It has been shown that computational hardness of cognitive tasks affects people's effort and ability to solve problems reliably. However, prior empirical studies lack generality. They quantify computational hardne...
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It has been shown that computational hardness of cognitive tasks affects people's effort and ability to solve problems reliably. However, prior empirical studies lack generality. They quantify computational hardness of tasks based on particular algorithms or for specific problems. Here, we propose a set of measures of computational hardness of individual instances of a task in a way that is independent of any algorithm or computational model and can be generalized to other problems. Specifically, we introduce two measures, typical-case complexity (TCC), a measure of average hardness of a random ensemble of instances, and instance complexity (IC), an instance-specific metric. Both measures are related to structural properties of instances. We then test the effect of those measures on human behavior by asking participants to solve instances of two variants of the 0-1 knapsack problem, a canonical and ubiquitous NP-hard problem. We find that participants spent more time on instances with higher TCC and IC, but that decision quality was lower in those instances. We propose that the study of mathematical properties of tasks related to computational hardness can contribute to the development of computationally plausible accounts of human decision-making, just like stochastic properties have proven to be critical to our understanding of human decisions in probabilistic tasks. (C) 2021 Published by Elsevier Inc.
Most theoretical definitions about the complexity of manipulating elections focus on the decision problem of recognizing which instances can be successfully manipulated rather than the search problem of finding the, s...
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Most theoretical definitions about the complexity of manipulating elections focus on the decision problem of recognizing which instances can be successfully manipulated rather than the search problem of finding the, successful manipulative actions. Since the latter is a far more natural goal for manipulators, that definitional focus may be misguided if these two complexities can differ. Our main result is that they probably do differ: If P not equal NP boolean AND coNP (Which itself is well known to hold if integer factoring is hard), then for election manipulation, election bribery, and sonic types of election control, there are election systems for which the problem of recognizing which instances can be successfully manipulated is polynomial-time solvable, yet the task of producing the successful manipulations cannot be done in polynomial time.
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