Bader and Ofer didn't invent the system; they simply proposed the adoption of a vote allocation system known as the Hagenbach-Bischoff method, which itself is a variation on the D'Hondt method (both named after 19th century physicists).

How does the method work? If I could easily insert equations into this blog, my life would be easier. I'll do my best.

Step 1: Let's call the total number of votes cast T. Any party that doesn't get at least T/50 votes is eliminated. (50 is an adjustable parameter; last time it was 66.666.) Let T' be the number of votes left after all the votes for eliminated parties are discarded.

Step 2: Let M = T'/120. This is the nominal cost of a single mandate. Let Xn be the number of votes received by party n. Then party n should in theory get Xn/M mandates. The problem is that Xn/M is almost certainly not a whole number but rather something like 13.356. So the first thing we do is give party n, the integer part ("floor") of Xn/M. In our example, that would be 13.

Step 3: Well, since the fractional mandates that each party theoretically got remain unallocated, the total number of mandates given out in Step 2 will be somewhat less than 120. In fact, since the fractional remainders should distribute normally around 1/2, the total number of unallocated seats should be equal to approximately 1/2 the number of parties that cross the threshold in Step 1. (This year that means around 5 mandates.) Now you might think that these should be distributed according to the size of the fractional remainder that each party got. But no! Rather, we use a recursive procedure in which the unallocated mandates are allocated one at a time in the following way:
Let Yn be the number of mandates allocated to party n so far. Initially, Yn is simply the integer part of Xn/M. (By the way, here is the difference between D'Hondt and Hagenbach-Bischoff; in D'Hondt, Yn is initialized to 0 rather than Xn/M.) Now (drum roll...here comes the main point) the next mandate is allocated to the party for which Xn/(Yn+1) is maximal.
The idea is that Xn/Yn is the average cost paid by party n for a mandate and this method keeps the range of such costs over the different parties as narrow as possible.
For example, suppose the total number of votes is 120,000 and party 1 gets 32,340 votes and party 2 gets 14,550 votes (for simplicity, let's ignore the other parties). The nominal cost of a mandate is 1000 so initially party 1 gets 32 mandates and party 2 gets 14 mandates. Now, we want to allocate the mandates lost to the fractional remainders. So we compare 32,340/33 (=980) to 14,550/15 (=970). The result is that party 1 gets the next mandate even though party 2 had a higher remainder.

(Note that the denominators in the above fractions are always 1 more than the current allocation for that party; the point is to determine the hypothetical cost of a mandate after a party were to get it. Another way to think about this is to let Qn be the number of votes that party n is missing to reach the next mandate. Instead of giving the mandate to the party for which Qn is minimal, we give it to the party for which Qn/(Yn+1) is minimal.)

Overall, as is evident in our example, this favors larger parties to a considerable extent (the larger Yn the better your chances of having the minimal Qn/(Yn+1)). For that reason, two parties can agree to be treated as a single party for the purpose of the above calculation. The entire procedure remains otherwise identical. Once, the combined party has been allocated its additional mandates, these are divided between the two consitutent parties using the same method. So when a party makes such an agreement, they should ideally do so with a like-minded party that is smaller than them but still big enough to cross the threshold in Step 1.

The main part of all this that should impact on how you vote is Step 1. If you vote for a party that doesn't make the cut in Step 1, the total number of votes that count (T') is diminished and therefore so is the nominal cost of a mandate (M). So all the other parties increase their mandates proportionally. The increase for larger parties is proportionally greater than for smaller ones. All the rest is just details.