You probably know the story of the boy who cried wolf. A boy is charged by his elders to watch their flock of sheep and to call them as soon as he sees a wolf approaching. The wolf supposedly would want to kill one of the sheep, and the boy’s cry of “wolf” would bring the elders running to fend of the wolf to protect their sheep. In the story the boy on two occasions cries wolf when there is no wolf, with the effect that the elders come running both times and being very upset at his “lying” (and the boy pleased). But when he does cry wolf for a third time, this time when there actually is a wolf, the elders do not believe him and stay away. This, of course, has the disastrous (?) effect that the wolf kills one of the sheep.
The nappy-changing game as I have written it down in my post on lying (which you may need to read before you can read this post) can also be seen as the game between the boy and his elders. There are two states of nature. Either there is a wolf or there is not. The boy, who is watching the sheep, knows which state it is and the elders, who are somewhere else, do not. The boy has four (pure) strategies: never say anything, be honest (cry wolf when there is one, be quiet when there is none), use “opposite speak”, and always cry wolf. The elders who listen to the boy’s cry also have four (pure) strategies: always come running, trust the boy, understand the boy as if he was using opposite speak, and never come running. Supposedly, the elder’s preferences are just as the parent’s are in the nappy-changing game. They would like to come running if there is a wolf, and they would like to keep doing whatever it is they are doing when there is no wolf. The boy’s preferences seem to be the same as Ernest’s in the nappy-changing game. If there is a wolf the boy would like to see his elders to come running to help, but the boy would like the elders to come running even when there is no wolf (he gets bored I suppose). The one slight difference between the two games seems to be that the assumed commonly known probability of a wolf appearing, is now less than a half (if we assume that the payoffs are still just ones and zeros). Well, what matters is that the ex-ante expected payoff of coming running is lower than the ex-ante expected payoff of staying put. We infer this from the elders’ supposed actions of staying where they are when they do not believe that there is a wolf. If the elders had found a wolf attack really disastrous and at the same time sufficiently likely, then after finding the boy not trustworthy, they would have decided to come always, that is to watch out for wolves themselves. The fact that they let the boy do the watching (and to then ignore his warnings – because they do not believe him) tells us that without further information about the likelihood of the presence of a wolf, they prefer to stay where they are (probably doing something important) and risk losing one sheep to a wolf over keeping constant watch for wolves.
In any case the same model as the nappy-changing game, but now with , now takes account of the supposed (long-run) behavior in this story. The game still has only two pure equilibria and they involve the boy either crying wolf in both states (or not doing so in both states), but now with the effect that the elders never come.