For "Game of Stopped Dice" the article states "we would keep our winnings on the first throw only if they are 5 or 6". Would it not be better to cash out even at $4 and play again using those winnings?
Only if you were playing with your last four dollars in the world. But in that case you might not want to risk it anyway.
Assuming you have more money to risk, you're better off with an expected $4.25 than a sure $4.
They may have realised they're playing a bad bet and won't let you play again.
I kid of course, but the implicit aim of these exercises is always to maximise expected value. The underlying assumption is that you can make a lot of independent bets and the results will average out thanks to the law of large numbers. Suppose you were playing this game 1000 times in parallel. Would you rather average zero profit at the end or $667?
