science

Penrose square root law

Penrose square root law is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Penrose square root law rather than just read about it. In short: In the mathematical theory of games, the Penrose square root law, originally formulated by Lionel Penrose, concerns the distribution of the voting power in a voting body consisting of N members. It states that the a priori voting power of any voter, measured by the Penrose–Banzhaf index ψ {\displaystyle \psi } scales like 1 / N {\displaystyle 1/{\sqrt {N}}} .

Key takeaways

Penrose square root law belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Penrose square root law to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Penrose square root law from memory before moving on to harder problems.

Reference excerpt

In the mathematical theory of games, the Penrose square root law, originally formulated by Lionel Penrose, concerns the distribution of the voting power in a voting body consisting of N members. It states that the a priori voting power of any voter, measured by the Penrose–Banzhaf index ψ {\displaystyle \psi } scales like 1 / N {\displaystyle 1/{\sqrt {N}}} . This result was used to design the Penrose method for allocating the voting weights of representatives in a decision-making bodies proportional to the square root of the population represented.

Short derivation To estimate the voting index of any player one needs to estimate the number of the possible winning coalitions in which his vote is decisive. Assume for simplicity that the number of voters is odd, N = 2j + 1, and the body votes according to the standard majority rule. Following Penrose one concludes that a given voter will be able to effectively influence the outcome of the voting only if the votes split half and half: if j players say 'Yes' and the remaining j players vote 'No', the last vote is decisive. Assuming that all members of the body vote independently (the votes are uncorrelated) and that the probability of each vote 'Yes' is equal to p = 1/2 one can estimate likelihood of such an event using the Bernoulli trial. The probability to obtain j votes 'Yes' out of 2j votes reads

P j = ( 1 2 ) 2 j ( 2 j ) ! ( j ! ) 2 . {\displaystyle P_{j}=\left({\frac {1}{2}}\right)^{2j}{\frac {\left(2j\right)!}{\left(j!\right)^{2}}}.}

For large N we may use the Stirling's approximation for the factorial j ! and obtain the probability ψ {\displaystyle \psi } that the vote of a given voter is decisive

ψ = P j ∼ 2 − 2 j ( 2 j / e ) 2 j 4 π j [ ( j / e ) j 2 π j ] 2 = 1 π j ∼ 2 π 1 N . {\displaystyle \psi =P_{j}\sim 2^{-2j}{\frac {(2j/e)^{2j}{\sqrt {4\pi j}}}{[(j/e)^{j}{\sqrt {2\pi j}}]^{2}}}\ =\ {\frac {1}{\sqrt {\pi j}}}\sim {\sqrt {\frac {2}{\pi }}}{\frac {1}{\sqrt {N}}}.}

The same approximation is obtained for an even number N. A mathematical investigation of the influence of possible correlations between the voters for the Penrose square root law was presented by Kirsch. Penrose law is applied to construct Penrose-like systems of two-tier voting, including the Jagiellonian Compromise designed for the Council of the European Union.

Worked examples

Example 1 — a first encounter with Penrose square root law

Start with the simplest possible case. Write down what Penrose square root law claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Penrose square root law before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Penrose square root law ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Penrose square root law

In research: Penrose square root law appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Penrose square root law in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Penrose square root law is common in secondary-school and first-year university syllabi. It links to neighbouring topics Game theory, so understanding it makes those chapters shorter.
In everyday life: Look for Penrose square root law outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Penrose square root law in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Penrose square root law means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Penrose square root law out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Penrose square root law in simple terms?

Why does Penrose square root law matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Penrose square root law?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Penrose square root law.