Does anyone know a mathematical formula for the power of political parties?

[Donald Ian Rankin]

Suppose you have a parliament in which each vote is a yes/no, decided by simple majority. This parliament is comprised of a number of parties, each controlling a fracion of the votes. Assume for ease of calculation that no set of parties controls exactly 0.5 of the votes, so draws are impossible.

I'm interested in devising a formula that would measure the "power" - as distinct from the proportion of votes wielded - of each party.

If three parties have 0.4, 0.4 and 0.2 votes each then any pair of them will pass a motion, so the formula should award all of them equal power - presumably "1/3", but it might make sense for that to be some other value, if we don't require that all the powers add up to 1.

If two parties have 0.51 and 0.49votes, one should have power 1 and the other power 0, obviously.

Can anyone suggest any mathematically nice formulas for doing this sort of thing?

[MrsBrady]

1. wouldn't you have to account

for procedural crap too?

[Donald Ian Rankin]

4. No, I'm interested in the mathematics, not the politics.

Politics is just a way of framing the problem.

[saras]

2. No such thing - it isn't possible. Study game theory.

The problem with the scenario you're describing is that the "power" isn't necessarily constrained to obey mathematical laws at all.

There are other approaches, though. Lots of good analysis, in relatively mathematical form, is available and can be done.

Personally I'd recommend the book Supercooperators, by Martin Nowak from Harvard and Roger Highfield from New Scientist magazine.

https://www.nytimes.com/2011/04/10/books/review/book-review-supercooperators-by-martin-a-nowak.html?_r=1&pagewanted=all

http://www.amazon.com/SuperCooperators-Altruism-Evolution-Other-Succeed/dp/1439100187

[Donald Ian Rankin]

3. What isn't possible? I haven't even formulated a formal problem!

But, for example "The probability that if you assign all the other parties to sides randomly, this party will hold the balance of power" and "The probability that if you assign the parties to sides randomly in a random order, this party will be the one that takes its side over 50%" are both measures of *something*.

Are you thinking of a specific nonexistance result? If so, what of?

[Jim__]

5. This paper describes an algorithm.

The paper is Manipulating the Quota in Weighted Voting Games. I don't think it will give you exactly what you want; but you may be able to use it. Here is the abstract:

Weighted voting games provide a popular model of decision making in
multiagent systems. Such games are described by a set of players, a list of
players’ weights, and a quota; a coalition of the players is said to be winning
if the total weight of its members meets or exceeds the quota. The
power of a player in such games is traditionally identified with her Shapley–
Shubik index or her Banzhaf index, two classical power measures that reflect
the player’s marginal contributions under different coalition formation scenarios.
In this paper, we investigate by how much the central authority can
change a player’s power, as measured by these indices, by modifying the
quota. We provide tight upper and lower bounds on the changes in the individual
player’s power that can result from a change in quota. We also study
how the choice of quota can affect the relative power of the players. From
the algorithmic perspective, we provide an efficient algorithm for determining
whether there is a value of the quota that makes a given player a dummy,
i.e., reduces his power (as measured by both indices) to 0. On the other hand,
we show that checking which of the two values of the quota makes this player
more powerful is computationally hard, namely, complete for the complexity
class PP, which is believed to be significantly more powerful than NP.

If you investigate the Shapley-Shubik index and Banzhaf index, they may also give you a pretty good start on what you want.

[Donald Ian Rankin]

7. Thank you very much indeed, that's exactly what I was after.

[BadgerKid]

6. Likely doable algorithmically.

I lead towards doubt there is a closed-form solution, but the back of my envelope suggests a solution like that would involve combinatorics and calculus.

[Ichingcarpenter]

8. The Theory of Games

Political science

The application of game theory to political science is focused in the overlapping areas of fair division, political economy, public choice, war bargaining, positive political theory, and social choice theory. In each of these areas, researchers have developed game-theoretic models in which the players are often voters, states, special interest groups, and politicians.
For early examples of game theory applied to political science, see the work of Anthony Downs. In his book An Economic Theory of Democracy (Downs 1957), he applies the Hotelling firm location model to the political process. In the Downsian model, political candidates commit to ideologies on a one-dimensional policy space.

The theorist shows how the political candidates will converge to the ideology preferred by the median voter.

A game-theoretic explanation for democratic peace is that public and open debate in democracies send clear and reliable information regarding their intentions to other states. In contrast, it is difficult to know the intentions of nondemocratic leaders, what effect concessions will have, and if promises will be kept. Thus there will be mistrust and unwillingness to make concessions if at least one of the parties in a dispute is a non-democracy (Levy & Razin 2003).


http://en.wikipedia.org/wiki/Game_theory#Political_science



`See also Rand Corporation

[bananas]

9. One of Clinton's appointees was railroaded - Lani Guinier

because she had written something favorable about proportional voting.

http://en.wikipedia.org/wiki/Lani_Guinier

Lani Guinier (play /ˈlɑːni ɡwɪˈnɪər/; born April 19, 1950) is an American lawyer, scholar and civil rights activist. The first African-American woman tenured professor at Harvard Law School, Guinier's work includes professional responsibilities of public lawyers, the relationship between democracy and the law, the role of race and gender in the political process, equity in college admissions, and affirmative action.

<snip>

Guinier is probably most well known as President Bill Clinton's nominee for Assistant Attorney General for Civil Rights in April 1993.<3><4> A combination of political factors led to her nomination being withdrawn in June 1993. Guinier was attacked by Clint Bolick of the Wall Street Journal Op-Ed page as one of "Clinton's Quota Queens".<5> (The title, some <6> said reminiscent of the denigrating term "welfare queen", was chosen not by author Bolick but by editors at the Wall Street Journal.)

Others described her views as "anti-constitutional" because of her views on using proportional representation in local elections. In addition, Democratic Senators such as David Pryor of Arkansas and Ted Kennedy of Massachusetts informed President Clinton that her interviews with Senators were going poorly and urged him to withdraw the nomination.<7>

According to Clinton's autobiography, Democratic Senator Carol Moseley-Braun of Illinois, the only African-American who was serving in the upper chamber at that time, also urged the President to withdraw Guinier's nomination.<8> President Clinton took the advice of these elected officials and withdrew her nomination, claiming he was unfamiliar with her writing and that he didn't realize that she advocated pure racial quotas as opposed to affirmative action, as opponents had charged. The charge was false; Guinier had many times explicitly rejected the use of racial quotas in her law review articles.<9>

Civil Rights Theories

Alternative Voting Systems

Guinier's theories were first presented in law-school publications. They were also aired in part with her 1994 publication, The Tyranny of the Majority. In this work and others, Guinier suggests various ideas to strengthen minority groups' voting power, and rectify what is, according to her, an unfair voting system. She claims that she is referring not only to racial minorities, but any numerical minority group, such as fundamentalist Christians, the Amish, or in states such as Alabama, Republicans; she also states that she does not advocate any single procedural rule, but rather that all alternatives be considered in the context of litigation "after the court finds a legal violation."<10>

Some of the ideas she considers are:

<snip>