US States name game is a terrible example 

Due to the limited number of states, this game would be over almost as quickly as it started.  Why?  Consider:
Over 40% (21 out of 50) of states end in the letter 'A'.  Of the four states that start in 'A', three (Alabama, Alaska, Arizona) also end in 'A', and the fourth, Arkansas, would lead you to either South Carolina or South Dakota, both of which end in 'A', and thus you'd be stuck in an endless loop of 6 states from which you cannot escape.  This loop is almost completely unavoidable.  The example given in the article is already stuck in this loop.
Almost as many (18 states) start with a letter that no other state ends in.  So the only way these states would ever come up in gameplay is if you started with them.  Of these states, we also have the only state ending in a 'T' (Connecticut) and the only state ending in an 'H' (Utah).  Thus, you can add Hawaii, Tennessee, and Texas to the list of states that will almost never come up.
The only way you won't end up stuck in the endless 'A'-loop is if you somehow land on one of the five states (Delaware, New Hampshire, Tennessee (unlikely as seen above), Kentucky, and New Jersey) that end in a letter no other state starts with, at which point the game would be over.
There are numerous other pitfalls; consider Ohio.  You could either loop endlessly on Ohio, or go to Oklahoma, thus falling into the 'A'-loop, or you could go to Oregon.  From there you could go to Nebraska, Nevada, North Carolina, and North Dakota, all falling again into the 'A'-loop.  Or you could hit New Hampshire or New Jersey, both game enders, or New Mexico, which just takes you back where you started, or finally New York.  New York then leads to Kentucky (ending the game) or Kansas (falling via South Carolina or South Dakota back into the 'A'-loop yet again.)
Maybe Nations of the World would be a better example.  There's more of them for one thing, though still quite a few ending or starting in 'A' (and again a large number of which both start and end in 'A', so the loop still exists, but it's larger and easier to get out of).  Also, most letters of the alphabet are represented at the beginning or end of at least one country, so you'd be less likely to get stuck, and there aren't so many of them which could never appear.  (A few exist - can't think of any countries ending in 'J', 'V' or 'Z' - sorry Jordan, Vietnam, Venezuela, and Zimbabwe, probably more but I'd have to do a more thorough search.)  Anyhow, maybe I'm crazy for noticing stuff like this, but I thought it was worth mentioning.