Rationalized Lies Do people cheat or lie when the cheating or lying is not easy to identify (such as filing of taxes)? A total of 2568 college-aged subjects from various countries throughout the world rolled a single six-sided die twice. The subjects were told that the first roll counted in determining a reward and the second roll was only to determine whether the die was “working properly.” Rewards were as follows: rolling a one meant earning 1 unit of the local currency (such as \$1), rolling a two meant earning 2 units, and so on—except that rolling a six meant earning nothing. The rolling was done unsupervised (although results were secretly recorded) with the subjects free to report the outcomes of their respective rolls of the die (thereby creating an opportunity to cheat or lie about the outcome). Source: Gachter, Simon and Schulz, Jonathan, “Intrinsic Honesty and the Prevalence of Rule Violations Across Societies,” Nature (24 March 2016) 531, 496–499.
a. If individuals do not lie about the outcome of the first roll of the die, what would you expect the distribution of outcomes to be?

Verified step by step guidance

1. Understand that the first roll of the die is a fair six-sided die with outcomes 1 through 6, each equally likely if there is no lying or cheating.
2. Since the die is fair and each face has an equal probability, the probability of each outcome (1, 2, 3, 4, 5, or 6) is \( \frac{1}{6} \).
3. The expected distribution of outcomes would therefore be uniform, meaning each outcome should appear approximately \( \frac{1}{6} \) of the time in the total number of rolls.
4. To express this mathematically, if \( N \) is the total number of subjects (or rolls), the expected count for each outcome \( k \) (where \( k = 1, 2, 3, 4, 5, 6 \)) is given by: \[ \text{Expected count for outcome } k = N \times \frac{1}{6} \]
5. Since rolling a six results in zero reward, but does not affect the fairness of the die, the expected distribution remains uniform regardless of the reward structure.

Key Concepts

Uniform Probability Distribution

When rolling a fair six-sided die, each face (1 through 6) has an equal chance of landing face up, with a probability of 1/6. If individuals report outcomes honestly, the distribution of reported results should reflect this uniform probability, meaning each number appears approximately equally often in a large sample.

Expected Value

The expected value is the average outcome you anticipate from a random process, calculated by multiplying each possible outcome by its probability and summing these products. For a fair die roll with rewards assigned to numbers 1 to 5 and zero for 6, the expected value helps understand the average reward if no cheating occurs.

Honesty and Reporting Bias in Experiments

In experiments where participants self-report outcomes, honesty affects data validity. If participants lie to increase rewards, the observed distribution deviates from the expected uniform distribution. Understanding this concept is crucial to interpreting whether reported results reflect true behavior or are biased by dishonesty.