Why do the Orthodox Jews have so much political power in Israel? Why are third parties in the U.S. so weak? These phenomenon can be explained by the Banzhof power index. The index is calculated as follows. Let us look at the Israeli election in 2003 for the Knesset. Here are the voting results of the top 3 parties:
| Party | Votes | % | Seats at end of session | |
| Likud | 925279 | 29.39% | 27 | |
| Labour | 455183 | 14.46% | 21 | |
| Shas | 258879 | 8.22% | 11 |
In the Knesset, the parties must form a coalitions which has at least a majority of the seats. In the case above, a majority would consist of a coalition with at least 30 seats. The Banzhof power index tells us the Shas (religious) party is equally powerful as the Likud (conservative) and Labour (liberal) parties even though the Shah party has less than half the number of seats as the Likud. Let us look at all the winning coalitions:
Likud–Labor; Likud–Shas; Labor–Shas; Likud-Labor-Shas
There are 4 possible coalitions. The groups which are pivotal are underlined in each coalition. We see that each group is pivotal 2 times, and thus each has a Banzhof power index of 2/6=1/3. This is why the religious parties in Israel are so powerful even though they receive a much lower percentage of the vote.
Why is this not a problem in the U.S? In the U.S. no coalitions need to be formed. A candidate must win a majority of electoral college votes and if they do not then there is a run off. The benefit of this system is that small third parties do not become more powerful than they deserve. On the other hand, however, the concerns of third party voters are often ignored due to the American election rules.