← Game Theory
第 2 章 · 20 分钟
Dominance and iterated deletion: how far without equilibrium
Each extra round spends one more layer of belief.
概念地图
Dominance
Comparing two strategies without knowing what anyone else will do.
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
von Neumann and Morgenstern · 1944 · the notion of dominance
Introduced dominance between strategies within the zero-sum frame as the first step in simplifying a game. They argued against enumerating outcomes one by one; the cost is that dominance removes very little in non-zero-sum games, and most games do not reduce to a unique solution.
Foundational
Bernheim and Pearce · around 1984 · rationalizability
Showed independently that what survives iterated deletion of strictly dominated strategies is exactly what can be optimal under some belief about opponents. This is what pairs each deletion round with one layer of belief, and it is the core of this chapter.
Turn
The experimental line · the beauty-contest game (two-thirds of the average)
Repeatedly finds that populations choose at the first to third layer of deletion rather than at the fully deleted solution. The most direct counterexample to the commitment that common knowledge runs to unlimited depth.
Counterexample
Bonanno · the chapters on dominance and iterated deletion
The subject of this course. The textbook is explicit about how strict and weak dominance differ in order-dependence, and this chapter uses that difference as the standard case of a swapped definition.
Subject of this course
机制02
Comparison without conditions
Some conclusions need no knowledge of what the opponent will do at all. The relation that licenses them, introduced in 1944 and used ever since as the first step in simplifying a game, is strict dominance. Its place in the argument is unique here: it is the only reasoning in this course that spends no belief, so a conclusion drawn from it holds whether the opponent is a genius or a dice roll. Everything after this chapter starts spending.
机制One strategy, through being strictly better against every opponent profile, removes another unconditionally.
可迁移性测试
Move it to choosing equipment: if option A costs less in total whatever the future load turns out to be, you do not have to forecast the load first. The isomorphism breaks in that engineering's 'whatever' usually covers a handful of listed cases, while the formal definition demands every element of the opponent's strategy set.
机制03
One layer per round
The first round of deletion uses only 'the opponent is rational'. The second needs 'the opponent knows I am rational', the third 'the opponent knows that I know that they are rational'. Every round of iterated deletion spends a layer of mutual belief, and layers are not free: real people usually go one to three deep. That correspondence is the structure this chapter delivers.
机制Each deletion round, through spending one more layer of mutual belief, narrows the surviving strategy set further.
可迁移性测试
Move it to pricing: round one needs only that a rival will not sell below cost, round two that the rival also thinks you will not. Each step is a stronger assumption and easier to get wrong. The isomorphism breaks in that pricing lets you calibrate depth against a rival's history.
Derivation04
From strict dominance to order independence
What is being proved: deletion by strict dominance is order independent while weak dominance is not — and the difference is one word in the definition.
Strategy sets are finite and deletion is by strict dominance
A hypothesis. Drop finiteness and the process may not terminate.
Strict dominance: strictly better against every opponent profile
A hypothesis. Change 'every' to 'at least one, and no worse elsewhere' and you have weak dominance, and the conclusion changes with it.
A deleted strategy no longer affects anyone's best response
A hypothesis, and the substance of the theorem: a strictly dominated strategy is chosen under no belief.
推导 · 0 / 4
Worked derivation05
Under weak dominance, order changes the answer
Why is deletion by strict dominance order independent while weak dominance is not? Give the smallest example.
Find a weak-dominance relation that depends on whether a particular opponent strategy is still present.
裂缝06
The boundary of this chapter's claims
争议地形07
本书主张
Iterated deletion of strictly dominated strategies is uncontroversial simplification: what it removes would be chosen under no belief, so no solution is lost.
另一种看法
The bounded-depth line holds that people perform one to three rounds, so the fully deleted solution usually predicts badly; depth should be modelled directly rather than treated as a deviation from an ideal.
分歧扎在
The disagreement is rooted in the modelling commitment about whether common knowledge runs to unlimited depth, not in the mathematics. Both accept the formal properties of deletion and differ on how many layers to attach when mapping it onto people.
什么证据能裁决
What would settle it is the choice distribution in games like the beauty contest, where the equilibrium solution and the layer-by-layer solutions differ visibly: see which layer the population concentrates at. Concentration at the fully deleted solution supports unlimited depth; concentration at shallow layers supports the depth model.
The evidence points to depth being finite and varying with population and experience: choices concentrate at shallow layers and move up after repeated play. What is missing is a way to separate a person's own depth from their estimate of others' depth — the observed behaviour is the composite.
Boundary and counterexample08
The one-layer-per-round commitment as a testable specification
The modelling commitment under test: reasoning depth in reality matches deletion rounds one for one, with no limit on depth.
Hypothesis dropped: mutual rationality is common knowledge to unlimited depth
Full deletion corresponds to reality only under this; with finite depth the fully deleted solution stops being a prediction.
The counterexample: beauty-contest choices concentrating on values from the first to third layer rather than the unique fully deleted solution
The unique solution and the shallow ones are numerically far apart and easy to distinguish.
The observable: the distribution of numbers chosen in a one-shot play
It must be the first play; repetition introduces learning.
What counts against it: the mode falls on a shallow-layer value, at a distance from the unique solution above a pre-registered threshold
The threshold is fixed in advance on the numerical gap.
Pre-registered failure condition: the mode falls at a shallow layer, or the distribution is clearly multimodal with the unique solution not the tallest peak
Either counts as the commitment failing.
推导 · 0 / 3
接口09
Which slot it hangs on
挂在哪个槽位The conclusion you already hold is probably 'to predict an opponent, assume they are as clever as you' — a judgement about opponent models. This chapter goes at its depth.
Does this chapter (a) replace 'assume they are as clever as me', or (b) constrain it by making you estimate a depth first and delete accordingly?
慢变量Register two slow variables: how many layers of belief your last strategic judgement used, and whether your estimate of the opponent's depth has any evidence behind it. Look at the second first.
小结10
本章小结
01Strict dominance is the only belief-free reasoning here; its conclusions hold against any opponent.
02'Strict dominance deletion is order independent' is the definition unpacked; the same reasoning fails for weak dominance.
03A theorem's robustness can rest on one quantifier: 'for every' versus 'for at least one'.
04Every extra deletion round spends one more layer of mutual belief, and layers are not free.
05The only step carrying observational risk is the commitment that depth is unlimited.
提取练习 · 合上书，先自己答一遍。
?Is 'deletion order does not matter under strict dominance' (a) true by definition or (b) a modelling commitment about reality?
?Does population concentration at shallow layers refute (a) the formal properties of deletion or (b) the unlimited-depth commitment?
Forced choice11
A · Identify the mechanism
Which of these is a mechanism?
二选一
Forced choice12
C · Locate the crack
In which situation does this framework give a confident and wrong answer?
二选一
读到这里，把它记为已读。