Common Knowledge of Rationality is
Self-Contradictory
Herbert Gintis
Santa Fe Institute
Central European University
Institute for New Economic Thinking (INET)
Princeton University
Press, 2009
Summary
The conditions under which rational agents play a Nash
equilibrium are demanding and often implausible,
even when each agent knows the other agents are
rational (Aumann and Brandenburger 1995).
Common knowledge of rationality (CKR), by contrast,
often implies agents play a Nash equilibrium because
CKR implies rationalizability in normal form games
(i.e., the iterated elimination of strongly dominated
strategies)
and subgame perfection in extensive form games (i.e.,
backward induction).
Summary
Game theorists consider CKR to be a strengthening of
mutual knowledge of rationality.
In fact, CKR is not a legitimate epistemic condition,
as rationality may imply the absence CKR.
Thus CKR is logically self-contradictory.
The failure of CKR is related to some well-known
antinomies of modal logic.
CKR in Normal Form Games
CKR implies that in a normal form game, agents will use
strategies that survive the iterated elimination of
strongly dominated strategies.
In many games, subjects do not conform to this behavior,
implying that CKR is violated.
Moreover, when players violate CKR, their payoffs may
be higher than under CKR, so that the rationality
assumption is not violated.
CKR in Normal Form Games:
The Traveler’s Dilemma
Two travelers incur equal expenses but do not have receipts.
Their boss tells them to report independently a number of
dollars between $2 and $n.
If they report the same number, each receives this amount.
If they report different numbers, each receives the smaller
amount, plus the low reporter will get an additional $2
(for being honest) and the high reporter will lose $2.
The Traveler’s Dilemma
For illustrative purposes, I will use a slightly perturbed Traveler’s
Dilemma with n = 5.
The Traveler’s Dilemma
It is easy to check that s5 is strongly dominated by s4.
The Traveler’s Dilemma
When s5 is dropped, s4 is strongly dominated by s3.
The Traveler’s Dilemma
When s4 is dropped, s3 is strongly dominated by s2.
The Traveler’s Dilemma
After dropping s3, only s2 remains.
The Traveler’s Dilemma
Thus the only rationalizable strategy, and hence the only
Nash equilibrium, is truth-telling.
This analysis is extended to all larger n in The Bounds of
Reason (Princeton University Press, 2009).
Of course, in reality players pick much higher amounts and
make much more money when n is large.
CKR in Extensive Form Games
Robert Aumann (1995) proved that in extensive form games
of perfect information,
CKR implies that only strategies that will be chose are
those survive backward induction
which is the same as the iterated elimination of weakly
dominated strategies
and equivalent to subgame perfection.
However, this is not how rational players behave.
For example:
Repeated Prisoner’s Dilemma
Suppose Alice and Bob play a Prisoner's
Dilemma 100 times, with the condition
that the first time either player defects,
the game terminates.
Common sense tells us that players will cooperate for at
least 95 rounds, and this is indeed supported by
experimental evidence (Andreoni and Miller 1993).
However, a backward induction argument indicates that
players will defect in the very first round.
It follows that CKR implies that defection will take place on
round one.
Repeated Prisoner’s Dilemma
Suppose however, that Alice and Bob are rational,
have subjective priors concerning each other's play,
and maximize their payoffs subject to these priors.
Specifically, suppose Alice believes that Bob will cooperate
up to round k and then defect, with probability gk for
k=1,…,100.
Then Alice will choose a round m to defect in that
maximizes the expression
Repeated Prisoner’s Dilemma
In the above equation, is the payoff to Alice when defecting on
round m. R = 3 is the payoff if both cooperate, P = 1 is the payoff if
both defect, T = 4 is the payoff to a defector when the opponent
cooperates, and S = 0 is the payoff to a cooperator whose opponent
defects.
The first term in this expression is the payoff if Bob defects first,
the second term is the payoff if they defect simultaneously,
and the final term is the payoff if Alice defects first.
Repeated Prisoner’s Dilemma
For instance, suppose gk is uniformly distributed in the
rounds m =1,…,99.
Then it is a best response to cooperate up to round 98.
Indeed, suppose Alice expects Bob to defect in round one
with probability 0.95 and otherwise defect with equal
probability on any round from two to 99.
Then it is still optimal for her to defect in round 98.
Backward induction is not plausible, and does not follow
from rationality.
Rather, it follows from CKR, which is contradictory.
For example:
CKR is Contradictory
Bob writes three distinct whole numbers of his choosing
between 1 and 1000 on three slips of paper.
Alice chooses one of these slips at random.
Alice can Play or Pass.
If she Plays and she chose the largest of the three numbers,
Bob pays her $10;
otherwise she pays Bob $10,000.
If she Passes, she pays Bob $1.
CKR is Self-Contradictory
Let Bob's Random Strategy be to chose the three distinct
integers randomly.
We do not assume that Bob's Random Strategy is optimal.
Alice's best response to Bob's Random Strategy is to Pass
unless her number is 1000.
To see this, note that if it a best response to Pass if her slip
shows 999, then it also a best response to pass if her slip
shows any number lower than 999.
So let us assume her slip says 999.
CKR is Contradictory
Alice loses choosing Play only if Bob chose the three
numbers m, 999, 1000 where 1 < m < 999.
Conditional on the fact that Alice chose 999, the probability
that Bob choses m, 999, 1000 is p=2/999.
Alice's payoff to is -$10000p + $10(1-p) = -$10.04.
Thus Alice's best response is to Play if her slip shows 1000,
and to pass otherwise.
The probability that Alice chooses Play is then
(1- 3/1000)(1/3) = 0.001.
Thus the payoff to Bob from using the Random Strategy is
(0.999) x $1 - (0.001) x $10 = $0.989.
CKR is Contradictory
Assuming CKR, we can show that the payoff to the game
for Alice is strictly positive, and since this is a zero sum
game, the payoff to Bob is strictly negative.
Therefore CKR implies Bob is not rational, as the Random
Strategy has a strictly higher payoff.
CKR is Contradictory
Proof:
Since Bob can choose the three numbers any way he wishes,
we can see that a rational Bob would never include 1000
in his three numbers.
For if Bob did choose 1000, then Alice will win the $10
with probability 1/3 and lose $1 with probability 2/3,
giving Bob a loss of $8.
Thus Bob's payoff to including 1000 is strictly negative, and
including 1000 is dominated by Bob's Random Strategy.
CKR is Contradictory
Because Alice knows that Bob is rational, she knows he will
not include 1000 among his three numbers.
But Bob knows that Alice knows he is rational, so if he
includes 999 among the three numbers, he knows Alice
will know that if she picks 999, she will guess that it is
the highest.
Thus including 999 among the three numbers is dominated
by the Rational Strategy.
Continuing to iterate this argument, assuming CKR, Bob
must choose numbers 1, 2, and 3.
But then Alice knows this, so if her slip says 3, she will
guess correctly that it is the highest number.
CKR is Contradictory
It follows that if we assume CKR, then Bob cannot play his
Random Strategy, and hence his strategy choice does not
maximize his payoff.
But this contradicts Bob's rationality, and hence also CKR.
The implication of this reasoning is that CKR is
contradictory.
CKR does not describe a condition of knowledge that can be
assumed under any conditions.
Unknowable Truths
The misleading attractiveness of CKR flows from the
common sense notion that if something is true, then
at least ideally, it should be possible to know that this
is the case.
It appears obvious that if you are rational, then I should
be able to know that you are rational, you should be
able to know that I know that you are rational, and so
on.
Thus if CKR fails. there is some level of mutual
knowledge of rationality that obtains but that cannot
be known.
Unknowable Truths
The following Surprise Examination problem shows
how backward induction type arguments can be
invalidated when truth does not imply the possibility
of knowing the truth.
Consider a class that meets daily, Monday through
Friday.
The instructor, Professor Alice, announces that there will
an exam one day next week,
but students, of whom Bob is one, will not know
before they see the test that it will be given that day.
Unknowable Truths
Bob reasons, “Suppose Professor Alice is telling the
truth. Then the exam cannot be given on Friday
because then we would know beforehand.”
Bob then notes that a similar argument shows that the
exam could not be given on Thursday.
And so on.
He concludes that such an exam cannot be given.
On the following Wednesday, Professor Alice gives an
exam, and Bob did not know that this would occur.
Thus Professor Alice’s statement was true but could not
be known by Bob.
Unknowable Truths
Of course, Bob may realize this as well---he “knows”
that Alice’s statement may be true.
However he cannot apply the laws of the modal logic of
knowledge to reasoning about the truth of Alice’s
statement!
Unknowable Truths
For an overview of the many proposed solutions to the
Surprise Examination by philosophers and logicians
(I have read at least twenty papers on the subject, and
there are some that I have not read) see Margalit and
Bar-Hillel (1983) and Chow (1998).
I will follow Binkley (1968), who assumes there are only
two days, Monday and Tuesday, and a single student with
knowledge operator k. We assume for any knowledge
operator that
Unknowable Truths
Unknowable Truths
Unknowable Truths
Unknowable Truths
Unknowable Truths
Common Knowledge in
Interactive Epistemology
The characteristics of rationality are shared by several other
important modal attributes,
including truthfulness and logicality.
Interactive epistemology often assumes, explicitly or implicitly, that
these attributes are commonly known.
However, in some cases this assumption may be self-contradictory or
simply false.
It is easy to say, for instance, “we assume all agents report truthfully,
and that this is common knowledge.”
However, it could be that this assumption leads to inconsistencies.
Common Knowledge in
Interactive Epistemology
This is a topic for future study.