Power index

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A power index is a tool used to measure power.

The most well-known power indices, which are presented below, measure a very special form of power: the so-called “ voting power” . This means the decision-making weight of individual members of a committee in the case of majority decisions .

A decision in a panel is made when the proponents of an alternative reach a quorum . Examples of a quorum are an absolute majority of more than 50 percent, the qualified majority of two-thirds, which is required in many countries for constitutional amendments, or 10 percent for the implementation of referendums. The power indices explained below show the proportion of all possible coalitions of the parties involved in which a voter can direct the decision on the alternative he or she wants.

Example of Luxembourg in the EEC

In the European Economic Community of 1958, the large states ( Germany , France , Italy ) each had four votes, the medium-sized states ( Belgium , Netherlands ) two each and Luxembourg , which represented almost 0.2% of the population, one vote. The two-thirds majority required for a resolution was achieved with 12 out of 17 votes. Since votes from states without Luxembourg cannot be added to 11, there was no constellation in which the one vote from Luxembourg was significant for the outcome of the vote. The power index of Luxembourg, whatever its definition, was therefore zero.

Two power indices for voting power

The calculation of the standardized power index according to Banzhaf ( Lionel Penrose 1946, John Banzhaf 1965) and that according to Shapley-Shubik ( Lloyd Shapley / Martin Shubik 1954) provide similar results. (What exactly these results say and whether they really measure what they claim to be measuring, however, is controversial.)

Shapley-Shubik index

In the Shapley-Shubik index, the power of a member is determined by the number of orders of all members in which the quorum is reached with precisely this member. This will weight each coalition by the factor , where is the number of its members, the number of individual members that can cause it to fail, and the number of all members.

Example: Three members A, B and C with a voting weight of 50, 49 and 1 make decisions with a quorum of 51 percent. You consider all possible triple permutations and note from which member the quorum has been reached.

The quorum is only achieved when member B joins A: B decides. The same goes for C when it merges with A, because A and C also achieve quorum. The other cases are analogous:

  • ABC: B decides (quorum reached with AB)
  • ACB: C decides (quorum reached with AC)
  • BAC: A decides (quorum reached with BA)
  • BCA: A decides (quorum reached with BCA)
  • CAB: A decides (quorum reached with CA)
  • CBA: A decides (quorum reached with CBA)

Of the six possible permutations, A decides four, B and C each decide one. The Shapley-Shubik index is the normalized value:

  • A has a power index of 2/3 ≈ 67 percent
  • B has the power index 1/6 ≈ 17 percent
  • C has a power index of 1/6 ≈ 17 percent

Member C with only one vote has the same power index as Member B with 49 votes. If the quorum is increased by one vote to 52, C loses all decision-making power, as is the case with a quorum of 49 votes:

quorum Power index of

A (50 votes)

Power index of

B (49 votes)

Power index of

C (1 vote)

100 1/3 1/3 1/3
52-99 1/2 1/2 0
50-51 2/3 1/6 1/6
2-49 1/2 1/2 0
1 1/3 1/3 1/3

Banzhaf index

In the Banzhaf Index, a member's power is determined by counting the number of victorious coalitions in which they make a significant contribution to victory. In this way, each coalition is weighted with the number of individual members who can cause it to fail. This results in the above example:

  • A is crucial for the coalitions AB, AC and ABC
  • B is crucial for the coalition AB
  • C is critical to the AC coalition

This gives the Banzhaf power as the sum of the victorious essential coalitions of A, B and C: 3 + 1 + 1 = 5. The Banzhaf index is defined as the normalized Banzhaf power:

  • A has the power index 3/5 = 60 percent
  • B has the power index 1/5 = 20 percent
  • C has the power index 1/5 = 20 percent

This means that although C has only a fraction of the votes of B, it still has the same banzhaf power.

quorum Power index of

A (50 votes)

Power index of

B (49 votes)

Power index of

C (1 vote)

100 1/3 1/3 1/3
52-99 1/2 1/2 0
50-51 3/5 1/5 1/5
2-49 1/2 1/2 0
1 1/3 1/3 1/3

More power indices

  • Non-normalized Banzhaf index
  • Johnston Index
  • Deegan Packel Index
  • Public Good Index

Forerunner: Martin Index

The first known method of quantifying voting power comes from the American anti-federalist Luther Martin (1748–1826). By counting the possible minimum profit coalitions , Martin tried in 1788 to prove that the distribution of the seats in the House of Representatives in proportion to the population favors populous states in the constitution of the United States , which was then just passed . In the literature it is controversial whether Martin thus introduced an early form of the Banzhaf, the Deegan-Packel or the Public Good index. At the time, his calculations did not attract much attention.

See also

literature

  • Steven J. Brams: Game theory and politics. Free Press, New York 1975, ISBN 0-02-904550-9 , pp. 157-198 (English).
  • Brian Barry : Is it better to be powerful or lucky? In: Brian Barry: Democracy and Power. Essays in political theory. Clarendon Press, Oxford 1991, ISBN 0-19-827297-9 , pp. 270-302 (English; first published 1980).
  • Alan D. Taylor: Mathematics and politics. Strategy, voting, power and proof. Springer, New York 1995, ISBN 0-387-94391-9 , pp. 63-95, 205-240 (English).
  • Dan S. Felsenthal, Moshé Machover: The measurement of voting power. Theory and practice, problems and paradoxes. Edward Elgar, Cheltenham 1998, ISBN 1-85898-805-5 (English).
  • Manfred J. Holler, Gerhard Illing: Introduction to game theory. 6th edition, Springer, Berlin 2006, ISBN 3-540-27880-X , pp. 304–338.

Web links

Individual evidence

  1. ↑ Werner Kirsch : Europe, recalculated. In: The time. June 9, 2004.
  2. ^ Lionel S. Penrose: The elementary statistics of majority voting. In: Journal of the Royal Statistical Society. Volume 109, 1946, pp. 53-57.
  3. ^ A b John F. Banzhaf: Weighted voting doesn't work. A mathematical analysis. In: Rutgers Law Review. Volume 19, 1965, pp. 317-343.
  4. ↑ Lloyd S. Shapley, Martin Shubik: A method for evaluating the distribution of power in a committee system. In: American Political Science Review. Volume 48, 1954, pp. 787-792.
  5. ↑ See Barry 1991.
  6. ↑ Ronald J. Johnston: On the measurement of power. Some reactions to Laver. In: Environment and Planning A. Volume 10, 1978, pp. 907-914.
  7. ^ John Deegan, Edward W. Packel: A new index of power for simple n-person games. In: International Journal of Game Theory. Volume 7, 1978, pp. 113-123.
  8. ↑ Introduced in Manfred J. Holler: A priori party power and government formation. In: Munich Social Science Review. Volume 1, 1978, pp. 25-41; axiomatized in Manfred J. Holler, Edward W. Packel: Power, luck and the right index. In: Journal of Economics. Volume 43, 1983, pp. 21-29.
  9. ^ In addition, William H. Riker: The first power index. In: Social Choice & Welfare. Volume 3, 1986, pp. 293-295; Dan S. Felsenthal, Moshé Machover: The measurement of voting power. Theory and practice, problems and paradoxes. Edward Elgar, Cheltenham 1998, ISBN 1-85898-805-5 , pp. 486 f.