**1.0 Introduction**
This post presents a perhaps surprising example of results from measuring political power in a system with weighted voting. I provide examples in which the weight of a person's vote is increased. Yet that voter, in some cases, gains no additional power, in some sense. In one case, by the measures of voting power considered here, the additional weight has no effect on the power of any voter. In another case, another player, with unchanged weight to his vote, is elevated in power with the voter whose weight is increased.

I find these results to be an interesting consequence of power measures. I have not yet found a simple example where the effect on the ranking of voting power is different for the three indices considered here. Nor have I found an example where a voter declines in power with an increase in the weight of his vote.

**2.0 An Example of a Voting Game**
A voting game is specified as a set of players, the number of votes needed to enact a bill into law (also referred to as passing a proposition), and the weights for the votes of each player. In considering voting games with a small number of players and weighted, unequal votes, one might think of such a game as describing a council or board of directors, where members represent blocs or geographic districts of varying sizes.

As example, consider a set, *P*, of four players, indexed from 0 through 3:

*P* = The set of players = {0, 1, 2, 3}

A common way to indicate the remaining parameters for a voting game is a tuple in which the first element is followed by a colon and the remaining elements are separated by commas:

(6: 4, 3, 2, 1)

The positive integer before the colon indicates the number of votes - six, in this case - needed to pass a proposition. The remaining integers are the weights of players' votes. In this case, the weight of Player 0's vote is 4, the weight of Player 1's vote is 3, and so on.

**3.0 Two Power Indices**
Consider all 16 possible subsets of the four players. These subsets are listed in the first column of Table 1. A subset of players is labeled a coalition. The second column indicates whether or not the coalition for that row has enough weighted votes to pass a proposition. If so, the characteristic function for that coalition is assigned the value unity. Otherwise, it gets the value zero. A player is decisive for a coalition if the player leaving the coalition will convert it from a winning to a losing coalition. The last four columns in Table 1 have entries of unity for each player that is decisive for each coalition. The last row in Table 1 provides a count, for each player, of the number of coalitions in which that player is decisive. The Penrose-Banzhaf power index, for each player, is the ratio of this total to the number of coalitions.

**Table 1: Calculations for Penrose-Banzhaf Power Index**
**Coalition** | **Characteristic** Function | **Player** |

**0** | **1** | **2** | **3** |

{} | *v*( {} ) = 0 | 0 | 0 | 0 | 0 |

{0} | *v*( {0} ) = 0 | 0 | 0 | 0 | 0 |

{1} | *v*( {1} ) = 0 | 0 | 0 | 0 | 0 |

{2} | *v*( {2} ) = 0 | 0 | 0 | 0 | 0 |

{3} | *v*( {3} ) = 0 | 0 | 0 | 0 | 0 |

{0, 1} | *v*( {0, 1} ) = 1 | 1 | 1 | 0 | 0 |

{0, 2} | *v*( {0, 2} ) = 1 | 1 | 0 | 1 | 0 |

{0, 3} | *v*( {0, 3} ) = 0 | 0 | 0 | 0 | 0 |

{1, 2} | *v*( {1, 2} ) = 0 | 0 | 0 | 0 | 0 |

{1, 3} | *v*( {1, 3} ) = 0 | 0 | 0 | 0 | 0 |

{2, 3} | *v*( {2, 3} ) = 0 | 0 | 0 | 0 | 0 |

{0, 1, 2} | *v*( {0, 1, 2} ) = 1 | 1 | 0 | 0 | 0 |

{0, 1, 3} | *v*( {0, 1, 3} ) = 1 | 1 | 1 | 0 | 0 |

{0, 2, 3} | *v*( {0, 2, 3} ) = 1 | 1 | 0 | 1 | 0 |

{1, 2, 3} | *v*( {1, 2, 3} ) = 1 | 0 | 1 | 1 | 1 |

{0, 1, 2, 3} | *v*( {0, 1, 2, 3} ) = 1 | 0 | 0 | 0 | 0 |

**Total:** | 5 | 3 | 3 | 1 |

The Shapley-Shubik power index considers the order in which players enter a coalition. For the example, one considers all 24 permutations for the players. The first column in Table 2 lists these permutation. For each row, a player gets an entry of unity in the appropriate one of the last four columns if including that player in a coalition, reading the entries in a permutation from left to right, creates a winning coalition. The Shapley-Shubik power index, for each player, is the ratio of the totals of each of the last four columns to the number of permutations.

**Table 2: Calculations for the Shapley-Shubik Power Index**
**Permutation** | **Player** |

**0** | **1** | **2** | **3** |

(0, 1, 2, 3) | 0 | 1 | 0 | 0 |

(0, 1, 3, 2) | 0 | 1 | 0 | 0 |

(0, 2, 1, 3) | 0 | 0 | 1 | 0 |

(0, 2, 3, 1) | 0 | 0 | 1 | 0 |

(0, 3, 1, 2) | 0 | 1 | 0 | 0 |

(0, 3, 2, 1) | 0 | 0 | 1 | 0 |

(1, 0, 2, 3) | 1 | 0 | 0 | 0 |

(1, 0, 3, 2) | 1 | 0 | 0 | 0 |

(1, 2, 0, 3) | 1 | 0 | 0 | 0 |

(1, 2, 3, 0) | 0 | 0 | 0 | 1 |

(1, 3, 0, 2) | 1 | 0 | 0 | 0 |

(1, 3, 2, 0) | 0 | 0 | 1 | 0 |

(2, 0, 1, 3) | 1 | 0 | 0 | 0 |

(2, 0, 3, 1) | 1 | 0 | 0 | 0 |

(2, 1, 0, 3) | 1 | 0 | 0 | 0 |

(2, 1, 3, 0) | 0 | 0 | 0 | 1 |

(2, 3, 0, 1) | 1 | 0 | 0 | 0 |

(2, 3, 1, 0) | 0 | 1 | 0 | 0 |

(3, 0, 1, 2) | 0 | 1 | 0 | 0 |

(3, 0, 2, 1) | 0 | 0 | 1 | 0 |

(3, 1, 0, 2) | 1 | 0 | 0 | 0 |

(3, 1, 2, 0) | 0 | 0 | 1 | 0 |

(3, 2, 0, 1) | 1 | 0 | 0 | 0 |

(3, 2, 1, 0) | 0 | 1 | 0 | 0 |

**Total:** | 10 | 6 | 6 | 2 |

**4.0 Three Power Indices for Three Voting Games**
Table 3 summarizes and expands on the above calculations. The Penrose-Banzhaf power index need not sum over the players to unity. Accordingly, I break this index down into two indices, where the second index is normalized. The Shapley-Shubik power index is guaranteed to sum to unity. I introduce two other voting games, with corresponding power indices, presented in Tables 4 and 5.

**Table 3: Power Indices for (6: 4, 3, 2, 1)**
**Player** | **Penrose-Banzhaf Power Index** | **Shapley-Shubik** Power Index |

**Index** | **Normalized** |

0 | 5/16 | 5/12 | 10/24 = 5/12 |

1 | 3/16 | 3/12 = 1/4 | 6/24 = 1/4 |

2 | 3/16 | 3/12 = 1/4 | 6/24 = 1/4 |

3 | 1/16 | 1/12 | 2/24 = 1/12 |

**Table 4: Power Indices for (6: 4, 2, 2, 1)**
**Player** | **Penrose-Banzhaf Power Index** | **Shapley-Shubik** Power Index |

**Index** | **Normalized** |

0 | 6/16 = 3/8 | 6/10 = 3/5 | 16/24 = 2/3 |

1 | 2/16 = 1/8 | 2/10 = 1/5 | 4/24 = 1/6 |

2 | 2/16 = 1/8 | 2/10 = 1/5 | 4/24 = 1/6 |

3 | 0 | 0 | 0 |

**Table 5: Power Indices for (5: 4, 2, 2, 1)**
**Player** | **Penrose-Banzhaf Power Index** | **Shapley-Shubik** Power Index |

**Index** | **Normalized** |

0 | 6/16 = 3/8 | 6/12 = 1/2 | 12/24 = 1/2 |

1 | 2/16 = 1/8 | 2/12 = 1/6 | 4/24 = 1/6 |

2 | 2/16 = 1/8 | 2/12 = 1/6 | 4/24 = 1/6 |

3 | 2/16 = 1/8 | 2/12 = 1/6 | 4/24 = 1/6 |

**5.0 Constitutional Changes**
Consider a change in the constitution, from one of the three voting games with tables in the previous section to another such game. The calculations allow one to measure the impact on voting power for any such change. To simplify matters, I consider only rankings of voting power. And, for these three voting games, the three power indices consider here happen to yield the same ranks, for any given voting game out of these three.

Accordingly, Table 6 shows changes in the rules (the "constitution") for these cases. The change to the rules on the right superficially strengthens Player 1, either by increasing the weight of Player 1's vote or requiring less votes to pass a resolution. As noted below, I am unsure what naive intuition might be for the second row. For the third vote, the number of votes needed to pass a proposition is altered such that a simple majority is needed before and after the change in weight.

**Table 6: Changing the Rules to Strengthen the Players?**
**Starting Game** | **Player Ranks** | **Ending Game** | **Player Ranks** |

(6: 4, 2, 2, 1) | 0 > 1 = 2 > 3 | (6: 4, 3, 2, 1) | 0 > 1 = 2 > 3 |

(6: 4, 2, 2, 1) | (5: 4, 2, 2, 1) | 0 > 1 = 2 = 3 |

(5: 4, 2, 2, 1) | 0 > 1 = 2 = 3 | (6: 4, 3, 2, 1) | 0 > 1 = 2 > 3 |

The first row shows a case where the weight of Player 1's vote increases, which might intuitively give him more power with respect to the apparently weaker Players 2 and 3. Yet this increase in weight also increases the power of Players 2 and 3, even though the weight of their votes does not change. And Player 1 remains equal in power to Player 2, both before and after the change. In fact, the change has no effect on the ranking of the players' voting power.

The second row shows a case where the votes needed to pass a measure declines, after the change in rules, from a super-majority to a simple majority, given the total of weighted votes. Would one expect such a constitutional amendment to strengthen the most powerful, or moderately powerful voters before the change? I find that this change raises the power of the weakest voter to the power of the middling voters. I am not sure this is counter-intuitive, unlike the other two rows.

The third row shows a case in which, like the first row, the weight of Player 1's vote increases. Both before and after the change, a simple majority, given the total of weighted votes, is needed to pass a proposition. This change makes Player 1 more powerful than the weakest player, as one might intuitively expect. But Player 2 is also made more powerful than the weakest player, despite the weight of his vote not varying. And Player 1 ends up no more powerful than Player 2. These effects on Player 2 seem counter-intuitive to me.

**6.0 Conclusions**
So my examples above have presented somewhat counter-intuitive results in voting games.

I gather that the Deegan-Packel and Holler-Packel are some other power indices I might find of interest. And Straffin (1994) is one paper that explains axioms that characterize some power index or other.

**References**
- Donald P. Green and Ian Shapiro (1996).
*Pathologies of Rational Choice Theory: A Critique of Applications in Political Science*, Yale University Press
- P. Straffin (1994). Power and stability in politics.
*Handbook of Game Theory with Economic Applications*, V. 2, Elsevier.