The Price of Independence in a Model With Unknown Dependence

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Peer-Reviewed Article

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How much does it cost a decisionmaker to base her payoff on interdependent, biased information sources? This question is relevant in economics, statistics, and politics. When there are many information sources, their dependence may be unknown or uncertain, which creates multivariate ambiguity. One approach to answer our leading question involves decoupling inequalities from probability theory. We present new inequalities which hold for any type of dependence. We apply our method to a simple formalization of risky asset investment, and to a voting model where citizens face dependent political signals. For a given set of marginal information, the decoupling bound is the sup over all possible joint distributions connecting the marginals. The bound may therefore be useful in other contexts, when a decisionmaker faces unawareness about the joint distribution of information. Our method highlights a frontier which bounds the maximum value of the decisionmaker’s payoff from dependent multidimensional signals. Beneath the bound lies the set of possible payoffs one could obtain from the signals. In this setting, decoupling performs a somewhat similar function to that of the threshold of acceptance sets, in choice under uncertainty. We show that a conservative decisionmaker’s maximal payoff is approximately 50% more than if the signals were independent.