MEASUREMEIT OF INTUITIVE-PROBABILITY BY A METHOD OF GAME

We have several methods for the measurement of intuitive-probability, which are, however, all unsatisfactory in the strict sense. For instance, take the method in the conmonest use, i.e. the statistical nethod. This method consists of the following steps: First, an experimenter provides an exhaustive set of two exclusive events; Second, he asks subjects of a sufficiently large number to anticipate which one of the two alternative events would occur more certainly. Finally, he calculates the relative ratio of the number of the subjects having preferred a particular one, which is the measure of the intuitive degree of certainty as to the occurrence of that event. So far as the result of this measurement depends upon a particular distribution of intuitive-probabilities among the subjects, it is inevitable that substances derived from it are in general exceedingly poor. For instance, even in the extreme case, when 100% is obtained, we could say by no means that the values of their intuitive-probabilities are all 100%, but could only say they exceed 50%.Here we propose a new method for the strict measurement of intuitive-probability free from the defects comprised in the above-mentioned methods, which is accomplished mainly by means of putting subjects into a situation of game. The characteristics of our game display itself in the following rules:1) An umpire and two players constitute the members of the game.2) One of the two players plays a role denoted by X (estimater), the other Y (opponent). The roles are never interchanged throughout the game.3) In advance of the play, the umpire provides an exhaustive set of two exclusive events, say A and B, about whose order of occurrence the two players should be informed alike.4) The whole play consists of a sufficiently large number of sets. Points of the players are given at each finish of the sets, and the final gain and loss of the players are determined in proportion to their total points after the whole play is over.5) Each set goes on in the following manner:(a) X chooses an arbitrary positive number “a” (≤1) and presents it to Y.(b) Y chooses one between the next two al ternative cases:Case (1); To make “a” X's point.Case (2); To make “a” Y's point.(c) After observing which one of the events has occurred, if the event A is act ually got, in Case (1), “1” is scored for Y's point, in Case (2), “1” is scored for X's point;and if the event B is actually got, in Case (1), “0” is scored for Y's point, in Case (2), “0” is scored for X's point.For conveniences, we used a set of cards printed “1” or “0” as the required set of exclusive events. Now we will explain the meanings of these rules in accordance with our example of using the cards.1), 2) The experimenter should be the umpire and the two subjects are players.3) The experimenter provides asufficiently large number of cards mentioned above and arranges them into the order wanted, about which the subjects are not informed or informed just alike.4) The play consists of as many sets as the number of the cards. It is just this character of the game that makes us possible to study the laws governing the changes of the value of intuitive-probability following the accumulation of passed experiences.5) X presents an arbitrary positive number “a” (≤1) to Y, and Y chooses one for his point between the number “a” and the number printed on the card, without knowing yet whether it is “1” or “0”.Now let us consider which one of the alternatives is better to choose in a given situation. Suppose Y's value of intuitive-probability for the event the card being “1” is “b”. This means that he thinks he would have “b” as the mean value of his points if he had resolved to