There is a German saying about lying: “Wer einmal lügt, dem glaubt man nicht, und wenn er auch die Wahrheit spricht.” The closest corresponding idiom in English is probably this: “A liar is not believed even when he speaks the truth.” This is good enough for the moment but there is a little bit more information in the German saying than in the English one and this little bit more will become interesting in my discussion further below.
Both statements are sufficient for a first quick side discussion I want to provide here as they both contain “even when he speaks the truth.” As a child, I have been made aware of this idiom on a few occasions. While I recall that I always understood it to mean that I should not lie, I also recall that the statement in itself puzzled me. I thought that if this liar speaks the truth then of course I will believe him. It took me some time to realize that there is a specific information structure assumed in this statement that is not made explicit. It should really say that “a liar is not believed even when he speaks the truth, and the truth is not known by the listener”. This addition was probably omitted for two reasons, one it makes the statement shorter, and two it should be obvious that this is what is meant. In other words, any statement made by someone generally known to be a liar will not be taken at face value. It will be ignored. This means that after a liar makes a statement we know as much as before, no more and no less. Note that this is true in the nappy-changing game between my child, Oscar, and I that I described in my previous post. Here is a brief summary of this game. I ask Oscar if his nappy is full (after some initial but uncertain evidence pointing slightly in this direction). Oscar can make his answer depend on the true state of his nappy (full or clean) and this answer can either be “yes” or “no”. I then listen to his answer and make my decision whether to check the state of his nappy or not a function of what answer he gave. Let me reproduce the normal form depiction of this game again here (with ).
We found that the only equilibrium of this game is that Oscar lies (in that he either always says yes or always says no – regardless of the state of his nappy) and that I do not believe him and always check his nappy. Now note that this equilibrium is bad for both Oscar and me. Oscar is faced with the reality that I ignore his answer and check him no matter what he says, which is very annoying to him. I am faced with the reality that I cannot trust Oscar and have to check his nappy even in those cases when it is clean. Thus we have that this little liar (a bit too strong a term really for my little son) is not believed even when he speaks the truth, that is, even when his nappy is not full.
Looking at the matrix we can see that we here have a situation that is somewhat reminiscent of the prisoners’ dilemma. There is a potential outcome in this game that is a Pareto improvement, that means it is better for both of us Oscar and me, than the equilibrium outcome. If Oscar was truthful and I could trust him we would both be better off. I would not have to check his nappy when it is clean and Oscar would now only be bothered when the nappy is clean. In the matrix this can be seen as the payoffs in this case are instead of .
Isn’t there some way of getting these payoffs and making Oscar honest and me trusting? Well, there is hope. The nappy changing game is one that Oscar and I play many times. It is really what the literature calls a repeated game. True, the is not always the same – sometimes I have stronger suspicions that the nappy is full than at other times – but this is not so important for the discussion. The big question in this repeated game is the question of how forward looking the two players are. Well, as a grown-up I am very forward looking. This means my discount factor, with which I discount the future relative to the present, is very close to one. I value payoffs in the future almost as much as in the present. For Oscar this is unclear. In fact I believe that the older he gets the higher his discount factor becomes. As a very young child he did not seem to care one bit about what happens even in one hour. The now was everything. More recently (he is already six years old now and we have not played the nappy game in a very long time – but we do play similar games) he can be easily incentivized to do something now with a promise or a threat about tomorrow or next week or even xmas when it is quite far away.
You will see that the discount factor plays an important role in the possibility of achieving higher payoffs in the nappy changing game. Let us see what we can do in the repeated game. Note first that in this game I will always learn the true state of the nappy eventually. So I can always check later at some point whether Oscar was truthful or not. This is very important of course. Lying is much easier when there is no chance of being detected. This would be an interesting topic for another blog post.
Recall that I said that there was more information in the German saying than in the English one. But clearly both statements are to be understood as a threat. If you lie you will be called a liar and liars won’t be believed. This is supposedly a bad thing also for the liar, as it is in my nappy-changing game. The German saying is more explicit about what induces people to call you a liar. In fact, according to the German saying, you only have to lie once to be called a liar. Literally translated it says “He who has lied once will not be believed even when he speaks the truth.” The German saying prescribes a strategy in the repeated game that the literature calls the “grim trigger” strategy. It is essentially as follows. I trust Oscar as long as he was always truthful in the past. If he was not truthful even once (and no matter how long ago this was) I will never believe him anymore and I will always check his nappies from then on. Oscar’s strategy is to be truthful at all times unless I have, at one point, not been trusting.
Under what circumstances is this strategy a Nash equilibrium in the repeated game? If Oscar is always truthful then I am always trusting and Oscar gets a payoff of every time. If he lies at one point by saying no even though the nappy is full he gets a payoff of one once and then zero ever after. With (the usual) exponential discounting and with denoting the discount factor, this means that Oscar prefers to be truthful if or, equivalently, if . Recall that . So if Oscar is sufficiently forward looking, the grim trigger strategy described in the German saying would indeed incentivize Oscar to be truthful at all times.
I think that there is one lesson we can take from this discussion. If we want to teach our kids to be truthful we may have to wait until they are old enough to be sufficiently forward looking. But on the issue whether the grim trigger strategy really works, and whether this is really a feasible way to teach honesty, I have more to say in my next blog post.