Nash Equilibrium and Predicting Human Response
If you were asked to guess ⅔ the average value of all other guesses, from a range of 0 to 100, what would you guess? The question itself requires some rereading – essentially people were asked to calculate ⅔ of the mean from a pool of other people’s guesses about what ⅔ of the mean would be, where the values were in a range from 0 to 100. The best way to look at this situation is to try out some test cases. For example, if everyone guessed the number 100, then the mean value of all guesses would be 100, which means that ⅔ of the mean would be 66.666. However, if one person calculated this ahead of time, then instead of guessing 100, they would guess 66.666. Now, the game runs on the idea of “common knowledge”, meaning that everyone participating knows the same thing about the game as everyone else, and everyone knows that everyone knows. Therefore, if one person figured that ⅔ of the mean 100 would be 66.666, then that means everyone else would have also guessed 66.666, which then would make the mean = 66.666. This means that ⅔ the mean will not be 66.6666, but in fact will be 44.444. Essentially,
- If everyone guesses 100, ⅔ of the mean will be 66.666
- If everyone discovers that 66.666 will be the correct guess, then everyone will guess 66.666, which means that the mean will be 66.666
- Therefore, ⅔ of the mean will be 44.444
- If everyone knows this, then everyone will guess 44.444, and then ⅔ of the mean will become 29.627, which means the correct answer will be 29.672 …..
- …..
At some point, the mean will be 0, which would then make ⅔ of the mean zero as well. When the mean reaches a value of zero, then Nash’s equilibrium has been reached. When this happens, there is no better option than to guess 0 – it is the best possible strategy for all participants.
However, we have to acknowledge that models do not always represent the real world. A Danish newspaper, Politiken, decided to test out the model and asked its readers to submit a guess. Politiken received 19,000 responses and discovered that the average of the guesses came out to be in the range of 22, which made the correct answer, ( ⅔ of the mean) 14. Ted-Ed also carried out the same experiment and the average of those guesses came to be 31.3, so the correct answer (⅔ of the mean) was 21.
There is a name for this phenomenon: k-level reasoning. From the way it is described, it seems like k-level reasoning is just another way of finding the Nash equilibrium in the sense that in k-level reasoning, participants assume what other participants are planning to do and plan accordingly. The larger the k, the “wiser” the decisions made by that participant, since k represents the “rounds” of reasoning made by the participant. Another way to look at it is to say that the k value represents the experience of the participant in that particular environment.. For example, a participant at level k=0 will reason that all other participants will guess randomly, without thinking of the dynamics of the game. At higher levels of k, participants will assume that other participants are on level k-1 of reasoning. For example, a participant at level k=1 will assume that everyone will use level k=0 reasoning (so everyone is guessing random numbers between 0 and 100), so the participant will expect the average to be 50 and will therefore assume that ⅔ of the mean will be 33. Eventually, the highest level k reasoning will lead participants to guess 0, which we concluded to be the final result in the model.
All in all, the message seems to be that although the model was not correct, humans do implement some elements of Nash equilibrium in their lives in various ways. We tend to assume what others do based on our presumptions of their knowledge and experience. We then use these assumptions to build our own conclusions. No matter which k-level of reasoning one finds themselves to be on, everyone is always trying to find the best possible strategy in response to what others are likely to do.
Link to Ted-Ed video: https://www.youtube.com/watch?v=MknV3t5QbUc