A note about notation

Leibniz notation in his own hand. Picture from Stephen Wolfram.

Peter Fielitz and Guenter Borchardt kindly sent me a review of my draft paper and make excellent contributions (thanks also to Tom Brown for a thorough read and commentary). Among other things, they suggest the more pedagogical notation:

$$
P : I_{D} \rightleftarrows I_{S}
$$

which is very good -- and makes more apparent a connection with chemical reactions. The double arrow notation has a unicode representation (however the subscripts D and S are not commonly available unicode).

P : I(D) ⇄ I(S)

The information notation I(D) or $I_{D}$ (for information in process variable D) would get a bit more unwieldy were we to use variables like I(NGDP) or $I_{NGDP}$. The unicode representation was one reason for using the single arrow notation

P : D → S

despite the notation being overloaded (category theory, functions, maps, fiber bundles). The other thing I did want to make clear was the relationship between the variables in the case of non-ideal information transfer where I(D) ≥ I(S). However, that relationship is evident in the D preceding the S.

My tentative choice is to change to

P : D ⇄ S

potentially keeping the single arrow notation for non-ideal information transfer. For example, the price level model is:

P : NGDP ⇄ M0

One thing this notation helps avoid is concerns about the direction of information flow (source/destination designation) in the case of information equilibrium. Commenter M previously mentioned the lack of symmetry in the single arrow notation as being unclear.

It also helps avoid the overloading problem of the single arrow notation -- so that all we now have to worry about is the overloaded term "information". I thought this post could serve as a reference on information transfer-/information equilibrium- specific notation as well.