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.
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Santa Fe Institute
Central European University
Institute for New Economic Thinking (INET)
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).
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 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.
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.
For illustrative purposes, I will use a slightly perturbed Traveler’s Dilemma with n = 5.
It is easy to check that s5 is strongly dominated by s4.
When s5 is dropped, s4 is strongly dominated by s3.
When s4 is dropped, s3 is strongly dominated by s2.
After dropping s3, only s2 remains.
Thus the only rationalizable strategy, and hence the only Nash equilibrium, is truth-telling.
This analysis is extended to all larger nin 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.
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.
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.
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 gkfor k=1,…,100.
Then Alice will choose a round m to defect in that maximizes the expression
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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!
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
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.
This is a topic for future study.