Anonymity in Predicting the Future

Bajpai, Dvij; Velleman, Daniel J.

doi:10.4169/amer.math.monthly.123.8.777

Cited by 4 publications

(26 citation statements)

References 1 publication

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“…• Is it possible to extend Theorems 1, 3, and 4 to the case of the trader without a synchronized watch (see [7,Section 7.7] or [1])?…”

Section: Discussionmentioning

confidence: 99%

Getting rich quick with the Axiom of Choice

Vovk

2016

Preprint

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This paper proposes new get-rich-quick schemes that involve trading in a financial security with a non-degenerate price path. For simplicity the interest rate is assumed zero. If the price path is assumed continuous, the trader can become infinitely rich immediately after it becomes nonconstant (if it ever does). If it is assumed positive, he can become infinitely rich immediately after reaching a point in time such that the variation of the log price is infinite in any right neighbourhood of that point (whereas reaching a point in time such that the variation of the log price is infinite in any left neighbourhood of that point is not sufficient). The practical value of these schemes is tempered by their use of the Axiom of Choice.

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“…• Is it possible to extend Theorems 1, 3, and 4 to the case of the trader without a synchronized watch (see [7,Section 7.7] or [1])?…”

Section: Discussionmentioning

confidence: 99%

Getting rich quick with the Axiom of Choice

Vovk

2016

Preprint

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“…-Is it possible to extend Theorems 2.2, 3.3 and 3.6 to the case of the trader without a synchronized watch (see [5,Sect. 7.7] or [1])? -The construction in Sect.…”

Section: Resultsmentioning

confidence: 99%

The role of measurability in game-theoretic probability

Vovk¹

2017

Finance Stoch

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This paper argues that the requirement of measurability (imposed on trading strategies) is indispensable in continuous-time game-theoretic probability. The necessity of the requirement of measurability in measure theory is demonstrated by results such as the Banach-Tarski paradox and is inherited by measure-theoretic probability. The situation in game-theoretic probability turns out to be somewhat similar in that dropping the requirement of measurability allows a trader in a financial security with a non-trivial price path to become infinitely rich while risking only one monetary unit.

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“…Lemma 12 (analogue of Lemmas 2 and 3 of [1]). Suppose F ∈ I S, ϕ ∈ Homeo + (I), and F is ϕ-invariant before t. Then there is an H : I → S that extends F | (−∞,t) and is ϕ-invariant.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…So, U -anonymity of P means that P is insensitive to distortions in time caused by members of U . The larger the set U , the more robust is the predictor P. We already mentioned that Hardin and Taylor produced good predictors that were independent of horizontal shifts; using the terminology above, their theorem can be rephrased as: for every set S, there is a good S-predictor that is anonymous with respect to the group of shift functions (i.e., the 1 We do not necessarily assume U is a group; e.g., the set C ∞ (R) ∩ Homeo + (R) in Bajpai-Velleman's Theorem 4 below is not a group under composition, because it is not closed under inverses.…”

Section: Introductionmentioning

confidence: 99%

“…The proof of Theorem 5 relies heavily on the ideas of the proofs in Bajpai-Velleman [1] and Hardin-Taylor [5]. In fact, Theorem 5 is essentially the result of noticing that the Hardin-Taylor theorem about shift-anonymous predictors was essentially due to the classic group-theoretic Hölder's Theorem discussed in Section 2, and that the Bajpai-Velleman strengthening was essentially about about fixed points and commutator subgroups.…”

Section: Introductionmentioning

confidence: 99%

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How robustly can you predict the future?

Cox

ELPERS²

2022

Can. J. Math.-J. Can. Math.

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Hardin and Taylor [5] proved that any function on the reals-even a nowhere continuous one-can be correctly predicted, based solely on its past behavior, at almost every point in time. They showed in [6] that one could even arrange for the predictors to be robust with respect to simple time shifts, and asked whether they could be robust with respect to other, more complicated time distortions. This question was partially answered by Bajpai and Velleman [1], who provided upper and lower frontiers (in the subgroup lattice of Homeo + (R)) on how robust a predictor can possibly be. We improve both frontiers, some of which reduce ultimately to consequences of Hölder's Theorem (that every Archimedean group is abelian).If f : (−∞, t f ) → S is a member of R S, we could view P(f ) as an attempt to predict which member of S (which "state") the function f would/should/will pick out at "time" t f , if t f were in its domain. We gauge how well P makes predictions by asking: for which F ∈ R S and which t ∈ R does P correctly predict F (t), based solely on F | (−∞,t) ? I.e., for which F : R → S and t ∈ R does the equality (*)hold?If we restricted our attention to continuous F : R → S (with respect to some nice topology on S) then the problem would trivialize, since P F | (−∞,t) could simply pick out lim z t F (z), which depends only on F | (−∞,t) . But if |S| ≥ 2 then it is impossible to find a P such that the equation (*) holds for every function F : R → S and every t ∈ R. Simply fix any f : (−∞, 0) → S; since |S| ≥ 2 there is a (possibly non-continuous) function Hardin and Taylor [5] considered what happens when we require the equality (*) to merely hold for almost every t. We will say that an S-predictor P is good if, for all total

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Anonymity in Predicting the Future

Abstract: Consider an arbitrary set S and an arbitrary function f : R → S.

Cited by 4 publications

References 1 publication

Getting rich quick with the Axiom of Choice

Getting rich quick with the Axiom of Choice

The role of measurability in game-theoretic probability

How robustly can you predict the future?

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