In 1976, publishing in the Annals of
Statistics, Robert J Aumann made a contribution that perhaps is not just
significant but provokes certain debate. He propounded what has to come be
known as the Aumann agreement theorem. It is about agreeing to disagree or
rather there would be no agreeing to disagree. To Aumann, two people let’s say
1 and 2 are said to ha have common knowledge of an event E if both know it, 1
knows that 2 knows about it and 2 knows 1 knows about it and 1 knows that 2
knows 1 knows about it and so on. Implied is the knowledge about the event is
public with no secrecy. Carrying on, the theorem suggests, if two people have
the same priors and their posteriors for an event A are common knowledge, then
there posteriors are equal. If one were to decode the meaning, it implies given
the absence of information asymmetry about an event, the two people in knowledge
of the events would hold the same views even though their source of information
would have differed in quantity and quality. Some examination of this assertion
would be in order. The piece will just delve into the basics of the theorem
rather than making a value judgment in terms of its applications or otherwise.
The first assumption is that the
agents are rational and they have common knowledge. Rational agents are those
who have clear preferences. As one is aware preferences are complete, clear and
transitive. Given their preferences are unambiguous, they would base their
decisions on the expected cost benefit analysis of the outcomes they have
modeled. The outcomes might be uncertain, yet uncertainty leads to probability
being attached to each of the outcomes that are possible or anticipated. They assign
the expected values to each outcome possible and thus based on the optimal
expected outcome of the value, they exercise their choice. Common knowledge is
a framework in economics and decision sciences wherein a group of agents have
the knowledge of an event. Furthermore, as stated above, not only they know
that they know the event but also everyone knows everyone knows about the
event. In such a circumstances, they can be described to have common priors.
Prior in statistics is about the probability distribution that would express
one’s beliefs about some uncertain quantity before some evidence is taken into
account.
Since they have common knowledge
about their posterior probabilities, the posteriors would be equal and thus no agreement
is possible on disagreement. Posterior probability of any random event is the
conditional probability assigned to the event after the context is factored
into the decision making. Posterior is something ex-post information about the
event once the background is examined at certain length. Aumann assumes that it
does not matter how information was obtained. In other words, each of the
agents might have gained the knowledge of the event through different bits of
information, but as long as the rational agents had the common knowledge, it
was immaterial about the means of information being available. The final
outcome as Aumann would project was there would be agreement between the agents
irrespective of the number of agents given the conditions assigned above.
Yet, there would arise questions
about the inter-temporal matters- whether the time frame is relevant in the
context of the agreements being arrived at. There would also be the context to
be examined about the efficiency of the process in the mathematical landscape. There
are however many issues that might arise including the presence of priors not
being common. If the priors are not common, then there would exist a need to measure
the distance of the priors being held by the agents.
The blog “Marginal Revolution’ does
of course present a nuanced view on the applicability of Agreement Theorem. In the
views of the blogger, it is the common sense morality that prevails while
related to self, yet when it comes to political standing, there would be the
differences that would stand out. In fact, in the context of the societal
understanding, putting up an individual point however unpopular it might be
would perhaps be in self-interest. Yet, in the context where decisions are self-related,
it might be common sense to accept the agreement. It gives an example of a
hotel offering room service which everyone knows and agrees. There is yet a
choice of travelling to another restaurant but there is disagreement over its
location. Even if one has a feel about its location, the cost benefit analysis
would naturally tend to skew in favor of opting for the room service over which
there is general consensus. In some ways, in this context, the self-interest
actually would dictate sort of Prisoner’s Dilemma which would result into the agreement.
There is of course can be
speculation on its application in the legal context. Let us say there are two
people who are witness to an event. They both know that the other knows about
the event. If the Aumann were to be true, then both the witnesses would have to
present a similar conclusion to the judge. Would the witnesses differ is
something that has to be pondered about. As the eyewitnesses differs in their interpretation
of the events, it begs to be questioned about the proposition put forth by
Aumann. In this context, it might be argued that witness credibility would be
in question. There is of course the question of each trusting the other.
Moreover, it would be open to the question of examining the veracity of both
the stories. Leave alone agreement or disagreement, there would arise the need
to verify the correctness of the narrative. In this context, there is a
possibility that the theorem might find its limitations.
The proposition does offer quite a
bit of interesting insights into the decision making under conditions of common
knowledge. The common knowledge might not exist to perfection but certainly
offers a starting point for analysis of consensual decision making. There are
bound to be disagreements within the organization or the society. The final
decision is often the product of contestations around differing points of view.
The theorem suggests the possibility of consensus under certain conditions. The
stage for final analysis would however be the relaxation of those conditions
and subsequent examination of decision making under those relaxations.
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