← Game Theory
第 5 章 · 20 分钟
Information: information sets and common knowledge
Perfect information and complete information are two different things.
概念地图
Information set
When it is your turn, you cannot tell which of these nodes you are at.
深色为本章概念，浅色为其他章
陪练模式
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Locus01
延伸阅读
Kuhn · 1950s · information sets and behaviour strategies
Wrote 'you cannot tell which node you are at' into the formalism as an information set, and proved that with perfect recall mixed and behaviour strategies are equivalent. He argued against a notation that could only handle perfect information; the cost is that perfect recall does not hold when modelling agents with limited memory.
Foundational
Harsanyi · 1960s · the transformation for incomplete information
Used a fictitious move by nature drawing 'types' to turn a game where you do not know the other's preferences into one of imperfect information. This made incomplete information tractable; the cost is that the type distribution must be common knowledge, which is a strong requirement in itself.
Turn
Aumann · 1976 · common knowledge and agreeing to disagree
Formalised common knowledge and proved that two people with a common prior cannot agree to disagree about a posterior. The sharpest counterexample to the everyday idea that we can simply keep our own views.
Counterexample
Bonanno · the chapters on information sets and incomplete information
The subject of this course. The textbook keeps perfect and complete information apart, and this chapter explains why the two words get confused and what goes wrong once they do.
Subject of this course
机制02
Which node you cannot tell
When it is your turn, you may not know where in the tree you stand. The object Kuhn introduced in the 1950s to hold that fact is the information set. Its place in the argument is unyielding: a strategy specifies an action per information set, not per node — and that alone decides whether chapter 4's backward induction can proceed node by node.
机制Indistinguishable nodes, through being grouped into one information set, force a single action on all of them.
可迁移性测试
Move it to blind review: a reviewer cannot tell which lab a manuscript came from and so must apply one standard to both. The isomorphism breaks in that blind review's indistinguishability is manufactured by a procedure and can be unblinded, while an information set is part of the model.
机制03
Two kinds of not knowing
Two words in the textbook are easily run together. perfect information is about whether you know where you are; complete information is about whether you know what the other wants. Chess is both, poker is complete but imperfect, and bargaining is often perfect but incomplete. Run the two together and chapter 4's theorem gets applied where it does not hold.
机制Two kinds of not knowing, through landing on different parts of the model, leave a game complete on one axis and deficient on the other.
可迁移性测试
Move it to a clinical decision: a doctor may hold the full case history (knows where they are) and still not know how the patient weighs quality of life against survival. The isomorphism breaks in that the doctor can simply ask, while in a game asking is itself a move in the tree.
Derivation04
From incomplete information to the Harsanyi transformation
What is being proved: a game where preferences are unknown can be rewritten as one of imperfect information, and where the cost of that rewriting is booked.
Each player's preferences are parameterised by a type (read: i's type)
A hypothesis. Types must be enumerable or nature's action set cannot be defined.
The joint distribution of types is common knowledge
A hypothesis, and the entire cost of the rewriting: 'not knowing what the other wants' is traded for 'knowing the distribution of what the other wants'.
Each player knows their own type
A hypothesis. Relaxing it gives richer models the textbook does not cover.
推导 · 0 / 4
裂缝05
The boundary of this chapter's claims
争议地形06
本书主张
The Harsanyi transformation reduces incomplete information to imperfect information, letting the standard solution concepts be used directly; the common prior is the price, and it buys the whole toolkit.
另一种看法
The heterogeneous-priors line holds that real disagreement is often not about different information but about different priors; forcing a common prior records 'different views' as 'different information', and mechanisms designed on that basis fail in real markets.
分歧扎在
The disagreement is rooted in whether disagreement comes from information or from priors. Both accept the mathematics of the transformation; they differ on whether the second hypothesis holds when it is applied to a real interaction.
什么证据能裁决
What would settle it is behaviour after disclosure: give both sides the same information and see whether the disagreement converges. Under a common prior, once the information is common knowledge the posteriors must agree (Aumann's result); if disagreement persists and both know the other still disagrees, the common prior is refuted.
The evidence points to both being present: disclosure converges part of the disagreement and part persists. What is missing is a way to separate 'different priors' from 'different rules for reading the same information' — the two are hard to tell apart behaviourally.
Boundary and counterexample07
The common-prior commitment as a testable specification
The modelling commitment under test: everyone's beliefs about everyone's types are representable by one distribution, and this is common knowledge.
Hypothesis dropped: priors are identical and this is common knowledge
The fulcrum of the Harsanyi transformation; without it the rewritten game no longer represents the original interaction.
The counterexample: both sides still disagree after the same information is disclosed, and each knows the other disagrees
Aumann's result says this cannot happen under a common prior.
The observable: the posterior probabilities each side reports before and after disclosure
They must be comparable numbers, not a yes-or-no answer about agreement.
What counts against it: the posterior gap after disclosure still exceeds a pre-registered threshold, with both sides confirming they know the other's report
Both conditions are needed; a gap without mutual awareness is not a counterexample.
Pre-registered failure condition: gap above threshold with mutual awareness, or a gap that does not converge over disclosure rounds
Either counts as the commitment failing.
推导 · 0 / 3
接口08
Which slot it hangs on
挂在哪个槽位The conclusion you already hold is probably 'put the information on the table and the disagreement should go away' — a judgement about communication. This chapter goes at its condition of application.
Does this chapter (a) replace 'disclosure removes disagreement', or (b) constrain it to hold only when the two sides share a prior?
慢变量Register two slow variables: how many of your disputes converged once material was shared, and how many persisted after both sides had read the same material. Look at the second first.
小结09
本章小结
01An information set is a group of indistinguishable nodes; strategies specify actions per set, not per node.
02Perfect information is about knowing where you are; complete information is about knowing what the other wants.
03The Harsanyi transformation is a construction whose price is the strong common-prior hypothesis.
04Aumann's result shows that under a common prior and common knowledge, two people cannot agree to disagree.
05If types may be assigned after the fact, the common-prior commitment carries no observational risk.
提取练习 · 合上书，先自己答一遍。
?Is 'adding a nature move at the root to draw types' (a) a construction or (b) a modelling commitment about reality?
?Is poker (a) complete but imperfect or (b) perfect but incomplete?
Forced choice10
B · Identify the tag
'Add a fictitious player at the root who draws types from a distribution' — what kind of step is that?
二选一
Forced choice11
D · Judge the interface
Which one hangs on a slot in your existing structure?
二选一
读到这里，把它记为已读。