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Multi-stage game

Multi-stage 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 Multi-stage game rather than just read about it. In short: In game theory, a multi-stage game is a sequence of several simultaneous games played one after the other. This is a generalization of a repeated game: a repeated game is a special case of a multi-stage game, in which the stage games are identical.

Multi-stage game — main illustration
Multi-stage game — illustration

Key takeaways

  • Multi-stage 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 Multi-stage game to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Multi-stage game from memory before moving on to harder problems.

Reference excerpt

In game theory, a multi-stage game is a sequence of several simultaneous games played one after the other. This is a generalization of a repeated game: a repeated game is a special case of a multi-stage game, in which the stage games are identical.

Multi-Stage Game with Different Information Sets As an example, consider a two-stage game in which the stage game in Figure 1 is played in each of two periods:

The payoff to each player is the simple sum of the payoffs of both games. Players cannot observe the action of the other player within a round; however, at the beginning of Round 2, Player 2 finds out about Player 1's action in Round 1, while Player 1 does not find out about Player 2's action in Round 1. For Player 1, there are 2 3 = 8 {\textstyle 2^{3}=8} strategies. For Player 2, there are 2 5 = 32 {\textstyle 2^{5}=32} strategies. The extensive form of this multi-stage game is shown in Figure 2:

In this game, the only Nash Equilibrium in each stage is (B, b). (BB, bb) will be the Nash Equilibrium for the entire game.

Multi-Stage Game with Changing Payoffs In this example, consider a two-stage game in which the stage game in Figure 3 is played in the first period and the game in Figure 4 is played in the second:

The payoff to each player is the simple sum of the payoffs of both games. Players cannot observe the action of the other player within a round; however, at the beginning of Round 2, both players find out about the other's action in Round 1. For Player 1, there are 2 5 = 32 {\textstyle 2^{5}=32} strategies. For Player 2, there are 2 5 = 32 {\textstyle 2^{5}=32} strategies. The extensive form of this multi-stage game is shown in Figure 5:

Each of the two stages has two Nash Equilibria: which are (A, a), (B, b), (X, x), and (Y, y). If the complete contingent strategy of Player 1 matches Player 2 (i.e. AXXXX, axxxx), it will be a Nash Equilibrium. There are 32 such combinations in this multi-stage game. Additionally, all of these equilibria are subgame-perfect.

References

Fudenberg, Drew; Tirole, Jean (1991). Game Theory. Cambridge, Massachusetts: MIT Press. ISBN 9780262061414. Book preview. Watson, Joel (2013). Strategy: an introduction to game theory (Third ed.). New York. ISBN 978-0-393-91838-0. OCLC 842323069.{{cite book}}: CS1 maint: location missing publisher (link)

Illustrations

Multi-stage game: Figure 2
Figure 2
Multi-stage game: Figure 3
Figure 3
Multi-stage game: Figure 4
Figure 4
Multi-stage game: Figure 5
Figure 5

Worked examples

Example 1 — a first encounter with Multi-stage game

Start with the simplest possible case. Write down what Multi-stage 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 Multi-stage 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 Multi-stage 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 Multi-stage game

In research
Multi-stage 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 Multi-stage 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
Multi-stage game is common in secondary-school and first-year university syllabi. It links to neighbouring topics Economic theories stubs, Game theory game classes, Microeconomics stubs, so understanding it makes those chapters shorter.
In everyday life
Look for Multi-stage 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.

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How to study Multi-stage game in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Multi-stage game 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 Multi-stage game out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Multi-stage game in simple terms?

In game theory, a multi-stage game is a sequence of several simultaneous games played one after the other. This is a generalization of a repeated game: a repeated game is a special case of a multi-stage game, in which the stage games are identical.

Why does Multi-stage 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 Multi-stage 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 Multi-stage game.

Tags

  • Economic theories stubs
  • Game theory game classes
  • Microeconomics stubs

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