Quantification of Performance in a Logical Task With Uncertainty 233 



but unfortunately this is not the case, as will be seen in the following example. 



Suppose we present the subject with the famous coin-weighing problem. 

 One of twelve coins is of odd weight. It is required to detennine the coin and 

 whether it is lighter or heavier than the rest in a minimum number of weighings 

 on a balance using only the coins for weights. 



Information theory not only reveals that three weighings are necessary and 

 sufficient but also indicates the strategy. Obviously there are loga 24 = 4.59 

 bits of information (uncertainty) in the problem. A weighing can yield a 

 maximum of logo 3 = 1.59 bits. Therefore, at least three weighings are necessary 

 and may be sufficient. Further analysis shows that the first weighing can yield 

 the full 1.59 bits and that only if four coins are weighed against four. The 

 second weighing must involve six coins chosen in such a way that the three 

 outcomes have probabihties 3/8, 3/8, 2/8, which yields 1.55 bits. The third 

 weighing will therefore involve three coins, that is, 1.59 bits, in three 

 fourths of the cases and two coins or 1 bit in one fourth of the cases, i.e. 

 an average of 1.45 bits. The total information is 4.59, exactly equal to the 

 initial uncertainty. 



Now the uncertainty in the problem as it is presented is clearly perceived. 

 At least it is easy to recognize that there are initially twenty-four possibilities. 

 It takes some effort to determine the remaining uncertainty after each weighing, 

 but it is none too difficult to do so. We may therefore suppose that in most 

 instances the 'uncertainty' of the problem is perceived by a fairly intelligent 

 subject correctly, that is, in accordance with the 'objective' assignment of 

 uncertainty. However, it is by no means true that the majority of subjects 

 proceed to the solution in the optimal way. That is, they cannot deduce the 

 'correct' strategy, even when they perceived the 'actual' amount of uncertainty 

 in the problem. 



It appears, therefore, that it is too much to expect to be able to deduce the 

 subject's personal evaluation of uncertainty from his strategy in the solution 

 of a problem in which both the deductive process and resolution of actual 

 uncertainty must operate. However, this circumstance only reveals the situa- 

 tions to be more 'psychological' than they appear in the light of the personal 

 evaluation of uncertainty. Not only is this evaluation personal but also the 

 choice of strategy is, and the latter is by no means always optimal relative to 

 the uncertainty perceived. We are reminded of a similar difficulty in the 

 psychology of decisions in which subjective estimates of probabilities and 

 subjective utility functions are intimately intertwined. 



As pointed out, ours is a similar problem. Assuming that the solution of 

 a logical task with uncertainty will be determined by two 'subjective' characteris- 

 tics, namely, (a) the amount of uncertainty perceived by the subject at each 

 step, and (b) his preference of strategy for a given amount of perceived un- 

 certainty, then our problem is to determine these subjective characteristics in 

 the course of a solution of a problem. It should be mentioned that some obvious 

 techniques for detennining subjective uncertainty are in most cases unusable. 

 If, for example, the solving process is interrupted to ask the subject what he 

 does or does not know, the subject may through these questions become 

 aware of relations he had not been aware of or he may doubt some assumptions 

 he had been making correctly but with insufficient justification. 



