Adaptive uses of random criterion: The largest number problem, the two-envelope problem, and the anchoring and adjustment heuristic

Many cognitive processes appear to incorporate threshold criteria, but when criteria are know to be random their use may appear irrational. For example, when people’s estimates are influenced by random anchors (Tversky & Kahneman, 1974). However Cover (1987) showed that choosing whether a seen or unseen number was greater is improved by using a random number as a criterion. Such Cover functions are also the basis for solving the two-envelope problem. This solution suggests that people’s responses should be influenced by where a value falls in its distribution, a hypothesis supported empirically. The anchoring and adjustment heuristic can also be seen as application of a Cover function. Simulation can demonstrate that adjustment towards a random anchor from an initial random estimate will on average improve the final estimate. Anchoring and adjustment is an example of how Cover functions can contribute to understanding cognitive phenomena, and may have wide applicability.