# What is the significance of the findings in Nagel's 1995 paper on Learning Theory?

This post concerns the findings in Nagel (1995). It is a bit outdated but nevertheless still relevant. She examines the Guessing Game, and how individual study-subjects behave in repeated games.

In her paper, she explains the behavior of the study-subjects in two different ways;

1. Behavior based on $$\bar{X}$$ from the previous games.
2. Behavior based on individual experience, under the umbrella term Learning-Direction Theory.

Where $$\bar{X}$$ is the average guess.

She formalizes her claim in the second explanation, in the following way;

$$\begin{equation} \text{a}_{it}= \begin{cases} \frac{X_{it}}{50}, \text{for } t = 1\\ \frac{X_{it}}{\bar{X}_{t-1}}, \text{for } t = 2,3,4 \end{cases} \end{equation}$$

Where $$a_{it}$$ is called the adjustment factor. She then goes on, to define the optimal adjustment factor that comes from retrospective reasoning;

$$\begin{equation} \text{a}_{opt,t}= \begin{cases} \frac{p\bar{X}_{t}}{50}, \text{for } t = 1\\ \frac{p\bar{X}_{t}}{\bar{X}_{t-1}}, \text{for } t = 2,3,4 \end{cases} \end{equation}$$

where $$p$$ is the factor used to adjust the mean. Ie. Guess $$\frac{2}{3}$$ of Mean.

This claim, is merely a descriptive model that explains how the study-subjects learn. And how they eventually adapt. She finds empirical support from the experiments she makes.

This claim, insofar, is rather simple to grasp and is an important finding.

What I do not understand, is the importance of her first claim. In essence, what she finds is that there is a tendency, as the game is repeated, that the average guess moves towards equilibrium with differing rates depending on $$p$$.

• How is this even an important finding? This is already the predicted Nash-Equilibrium, albeit, it is reached later than expected.
• What do you consider her "first claim" here? Feb 10, 2021 at 15:10
• As far as I understood her paper, the first claim is that an equilibrium is achieved because the players adjusts their guess according to previous game. But she also notes that the level of reasoning does not increase; however, if the Guessing Game is repeated equilibrium will eventually be reached in any case if individuals adapt to the previous game. The order of reasoning should not matter. Feb 10, 2021 at 15:16
• I understand it more as pointing out that these hypothesized "levels of reasoning" are clearly visible in the data, and that they seem not to increase under repetition. Convergence towards Nash equilibrium is also noted, but that's not the interesting part in my opinion. Feb 11, 2021 at 12:57
• Game Theory is not my force, granted. If the game is played once it would require a 12th level of reasoning to reach equilibrium! So, as of now, my thoughts are that she uses the second claim that players adapt according to a learning-strategy rather than playing at some level of reasoning that would eventually be needed to reach equilibrium. Feb 11, 2021 at 14:15
• I think the story is that they play between 0 and 3 levels in each game, but upon each iteration of the game $p\bar X$ from the last round serves as the new round's level 0. This then drives convergence to NE. Feb 12, 2021 at 15:14