Economists in a variety of fields are interested in games of incomplete information, including industrial organization, public economicss, mechanism design, and contract theory.1 In the usual framework, originally in Harsanyi [1967, 1968a,b], there exists a set of possible states of the world, one of which is the true state of the world. A state of the world is a description of all aspects of the game, including payoffs to the players, and, potentially, the actions available to the players. Players have private information regarding which state is the true state, leading to differing posterior probabilities, or beliefs, about which state is real.
When building games of incomplete information, an almost universal assumption is that players have a common prior, or distribution, over the state space. Differences in players’ beliefs are then driven entirely by differences in the information to which they have access. This assumption, while apparently innocuous, leads to a host of No Trade Theorems.2 If players have identical underlying beliefs, and equivalent utility formulations, and differ only in their private information, then players will always choose not to trade. Given that players do appear to engage in speculative trade, it may be useful to know what exactly is being required by the common prior assumption.
Determining whether a given set of posterior beliefs could have been generated by a common prior is the Harsanyi Consistency Problem. Morris [1994] provides a solution to the Harsanyi Consistency Problem based on feasible trades. Samet [1998b] proves common prior existence using iterated expectations of random variables, and Samet [1998a] finds the common prior using appropriate seperation theorems on convex sets. The solution of Rodrigues-Neto [2009] constructs a graph called the meet-join diagram. For each cycle of this graph there is a cycle equation, and a common prior exists if and only if all cycle equations hold.
The cycles approach of Rodrigues-Neto [2009] is refined by Hellwig [2013] and Rodrigues-Neto [2014] for particular restrictions on the geometry of the information structure. Rodrigues-Neto [2014] assumes that players’ partitions are connected with respect to a linear order over the state space, and shows that a common prior exists if
1See Mas-Colell et al. [1995] for a textbook treatment.
2Classical references include Rubinstein [1975], Kreps [1977], Milgrom and Stokey [1982], and Tirole
and only if cycles of length at most two are consistent. Hellwig [2013] assumes that players’ information is independent, in the sense that the information signal received by player j does not allow her to infer which signals might have gone to player j0. Under this asusmption, Hellwig [2013] shows that a common prior exists if and only if cycles of length at most four are consistent.
This work refines the cycle approach of Rodrigues-Neto [2009] without placing ad- ditional restrictions on the geometry of the information structure. For any information structure, a common prior exists if and only if all cycles of length at most 2d+ 1are consistent, where dis the diameter of the meet-join diagram. This is further refined to2dfor two-player models. The result of Hellwig [2013] for two-player models is a special case of our result.
Section 6.4 considers comparisons between models which are ordered by the amount of knowledge held by the players. Rodrigues-Neto [2012] gives a result which restricts the number of cycles which need to consistent in order to have a common prior, where this number is sensitive to the relative informativeness of the players’ information structures. In particular, in models with less informed players, more cycles need to be checked for consistency to show a common prior exists. By contrast, in the results of this chapter, when players are less informed we need only check shorter cycles relative to the case when players are more informed.
6.2
Model Setup
Consider a finite non-empty state spaceΩwith typical stateω ∈ Ω. LetJ be a finite, non-empty set of players. Playerj’s knowledge is characterized by her partitionΠj of spaceΩ. When the true state of the world is someω∗, then playerjknows the true state is some ω ∈ πj(ω∗), where πj(ω∗) ∈ Πj is the unique element of playerj’s partition that contains stateω∗. That is, whenω∗ is the true state, player j rules out all states
ω /∈πj(ω∗).
A posterior for player j is a vectorθj = (θj 1, . . . , θ
j
|Ω|), where, for eachω ∈ Ωand
j ∈ J, 0 < θωj ≤ 1denotes player j’s conditional belief that the true state is ω given that the true state belongs to πj(ω).3 At each ω∗ ∈ Ω
, every player j ∈ J assigns probability 1 to eventπj(ω∗), the element of her partition that contains the state ω∗;
3Chapter 5 considers the common prior problem when posteriorsθj
formally,P ω∈πj(ω∗)θjω = 1. A ModelM is a tupleM = Ω, J,{Πj, θj} j∈J .
Consider a probability measureµj =µj1, . . . , µj|Ω|over the spaceΩwith full sup- port. Let µj(B) = P
ω∈Bµ j
ω, for anyB ⊆ Ω. The measureµj is a prior for playerj if
θωj =µjω/(µj(πj(ω))), for allω∈Ω. A ModelMadmits a common prior if there is some probability measureµwhich is a prior for all players inJ.
A directed multigraph G = (V, E)is a tuple consisting of a set of nodesV, and a set of (directed) edgesE, where each edgee∈E has an initial node and a terminating node. In this work, all graphs will be directed multigraphs. An edgeegoes from node
v1 to node v2 if v1 is the initial node of e, and v2 is the terminating node of e. It is
possible for an edge to go fromv0 tov0. It is possible for more than one edge to go from
v1 tov2. Edgee2is consecutive to edgee1 if the terminating node ofe1is the initial node
ofe2. A pathpis a list of consecutive edgesp= (e1, e2. . . , en). A pathc= (e1, e2. . . , en)
is a cycle if the terminating node ofenis the initial node ofe1. A multigraphG= (V, E)
is connected if for allv1, v2 ∈V, there exists a path fromv1tov2.
The meet-join diagram of a ModelM is a directed multigraphG, defined as follows: (i) the nodes of Gare the states; i.e., elements ofΩ; (ii) a j-edge is any ordered triple hj;ω1, ω2i, withω1,ω2 ∈Ω, composed of a playerj ∈J and an ordered pair of statesω1
andω2belonging to the same element ofj’s partition,πj(ω1) =πj(ω2). The set of edges
ofGis composed of allj-edges, for allj ∈J. A ModelM is connected if the associated meet-join diagram is connected.
IfGis the meet-join diagram of some ModelM, define the cycle equation of a cycle
c= (hj1;ω1, ω2i,hj2;ω2, ω3i, . . . ,hjn;ωn, ω1i)as: θj1 ω2 ·θ j2 ω3· · ·θ jn−2 ωn−1 ·θ jn−1 ωn ·θ jn ω1 =θ j1 ω1 ·θ j2 ω2 ·θ j3 ω3· · ·θ jn−1 ωn−1·θ jn ωn. (6.1)
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
, posteriors are consistent with a common prior if and only if all cycle equations hold.
Hellwig [2013] shows that given specific assumptions on the information structure, a common prior exists if and only if all cycle equations for cycles of length four or less are satisfied.
In Hellwig [2013], the assumptions on partitions are designed to model the case where players have private types. Players are informed about their own type, but not about the type of their opponent. Specificallyπj(ω)∩πj0(ω0)6=∅for all playersj, j0 ∈J
and statesω, ω0 ∈Ω. This is almost equivalent to the requirement that the state space is the Cartesian product of the possible types of each playerΩ = Ω1 × · · · ×Ω|J|, where
Ωj is the set of types for playerj.4
Definition 6.1. A ModelM =Ω, J,{Πj, θj} j∈J is a Hellwig model ifπj(ω)∩πj0(ω0)6=∅ , for allω, ω0 ∈Ω,j, j0 ∈J.