In my previous post I argued that a person can be kept truthful (in a repeated setting) by the threat of never believing this person again once this person has been caught lying even once. This is a strategy that many proverbs suggest.
In this post I want to ask the question whether this threat is a credible one. I will have two answers to this question. Yes and no.
Haha. Well, it depends on what you call a “credible” threat. The most commonly known notion of a non-credible threat is due to Reinhard Selten. Paraphrasing his work somewhat, a threat is not credible if, once asked to actually go through with it, people do not find it in their own interest to do so. Reinhard Selten then defined a Nash equilibrium to be free of non-credible threats if it is a subgame perfect equilibrium, that is a Nash equilibrium that is also a Nash equilibrium in every subgame. What does that mean? Well this means the strategy profile that both players in the game play is such that no matter what has happened in the game so far the strategy profile from any point onwards is still so that no player would want to deviate from it if they believe the other player to follow their part of this strategy profile.
Let us come back to the nappy-changing game as described in my first post in this series. Here is a brief summary of this game. You are a parent and ask your toddler son Ernest if his nappy is full (after some initial but uncertain evidence pointing slightly in this direction). Ernest can make his answer depend on the true state of his nappy (full or clean) and this answer can either be “yes” or “no”. You then listen to his answer and make your decision whether to check the state of his nappy or not as a function of what answer he gave. Let me reproduce the normal form depiction of this game again here:
The one-shot game only has (Bayes Nash) equilibria in which you do not trust Ernest’s statement and always check his nappy. This is bad for both of you. It would be better for both of you if Ernest was truthful and you believed him. This, I then argued in the previous post in this series can be made an equilibrium outcome if the two players play the grim trigger strategy as suggested by all these proverbs. Note that I keep assuming that you (as a player in this game) can always find out about the true state of the nappy sooner or later. [One could here talk about the more recent literature on repeated games with imperfect monitoring, but I will refrain from doing so at this point – the reader may want to consult the 2006 book by Mailath and Samuelson on Repeated Games and Reputations]
The grim trigger strategy is as follows. You believe Ernest as long as he was always truthful in the past (and as long as you were always trusting in the past). Once you catch Ernest lying (or once you have not trusted Ernest) you never again believe him and always check his nappy from then on. Ernest is truthful as long as you have always trusted him (and as long as he has always been truthful). Once he catches you not trusting him (or once Ernest himself was untruthful) Ernest will always do so from that point on. The statements in brackets probably seem strange to someone not used to game theory, but they are needed for the statements below to be fully correct.
This grim trigger strategy, then, is a subgame perfect equilibrium, provided Ernest’s discount factor . Why? I have already argued that, in this case, Ernest would not want to deviate from the equilibrium path of being truthful, because lying would lead to you never trusting him again and this is sufficiently bad for him if his discount factor is sufficiently high. You certainly have no incentive to deviate from this equilibrium path, because you get the best payoff you can possibly get in this game. But subgame perfection also requires that the threat, when we are asked to carry it out, is in itself also equilibrium play. So suppose Ernest did lie at some point and you (and Ernest) are now supposed to carry out the threat. What do you do? Well, you now play the equilibrium of the one-shot game forever and ever. But this is of course an equilibrium of the repeated game, so the grim trigger strategy described here does indeed constitute a subgame perfect equilibrium.
So according to Reinhard Selten’s definition the punishment of never ever believing Ernest again is a credible threat.
But others have argued that Reinhard Selten’s notion of a credible threat is only a minimal requirement and further requirements may be needed in some cases. I do not know what Reinhard Selten thought of this, but I guess he would have agreed. So what is the issue?
You and Ernest, when you look at this game, should realize that you would both like to play the truthful and trusting equilibrium path. To incentivize you to do so, especially Ernest in this case, you need to use this threat of you never again believing him if you catch Ernest lying. But suppose we are in a situation in which you have to carry out this threat. Then you would both agree that you are in a bad equilibrium and that you would want to get away from it again. In other words, you both would want to renegotiate again. But if Ernest foresees this, that you would always be willing to renegotiate back to the truthful and trusting outcome of the game, then his incentives to be truthful are greatly diminished.
With something like this in mind Farrell and Maskin (1989) and others have put forward different versions of equilibria of repeated games that are renegotiation-proof, that are immune to renegotiation.
They call a strategy profile of a repeated game a weakly renegotiation proof equilibrium if the set of all possible prescribed strategy profiles (after any potential history) are Nash equilibria in the subgame and cannot be Pareto-ranked. This means that for any two possible prescribed strategy profiles (or continuation equilibria as they call it) must be such that one person prefers one of the two while the other person prefers the other. Note that this is not the case in the grim trigger equilibrium of the repeated nappy changing game. In this subgame perfect equilibrium both you and Ernest prefer the original equilibrium of the game over the one carried out after Ernest is lying.
So what is weakly renegotiation proof in the repeated nappy-changing game? Well, I have made some calculations and the best weakly renegotiation proof equilibrium for you (in terms of your payoffs) that I could find is this: On the equilibrium path Ernest is truthful and you randomize between trusting Ernest with a probability of and with a probability of you play “do not check (regardless of what Ernest says)”. For this to work you have to randomize using dice or something like this in such a way that Ernest can verify that you correctly randomize. If Ernest ever deviated you then verifiably (to Ernest) randomize by playing “check nappy regardless of Ernest’s answer” with a probability of and trusting Ernest with a probability of . Ernest then continues to be truthful. Ernest incentivizes you to behave in this way (after all you are letting Ernest do what he wants sometimes, which is not what you would want) by punishing you, if you ever deviate, with the following strategy. He would continue to be truthful but ask you to play “do not check (regardless of what Ernest says)”. This complicated “construction” works only if the punishment is not done forever, but only for a suitably long time after which you go back to the start of the game with the equilibrium path behavior. Whenever one of you deviates from the prescribed behavior in any stage you simply restart the punishment phase for this player.
This probably all sounds like gobble-dee-gook and I admit it is a bit complicated. Before I provide my final verdict of this post let me just clarify this supposedly best-for-you renegotiation proof equilibrium. Suppose is essentially one half. Then in the prescribed strategy profile, on the equilibrium path, you are randomizing between trusting Ernest (which you find best given that he is truthful) with a probability of 2/3, that means two thirds of the time. But one third of the time you leave Ernest in peace even when he tells you his nappy is full. As this happens one half of the time you actually leave him in peace despite a full nappy in one sixth of all cases. You have to do this in order for your punishment of him to be actually preferred by you over the equilibrium path play. In the punishment you mostly switch probability from leaving Ernest in peace to checking him always, which is something that you do not find so bad, but that Ernest really dislikes.
Well, I do not know if you find this very convincing, but I do think that the grim trigger strategy is not really feasible when it comes to teaching kids not to lie. What I would actually use in real life is a simple trick. I would not punish behavior only within the nappy-changing game itself. I would use television watching rights, something outside the game I just described. The great thing about this is that while, I assume, Ernest does not like it when his television time is reduced you are, I assume, actually quite happy when he watches less television and so this works as a renegotiation proof punishment. But the fact that you do this, if you do, can be explained by the failure of the grim trigger strategy to be renegotiation proof.