Jo˜ao Correia-da-Silva

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Research Papers in Economics

Agreeing to disagree without
the countable additivity axiom?
Jo˜ ao Correia-da-Silva
Faculdade de Economia. Universidade do Porto. PORTUGAL.
July 26, 2006
Abstract. An example is given in which agents “agree to disagree”. It relies on
the absence of the countable additivity axiom.
Keywords: Agreeing to disagree, Interactive epistemology.
JEL Classiﬁcation Numbers: D82, D84.
?Jo˜ ao Correia-da-Silva (joao@fep.up.pt) acknowledges support from CEMPRE, Funda¸ c˜ ao
para a Ciˆ encia e Tecnologia and FEDER.
1The fable
Ann and Bob face a countable set of possible states of nature, that we denote
by Ω = {1,2,...}. Knowing nothing about probability theory, they consider all
states to be equally probable.
Ann and Bob have diﬀerent observation capabilities, described by partitions of Ω.
These are such that what is observed is the element of the partition that contains
the actual state of nature. Their private information partitions are, respectively:
PA = {{1,2,3},{3 + 1,...,22 · 3},...,{22k · 3 + 1,...,22k+2 · 3},...};
PB = {{1,...,2 · 3},{2 · 3 + 1,...,23 · 3},...,{22k−1 · 3 + 1,...,22k+1 · 3},...}.
Ann and Bob are curious about whether the event described as follows occurred:
E = {1,2} ∪ {2 · 3 + 1,22 · 3} ∪ ... ∪ {2(2k+1) · 3 + 1,2(2k+2) · 3} ∪ ... ;
∼ E = {3,2 · 3} ∪ {22 · 3 + 1,23 · 3} ∪ ... ∪ {2(2k) · 3 + 1,2(2k+1) · 3} ∪ ... .
All this is common information. The space of states, the information structures,
and the event are described in the picture below. Notice that in every set of
Ann’s information structure, the event E occurs in two thirds of the states of
nature; while in every set of Bob’s information structure, E occurs in only one
third of the states.
Having received their private information (that is, knowing in which set of their
partition of information is the actual state of nature), Ann will estimate that the
2probability of occurrence of E is 2/3, and Bob will estimate that it is 1/3. This
is common knowledge, since it is independent of the state of nature that occurs.
For the same reason, truthful exchange of information about their estimates does
not lead to any update of their posteriors.
Pooling their information, Ann and Bob are always able to ﬁnd out whether E
occurred or not. Being restricted to truthful communication of their estimates,
Ann and Bob always agree to disagree on the probability of occurrence of E.
References
Aumann, R.J. (1976), “Agreeing to Disagree”, Annals of Statistics, 4, pp. 1236-1239.
3

Agreeing to Disagree”,

Agreeing to disagree without the countable additivity axiom

http://www.dklevine.com/archive/refs4321307000000000219.pdf

Agreeing to disagree without the countable additivity axiom

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