Game Theory Tuesday: Voting Power in Israeli Judge Selection and the Shapley-Shubik Index

source: ninjapoodles via flickr

How many votes should be necessary for a decision? Is a simple majority sufficient, or should a higher standard be applied, such as a consensus?

These are questions and circumstances we all face, whether as a voter or an organizer. Understanding voting theory can help you understand the incentives and dynamics of elections. A recent event can help illustrate the theory.

Israel judge selection

Voting issues took center stage in the Knesset Law Committee, part of Israel’s legislative body. The Committee passed a law requiring a seven member majority when appointing Supreme Court judges.

Previously, the nine-member selection committee only needed to reach a simple majority of five votes. The law likely seeks to encourage discussion and guarantee a higher standard for agreement.

But good intentions do not always translate to good laws because of loopholes. What incentives does the law create? How does the law affect the strategy of voting?

To address this topic, the Law Committee invited Nobel Prize laureate Yisrael Aumann to discuss the strategic implications. (Side note: I think this is the same Aumann I discussed two weeks earlier who used game theory to solve a religious mystery).

Aumann points out the law may not exactly serve its purpose, as reported in The Jerusalem Post

…Aumann told the committee that if three out of nine committee members always voted as a bloc, while the other six voted individually, their “power index” increased from one third to 43 percent. He did not explain why, saying the matter was too complicated. Aumann added that it would make no difference as far as the power index was concerned whether the required majority were increased to seven, or even six…

At first glance, Aumann’s claims are both surprising and perplexing. Why doesn’t raising the majority change from five to seven (or six) affect the voting power index? How do you even measure voting power?

I recently researched these questions and have come up with some answers. In this article, I wish to offer my understanding of Aumann’s claims. The final analysis offers counterintuitive results that anyone running an election should consider.

This article is divided into four sections motivated by the following questions:

1. What is voting power?
2. How can one measure voting power?
3. How can groups of players increase voting power?
4. Why doesn’t voting power change when the majority is increased from five to seven (or six)?

What is a voting power?

• Roughly speaking, voting power is the ability of a voter to impact an election.

For example, dictators always get their way and have complete voting power. In contrast, underage children are not allowed to vote, and hence have no voting power.

Most elections lie somewhere in the middle: every voter has some ability to affect the outcome. Loosely speaking, some votes are especially important because they cause a decision (like a tie-breaking vote). Such a vote is appropriately called a decisive vote (or in some contexts, a pivotal vote).

• To pin down the analysis, voting power can be quantified in an index that ranges from 0 (no voting power) to 1 (absolute voting power).

The middle values have a probabilistic interpretation. For the index I will discuss, a person with a voting power index of 0.5 is someone that has the decisive vote 50 percent of the time.

• Voting power is a different concept from voting size or number of votes.

Someone with a large number of votes might not necessarily hold any more voting power than someone with fewer votes.

To see this, consider an investing club that makes buying decisions based on a minimum 50 percent approval. If three members have voting weights based on ownership stakes of 15 percent, 39 percent, and 46 percent, how much power does each voter have?

The first thing to observe is that at least two voters are needed to reach the 50 percent approval mark. No single voter can split off and make decisions. Furthermore, no single voter can block decisions of the other two. Every winning measure contains two or more voters.

All of these facts mean the 15 percent voter has as much power to block or ratify measures as the others. The game is equivalent to one in which each voter has a single vote.

Since each voter has equal voting power, one could assign a power index of one-third to each voter.

(As a side note, I want to point out that there can be other ways to calculate voting power. If you want more information, you can browse this excellent voting power index website, operated by Anitti Pajala at the University of Turku in Finland.)

What did Aumann do to measure voting power?

I limit today’s discussion to the measure I think Aumann used, the Shapley-Shubik index. This index was developed by Lloyd Shapley and Martin Shubik in the 1950s.

• Informally speaking, the Shapley-Shubik index for a voter is the probability the voter can cast a decisive vote, when considering all possible voting orders.

The index is based on the following thought process. Assume that voting takes place sequentially in a randomly assigned order. For each sequence, find out which voter occupies a decisive position. That is, find the first voter in each sequence that raises the cumulative sum of votes above the passing mark. Powerful voters will be decisive in more sequences.

I’ll go through two examples to illustrate the index and then I’ll apply it to the Israeli judge problem.

• Example 1: shareholder voting

Consider voters A, B, C with votes of 15, 39, and 46, as above who need a majority vote of 50.

There are 6 possible orders for the votes:

A B C
A C B
B A C
B C A
C A B
C B A

Note that the second voter always raises the cumulative vote total above 50. Hence, regardless of the voting order, the second voter is always the decisive position.

The voters A, B, and C each hold the decisive position in two of the possible six voting orders.

Hence, each voter has a Shapley-Shubik power index of 2/6, or one-third. This outcome matches our intuition that each voter has equal power.

• Example 2: three voters, not equal power

Consider voters A, B, C with votes of 3, 2, and 1, who need a majority vote of 4.

Again, there are 6 possible orders for the votes. I have bolded the voter in the decisive position:

A B C
A C B
B A C
B C A
C A B
C B A

In this election, the decisive position is either the second or third position, depending on how the votes are cast.

Voter A has the most votes and holds the decisive position in 4 of the 6 possible cases, thus having a voting power index of 4/6, or two-thirds.

Voters B and C are each decisive in one of the voting orders. Thus, despite holding one more vote, voter B has the same voting power as voter C of 1/6.

• Example 3: Israel judge selection

There are nine voters each with a single vote. A majority of seven is needed.

In this example, it’s not necessary to write out all voting orders (this is fortunate as there are 9! = 362,880 possible orders). Since each person has one vote, it will always be the voter in the seventh position that is decisive.

Each voter is equally likely to occupy the seventh position, so each voter has a voting power index of 1/9. It’s good to know that political votes are equal.

How can groups of players increase voting power?

In the Israel judge selection committee, consider any three voters as a group. Let’s examine how much power the three combined would have.

For a moment consider that the three voters act separately. How much power does the group hold? Since each person is the decisive vote in 1/9 of the cases, the three combined would influence 3/9 of the cases. This means the group has a power index of one-third. This is one of the claims Aumann made.

Now let’s tackle the next claim that the group increases its power index when voting as a bloc.

To do this, we use a trick and reformulate the question. If the three voters always acted together, then we can consider the group as a single person with three votes.

The voting committee could be seen as six individual voters, plus one bloc of 3 votes.

Let’s go through this analysis of the group’s power index.

• Example 4: Israel judge selection, voting bloc of three

There are seven voters, six with a single vote and one with three votes. A majority of seven is needed.

What is the power index of the bloc?

Again, it’s not necessary to write out all possible voting orders. The bloc holds three votes, so it will tip the majority when the cumulative total is already 4, 5, or 6. This means the bloc is decisive in voting positions 5, 6, or 7.

The bloc is decisive in 3 of the 7 equally likely positions it can occupy, and hence it has a power index of 3/7, or 43 percent. This verifies Aumann’s claim.

(The remaining voters each have one vote and are equally likely to be decisive. This means the remaining 57 percent is split equally to them, so each has a voting power index of 9.5 percent).

Why doesn’t voting power change when the majority is increased from five to seven?

Let’s work out the majority of five to see why power is unchanged.

• Example 5: Israel judge selection, voting bloc of three, majority of 5

There are seven voters, six with a single vote and one with three votes. A majority of five is needed.

What is the power index of the bloc?

The bloc holds three votes, so it will tip the majority when the cumulative total is already 2, 3, or 4. This means the bloc is decisive in voting positions 3, 4, or 5.

Again, the bloc is decisive in 3 of the 7 equally likely positions it can occupy, and hence it has a power index of 3/7, or 43 percent. This verifies Aumann’s claim that the power index does not change.

(The same analysis holds if the majority were six.)

Some implications

Here is what you can take away when creating your own voting structures:

1. Vote size does not equate to voting power
2. Smaller voters can still hold great power
3. Voters can increase power through voting blocs
4. Raising a majority might not diminish the power of a voter or bloc