Consider a finite non-empty state spaceΩwith typical stateω ∈ Ω. LetJ be a finite, non-empty set of players. Player j’s knowledge is characterized by her partitionΠj
of space Ω, where player j can distinguish ω and ω0 if and only if ω /∈ πj(ω0). Let
πj(ω)∈Πj denote the unique element of playerj’s partition that contains stateω.
A posterior for player j is a vectorθj = (θj
1,· · ·, θ j
|Ω|), where, for each ω ∈ Ωand
j ∈ J, θjω ≥ 0denotes playerj’s conditional belief that the true state isω given that the true state belongs toπj(ω). At eachω∗ ∈Ω
, every playerj ∈J assigns probability
1 to event πj(ω∗), the element of her partition that contains the state ω∗; formally,
P ω∈πj(ω∗)θjω = 1. A modelM is a tupleM = Ω, J,{Πj, θj} j∈J . Consider a probability measureµj =µj
1,· · · , µ j
|Ω|
over the spaceΩ. Letµj(B) =
P
ω∈Bµ j
ω, for any B ⊆ Ω. The measure µj is a prior for player j if for any ω ∈ Ω,
wheneverµj(πj(ω)) > 0, thenθj
ω = µjω/µj(πj(ω)). The support ofµj is the subset of
statesω∈Ωsuch thatµj
ω >0. A prior may fail to have full support; that is, the prior’s
support may be a proper subset ofΩ. A modelM admits a common prior if there is some probability measureµwhich is a prior for all players.
The meet-join diagram of a modelM is a directed multigraphG, defined as follows: (i) the nodes ofGare the states; i.e., elements ofΩ; (ii) aj-edge is any triplehj;ω1, ω2i,
withω1,ω2 ∈ Ω, composed of a playerj ∈ J and an ordered pair of statesω1 andω2
belonging to the same element ofj’s partition. The set of edges ofGis composed of allj-edges,j ∈J. MultigraphGis typically not a complete graph. In many cases,Gis not even connected.
Thej2-edgehj2;ω3, ω4iis consecutive to thej1-edgehj1;ω1, ω2iif and only ifω2 =ω3.
Define the opposite of edgehj;ω1, ω2iashj;ω2, ω1i; that is, the edge that switches the
order of the states.
{{1,2,3},{4,5}}, ΠB = {{1,4},{2},{3,5}}. The meet-join diagram for this model is shown in Figure 5.1. In this figure, the edges for playerAare in blue and dashed, and the edges for playerB are in red.
1
2
3
4
5
Figure 5.1:Meet-Join Diagram
A pathpis a collection of consecutive edges
p={hj1;ω1, ω2i,hj2;ω2, ω3i,· · · ,hjk−1;ωk−1, ωki,hjk;ωk, ωk+1i}
Let−pdenote the opposite path ofp; that is, the path obtained frompby considering the opposites of the edges ofpin reverse order. A path
c={hj1;ω1, ω2i,hj2;ω2, ω3i,· · · ,hjk−1;ωk−1, ωki,hjk;ωk, ωk+1i}
is a cycle if and only if: (i)cis a closed path; that is,ωk+1 =ω1; and (ii) there exists an
edgehj;ω1, ω2ibelonging to pathcsuch that its opposite,hj;ω2, ω1i, does not belong to
c.
Define the cycle equation of a cyclec={hj1;ω1, ω2i,hj2;ω2, ω3i,· · · ,hjk;ωk, ω1i}as
θωjk 1 ·θ j1 ω2 ·θ j2 ω3· · ·θ jk−2 ωk−1 ·θ jk−1 ωk =θ j1 ω1 ·θ j2 ω2 ·θ j3 ω3· · ·θ jk−1 ωk−1 ·θ jk ωk.
A cycle is consistent (respectively, inconsistent) if its cycle equation holds (does not hold).
Harsanyi’s Consistency Problem asks if a collection of posteriors, one for each player, admits a common prior. Rodrigues-Neto [2009] proves that, given any model
Ω, J,{Πj, θj} j∈J
such that all θj
ω > 0, for every j ∈ J and ω ∈ Ω, posteriors are
consistent with a full support common prior if and only if all cycle equations are satisfied. Now, suppose that posteriors are not consistent with a full support common
prior; that is, there is an inconsistent cycle, or θjω = 0for some playerj ∈ J and state
ω ∈ Ω. Can a common prior still exist? It would have to be a common prior without full support.
LetΩ0be the subset of statesω ∈Ωsuch thatθjω = 0for some playerj. Then,Ω\Ω0
is the subset of statesω∈Ωsuch thatθj
ω >0for everyj ∈J.
A pathp={hj1;ω1, ω2i,hj2;ω2, ω3i,· · · ,hjk−1;ωk−1, ωki,hjk;ωk, ωk+1i}is a positive
path ifθjt
ωt+1 >0for allt ∈ {1,· · · , k}. LetΩP denote the subset of all statesω∈Ωsuch that there is a positive path fromωto someω0 ∈Ω0.
LetC denote the collection of all cycles of multigraphGof modelM. Decompose C asC =CC ∪ CI, whereCC denotes the subcollection of all cycles ofM which are con-
sistent, andCI denotes the subcollection of all inconsistent cycles ofM. By definition,
CC ∩ CI =∅, so the decompositionC =CC ∪ CI is uniquely defined.
LetΩI denote the subset of all statesω ∈Ωsuch that there is a cyclec∈ CI having
an edge which contains state ω; that is, an edge of the form hj;ω, ω0ior hj;ω0, ωi, for somej ∈J,ω0 ∈Ω. In other words,ΩI contains the states that lie in an edge belonging
to an inconsistent cycle. Let ΩN = ΩI ∪ΩP ∪Ω0. LetΩY be the complement of ΩN.
SubsetsΩY andΩN decompose the state space: Ω = ΩY ∪ΩN andΩY ∩ΩN =∅.