If this question is selected for payment, there are two possible options: The Lottery and The Bet on The Average.
First, we will draw an integer at random between 1 – 100. Each integer (1,2,3,4,…, 96, 97, 98, 99, 100) is equally likely to be chosen. We’ll call the number that’s chosen the “Threshold”. Then, we will draw a second an integer at random between 1- 100. Again, each integer (1,2,3,4,…, 96, 97, 98, 99, 100) is equally likely to be chosen. We’ll call this number that’s chosen the “Draw”. If the “Draw” number is less than or equal to the “Threshold” number, the lottery will pay you CZK 200. If not, that is, if the “Draw” number is more than the “Threshold” number, the lottery will pay you nothing. For example, suppose the “Threshold” is 35. Then, if the “Draw” is 15 (or anything less than or equal to 35), you would win CZK 200. If not, you would win CZK 0. Thus, if the “Threshold” is 35, you have a 35% chance of winning the lottery.
We will pay you CZK 200 if the average amount transferred by Person Bs in this session lies between in the interval you guessed.
Your answer to the question above, that is, your “Percentage” tells us for which possible “Thresholds” you would prefer to be paid according to The Lottery, and for which possible “Thresholds” you’d rather be paid according to The Bet on The Average.
We will randomly draw the “Threshold” we described above.
Suppose you believe the true chance of the average transfer of Person B’s lying in that interval is 60%. Then, to maximize your payment, you should tell us your percentage is 60%. Why? Because if you tell us your percentage is 60%, then if we draw a “Threshold” less than or equal to 60, by telling the truth you will be choosing to be paid according to The Bet on The Average. This is good for you: for “Thresholds” less than or equal to 60, your chances of winning the lottery are less than 60%. For instance, suppose the threshold drawn is 55. You would have a 55% chance of winning The Lottery . But, according to your percentage, you have a 60% chance of winning the The Bet on The Average. So, for all thresholds less than or equal to 60, you’d rather be paid according to the The Bet on The Average. By telling the truth, you guarantee you get paid according to the option that’s better for you.
Similarly, if we draw a “Threshold” greater than 60, by telling the truth you will be choosing to be paid according to The Lottery . This is good for you: for “Thresholds” greater than 60, your chances of winning the lottery are greater than 60%. For instance, suppose the threshold drawn is 75. You would have a 75% chance of winning The Lottery. But, according to your percentage, you’d have only a 60% chance of winning The Bet on The Average. So, for all thresholds greater than 60, you’d rather be paid according to The Lottery . By telling the truth, you guarantee you get paid according to the option that’s better for you.
To summarize: by giving us your percentage, you are telling us, for different possible thresholds, whether you prefer to be paid according to The Lottery or The Bet on The Average. You have no control over what “Threshold” is chosen or what “Draw” will be chosen. But, by telling the truth, you can guarantee that no matter what the threshold and draw ends up being, you will get paid according to the option that has the higher probability of paying you CZK 200.