← Game Theory
第 6 章 · 22 分钟
Probability enters: expected utility and mixed strategies
The price of the existence theorem, and how to read a mixture.
概念地图
Mixed strategy
A distribution over your own strategies; existence holds here.
深色为本章概念，浅色为其他章
陪练模式
不想只是读？让它带你走一遍。
导师把这一章拆成小步，每一步先问你，再讲；你用自己的话答，它先认下对的部分再纠偏。用你自己的 AI，对话只存在这台设备上。
Locus01
延伸阅读
von Neumann and Morgenstern · 1944 · the axiomatisation of expected utility
Gave axioms on preferences over lotteries under which preferences are a probability-weighted average of outcome utilities. They argued against importing cardinal utility as a psychological quantity; the cost is that utility now has a scale resting entirely on those axioms, one of which — independence — has been attacked ever since.
Foundational
Nash · around 1950 · existence on mixed strategies
Extended the strategy set to probability distributions and obtained existence by fixed point. This bought universal existence at the price of the axioms above: without expected utility the payoff of a mixed strategy has no definition.
Turn
Harsanyi · 1970s · the purification interpretation
Read a mixed equilibrium as the limit of a family of complete-information games with small private perturbations, in each of which everyone plays a pure strategy. The most substantive answer to 'do people really roll dice', at the cost of requiring the perturbation structure to be plausible.
Revision
Bonanno · the chapters on expected utility and mixed strategies
The subject of this course. The textbook keeps axioms and theorems clearly apart; what this chapter handles separately is what the probability in a mixed strategy actually represents, a step textbooks usually pass over.
Subject of this course
机制02
The price of a scale
Chapter 1 said utility is only a ranking, and any order-preserving transformation gives the same preferences. The moment you want to speak of 'half a chance of A and half of B', a ranking is not enough: two lotteries must be comparable. The representation von Neumann and Morgenstern supplied for that is expected utility. It turns ordinal utility into a quantity with a scale, and that scale was bought by the axioms rather than given for free.
机制Axioms on preferences over lotteries, through granting utility a scale, make probability weighting a defined operation.
可迁移性测试
Move it to temperature: saying today is hotter than yesterday is a ranking, and saying it is twice as hot requires a chosen scale — Celsius and Kelvin give different multiples. The isomorphism breaks in that temperature's scale rests on a physical convention while utility's rests on preference axioms, which behaviour can violate directly.
机制03
Three readings of a mixture
What the mixed strategy in an equilibrium actually stands for admits at least three different answers: that the person really randomises; that the number is the opponent's belief about them; or Harsanyi's purification, where everyone in a population plays pure and the probability is a population share or the distribution of a private perturbation. The three give the same numbers and correspond to entirely different modelling commitments, so different observations refute them.
机制One set of equilibrium probabilities, through being given different interpretations, maps onto entirely different observables.
可迁移性测试
Move it to spot checks: a ten per cent inspection rate can mean a die really thrown each time, the inspected party's estimate, or one inspector in ten. On paper all three are one tenth, and the deterrent effect differs. The isomorphism breaks in that inspection logs can be audited directly.
Derivation04
From the axioms to 'equal payoffs across used pure strategies'
What is being proved is the test for a mixed equilibrium, and which hypotheses it rests on.
Preferences satisfy the expected utility axioms (completeness, transitivity, continuity, independence)
A hypothesis. Independence is systematically violated by Allais-type choices; drop it and the weighted-average form does not hold.
Strategy sets are finite and any distribution over them is allowed
A hypothesis. Drop mixing and rock–paper–scissors has no equilibrium.
Equilibrium is defined as mutual best response on mixed strategies
A hypothesis, and the definition itself.
推导 · 0 / 4
Worked derivation05
Your mixture is set by the opponent's payoffs
In a two-person zero-sum game, why is my equilibrium mixture determined by the opponent's payoff numbers rather than my own?
Solve the condition 'the opponent's pure strategies pay the same' backwards.
裂缝06
The boundary of this chapter's claims
争议地形07
本书主张
Mixed strategies are the natural extension of the strategy set: they buy universal existence and their test condition is clear and computable.
另一种看法
The purification line holds that people do not roll dice; an observed mixture should be read as the limit of a family of games with small private information in which everyone plays pure. On that account the equilibrium probability is a distribution over a population or over types, not any individual's randomising device.
分歧扎在
The disagreement is rooted in the modelling commitment about whose what the probability represents, not in the mathematics. The numbers computed are identical; what is at issue is which real quantity they map onto.
什么证据能裁决
What would settle it is individual-level sequences: record one participant's choices over repeated play and run a randomness test. Individual sequences failing the test while population frequencies sit near the equilibrium probability supports purification; individual sequences close to independent and identically distributed supports the randomising-device reading.
The evidence points toward purification: individual sequences typically show marked autocorrelation and pattern, while population frequencies are closer to the equilibrium values. What is missing is a way to separate a deliberate pattern from an inability to randomise — people are poor random generators, and the two look the same in behaviour.
Boundary and counterexample08
The 'mixture means individual randomising' commitment as a testable specification
The modelling commitment under test: the equilibrium mixture means the person randomises according to that distribution.
Hypothesis dropped: the individual draws each round independently from the same distribution
That is the entire content of the randomising reading; the population-share reading does not need it.
The counterexample: one participant's sequence showing marked autocorrelation, such as rarely repeating an action
Independent draws do not systematically avoid repetition.
The observable: autocorrelation and run statistics of individual choice sequences
It must be individual sequences; population frequencies cannot distinguish the three readings.
What counts against it: autocorrelation or run statistics outside the pre-registered interval for independent draws
The interval is computed in advance from the sequence length.
Pre-registered failure condition: individual sequences failing the randomness test, or marginal frequencies differing across participants far beyond sampling noise
The second points to the population-share reading.
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 stay unpredictable, act randomly' — a judgement about unpredictability. This chapter goes at where the parameter comes from.
Does this chapter (a) replace 'act randomly', or (b) constrain it: the proportion is set by the opponent's payoffs, not your own?
慢变量Register two slow variables: how many of the randomised proportions you use were computed from the opponent's payoffs, and how many were set by feel. Look at the second first.
小结10
本章小结
01To speak of probability you must trade ordinal utility for expected utility, and the scale is bought by axioms.
02'Used pure strategies pay the same in a mixed equilibrium' is the extremal property of a linear function on a simplex.
03The equilibrium mixture is set by the opponent's payoffs, not your own.
04A mixture has three readings — individual randomising, opponent's belief, population share — with different observables.
05Using a mixed equilibrium without naming the reading carries no observational risk at all.
提取练习 · 合上书，先自己答一遍。
?Is 'pure strategies with positive probability must pay the same' (a) a result of linear algebra or (b) a modelling commitment about reality?
?Do patterned individual sequences with population frequencies near the equilibrium support (a) the individual-randomising reading or (b) the population-share reading?
Forced choice11
B · Identify the tag
'The optimum of a mixed strategy is attained at some pure strategy' — what kind of step is that?
二选一
Forced choice12
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。