Connectedness and disconnectedness are important structural properties of graphs, and are of interest in studying the common prior problem. This section considers connectedness as it applies to the common prior assumption, with particular refer- ence to the differences between full support, and non-full support common priors for connected and disconnected models.
LetGbe a meet-join diagram for some modelM =Ω, J,{Πj, θj} j∈J
. The graph
Gis connected if for anyω, ω0 ∈Ω, there exists a path (not necessarily positive) fromω
toω0.4 GraphGis disconnected if it is not connected. A graphGe= (V ,e E)e , with nodes e
V and edgesEe, is a subgraph ofG = (V, E)ifVe ⊂V, and for any edgee ∈E both of
4A distinction is usually made between strongly connected, which requires directed paths, and
weakly connected, which ignores path direction. As the opposite edge of any edge inGis also inGfor meet-join diagrams, these concepts are identical, and any distinction will be ignored.
whose endpoints are inVe, thene∈Ee. That is, a subgraph takes a subset of the nodes,
and all edges connecting those nodes.
A graph Gb = (V ,b E)b is a connected component of G = (V, E) if Gb is connected, b
V ⊂ V,Eb = E|
b
V and for allω ∈ V \Vb andωb ∈ Vb, there is no path fromωb toω. That
is, Gb is a connected component if it is a maximal connected subgraph. In the case
of meet-join diagrams, as the set of nodesΩis finite, there will be a finite number of connected components.
Definition 5.1. LetM = Ω, J,{Πj, θj}j∈J, andGthe associated meet-join diagram. Let
G1, . . . , Gn be the connected components of G. For each k = 1, . . . , n, define the maximal
connected submodel Mk= Ωk, J, Πj,k, θj,k j∈J
whereΩkis the set of nodes ofGk, the partitionΠj,k is the restriction of partitionΠj toΩk, and
each vectorθj,k is the restriction ofθj toΩ k.
For each πj ∈ Πj, the partition element πj lies entirely in exactly one of the Ω k.
That is, the maximal connected submodels do not break any partition elements. This is why the restricted posteriorsθj,kcan be just the restriction ofθj, and not require any
renormalization to ensure that the posteriors sum to 1 inside eachπj.
When working with full support posteriors and common priors on disconnected models, guaranteeing the existence of a common prior requires that every maximal connected submodel can support a common prior. The common prior for the full model is then any strictly positive convex combination of the submodel common priors. This is not the case once common priors without full support are permitted. If allowing common priors without full support, it suffices that only one of the maximal connected submodels admits a common prior to guarantee a common prior for the full model. Proposition 5.3. LetM =
Ω, J,{Πj, θj}j∈J. The modelM admits a common prior if and only if at least one maximal connected submodel ofM admits a common prior.
Proof. SupposeM admits a common prior µ. Let Mk =
Ωk, J, Πj,k, θj,k j∈J be the maximal connected submodels ofM. Asµ(Ω) = 1, there is some submodel Mk such
that µ(Ωk) > 0. Suppose, without loss of generality, that µ(Ω1) > 0. Construct a
measure over submodelM1, denotedbµ, by
b
µω =
µω
Measureµbis a common prior overM1if, wheneverµ(πb j,1)>0, thenθj,1 ω =µbω/µ(πb j,1(ω)). Asπj,1(ω) = πj(ω)for allω∈Ω 1, b µω b µ(πj,1(ω)) = µω µ(πj(ω)) =θ j ω =θ j,1 ω
Sobµis a common prior overM1, as required.
Conversely, supposeMkadmits a common prior for somek = 1, . . . , n, and without
loss of generality, letk = 1. Letµbbe a common prior overM1. Define a measureµover
Ωby µω = ( b µω ; forω∈Ω1 0 ; otherwise
Forω /∈Ω1, asπj(ω)∩Ω1 =∅, thereforeµ(πj(ω)) = 0, so to determine ifµis a common
prior there is nothing to check. Forω ∈Ω1there are two cases. Ifµ(πj(ω)) = 0we are
done. Ifµ(πj(ω))>0, asπj(ω)⊂Ω 1, andπj(ω) =πj,1(ω), then µω µ(πj(ω)) = b µω b µ(πj,1(ω)) =θ j,1 ω =θ j ω
Soµis a common prior overM, as required.
Definition 5.2. A model M = Ω, J,{Πj, θj} j∈J
is connected (resp. disconnected) if the associated meet-join diagramGis connected (resp. disconnected).
This work considers two extensions to the usual framework: allowing priors with- out full support and allowing posteriors without full support. The following corollaries consider the case where posteriors are required to have full support, but priors poten- tially may not. Corollary 5.2 shows that if a model has strictly positive priors and is connected then a common prior exists if and only if a common prior with full support exists, if and only if, all cycle equations are satisfied. Corollary 5.3 shows that a model with positive posteriors which admits a common prior is connected if and only if it admits a unique common prior with full support. Together these corollaries show that the extension of allowing priors without full support extends the framework in a non-trivial manner exactly when the modelM is disconnected.
Corollary 5.2. Let M =
Ω, J,{Πj, θj}j∈J. LetM be connected, andθjω > 0for all j ∈
J, ω∈Ω. The following are equivalent:
ii) ModelM admits a common prior with full support, and does not admit a common prior without full support.
iii) ModelM admits a unique common prior.
iv) All cycle equations are satisfied
Proof. [(i) =⇒ (ii)]: Letµbe a common prior. Suppose, for the purposes of contradic- tion, thatµω0 = 0for someω0 ∈Ω. Letω
0 ∈Ω
. AsM is connected, there exists a pathp
fromω0toω. Asθj
ω >0for allj ∈J, ω∈Ω, the pathpis a positive path. So, by Lemma
5.2,µω0 = 0. This is true for allω0 ∈Ω, soµ(Ω) = 0, which is a contradiction. Therefore,
any common prior must have full support.
[(ii) =⇒ (iii)]: Letµ, µbbe common priors with full support. Suppose, by way of contradiction, thatµ 6= µb, and, without loss of generality, that µω1 > bµω1 for some
ω1 ∈Ω. Asµandbµare common priors overM, for anyj ∈J andω2 ∈π
j(ω 1), µω1 µω2 = µbω1 b µω2 = θ j ω1 θjω2
Asµω1 >µbω1 this givesµω2 >µbω2. Proceeding in this manner,µω >µbωfor anyωwith
a positive path toω1. As M is connected, allω have a positive path toω1. Therefore
µω >µbωfor allω∈Ω. Asµ, µbare measures,
1 =X ω∈Ω µω > X ω∈Ω b µω = 1
which is a contradiction. Therefore M admits a unique common prior with full sup- port. AsM does not admit a common prior without full support,M admits a unique common prior.
[(iii) =⇒ (iv)]: Letµbe a common prior with full support. AsM admits a common prior with full support, by Proposition 5.1, ΩY = Ω, soΩI = ∅. That is, there are no
inconsistent cycles.
[(iv) =⇒ (i)]: This is a special case of Proposition 5.2.
Corollary 5.3. LetM =Ω, J,{Πj, θj}j∈Jwithθωj > 0for allj ∈ J, ω ∈ Ωsuch thatM
admits a common prior. ModelM is connected if and only ifM admits a unique common prior with full support.
Proof. Suppose ModelM is connected. Asθj
ω >0for allj ∈J, ω∈Ω, andM admits a
Conversely, suppose ModelM is disconnected. We want to showM does not admit a unique common prior with full support. IfMdoes not admit any common prior with full support, we are done. Therefore, letbµbe a common prior with full support.
LetM1, . . . , Mnbe the maximal connected components ofM. AsM is disconnected,
n >1. LetΩkbe the state space of ModelMk, and letk(ω)be the index such thatω∈Ωk.
Asµbis a common prior with full support, andM is disconnected, any distributionµe
overM such thatµeω0/eµω00 =µbω0/bµω00 for allk(ω
0) =k(ω00), will also be a common prior
for ModelM. Letλ1, . . . , λn∈(0,∞)be weights, not all 1, and let
e
µω =
λk(ω)µbω P
ω∈Ωλk(ω)bµω
As all states in the same connected componentΩkhave been re-weighted by the same
proportion, the relative weight between them remains unchanged. Therefore µe is
common prior with full support of ModelM. Aseµ6=µb, thus ModelM admits multiple
common priors with full support.
Example 5.2. Let Ω = {1,2,3,4,5,6,7}, J = {A, B}, ΠA = {{1,2},{3,4},{5,6},{7}}, andΠB = {{1,3},{2,4},{5,7},{6}}. A visualization of these partitions is given in Figure
5.4 with playerAin blue and dashed, and playerB in red.
1
2
3
4
5
6
7
Figure 5.4:Version for Example 5.2
Consider model M = (Ω, J,{Πj, θj}
j∈J) whereΩ,J,ΠA, andΠB are as above, and the
values ofθA, θBare yet to be specified. This model admits two maximal connected submodels,
the restriction to {1,2,3,4} and the restriction to {5,6,7}. Call these models M1 and M2
respectively. By Proposition 5.3, modelM admits a common prior if and only if eitherM1 or
M2 admit a common prior. Consider submodel M2. Ifθωj,2 > 0for allj ∈ J andω ∈ Ω2, as
there are no cycles inM2, then by Corollary 5.2,M2 admits a common prior with full support.
In conclusion, regardless of the posterior values, submodelM2 will always admit a common
prior. Therefore by Proposition 5.3 the modelM will always admit a common prior.
5.5
Conclusion
This chapter revisits the classic Harsanyi Consistency Problem (Harsanyi [1967, 1968a,b]). It extends the cycles approach to the case where the supports of priors and posteriors can be any non-empty subset of the state space, and characterizes models that admit a common prior with and without full support. Checking all cycle equations is still a sufficient condition for the existence of a common prior, but it is no longer necessary.
Fiorini and Rodrigues-Neto [2017] extends the cycles approach to non-partitional models, studying consistency and self-consistency. Future work on this problem could focus on the supports of priors and posteriors in non-partitional models. We conjec- ture that the graph techniques and results used here will lead to analogous/similar propositions in the more general framework of non-partitional models.
Atomic Cycles and Common Prior
Existence
Harsanyi’s Consistency Problem (Harsanyi [1967, 1968a,b]) asks whether a given set of posteriors is consistent with a common prior. Rodrigues-Neto [2009] pro- poses a solution to this problem using a multigraph built from the players’ infor- mation partitions called a meet-join diagram. A common prior exists if and only if for each cycle of the meet-join diagram an equation called the cycle equation is consistent. This work shows that a common prior exists if and only if the cycle equations for atomiccycles are consistent. It is necessary and sufficient to check cycles of length2d+ 1or less, wheredis the diameter of the meet-join diagram, or length 2dor less in the two-player case. In the two-player case, this general- izes the result of Hellwig [2013] to a much broader class of partitional information structures.