On Lying, I

There are many forms of lying, from so called white lies that are really just a form of politeness to deliberate attempts to misrepresent the truth to fashion policy (of some institution) in your own interest. I am here interested in something somewhere in the middle of the lying spectrum, children lying about something to avoid a slightly unpleasant duty. We all know that a child’s answer to “Have you brushed your teeth?” is not always necessarily completely truthful.

In this and the next two blog posts, using the language of game theory, I want to discuss the incentives to lie and how one could perhaps teach children not to lie.

I don’t think I need to provide empirical evidence that children lie on occasion. In case you forgot your own childhood you may want to look at Eddie Izzard’s treatment of this subject.

To fix ideas let me tell you about a game parents play with their kids when they are very little. I call it the nappy-changing game. The situation is always more or less as follows. You are on nappy duty and one of your little ones, let’s call him Ernest, is busy playing. Walking past him, you get a whiff of an interesting smell. You ask Ernest “Is your nappy full?” and Ernest invariably answers with a loud “No“.

How can we rationalize this “data”?  First I need to describe the game between the parent and the child. The game, crucially, is one of incomplete information. While I believe it is safe to assume that Ernest knows the state of his nappy, you do not. This is the whole point of the game of course. If you already knew everything there would be no point in Ernest lying. No point in asking him, in fact. And if Ernest does not know the state of the nappy himself, one could also hardly call his behavior lying. It would just be an expression of his ignorance. But suppose you are pretty sure Ernest knows the state of his nappy. So let us assume that Ernest’s nappy can only be in either of two states: full or clean.

A game, to be well defined, needs to have players, strategies, and payoffs (or utilities). The players are obvious, Ernest and you. The strategies, taking into account the information structure, are as follows. You always ask the question, so let this not be part of the game. Then Ernest can say “Yes” or “No” and can make his choice of action a function of the state of the nappy. This means he has four (pure) strategies: always say yes (regardless of the state of the nappy), be truthful (say yes when the nappy is full and say no otherwise), use “opposite speak” (say no when the nappy is full and say yes otherwise), and always say no. You listen to Ernest’s answer and have four (pure) strategies as well: always check Ernest’s nappy (regardless of Ernest’s answer), trust Ernest (check the nappy if he says yes, leave Ernest in peace if he says no), understand Ernest’s answer as opposite speak (check nappy if he says no, leave Ernest in peace if he says yes), and always leave Ernest in peace.

Let us now turn to the payoffs in this game. Your payoffs are as follows. You want to do the appropriate thing given the state of the nappy. So let’s say you receive a payoff of one if you check Ernest’s nappy when it is full and also if you do not check Ernests’s nappy when it is clean (you will find out eventually!). In the other (two) cases you receive a payoff of zero. You, thus, receive a zero payoff when you check the nappy when it is not full and also when you do not check the nappy when it is full (as I have said, you will find out eventually!). One could play with those payoffs but nothing substantially would change as long as we maintain that you prefer to check the nappy over not checking when it is needed, and you prefer not checking over checking when it is not needed. What about Ernest’s payoffs? I think it is fair to assume that he always prefers not checking, i.e. that you leave him alone. I am sure he would eventually also want you to change him, but much much later than you would want to do it, and you will eventually find out and change him. So I think it is ok to assume that Ernest prefers to be left in peace at the moment of you asking, regardless of the state of the nappy. So let us give him a payoff of one when he is left alone and a payoff of zero when you check his nappy (in either state).

There is one thing I still need to do with this model. I need to close it informationally. The easiest way to do this is to assume that ex-ante there is a commonly known (between Ernest and yourself) probability of the state of the nappy being full. Let us call it  \alpha and let us assume (recall the whiff I got) that  \alpha > \frac12 . Now the assumption of a commonly known probability of the nappy being full is a ridiculous one, it is I am sure never true. But it allows me to analyze the game more easily, and I believe that in the present case, it is not crucial. I believe that the eventual equilibrium of this game will be quite robust to slight changes in the informational structure. I leave it to the readers to think about this for themselves.

With all this I can write this game down in so-called normal form, as a 4 by 4 bi-matrix game.

 \begin{array}{c|cccc} & \mbox{always c} & \mbox{trust} & \mbox{opposite} & \mbox{never c} \\ \hline \mbox{always yes} & 0,\alpha & 0,\alpha & 1,1-\alpha & 1,1-\alpha \\ \mbox{truthful} & 0,\alpha & 1-\alpha,1 & \alpha,0 & 1,1-\alpha \\ \mbox{opposite} & 0,\alpha & \alpha,0 & 1-\alpha,1 & 1,1-\alpha \\ \mbox{always no} & 0,\alpha & 1,1-\alpha & 0,\alpha & 1,1-\alpha \\ \end{array}

Ernest chooses the row, you choose the column, and the numbers in the matrix are the ex-ante expected payoffs that arise in the various strategy combinations. In each cell of the matrix Ernest’s payoff is the first entry, your’s the second. Once all this is in place it is easy to identify equilibria of this game. Note that as  \alpha > \frac12 your strategy to never check (never c) is strictly dominated by your strategy to always check (always c). Your ideal point would be that Ernest is truthful and you can trust him, but Ernest in that case has an optimal deviation to always say no. In that case you better do not trust him and instead always check his nappy. This is indeed the only pure strategy equilibrium of this game. Well there is also one in which Ernest always says yes and you always check him, but this is really the same. Note that language has no intrinsic meaning in this game. The meaning of language in this game could only potentially arise in equilibrium.

So what did we learn from this so far? Clearly Ernest’s behavior (of lying) is not irrational (it is a best reply to your behavior). But it has the, from his – and also your – point of view, unfortunate side effect that you do not trust him, so his lying does not fool you. This game is, by the way, an example of a sender-receiver game. See Joel Sobel’s paper on Signaling Games.  In fact it is an example of a special class of sender-receiver games: so called cheap-talk games. See Joel Sobel’s paper on Giving and Receiving Advice for further reading. In the language of these games the lying equilibrium between Ernest and you is called a pooling equilibrium. It is called so, because the two kinds of Ernest, the one with a full and the one with a clean nappy, both send the same “message”. The two Ernests play this game in such a way that you cannot differentiate between them. Hence the term pooling.

In the next post I will take this game up again and consider what can happen if you play this game over and over again, as most parents do with their kids.

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