The random thickness of indifference

Standard random utility models can account for stochastic choice. However, a common implication is that the realized utilities are equal with probability zero. This knife-edge aspect implies that indifference is thin because arbitrarily small changes in utility will break indifference. Semiorders can represent preferences where indifference is thick, however, choice is not random. We design an incentivized binary line length judgment experiment to better understand how indifference can be both thick and random. In the 2-choice treatment, subjects select one of the lines. In the 3-choice treatment, subjects select one of the lines or can express indifference, which directs the computer to "flip a coin" to decide. In every trial, there is a longer line and subjects were told this fact. For each of our line pairs, subjects make 5 decisions in the 2-choice treatment and 5 decisions in the 3-choice treatment. In the line pair with the smallest length difference, 49.7% of 2-choice treatment trials are optimal. For this line pair in the 3-choice treatment, only 1 out of 113 subjects selected indifference on all 5 available trials. There are well-known predictions that optimal choices will have shorter response times than suboptimal choices (Fudenberg, Strack, and Strzalecki, 2018) and we find evidence of this in our dataset. However, not much seems to be known about the response times and indifference. In the 3-choice treatment, we find that indifference choices have longer response times than suboptimal choices. We find that indifference choices are associated with risk aversion and a measure of the beliefs of the favorability of the coin flip. We do not find that indifference choices become more likely across trials, however we find the likelihood of selecting the longer line--in both 2-choice and 3-choice treatments--are decreasing across trials. We hope that the results of our experiment can help inform models of choice where indifference is both thick and random.