Bundle_Gerbe comments on Game Theory As A Dark Art - Less Wrong
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The specific problem with calling the last game a "prisoner's dilemma" is that someone learning about game theory from this article may well remember from it, "there is a cool way to coordinate on the prisoner's dilemma using coin flips based on correlated equilibria" then be seriously confused at some later point.
Of course, by changing the payoff matrix, Nick also changed the game, so after him putting in some more of his stakes, it wasn't Golden Balls / PD anymore but a game which had the structure Nick favoured. What is to be learned from this article is how to design games to your own profit - whether you are watching from the outside or playing from the inside.
Apparently I didn't quite understand what you wanted to tell me - I'm sorry! Yes, as an introduction to game theory, this indeed is a problem. Example #6 is a bit out of place for that, as game theory here didn't work in practice in the sense that it didn't make accurate predictions.