Board games sponsored for an agenda?
[Note: This answer focuses on the history of this subject, citing some famous examples, as opposed to contemporary propagandistic games, which I have no doubt exist, although I can't comment on the anonymity issue. Generally, designers are proud to be associated with such games, but we do seem to be entering a new era of propaganda, with definite emphasis on obfuscation of sources.]

You will want to look into the predecessor of Monopoly, known as The Landlord's Game.  

There is a delicious irony here in that, while the game was designed to teach about the dangers of certain aspects of capitalism, the later Monopoly, the most commercially successful boardgame in US history, glorifies practices that Landlord's inventor, Elizabeth Magie, was cautioning against. 

From the wiki:


  The game was created to be a "practical demonstration of the present system of land grabbing with all its usual outcomes and consequences". She based the game on the economic principles of Georgism, a system proposed by Henry George, with the object of demonstrating how rents enrich property owners and impoverish tenants. She knew that some people could find it hard to understand why this happened and what might be done about it, and she thought that if Georgist ideas were put into the concrete form of a game, they might be easier to demonstrate. Magie also hoped that when played by children the game would provoke their natural suspicion of unfairness, and that they might carry this awareness into adulthood.


There were also earlier boardgames designed to teach moral lessons to children and reinforce religious beliefs, including Snakes &amp; Ladders, Mansions of Happiness, and The Game of Pope or Pagan (aka The Siege of the Stronghold of Satan by the Christian Army.)

Nuclear War, the card game, is a famous recent example.  Created during the height of the Cold War, it demonstrated the principle of Mutual Assured Destruction in that the game very often results in all players losing. 



Also of note is utilization of games by think tanks such as the Rand Corporation. John Von Neumann, the originator of Game Theory, was a consultant at Rand starting in the late 1940's.