The ability for an app to post to users' News Feed is not meant for app to user communication such as daily updates to users, but instead for users to share interesting and relevant content with friends. We have built specific products which allow apps to communicate directly to their user base in an appropriate manner. Below are examples of "app to user" communication tools.
Unlike user to user requests, app-generated requests can be initiated by the app via the Graph API which updates the bookmark count. This is a great way to encourage re-engagement and prompt users to take an action such as moving a chess piece, tending to their farm, or viewing daily updates such as sports scores, horoscopes, and news articles.

Note: This feature is only available for canvas apps or mobile apps.
Using your app Page to communicate with your users is a great way to drive re-engagement with your app and keep users informed of updates to the app such as a new level or items for purchase. Similar to a regular Facebook Page, admins can comment on users’ posts, upload photos, and directly engage with users.
You can create a page for your app from the "Advanced" tab of your app settings.

The Page for an App has a "Go To App" button on the top right and other App-related links on a context menu also accessed from the top right.

See also the Like Box and Like Plugin to help grow the number of users that like your new page.
Including Open Graph tags on your Web page makes your page equivalent to a Facebook Page. This means that when a user clicks a Like button on your page, a connection is made between your page and the user. Your page will appear in the "Likes and Interests" section of the user's profile, and you have the ability to publish updates to the user which will be displayed in the News Feed.

Email is the best communication method for app-to-user communication. You can request the user’s email via the email extended permission or by having the user directly provide their email address to you in a field that you provide.