Proof Discontinuities and Civil Settlements

This Article explores settlement incentives under three different burden of proof rules. The conventional burden of proof is a discontinuous step-function, jumping from no damages to full damages at the 0.5 jury confidence level. Continuous burdens of proof, by contrast, would permit sanctions to steadily increase as juror confidence rises from 0 to 1, with no discontinuity. Linear burdens, which have received extensive attention in prior literature, escalate sanctions steadily across the whole range of confidence levels, while the logistic burden takes a nonlinear form. Using a data simulation approach guided by the empirical realities of American civil litigation, I consider the incentives that each of these rules creates for parties contemplating settlement, using a model in which parties make divergent forecasts of their expected outcomes at trial due to optimism bias. Based on this analysis, I conclude that a linear burden would likely raise our settlement rate by a modest amount, except in very large cases and in “easy” cases, in which an unbiased person would predict that a trial factfinder would have a level of confidence in liability quite close to either zero or one. I also compare the expected error rate of the settlements that each rule produces, and find that the linear rule modestly lowers the expected error rate of settlement overall, although this benefit does not hold for easy cases or those with very high damages. Lastly, I conduct a similar analysis for the logistic burden, finding that it induces a similar quality and quantity of settlements as we currently achieve using conventional burdens.

To perceive or not to perceive? Individual differences in perception predict rule induction and response generalization

When novel stimuli trigger a previously learned response, this can be due to failure to perceive the novel stimulus as different from the trained stimulus (perception), or active extrapolation of learned properties from the trained stimulus (induction). To date, there has been little investigation of how individual differences in perceptual ability relate to differences in induction. In this paper, we perform cluster analysis in six datasets (four published datasets and two unpublished datasets, N = 992 total) to examine the relationship between individual differences in perception and induction, as well as the utility of perception in predicting generalization gradients. The datasets were obtained from predictive learning tasks where participants learned associations between different colored cues and the presence or absence of a hypothetical outcome. In these datasets, stimulus perception and response generalization (expectancy ratings) were assessed in separate phases. Using cluster analyses, we identified similar subgroups of good and bad perceivers in all six datasets, with distinct patterns of response generalization between these subgroups. Based on the differences in stimulus perception, we could predict where across the stimulus range generalized responses would differ between subgroups as well as the direction of the difference. Furthermore, participants classified as good perceivers were more likely to report a similarity generalization rule than a relational or linear rule, providing evidence that individual differences in perception predict differences in induction. These findings suggest that greater consideration should be given to inter-individual variability in perception and induction and their relationship in explaining response generalization.

Logarithmic Versus Linear Change in Step Size When Using an Adaptive Threshold-Seeking Procedure in a Frequency Discrimination Task: Does It Matter?

Different rules for changing step sizes (e.g., logarithmic, linear) are alternately used in adaptive threshold-seeking procedures, with no clear justification. We hypothesized that the linear rule may yield more accurate thresholds for poor performers because the step sizes are predetermined and fixed across listeners and thus can be small, in contrast to the logarithmic rule, in which step sizes are changed with respect to the listener's performance. Purpose The aim of this study was to test the effect of logarithmic and linear rules on frequency discrimination (FD) thresholds. Method Three experiments involving human subjects and Monte Carlo computer simulations were designed and conducted. In the 1st experiment, FD thresholds were estimated in 40 young adults with either 3-interval 2-alternative forced choice (3I2AFC; n = 19) or 2-interval 2AFC ( n = 21) in a within-subject design. In the 2nd experiment, thresholds were estimated in 16 children (7–8 years old) in a within-subject design, using 3I2AFC. In the 3rd experiment, thresholds were estimated in 30 young adults in a between-subjects design using 3I2AFC. Results No significant differences were shown between the 2 rules, regardless of age group, method, or level of FD performance. Computer simulations supported the empirical findings, predicting similar FD thresholds for both rules in the majority of runs. However, they also yielded more accurate thresholds with the linear rule, but with a larger number of outliers, which increased as the listener's attention level decreased. Conclusion Overall, the use of a particular rule has little influence on FD thresholds. Possible outliers may be minimized by monitoring the participant's attention at the beginning of the run.

Linear Voting Rules

How should a society choose between two social alternatives if participation in the decision process is voluntary and costly, and monetary transfers are not feasible? Assuming symmetric independent private values, we show that it is utilitarian‐optimal to use a linear voting rule: votes get alternative‐dependent weights, and a default obtains if the weighted sum of votes stays below some threshold. Any combination of weights and threshold can be optimal. A standard quorum rule can be optimal only when it yields the same outcome as a linear rule. A linear rule is called upper linear if the default is upset at every election result that meets the threshold exactly. We develop a perturbation method to characterize equilibria of voting rules in the case of small participation costs and show that leaving participation voluntary increases welfare for any two‐sided upper linear rule that is optimal under compulsory participation.

LINEAR VERSUS NONLINEAR ALLOCATION RULES IN RISK SHARING UNDER FINANCIAL FAIRNESS

In a risk exchange, participants trade a privately owned risk for a share in a pool. If participants agree on a valuation rule, it can be decided whether or not, according to the given rule, these trades take place at equal value. If equality of values holds for all participants, then the exchange is said to be “financially fair”. It has been shown by Bühlmann and Jewell (1979) that, under mild assumptions, the constraint of financial fairness singles out a unique solution among the set of all Pareto efficient risk exchanges. In this paper, we find that an analogous statement is true if we limit ourselves to linear exchanges. Conditions are provided for existence and uniqueness of linear sharing rules that are both financially fair and Pareto efficient among all linear sharing rules. The performance of the linear rule is compared to that of the general (nonlinear) rule in a number of specific cases.