# A side remark on lying: the boy who cried wolf

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 mine 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 Oscar’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, $\alpha$ 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 $\alpha < \frac12$, 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.

# On Lying, II

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.

# 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.

# Me, Myself and Economics: Disequilibrium

I considered to choose ‘A Non-Equilibrium Approach’ as a subtitle of my dissertation thesis. About at the same time a colleague of mine stated that ‘disequilibrium economics’ are a ‘logic implausibility’ as an equilibrium in economics is not much more than a consistency condition – different to the notion in physics where it mainly refers to a state where the described system is at rest. I have to disagree with this maybe unintentional attempt to whitewash a bunch of approaches which are – as probably every other approach – criticized for good reason.

Just think of basic micro or macro and the definition of a market or an economy in equilibrium. There the term is not used to describe consistency in the derivation of the outcome, but mainly refers its characteristics – for example that supply and demand are balanced. Go further in the curriculum and think of an equilibrium in game theory. While it is also derived in a way which is consistent with the stated assumptions, its description states more than that – for example that it is a combination of strategies for which no individual has an incentive to unilaterally deviate.

Therefore, equilibrium approaches in my opinion go beyond detecting an outcome that is logically implied by assumptions and step-by-step analytics. They also tend to presume an outcome of a certain type and thereby risk the neglect of other outcomes, strategies, behaviour, and thereby even whole issues that may be highly relevant in reality.

In case my concern is not clear, a discussion of Rubinstein’s famous e-mail game may help. The e-mail game may be described as the following: A couple wants to meet and prefers being together over being separated. However, if it rains they prefer to meet inside, otherwise the prefer to meet outside. Whether it rains or not is determined by nature and only one person, let’s assume the woman, knows the weather for sure. If it will rain, she sends an e-mail to the man. Every received e-mail is read and automatically triggers a response, but every e-mail also gets lost with a certain small probability. That means that the e-mail conversation may last for a long time and even forever, but the probability for the latter case tends to be zero.

Because of the small but nevertheless positive probability for an e-mail to get lost, both parties will never know for sure how many e-mails have been sent. The woman knows whether she sent an e-mail or not, but she is confused about the state where one or two e-mails were sent (captured by the partition Pw). While it may be that the second e-mail – sent by automatic response from the man’s account – got lost, it also may be the case that her e-mail did not pass through in the first place. The moment the second e-mail passes through, the third e-mail is triggered automatically and she can distinguish that state from the ones before. However, she again cannot distinguish between the state of three and four e-mails sent – because if she would know about the fourth e-mail, she would have automatically sent the fifth, being in another state. The man faces a similar incompleteness of information (captured by the partition Pm). He in turn is confused about whether none or one e-mail was sent, just like he is confused about whether two or three e-mails were sent and so on.

Rubinstein thereby shows that the strictly formal approach does not lead to an equilibrium in which they meet outside in the nearby game even if there is a high probability for the information to pass on. In fact, the formal result of the game described above is that none of the two will risk to go outside as there is no state (described in terms of e-mails sent) about whose appearance exists common knowledge. However, the example not only shows how easy simple games may get complicated in formal term, but also shows how misleading the strictly formal conclusion can be with regard to an underlying issue. It was about a couple who wants to meet, inside on rainy days, outside otherwise. They both know their preferences. They differ only in the information they have – first about the state of nature and second about how many e-mails are sent. The second issue however should not be the one of primary interest. Instead a social scientist and therefore economist should just ask: how many e-mails have to be sent that they both know that they both know about the weather and therefore human beings of these days will coordinate for the preferred equilibrium.

One e-mail sent just states that it is rainy and the woman knows about it. Two e-mails sent means that the man received this important information, but the women does not know that yet. Three e-mails sent means that the woman knows that the man knows. Four e-mails sent means that the man now knows that the woman knows that he knows. Five e-mails mean that the woman now knows that the man knows that the women knows that the man knows. At the latest after the sixth and seventh e-mail both know that they reached the aspired situation where both know that they both know.

While they can never be sure that their last e-mail passed through, they reach a state where human beings of these and thereby the economic agents of interest will not care about it. Agents may differ with regard to the number of e-mails they require in order to believe in a successful coordination, but I claim that there are not much of them who require more than the five to seven e-mails.

So, while the formal equilibrium approach provides some insights in favour of a theoretical statement about mutual and common knowledge, it risks to draw too much attention towards the wrong issue or at least away from non-equilibrium outcomes that may be highly probable in reality. I think that this is a general issue of equilibrium economics, which are worthwhile and helpful in many regards, but always have to be done as well as interpreted with caution.

# Schnupperuni 2017

Ein Beispiel einer Vorlesungsstunde für potentielle angehende Studierende. Es geht um rationales Herdenverhalten. Wenn ein Produkt von vielen Menschen gekauft wird, heißt das dann, dass es ein gutes Produkt ist? Warum gehen wir lieber in das volle Lokal? Und was bedeutet das für andere?

# Economics on the beach IV: welfare optimal pricing – a model

This post builds on the previous two, economics on the beach II and economics on the beach III. I have started this, so I need to finish this now. In this post I will finally try to build a small model in which it is true that “charging a perhaps even substantial price for beach access would be welfare improving for all potential beach goers”.

# Economics on the beach III: towards discussing welfare optimal pricing – first steps in building a model

I will try and build a small model in which it is true that “charging a perhaps even substantial price for beach access would be welfare improving for all potential beach goers”, a claim I made in my last post. In this post I will take a first few steps in this direction, first only demonstrating my claim that beaches potentially suffer from the “tragedy of the commons” before I will tackle the main question in the next post.