[sdenton4]

I think a closer-to-real example is the problem of playing a collection of blackjack tables. All the games are the same, but sometimes the state of the decks (ie, which cards have been discarded) will lead to better odds of winning. If you know the state of the decks at each table, you can always choose to play the table with the best odds of winning. This type of strategy has been used to win piles of money in Vegas, FWIW, though it leads to ejection if you're caught.

This is similar to the 'ratchet' examples in the wikipedia page - you play the game with the best odds, and use one game to 'cool off' until you're in the right state to win the second game again. The games in the wikipedia article are kinda unsatisfying, though - there's too much dependence on player state.

[hgsgm]

Market timing in the stock market (if you have a little insider info), and card counting in lotteries, are historical examples.