Prophet Inequalities Research Articles

An unexpected connection between branching processes and optimal stopping

A curious connection exists between the theory of optimal stopping for independent random variables, and branching processes. In particular, for the branching process Z n with offspring distribution Y, there exists a random variable X such that the probability P(Z n = 0) of extinction of the nth generation in the branching process equals the value obtained by optimally stopping the sequence X 1,…, X n , where these variables are i.i.d. distributed as X. Generalizations to the inhomogeneous and infinite horizon cases are also considered. This correspondence furnishes a simple ‘stopping rule’ method for computing various characteristics of branching processes, including rates of convergence of the nth generation's extinction probability to the eventual extinction probability, for the supercritical, critical and subcritical Galton-Watson process. Examples, bounds, further generalizations and a connection to classical prophet inequalities are presented. Throughout, the aim is to show how this unexpected connection can be used to translate methods from one area of applied probability to another, rather than to provide the most general results.

Prophet regions for independent random variables with simultaneous costs and discountings

Let Y = (Y1,Y2,…) be a sequence of independent [li,ri]-valued random variables, where (li), (ri), (di):=(ri-li) are non-increasing sequences, and all di>0. Considering the class of all such sequences, a complete comparison is made between M(Y), the expected gain of a prophet (an observer with complete foresight), and V(Y), the maximal expected gain of a gambler (unsing only non-anticipatory stopping times). The solution of this problem is a set in , the so-called prophet region, which is determined for finite and infinite sequences. These regions generalize earlier results [4], [6], [1] in a natural manner. Especially, they yield a variety of prophet inequalities, e.g. all results derived in [5], [6] and [1].

A statistical version of prophet inequalities

All classical “prophet inequalities” for independent random variables hold also in the case where only a noise-corrupted version of those variables is observable. That is, if the pairs $(X_1, Z_1),\ldots,(X_n, Z_n)$ are independent with arbitrary, known joint distributions, and only the sequence $Z_1 ,\ldots,Z_n$ is observable, then all prophet inequalities which would 1 n hold if the $X$’s were directly observable still hold, even though the expected $X$-values (i.e., the payoffs) for both the prophet and statistician, will be different. Our model includes, for example, the case when $Z_i=X_i + Y_i$, where the $Y$’s are any sequence of independent random variables.

Prophet regions for independent random variables with increasing bounds

Let X = (X1, …, Xn) be a sequence of independent, integrable[ai, bi]-valued random variables, where a1 ≤ … ≤ an, b1 ≤ … ≤ bn . Considering the class of all such sequences, a complete comparison is made between M(X), the expected gain of a prophet (an observer with complete foresight), and V(X) the maximal expected gain of a gambler (an observer using only non-anticipatory stopping rules). The solution of this problem is a set in , the ‘prophet region’, which is explicitly characterized. This region yields a variety of prophet inequalities, e.g. M(X) ≤ V(X)/2 if bn = 0, bn-1 = -1, an = -2 and M(X) - V(X) ≤ an/2 if an > 0, bn-1 = 2an, bn = 3an .

A prophet inequality for independent random variables with finite variances

It is demonstrated that for each n ≧ 2 there exists a minimal universal constant, cn, such that, for any sequence of independent random variables {Xr, r ≧ 1} with finite variances, , where the supremum is over all stopping times Τ, 1 ≦ Τ ≦ n. Furthermore, cn ≦ 1/2 and lim inf n→ ∞ cn ≧ 0.439485 · ··.

A prophet inequality for independent random variables with finite variances

It is demonstrated that for each n ≧ 2 there exists a minimal universal constant, cn, such that, for any sequence of independent random variables {Xr, r ≧ 1} with finite variances, , where the supremum is over all stopping times Τ, 1 ≦ Τ ≦ n. Furthermore, cn ≦ 1/2 and lim infn→ ∞cn ≧ 0.439485 · ··.

Some best possible prophet inequalities for convex functions of sums of independent variates and unordered martingale difference sequences

Let $\Phi (\cdot)$ be a nondecreasing convex function on $[0, \infty)$. We show that for any integer $n \geq 1$ and real $a$, $$E \Phi ((M_n - a)^+) \leq 2E \Phi ((S_n - a)^+) - \Phi (0)$$ and $$E(M_n \vee \mathrm{med} S_n) \leq E|S_n - \mathrm{med} S_n|.$$ where $X_1, X_2, \dots$ are any independent mean zero random variables with partial sums $S_0 = 0, S_k = X_1 + \dots + X_k$ and partial sum maxima $M_n = \max_{0 \leq k \leq n} S_k$. There are various instances in which these inequalities are best possible for fixed $n$ and/or as $n \to \infty$. These inequalities remain valid if $\{X_k\}$ is a martingale difference sequence such that $E(X_k \{X_i; i \not= k\}) = 0$ a.s. for each $k \geq 1$. Modified versions of these inequalities hold if the variates have arbitrary means but are independent.

Prophet inequalities for dependent random variables

A prophet inequality with constant a satisfying K-log K=2 is obtained for a class dependent random variables in the full information case, under the same assumptions the prophet constant 4 is obtained in the partial information case. Applications to sums of independent random variables are given

Prophet inequalities for products of non–negative random variables

Stopping time and supremum comparisons known as “Prophet inequalities” are made for products of finite sequences of non–negative integrable random variables under various restrictions on the class of distributions governing these random variables. for example it is shown that for X0= constant > 0, and non-nagative integrable random variables, that the expected maximum of the product variables, that the expected maximum of the product sequence is no more that n+1 times the value of the product sequence when stopped by non-anticipating stopping times

Ratio prophet inequalities for convex functions of partial sums

Let x,, x,,... be independent mean zero random variables and set S, =X, + . * * +X,, and S,* = maxi ~ k ~ ,,S: . We are interested in comparing Sz and S,* for large values of their ranges. Since we cannot compare their tail probabilities directly, we compare ENS,*) with E@(Sz), where @ is any non-decreasing non negative convex function on [O, m). For the convex functions Q,(x) = I x I ‘, square function inequalities are known to become proportionately worse and worse as P + 00. (See Hitczenko (1990) for a summary of what is known.) It may therefore seem surprising that for all @, n and Xi,. . . , X,,, E@(S,*) , Y) (3) for all y sufficiently large (perhaps depending on IZ and Xi, . . . , X,). This inequality is trivial if the variates are symmetric. But even in the i.i.d. case it is essentially false, as shown by B. Davis (for n = 2). To see this, let X, and X, be i.i.d. mean zero random variables with P(X, 'Y) = & and P(X,= -:)=A.

Optimal stopping in urn models with payoff depending on maximal observed element

An urn contains N balls, labelled 1,…, N. Optimal stopping rules are considered for payoff functions f( k, m) where f( k, m) is the reward when stopping after k draws, and the largest number seen by then is m. f( k, m) is assumed nondecreasing i m for each k. We show: (i) For any horizon n ⩽ N, under optimal stopping, sampling without replacement yields a larger expected value than sampling with replacement. (ii) A sufficient condition, both when sampling with or without replacement, for the optimal stopping rule to be of the form t = inf{ k: M k ⩾ q k } for some constants q k , where M k is the maximal label Δ( k, m) = f( k, m + 1) − f( k, m) be nonincreasing in k for each m. Better sufficient conditions are given, and several e such as reward minus cost of sampling, or discounted rewards, are considered. Some limiting results, as N →∞, and prophet inequality considerations are included for the example where the payoff is reward minus cost of sampling.

A note on dilations and martingales

The purpose of this note is to investigate the effect of dilations on martingales and to give conditions under which a dilated martingale will retain the martingale property. The technique of dilating the joint distribution of a sequence of random variables has applications in optimal stopping theory and the study of “prophet inequalities” where martingales play a significant role.

A Difference Prophet Inequality for Bounded I.I.D. Variables, with Cost for Observations

Let $X_i$ be i.i.d. random variables, $0 \leq X_i \leq 1$ and $c \geq 0$, and let $Y_i = X_i - ic$. It is shown that for all $n$, all $c$ and all such $X_i, E(\max_{i \geq 1} Y_i) - \sup_t EY_t < e^{-1}$, where $t$ is a stopping rule and $e^{-1}$ is shown to be the best bound for which the inequality holds. Specific bounds are also obtained for fixed $n$ or fixed $c$. These results are very similar to those obtained by Jones for a similar problem, where $0 \leq X_i \leq 1$ are independent but not necessarily identically distributed. All results are valid and unchanged also when $Y_i$ is replaced by $Y^\ast_i = \max_{1 \leq j \leq i} X_j - ic$.

Stochastic and convex orders and lattices of probability measures, with a martingale interpretation

Letμ be any probability measure on ℝ with ∫|x|dμ(x)<∞ and letμ* denote the associated Hardy and Littlewood maximal p.m., the p.m. of the Hardy and Littlewood maximal function obtained fromμ. Dubins and Gilat [6] showed thatμ* is the least upper bound, in the usual stochastic order, of the collection of p.m.’sν on ℝ for which there is a martingale (Xt)0≤t≤1 having distributions ofX1 and sup0≤t≤1Xt given byμ andν respectively. In this paper, a type of ‘dual representation’ is given. Specifically, letν be any p.m. on ℝ with lim supx→∞xν[x,∞)=0xν[x, ∞)=0 and finitex0=inf{z :ν(−∞,z]0}. Then there is a ‘minimal p.m.’νΔ which is the greatest lower bound, in the usual convex order, of the collection of p.m.’sμ on ℝ for which there is a martingale (Xt)0≤t≤1 having distributions ofX1 and sup0≤t≤1Xt given byμ andν respectively. To demonstrate existence and to obtain identification of these minimal p.m.’s, we use, in particular, a lattice structure on the set of p.m.’s with the convex order, and an equivalence between a convex order of p.m.’s and the stochastic order of their maximal p.m.’s. Consequences of these order results include sharp expectation-based inequalities for martingales. These martingale inequalities form a new class of ‘prophet inequalities’ in the context of optimal stopping theory.

On the prophet inequality for subadditive processes

A prophet inequality with constant ≤ 3.15 is obtained for subadditive processes. Applications to optimal stopping of sums of independent random variables are given

Prophet inequalities for bounded negatively dependent random variables

It is shown that if X k satisfy P( X k < a k ∣ X 1 < a 1,…, X k-1 < a k-1 ) is nondecreasing in a 1,…, a k-1 , a negative dependence condition slightly weaker than CDS, and 0 ⩽ X k ⩽ 1, then E[max X k ] ⩽ 2 V – V 2, where V = sup EX t , t, a stopping rule, holds both for finite and infinite sequences X 1, X 2,…. Actually, here V can be replaced by the optimal value attainable by threshold rules.

Orderings of optimal stopping values and prophet inequalities for certain multivariate distributions

Suppose you observe a finite sequence of random variables from some known joint distribution F, you can stop the process at any time and your profit is the last observed value. If an optimal stopping rule is used, denote the expected profit by V F . What kind of ordering on multivariate distributions F and G will guarantee V F ≤ V G ? The comparison depends on two combined aspects: the actual size of the observations and the structure of their dependence. Closely related to these comparisons is the prophet inequality which compares V F to the expectation of the maximum of the sequence which is a prophet's value. We study monotonicity properties of V F and related prophet inequalities in two classes of distributions: one consisting of negatively dependent PF 2 variables and the other of samples with partial replacement from a finite population.

Optimal stopping problems with generalized objective functions

Optimal stopping of a sequence of random variables is studied, with emphasis on generalized objectives which may be non-monotone functions of EXt, where t is a stopping time, or may even depend on the entire vector (E[X1I{t=l}], · ··, E[XnI{t=n}]), such as the minimax objective to maximize minj{E[XjI{t=j}]}. Convexity is used to establish a prophet inequality and universal bounds for the optimal return, and a method for constructing optimal stopping times for such objectives is given.

Optimal stopping problems with generalized objective functions

Optimal stopping of a sequence of random variables is studied, with emphasis on generalized objectives which may be non-monotone functions ofEXt, wheretis a stopping time, or may even depend on the entire vector (E[X1I{t=l}], · ··,E[XnI{t=n}]),such as the minimax objective to maximize minj{E[XjI{t=j}]}.Convexity is used to establish a prophet inequality and universal bounds for the optimal return, and a method for constructing optimal stopping times for such objectives is given.

Prophet inequalities for cost of observation stopping problems

Comparisons are made between the expected gain of a prophet (an observer with complete foresight) and the maximal expected gain of a gambler (using only non-anticipating stopping times) observing a sequence of independent, uniformly bounded random variables where a non-negative fixed cost is charged for each observation. Sharp universal bounds are obtained under various restrictions on the cost and the length of the sequence. For example, it is shown for X 1, X 2, … independent, [0, 1]-valued random variables that for all c ≥ 0 and all n ≥ 1 that E( max 1 ≤ j ≤ n(X j − jc)) − sup t ∈ T n E(X t − tc) ≤ 1 e , where T n is the collection of all stopping times t which are less than or equal to n almost surely.