What does this mean? The cartoon makes it very clear. You and a number of other advisors report your belief in some variable (say the probability of a phone being a good buy, a stock going up). The advisee knows each of the advisors and she does not trust them all equally. This can be rephrased as follows: she has assigned a prior weight that represents the amount of influence each advisor has on her. These weights are updated when information about the variable comes to light (the phone is shown to break down easily; the stock did actually go up). The updating process takes into account not only whether the advisor was correct, but also how vigorously he reported his belief. If you stated high belief in some previously ignored stock going up, and it actually does go up, your influence will show an increase. It will get the highest increase if the other advisors expressed only weak belief about the stock going up (e.g. they overlooked or discarded the possibility). However, if you are wrong, and if you stated your belief very strongly, as opposed to the other advisors, then your influence will suffer a dramatic fall.
Bayarri and DeGroot show in their paper that in order to increase their influence (posterior weight) over time, advisors should adapt their belief reporting strategy, rather than faithfully stating their beliefs. If you happen to be an advisor, your optimal strategy depends on your current influence (or weight). When your influence is low, you should exaggerate your beliefs (vigorously give a definite yes or a definite no).
This is what the left side of the figure illustrates. It shows how you can take advantage of situations in which other advisers report low belief, and the outcome agrees with your belief. However, if you are a person who has high influence to begin with, the optimal strategy is to be conservative, understating your belief, as shown on the right side of the figure. This strategy keeps your influence rating from collapsing when your advice turns out to be wrong.
The cartoon highlights the real life implications. Optimal strategies really depend on your current influence! Any mistake that Max makes will cost him dear. But Moritz does not have to worry about such cost. His influence can hardly go down any further.
Is this a rare case? Far from it. The process of giving advice, and any transmission of privately held information, is the basis of communication and cooperation. It includes a first step of establishing the private beliefs, either from perception or experience, and a second step of communicating these beliefs. In the first stage you have identified a stimulus and assessed the probability of a reward – but it also involves metacognitive abilities. Bayarri and DeGroot’s study shows that your beliefs are transformed according to the social context even before they are communicated. So giving advice is not just a case of identifying a stimulus, and communicating it to another person. You have to assess not only how confident you are in your judgement but have to factor in the other person’s likely opinion about you. This is how deeply our social nature affects our judgment as well as our presentation. It makes sense: if we are highly trusted already we can easily fall from grace with injudiciously worded advice. Likewise, if we were previously ignored, we can suddenly gain status if we hit the bull’s eye. If we were wrong, no matter, – you can’t sink even further. As the hedge fund managers say “always remember the value of your investments can go down as well as up”.
So yes, metacognition is critical for social communication.