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On (game theory)

On (game theory) 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 On (game theory) rather than just read about it. In short: In mathematics, specifically combinatorial game theory, on, written as on {\displaystyle {\textbf {on}}} , is the value of the loopy game where Left may only loop back the same position, while Right has no moves. on {\displaystyle {\textbf {on}}} may be represented by surreal form on = { on ∣ } . {\displaystyle {\textbf {on}}=\{{\textbf {on}}\mid \}.} Because of Right's inability to move, on {\displaystyle {\textbf…

On (game theory) — main illustration
On (game theory) — illustration

Key takeaways

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

Reference excerpt

In mathematics, specifically combinatorial game theory, on, written as on {\displaystyle {\textbf {on}}} , is the value of the loopy game where Left may only loop back the same position, while Right has no moves. on {\displaystyle {\textbf {on}}} may be represented by surreal form on = { on ∣ } . {\displaystyle {\textbf {on}}=\{{\textbf {on}}\mid \}.} Because of Right's inability to move, on {\displaystyle {\textbf {on}}} is positive, as Left automatically wins. Its negative counterpart, off, written as off {\displaystyle {\textbf {off}}} , may be represented by off = { ∣ off } . {\displaystyle {\textbf {off}}=\{\mid {\textbf {off}}\}.}

on {\displaystyle {\textbf {on}}} and off {\displaystyle {\textbf {off}}} are not additive inverses. Their sum, dud (deathless universal draw), written as dud {\displaystyle {\textbf {dud}}} , always loops back to itself, that is, dud = { dud ∣ dud } . {\displaystyle {\textbf {dud}}=\{{\textbf {dud}}\mid {\textbf {dud}}\}.}

dud {\displaystyle {\textbf {dud}}} always results in a draw, hence its name.

on {\displaystyle {\textbf {on}}} is greater than any ender G {\displaystyle G} , as in the difference game on − G {\displaystyle {\textbf {on}}-G} , Left may play solely in on {\displaystyle {\textbf {on}}} , while Right has no option but to play in − G {\displaystyle -G} . Because − G {\displaystyle -G} is an ender, play in − G {\displaystyle -G} will eventually end, and Left will win. Similarly, off {\displaystyle {\textbf {off}}} is less than any ender.

Value- on {\displaystyle {\textbf {on}}} games

Hackenbush In Hackenbush, a position with infinitely many blue edges on the ground has value on {\displaystyle {\textbf {on}}} , as chopping an edge still leaves infinitely many edges remaining. This example illustrates the fact that on + 1 = on {\displaystyle {\textbf {on}}+1={\textbf {on}}} ; adding a blue branch won't change the position.

Col The Col position to the left has value on {\displaystyle {\textbf {on}}} . Right cannot move, while Left may color any of the empty areas, while not changing the position.

Snort A similar Snort position has value on {\displaystyle {\textbf {on}}} , shown to the right. Again, Right cannot move, while Left may color any of the empty areas, which doesn't change the position.

See also Loopy game Surreal number

References

Illustrations

On (game theory): This Col position has value on {\displaystyle {\textbf {on}}} .
This Col position has value on {\displaystyle {\textbf {on}}} .
On (game theory): This Snort position has value on {\displaystyle {\textbf {on}}} .
This Snort position has value on {\displaystyle {\textbf {on}}} .

Worked examples

Example 1 — a first encounter with On (game theory)

Start with the simplest possible case. Write down what On (game theory) 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 On (game theory) 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 On (game theory) 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 On (game theory)

In research
On (game theory) 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 On (game theory) 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
On (game theory) is common in secondary-school and first-year university syllabi. It links to neighbouring topics Combinatorial game theory, so understanding it makes those chapters shorter.
In everyday life
Look for On (game theory) 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 On (game theory) in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what On (game theory) means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain On (game theory) out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is On (game theory) in simple terms?

In mathematics, specifically combinatorial game theory, on, written as on {\displaystyle {\textbf {on}}} , is the value of the loopy game where Left may only loop back the same position, while Right has no moves. on {\displaystyle {\textbf {on}}} may be represented by surreal form on = { on ∣ } . {…

Why does On (game theory) 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 On (game theory)?

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 On (game theory).

Tags

  • Combinatorial game theory

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