We introduce a new class of prophet inequalities-convex prophet inequalities-where a gambler observes a sequence of convex cost functions ci (xi ) and is required to assign some fraction 0 ≤ x_i ≤ 1 to each, such that the sum of assigned values is exactly 1. The goal of the gambler is to minimize the sum of the costs. We provide an optimal algorithm for this problem, a dynamic program, and show that it can be implemented in polynomial time when the cost functions are polynomial. We also precisely characterize the competitive ratio of the optimal algorithm in the case where the gambler has an outside option and there are polynomial costs, showing that it grows as θ(n^(p-1)/ℓ), where n is the number of stages, p is the degree of the polynomial costs and the coefficients of the cost functions are bounded by [ℓ,u]

Qin, Junjie

Rajagopal, Ram

Vardi, Shai

Wierman, Adam

Caltech Authors - Main

Convex Prophet Inequalities
Extended Abstract
Junjie Qin
UC Berkeley
qinj@berkeley.edu
Ram Rajagopal
Stanford University
ramr@stanford.edu
Shai Vardi
California Institute of Technology
svardi@caltech.edu
Adam Wierman
California Institute of Technology
adamw@caltech.edu
ABSTRACT
We introduce a new class of prophet inequalities—convex
prophet inequalities—where a gambler observes a sequence
of convex cost functions ci (xi ) and is required to assign some
fraction 0  xi  1 to each, such that the sum of assigned
values is exactly 1. The goal of the gambler is to minimize
the sum of the costs. We provide an optimal algorithm for
this problem, a dynamic program, and show that it can be
implemented in polynomial time when the cost functions are
polynomial. We also precisely characterize the competitive
ratio of the optimal algorithm in the case where the gambler
has an outside option and there are polynomial costs, showing
that it grows as  (np 1/`), where n is the number of stages, p
is the degree of the polynomial costs and the coe￿cients of
the cost functions are bounded by [`,u].
CCS CONCEPTS
• Theory of computation→ Online algorithms;
KEYWORDS
Prophet inequality, online algorithms, resource allocation, sto-
chastic control
ACM Reference Format:
Junjie Qin, Ram Rajagopal, Shai Vardi, and AdamWierman. 2018. Con-
vex Prophet Inequalities: Extended Abstract. In Proceedings of ACM
SIGMETRICS 2018 (SIGMETRICS’18). ACM, Irvine, CA, USA, 2 pages.
1 INTRODUCTION
Consider an online decision maker tasked with procuring
C > 0 units of a divisible commodity at minimal cost from n
suppliers that arrive online. Supplier i arrives with a strictly
positive, real-valued, independent convex cost function ci that
is drawn from a known distribution that may be chosen adver-
sarially. The decision maker must decide on a contract with
supplier i before supplier i + 1 arrives. As a motivating exam-
ple, consider electricity markets. When procuring generation
capacity in order to meet demand, load serving entities (LSEs)
make contracts with generators. Most of these contracts are
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SIGMETRICS’18, June 2018, Irvine, California, USA
© 2018 Copyright held by the owner/author(s).
made months or even years in advance, when future availabil-
ity of other options for generation is not known to the LSE
[5, 10–12]. Thus, LSEs face an online decision problem where,
given a forecast for the capacity needed, they must make con-
tracts with generators that arrive online over a span of years.
Contracts with generators are typically strictly convex. While
the precise form of the cost functions may be complicated,
it is often modeled as quadratic in analytic work [4, 9, 13],
which motivates us to focus on quadratic (or more generally,
polynomial) cost functions. Beyond electricity markets, online
packing problems with convex cost functions are also highly
relevant for other procurement problems, e.g., optimal con-
trol [3, 8], cloud computing [1, 6], and inventory management
[2, 7].
If the cost functions are linear, the problem is a general-
ization of the minimization version of the classical prophet
inequalities setting. In the classical (maximization) setting, a de-
cision maker must choose one of n items that arrive online and
the reward of each item is drawn from a known, non-negative,
real-valued distribution that may be chosen adversarially, with
the goal of maximizing the reward. The prophet inequality
bounds the reward obtainable by the decision maker as a func-
tion of the reward that can be obtained by a prophet who can
foresee the entire sequence and stop at the maximal value. In
the minimization version, the rewards are replaced by costs,
and the goal is to minimize the cost. This problem and ours
are analogous in the case of linear costs: the optimal strategies
of both the decision maker and prophet in the setting above
is to set xi = 1 for some i and x j = 0 for all j , i , which cor-
respond to the decision maker and prophet choosing a single
item. Classical prophet inequality are stopping problems, as the
decision maker sees the items in sequence, and chooses when
to stop and accept the current item.
When the cost functions are strictly convex and positive,
this is not a stopping problem since the prophet always pro-
cures a strictly positive amount of the commodity at each
stage, and thus never “stops”. Instead, this problem is better
thought of as an online packing problem. More speci￿cally,
this problem di￿ers from the setting of the classical prophet
inequality in three ways: (i) the cost functions are convex,
(ii) the decision maker can make real-valued (non-integral)
decisions, and (iii) the decision maker seeks to minimize cost
instead of maximize reward.
Performance Evaluation Review, Vol. 46, No. 2, September 2018 85
2 MODEL
We consider an online decision maker (gambler) tasked with
procuring a single unit of a divisible commodity at minimal
cost from n suppliers that are arriving online. Supplier i ar-
rives with a non-negative, real-valued convex cost function
ci , which is drawn from a known distribution Di . The dis-
tributions D1, . . . ,Dn are independent and possibly chosen
adversarially. The gambler must decide on a contract with
supplier i before supplier i + 1 arrives.
We compare the cost of the gambler to the cost of a prophet,
who can foresee the sequence of realized cost functions. Since
the prophet knows the cost functions, it can simply solve the
following convex program to identify the optimal allocation
OPT = min
x
n’
i=1
ci (xi )
s.t.
n’
i=1
xi = 1; 0  xi  1; i = 1, . . . ,n.
Since the cost functions are random, we are interested in the
expected performance, E[OPT].
Without the ability to foresee the future, the gambler has
to base decisions on the information that is available at stage i ,
denoted byHi . This information includes all the distributions,
the realized cost functions up to ci , and the amount of com-
modity that has already been obtained (denoted by si ). That is,
the decision of the gambler has to be causal, i.e., xi =  i (Hi ),
i = 1, . . . ,n, where  i is the gambler’s policy for determining
the amount to procure from supplier i given the information
at stage i .
Let   = ( 1, . . . , n ) be the sequence of policies the gambler
may use and let  be the set of admissible policies that includes
all the policies generating xi ’s such that xi   0 and Õi xi = 1.
Then, the expected cost of the gambler using an admissible
policy   2   is E[ALG  ] = E ⇥Õni=1 ci ( i (Hi ))⇤ . For conve-
nience, we write E[ALG] when the policy used is clear from
context. The gambler’s problem is then to design an admissible
policy   that minimizes his expected cost: inf  2  E[ALG  ].
For deriving convex prophet inequalities, we are interested in
obtaining the exact competitive ratio
sup
D
inf
  2 
E[ALG  ]
E[OPT] , (1)
where the maximization is over the space of all sequences of
independent distributions D = (D1, . . . ,Dn ).
Our goals are (i) to obtain explicit expressions of (1) for a
wide class of cost functions which are useful for bounding the
algorithm performance for suboptimal policy or for arbitrary
distributions, and (ii) to identify simple and e￿cient algorithms
for solving the online allocation problem with provably good
competitive ratios.
3 MAIN RESULTS
Consider the setting with polynomial cost functions ci (xi ) =
aix
p
i when there is an outside option. The existence of an
outside option is common in real world applications, as it is
typically possible to procure part of the desired quantity of the
commodity through channels outside of the online procure-
ment process. In our model, an outside option corresponds to
a normalization of the model where a1 = 1.
Our main results are summarized below:
Optimal algorithm. We derive optimal online algorithm that
is polynomial time implementable based on explicitly solving
dynamic programming.
Exact competitive ratio (prophet inequalities). We obtain the
exact value of (1) by characterizing worst case distributions:
T￿￿￿￿￿￿ 1. Consider polynomial costs of the form ci (x) =
aixp , p > 1, with ` < a1 = 1 < u. For any independent
distributions D = (D2, . . . ,Dn ) with Di supported on [`,u],
inf
  2 
EALG 
EOPT 
✓
1 + n   1
`1/(p 1)
◆p 1
.
T￿￿￿￿￿￿ 2. The bound
⇣
1 + n 1
`1/(p 1)
⌘p 1
in Theorem 1 is
tight.
Simple thresholding algorithm. We develop a thresholding
algorithm (B￿￿￿￿￿￿￿ T￿￿￿￿￿￿￿￿) that is much simpler than
dynamic program and has the following guarantees:
T￿￿￿￿￿￿ 3. Consider i.i.d. quadratic cost functions of the
form ci (x) = aix2 and B￿￿￿￿￿￿￿ T￿￿￿￿￿￿￿￿ parameterized
by p  2 (0, 1) such that   satis￿es Pr[ai    ] = p  and k =
np   
r
n log
⇣
2nu
`
⌘
. Then, the asymptotic competitive ratio of
B￿￿￿￿￿￿￿ T￿￿￿￿￿￿￿￿ (i.e., for n !1) is at most  `p  + o(1).
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86 Performance Evaluation Review, Vol. 46, No. 2, September 2018


Convex Prophet Inequalities

We introduce a new class of prophet inequalities-convex prophet inequalities-where a gambler observes a sequence of convex cost functions c_i(x_i) and is required to assign some fraction 0 ≤ x_i ≤ 1 to each, such that the sum of assigned values is exactly 1. The goal of the gambler is to minimize the sum of the costs. We provide an optimal algorithm for this problem, a dynamic program, and show that it can be implemented in polynomial time when the cost functions are polynomial. We also precisely characterize the competitive ratio of the optimal algorithm in the case where the gambler has an outside option and there are polynomial costs, showing that it grows as Θ(n^(p-1)/l), where n is the number of stages, p is the degree of the polynomial costs and the coefficients of the cost functions are bounded by [l, u]

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Convex Prophet Inequalities

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