In the theory of online algorithms and optimal stopping, a prophet inequality is a bound on the expected value of a decision-making process that handles a sequence of random inputs from known probability distributions, relative to the expected value that could be achieved by a "prophet" who knows all the inputs (and not just their distributions) ahead of time.[1][2] These inequalities have applications in the theory of algorithmic mechanism design and mathematical finance.[3]

Single item

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The classical single-item prophet inequality was published by Krengel & Sucheston (1978), crediting its tight form to D. J. H. (Ben) Garling. It concerns a process in which a sequence of random variables   arrive from known distributions  . When each   arrives, the decision-making process must decide whether to accept it and stop the process, or whether to reject it and go on to the next variable in the sequence. The value of the process is the single accepted variable, if there is one, or zero otherwise. It may be assumed that all variables are non-negative; otherwise, replacing negative values by zero does not change the outcome. This can model, for instance, financial situations in which the variables are offers to buy some indivisible good at a certain price, and the seller must decide which (if any) offer to accept. A prophet, knowing the whole sequence of variables, can obviously select the largest of them, achieving value   for any specific instance of this process, and expected value  . The prophet inequality states the existence of an online algorithm for this process whose expected value is at least half that of the prophet:  . No algorithm can achieve a greater expected value for all distributions of inputs.[3][4]

One method for proving the single-item prophet inequality is to use a "threshold algorithm" that sets a parameter   and then accepts the first random variable that is at least as large as  . If the probability that this process accepts an item is  , then its expected value is   plus the expected excess over   that the selected variable (if there is one) has. Each variable   will be considered by the threshold algorithm with probability at least  , and if it is considered will contribute   to the excess, so by linearity of expectation the expected excess is at least   Setting   to the median of the distribution of  , so that  , and adding   to this bound on expected excess, causes the   and   terms to cancel each other, showing that for this setting of   the threshold algorithm achieves an expected value of at least  .[3][5] A different threshold,  , also achieves at least this same expected value.[3][6]

Generalizations

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Various generalizations of the single-item prophet inequality to other online scenarios are known, and are also called prophet inequalities.[3]

Comparison to competitive analysis

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Prophet inequalities are related to the competitive analysis of online algorithms, but differ in two ways. First, much of competitive analysis assumes worst case inputs, chosen to maximize the ratio between the computed value and the optimal value that could have been achieved with knowledge of the future, whereas for prophet inequalities some knowledge of the input, its distribution, is assumed to be known. And second, in order to achieve a certain competitive ratio, an online algorithm must perform within that ratio of the optimal performance on all inputs. Instead, a prophet inequality only bounds the performance in expectation, allowing some input sequences to produce worse performance as long as the average is good.[3]

References

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  1. ^ Correa, Jose; Foncea, Patricio; Hoeksma, Ruben; Oosterwijk, Tim; Vredeveld, Tjark (2018), "Recent Developments in Prophet Inequalities" (PDF), ACM SIGecom Exchanges, 17 (1): 61–70
  2. ^ Hill, Theodore P.; Kertz, Robert P. (1992), "A survey of prophet inequalities in optimal stopping theory", in Bruss, F. Thomas; Ferguson, Thomas S.; Samuels, Stephen M. (eds.), Strategies for Sequential Search and Selection in Real Time: Proceedings of the AMS–IMS–SIAM Joint Summer Research Conference held at the University of Massachusetts, Amherst, Massachusetts, June 21–27, 1990, Contemporary Mathematics, vol. 125, Providence, Rhode Island: American Mathematical Society, pp. 191–207, doi:10.1090/conm/125/1160620, MR 1160620
  3. ^ a b c d e f Feldman, Michal; Kesselheim, Thomas; Singla, Sahil (2021), "Tutorial on Prophet Inequalities", EC'21
  4. ^ Krengel, Ulrich; Sucheston, Louis (1978), "On semiamarts, amarts, and processes with finite value", in Kuelbs, James (ed.), Probability on Banach Spaces, Advances in Probability and Related Topics, vol. 4, Dekker, New York, pp. 197–266, MR 0515432
  5. ^ Samuel-Cahn, Ester (1984), "Comparison of threshold stop rules and maximum for independent nonnegative random variables", Annals of Probability, 12 (4): 1213–1216, JSTOR 2243359, MR 0757778
  6. ^ Kleinberg, Robert; Weinberg, S. Matthew (2019), "Matroid prophet inequalities and applications to multi-dimensional mechanism design", Games and Economic Behavior, 113: 97–115, doi:10.1016/j.geb.2014.11.002, MR 3926869
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