There is a period of time each summer when many members of my wife’s fairly large family all come together in one large house. People take turns (to some degree) cooking dinner. In this post I describe what happens when dinner is ready and try to explain it using a bit of game theory.
You need to know some facts and some rules. First, dinner is not always ready at exactly the same time. In fact the time when dinner is ready is a random variable. Some of the family members may know more or less about this random “dinner is ready” time, but as you will see this is not so important. Second, dinner (that is eating) can of course only begin when everyone is present at the dinner table. Third, while everyone prefers to have dinner to not having dinner, most prefer to keep doing what they were just doing over waiting at the dinner table.
The problem we are therefore facing every evening is one of coordination. As long as at least one person other than me is not yet at the dinner table, I prefer not to be at the dinner table myself. If everyone else is at the dinner table, I also want to be at the dinner table. This is true for everyone in the family.
The problem of coordination, especially given the random nature of the “dinner is ready” time, requires communication. What would happen if we had no communication at all? I suppose in this case, strong pressure would be placed on whomever is cooking to have the food ready at, say, 8pm sharp, and everyone would aim to be at the dinner table at 8pm sharp. Social sanctions, such as sarcastic remarks about some people not being able to keep time or watches apparently not working properly, would be put in place to ensure that the arrival time at the dinner table does not gradually slide to sometime after 8pm with the effect that the food would get cold. Occasionally, dinner not being quite ready by 8pm, this would lead to inefficient waiting by all concerned.
But this is not our problem as we DO have communication. In the past few summers we have tried two forms of communication. The first is achieved by whoever is doing the cooking telling someone that dinner is ready and telling them to tell others and so on. The second, and I will explain why this is different in a few ways, is by means of sounding a dinner gong that can be heard easily in all parts of the house.
So, what happens when the house houses many people and we use the first communication protocol? So, imagine that dinner is ready and that whoever is doing the cooking informs the first person they see about this fact and also tells them to spread the word. Then after some time a few people start appearing at the dinner table. They then start asking each other: “Does so-and-so know?” When the others state that they “do not know whether so-and-so knows” one of them will then head off to try and tell so-and-so that dinner is ready. Then someone else decides that “given that not everybody seems to yet know, they still have a bit of time to quickly finish what they were doing before they came down just now”. After a while another person arrives at the doorstep of the dining room and, upon taking in the information that some people are missing (maybe the room is even empty), immediately goes off again with the goal to find them to tell them that dinner is ready.
In more game theoretic terms, the problem with this first form of communication is that, while it may fairly quickly lead to everyone knowing that dinner is ready, it does typically not very quickly lead to everyone knowing that everyone knows that dinner is ready. In fact even when it is actually the case that everyone knows that everyone knows that dinner is ready, there may still be people running about the house trying to find others. Why? For instance it could be that Person A knows that Person B knows that dinner is ready, but Person A is not sure that Person B knows that Person C also knows that dinner is ready, and if Person B does not know that Person C also knows that dinner is ready (Person C is perhaps already at the dinner table) Person A might be going through the house trying to find Person B to inform her that Person C already knows that dinner is ready and that Person B can stop trying to find Person C, while in fact Person B knows that Person C knows that dinner is ready and Person B “will be down in a sec”. You have to read this sentence very slowly many times, perhaps drawing a diagram. It may sound very convoluted, but I assure you that I have experienced this many times.
The dinner gong solves all this. The dinner gong, a distinct sound that can be heard loud and clear in every corner of the large house, is perhaps a little bit faster in making everybody know that dinner is ready, but more importantly it makes everyone know that everyone knows that dinner is ready and makes everyone know that everyone knows that everyone knows that dinner is ready, and so on ad infinitum (as game theorists like to say). In short, it achieves that the event that the dinner is now ready is common knowledge among all of the many family members and with this reassuring certainty we all come running down to dinner in next to no time.