Robust asymmetric information? Bergeman and Morris (2014) explain how the concept of Bayesian Correlated equilibrium contains all the robust predictions in games of incomplete information. In particular, their question is what are the possible Bayesian Nash equilibria of a game where we know each player has at least some information, but can potentially have more information. 

Does anyone know of a useful reference to think of refinements of the set of BCE by imposing further assumptions? For example, what if we were willing to assume that one of the players has at least as much information as every other player? What kind of refinements can we expect to maintain the nice linear properties of BCE?

Any thoughts will be greatly appreciated.
It seems to me based on the 2019 JEL review of the same authors that they had published more-or-less what you ask about in a 2015 paper about monopolies and price discrimination (as an application of their BCE theory); see section 5 (and 5.1 in particular) in the 2019 paper. Here the relevant information is the monopolist's knowledge of the consumers valuation. Here are some snippets:


  The set of
  BCE is precisely the set of outcomes that can
  arise with additional information for a given
  basic game and prior information structure. If there are properties that hold for all
  BCE, we have identified predictions that are
  robust to the exact information structure.
  Identifying the best or worst outcome that
  can arise under some information structure
  according to some objective function as criterion is the same as solving an information
  design problem where the designer is maximizing or minimizing that criterion. In this
  section, we will review two such economic
  applications of information design. We will
  highlight the implications of this approach in
  the context of third-degree price discrimination. 
  (Bergemann, Brooks, and Morris 2015) [...]
  
  A classic issue in the economic analysis
  of monopoly is the impact of discriminatory
  pricing on consumer and producer surplus.
  A monopolist engages in third-degree price
  discrimination if he uses additional information about consumer characteristics to offer
  different prices to different segments of the
  aggregate market. Bergemann, Brooks, and
  Morris (2015) characterize what could happen to consumer and producer surplus for all
  possible segmentations of the market. 
  
  One can provide some elementary bounds
  on consumer and producer surplus in any
  market segmentation. First, consumer surplus must be nonnegative as a consequence
  of the participation constraint: a consumer
  will not buy the good at a price above his valuation. Second, the producer must get at least
  the surplus that he could get if there was no
  segmentation and he had no additional information beyond the prior distribution. In this case, an optimal policy is always to offer the
  product with probability one at a given price
  to all buyers. We therefore refer to it as uniform monopoly price, and correspondingly,
  uniform monopoly profit. Third, the sum
  of consumer and producer surplus cannot
  exceed the total social value that is generated by the good, which is willingness-to-pay
  minus unit cost of production. The shaded
  right-angled triangle in figure 5 illustrates
  these three bounds.
  
  
  
  The main result in Bergemann, Brooks,
  and Morris (2015) is that every welfare outcome satisfying these constraints is attainable by some market segmentation. This is
  the entire shaded triangle in figure 5. If the
  monopolist has no information beyond the
  prior distribution of valuations, there will
  be no segmentation. The producer charges
  the optimal monopoly price and gets the
  associated monopoly profit, and consumers
  receive a positive surplus; this is marked by
  point A in figure 5. If the monopolist has
  complete information, then he can charge
  each buyer his true valuation, i.e., engage in
  perfect or first-degree price discrimination;
  this is marked by point B. The point marked
  C is where consumer surplus is maximized;
  the outcome is efficient and the consumer
  gets all the surplus gains over the uniform
  monopoly profit. At the point marked D,
  social surplus is minimized by holding producer surplus down to uniform monopoly
  profits and holding consumer surplus down
  to zero.
  The main result states that we can make
  only very weak predictions about producer
  and consumer surplus. It can be understood
  as the outcome of a set of metaphorical
  information design problems. If an information designer wanted to maximize consumer surplus, she would choose point C.
  If she wanted minimize consumer surplus,
  or producer surplus, or any weighted combination of the two, she could choose point
  D. Any other point on the boundary of the
  triangle is the solution to some maximization
  problem of the information designer defined
  by some preferences over producer and consumer surplus.
  
  The information design problem has a
  very clear literal interpretation in the case
  where the monopolist knows the consumer’s
  valuation. She can then achieve perfect price
  discrimination at point B. However, giving a
  literal information design interpretation of
  point C is more subtle. We would need to
  identify an information designer who knew
  consumers’ valuations and committed to
  give partial information to the monopolist
  in order to maximize the sum of consumers’ welfare. Importantly, even though the
  disclosure rule is optimal for consumers as
  a group, individual consumers would not
  have an incentive to truthfully report their
  valuations to the information designer, given
  the designer’s disclosure rule, since they
  would want to report themselves to have low
  values.
  
  [...] One can show
  that any point where the monopolist is
  held down to his uniform monopoly profits with no information beyond the prior
  distribution—including outcomes A, C,
  and D in figure 5—can be achieved with
  the same segmentation. In this segmentation, consumer surplus varies because the
  monopolist is indifferent between charging
  different prices. [They then give a numerical example, which I omit here.]
  
  [...] Roesler and Szentes (2017) consider a
  related information design problem in which
  a single buyer can design her own information about her value before she is facing a
  monopolist seller. While the analysis of the
  third-degree price discrimination proceeds
  as a one-player application, the arguments
  extend to many-player settings. Bergemann,
  Brooks, and Morris (2017a) pursue the question of how private information may impact
  the pricing behavior in a many-buyer environment. There, we derive results about
  equilibrium behavior in the first-price auction that hold across all common-prior information structures. The results that we obtain
  can be used for a variety of applications, e.g.,
  to partially identify the value distribution in
  settings where the information structure is
  unknown and to make informationally robust
  comparisons of mechanisms.