science

Silverman's game

Silverman's game 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 Silverman's game rather than just read about it. In short: In game theory, Silverman's game is a two-person zero-sum game played on the unit square. It is named for mathematician David Silverman.

Key takeaways

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

Reference excerpt

In game theory, Silverman's game is a two-person zero-sum game played on the unit square. It is named for mathematician David Silverman. It is played by two players on a given set S of positive real numbers. Before play starts, a threshold T and penalty ν are chosen with 1 < T < ∞ and 0 < ν < ∞. For example, consider S to be the set of integers from 1 to n, T = 3 and ν = 2. Each player chooses an element of S, x and y. Suppose player A plays x and player B plays y. Without loss of generality, assume player A chooses the larger number, so x ≥ y. Then the payoff to A is 0 if x = y, 1 if 1 < x/y < T and −ν if x/y ≥ T. Thus each player seeks to choose the larger number, but there is a penalty of ν for choosing too large a number. A large number of variants have been studied, where the set S may be finite, countable, or uncountable. Extensions allow the two players to choose from different sets, such as the odd and even integers.

References Evans, Ronald J. (April 1979). "Silverman's game on intervals". American Mathematical Monthly. 86 (4): 277–281. doi:10.1080/00029890.1979.11994788. Evans, Ronald J.; Heuer, Gerald A. (March 1992). "Silverman's game on discrete sets" (PDF). Linear Algebra and Its Applications. 166: 217–235. doi:10.1016/0024-3795(92)90279-J. Heuer, Gerald; Leopold-Wildburger, Ulrike (1995). Silverman's Game. Springer. p. 293. ISBN 978-3-540-59232-7.

Worked examples

Example 1 — a first encounter with Silverman's game

Start with the simplest possible case. Write down what Silverman's game 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 Silverman's game 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 Silverman's game 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 Silverman's game

In research: Silverman's game 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 Silverman's game 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: Silverman's game is common in secondary-school and first-year university syllabi. It links to neighbouring topics Economic theories stubs, Microeconomics stubs, Non-cooperative games, so understanding it makes those chapters shorter.
In everyday life: Look for Silverman's game 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.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Silverman's game” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Silverman's game in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Silverman's game 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 Silverman's game out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Silverman's game in simple terms?

In game theory, Silverman's game is a two-person zero-sum game played on the unit square. It is named for mathematician David Silverman.

Why does Silverman's game 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 Silverman's game?

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 Silverman's game.