The data used in the analysis of this paper are the hypotheses entered by players. We compare two data sets. One is Haruvy, Stahl and Wilson's (1996), of which we look at five games, and the other with 19 additional games and a modified experimental procedure.

Two restrictions were imposed on observations in both data sets to be included in the analysis:

1. The hypothesis by any player for any game must be consistent with the final choice entered by the player for that game.
2. Only the last hypothesis (that satisfied restriction 1) by a given player for a given game was admissible.

The above restrictions create potential problems in the analysis of the first data set, among which are:

1. Insufficient observations -- Discarding observations which did not satisfy the above conditions, we are left with 137 observations. This number of observations may not be sufficiently large to give a reasonable mean squared error for two dimensional kernel density estimates with more than one mode in the underlying density (19 observations would be sufficient for a unimodal density, according to Silverman, 1986).
2. Selection bias -- In the first data set, missing hypotheses could be disproportionally distributed over the hypothesis space. Furthermore, some players were represented in the data less than other players due to missing hypotheses. Players with missing data could be more likely to be of a particular type, hence causing an under-representation of that type in the data relative to other types.

 

In addition, the fives games of the first data set have been designed for discrete choice analysis and not for extraction of beliefs; hence, a potential third problem with the first data set:

 

3. The effectiveness of the scoring rule-- The five games of the first data set lack the steep payoff differences between the immediate neighborhoods of level-n hypotheses.

The second experiment was intended to remedy these shortcomings in the extraction of beliefs. For the second data set, we have observations on hypotheses for 22 players over 24 games. In this data set, three players entered the maximax hypothesis for 16, 17, and 19 games out of 24. By maximax hypothesis, we define any hypothesis which puts a probability greater than 0.8 on the column containing the maximax payoff. Due to expected difficulties with the maximax response, we delete these players from our data set. We are left with 19 players over 24 games, or 456 observations; enough to produce a sufficiently small mean squared error. This new data set is also superior due to no missing hypotheses, thus minimizing the potential for selection bias. Moreover, the steep payoff differences remedy the problems with the scoring rule in the first experiment.

Elements of each hypothesis vector x had to be rearranged for the purpose of this paper so that hypothesis vectors from different choice matrices can be compared. The new arrangement is:

p₁ = Probability assigned to column of the matrix corresponding to type 1 behavior;

p₂ = Probability assigned to column of the matrix corresponding to type 2 behavior;

p₃ = Probability assigned to column of the matrix corresponding to Nash behavior;

Just from the sorted data we can deduce some players' types. For example, in the second data set, three players entered the type 1 hypothesis more than 14 times. Three players entered the type 2 hypothesis more than 16 times, and four players entered the Nash hypothesis more than 12 times. However, to classify the rest of the players this way could prove misleading.

Given that the elements of each vector are linearly dependent, we ignore the redundant third element in each vector and our data is reduced to two-dimensional observations. This corresponds to a right triangle in the positive quadrant, or an equilateral triangle plane in three dimensions.

Data is sorted BY PLAYER, and then BY GAME: Player 1 game 1, player 1 game 2, player 1 game 3, . . . . . ., player 19 game 22, player 19 game 23, player 19 game 24.