2. A class experiment used the probability matching (PM) Veconlab software with

Question

2. A class experiment used the probability matching (PM) Veconlab software with payoffs of $0.20 for each correct prediction, an in-class earnings averaged several dollars. There were six teams of 1-2 students, who made predictions for 20 trials only. The more likely event was programmed to occur with probability 0.75. Calculate the expected earnings per trial for a team that follows perfect probability matching. How much more would a team earn per trial by being perfectly rational after learning which event is more likely?

3. Answer the two parts in question 2 for the case where a correct answer results in a gain of $0.10 and an incorrect answer results in a loss of $0.10.

Explanation / Answer

2. Let us call the most likely event with probability of occuring 0.75 as A and its complement as B.

Hence P(A) = 0.75 and P(B) = 0.25

Now in perfect probability matching without prior knowledge, A and B will be predicted with equal probability, 0.5.

Hence expected earning per trial = P(A) X P (predicting A) X 0.2 + P(B) X P(predictiing B) X 0.2

= 0.75 X 0.5 X 0.2 + 0.25 X 0.5 X 0.2 = 0.075 + 0.025 = 0.1 $

After learning which event is more likely, the team will always predict the more likely event A.

Hence expected earning per trial = P(A) X P (predicting A) X 0.2 + P(B) X P(predictiing B) X 0.2

= 0.75 X 1 X 0.2 + 0.25 X 0 X 0.2 = 0.15 $

3. In case correct result gives gain of 0.1$ and incorrect result gives loss of 0.1$

expected earning per trial in perfect probability matching without prior knowledge is :

P(A) X P (predicting A) X 0.1 + P(B) X P(predictiing B) X 0.1 +

P(A) X P (not predicting A) X -0.1 + P(B) X P(not predictiing B) X -0.1

= 0.75 X 0.5 X0.1 + 0.25 X 0.5 X 0.1 + 0.75 X 0.5 X -0.1 + 0.25 X 0.5 X -0.1

= 0.0375 + 0.0125 - .0375 - .0125 = 0 $

expected earning per trial after learning which is the more likely event and hence always predicting A is:

P(A) X P (predicting A) X 0.1 + P(B) X P(predictiing A) X -0.1

0.75 X 1 X 0.1 + 0.25 X 1 X -0.1 = 0.075 - 0.025 = 0.05 $

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