Abstract
This article provides an experimental analysis of two-armed bandit problems that have a different structure in which the first unsuccessful outcome leads to termination of the game. It differs from a conventional two-armed bandit problem in that there is no opportunity to alter behavior after an unsuccessful outcome. Introducing the risk of death into a sequential decision problem alters the structure of the problem. Even though play ends after an unsuccessful outcome, Bayesian learning after successful outcomes has a potential function in this class of two-armed bandit problems. Increasing uncertainty boosts the chance of long-term survival since ambiguous probabilities of survival are increased more after each successful outcome. In the independent choice experiments, a slim majority of participants displayed a preference for greater risk ambiguity. Particularly in the interdependent choice experiments, participants were overly deterred by ambiguity. For both independent and interdependent choices, there were several dimensions on which participants displayed within session rationality. However, participants failed to learn and improve their strategy over a series of rounds, which is consistent with evidence of bounded rationality in other challenging games.
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Notes
This formulation is equivalent to a more standard parameterization in which the prior is characterized as equivalent to having observed d successes and e failures, leading to a mean prior of d/(d + e), and a posterior after a successful outcome of (d + 1)/(d + e + 1). Thus, γ equals d + e, and q equals d/(d + e). The advantage of adopting the γ, q parameterization is that it is feasible to take derivatives with respect to γ to analyze the effect of increased precision, i.e., less uncertainty.
While we collected information on participants’ gender and major, these variables did not have a statistically significant effect on whether the participant made optimal choices.
The various measures of risk aversion derived from this part of the study never had statistically significant effects on decisions. The absence of such an effect does not necessarily imply that risk aversion is not influential since the Holt-Laury procedure may not always generate an accurate index of the subject’s underlying risk preferences.
Bayes’ theorem allows one to calculate the probability of a certain scenario based on conditional probabilities. Here, using Bayes’ theorem would allow participants to update their beliefs about being in Situation A after a successful outcome. The formula used to calculate this Bayesian updating is as follows:
P(A| B) = P(B| A)P(A)/P(B).
Applying the formula to this experiment results in the following:
\( {\displaystyle \begin{array}{c}P\left( Situation\ A| Winning\ Draw\right)=\\ {}P\left( Winning\ Draw| Situation\ A\right)P\left( Situation\ A\right)/P\left( Winning\ Draw\right).\end{array}} \)
Using known and updated probabilities allows one to calculate the probability of being in Situation A. The equation can continue to be used in successive rounds using the updated probabilities solved for in the prior round.
The same regression analysis that is reported in Table 4 was also conducted with the inclusion of individual fixed effects. The results remain quite similar with one exception—the coefficient of the session variable Treatment 2 in Panel B is statistically significant (p < 0.01) when including individual fixed effects.
Had the decisions been made in a group rather than on an individual basis, one would have expected even stronger performance on this dimension (Charness et al. 2007).
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The authors are indebted to Gary Charness for superb suggestions that greatly improved the manuscript.
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Kip Viscusi, W., DeAngelis, S. Decision irrationalities involving deadly risks. J Risk Uncertain 57, 225–252 (2018). https://doi.org/10.1007/s11166-018-9292-4
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DOI: https://doi.org/10.1007/s11166-018-9292-4