The Effects of Untruthful Bids on User Utilities and Stability in Computing
Sergei Shudler∗, Lior Amar†, Amnon Barak†
Department of Computer Science
The Hebrew University of Jerusalem
Jerusalem, 91904 Israel
Social and Information Sciences Laboratory
California Institute of Technology
Pasadena, California, USA 91125
Markets of computing resources typically consist of a
cluster (or a multi-cluster) and jobs that arrive over time
and request computing resources in exchange for payment.
In this paper we study a real system that is capable of pre-
emptive process migration (i.e. moving jobs across nodes)
and that uses a market-based resource allocation mech-
anism for job allocation. Specifically, we formalize our
system into a market model and employ simulation-based
analysis (performed on real data) to study the effects of
users’ behavior on performance and utility. Typically on-
line settings are characterized by a large amount of un-
certainty; therefore it is reasonable to assume that users
will consider simple strategies to game the system.
thus suggest a novel approach to modeling users’ behav-
ior called the Small Risk-aggressive Group (SRG) model.
We show that under this model untruthful users experi-
ence degraded performance. It is also shown that the k-
th price payment scheme, which is a natural adaptation of
the classical second-price scheme, discourages these users
from attempting to game the market. The preemptive ca-
pability makes it possible not only to use the k-th price
scheme, but also makes our scheduling algorithm superior
to other non-preemptive algorithms. Finally, we design a
simple one-shot game to model the interaction between the
provider and the consumers. We then show (using the same
simulation-based analysis) that market stability in the form
In recent years the area of High-Performance Computing
(HPC) has witnessed a shift towards market-based resource
allocation mechanisms.These mechanisms incorporate
†Supported in part by the EU IST program, grant 034286 “SORMA”.
market features into their architectures [12, 5, 17, 4, 3, 21]
and allocate scarce resources more efficiently, not only by
increasing utilization (in the traditional sense) but also by
maximizing users’ utilities . One of the central princi-
ples in these mechanisms is incentive compatibility; i.e., the
ability to discourage strategic users from gaming the mar-
ket to obtain better personal utility  for themselves. By
gaming the market these users disproportionally harm other
users who are best described as “conservative” in terms of
gaming attempts. Usually the goal of incentive compati-
ble mechanisms is to maximize the utility for all the users
such that gaming the market will not increase the utility of
any of the users. However, there are costs associated with
using incentive compatible mechanisms. For example, in
the combinatorial auctions field, the well-known Vickrey-
Clarke-Groves (VCG) mechanism is known to be incentive
computationally hard to use it in real systems. As a result,
many real world implementations use various approxima-
tions of VCG [14, 8, 17, 18] thus compromising incentive
compatibility in its game-theoretical meaning.
This paper is based on  in which it was shown that
using a type of preemptive job scheduler, specifically the
Online Greedy Migration (GM) scheduler, in a cluster re-
sulted in increased performance and increased fairness. The
GM scheduler and its preemptive capability are part of the
MOSIX  multi-cluster grid deployed at the Hebrew Uni-
versity. Our goal in this paper is to analyze the effects of
users’ untruthful behavior, suggest a way to discourage this
and to explore the stability of the market. To this end,
in Section 2, we transform the GM scheduler into a mar-
ket mechanism by allowing it to support various payment
schemes and introduce the Small Risk-aggressive Group
(SRG) untruthfulness model which provides a simple char-
acterization of users’ behavior. The study was performed
by running real-world workload traces in a simulated envi-
ronment. The rationale for performing simulations instead
of running workloads on a real environment was to save
time and resources, since normally workloads span months
(or even years) and assume clusters of hundreds of nodes.
Although there is a significant amount of theoretical work
in this field [13, 10, 14], the theoretical treatment of the
problem is complicated due to: (a) an on-line environment
in which the number and the arrival time of future jobs is
unknown; (b) no job run-time information is available; (c)
the preemptive capability of the scheduler. In Section 3
we show that untruthful users experience degraded perfor-
mance in that their slowdown is higher than for truthful
users. This section also verifies that compared to other non-
preemptive scheduling algorithms, our upgraded scheduler
retains the same properties as the GM scheduler. In Sec-
tion 4 we report the most significant results of this paper. By
dividing all of the jobs in a workload into three groups with
increasing run-times and analyzing each group separately,
we show that the k-th price payment scheme discourages
untruthful users from attempting to game the market. In ad-
dition to being robust, it also increases the utility of all the
users compared to the simple First price payment scheme.
These results are another strong case for preemptive migra-
tion ability in clusters since this ability is the key to using
the k-th price scheme within our mechanism. In Section 5
we explore the stability of the market by formulating a one-
shot game between the mechanism designer and untruth-
ful users. As it turns out, there are specific combinations
of strategies that produce equilibria in this game. Finally,
Section 6 discusses related work and Section 7 presents the
conclusions and ideas for future work.
2. Computing Market Models
2.1. The Basic Model
We now formalize our existing system into a somewhat
simpler model that allows us to specify the scheduling algo-
rithm. Consider a set of n homogeneous (same architecture
and same speed) computers (a cluster or a multi-cluster) in
which an on-line bidding scheduling algorithm is used to
assign incoming sequential jobs to computers (nodes). As-
sume that each node runs only one job at a time and that the
system supports job preemptions; i.e., a job can be stopped
at any stage of its execution and be resumed later, not nec-
essarily in the same node. Before starting a job, each user
must determine his or her private bid, which is the maximal
amount that the user is willing to pay per unit of run-time.
Note that the private bid is not known to the other users.
Upon submitting a job each user states a bid, called the re-
ported bid, which might be lower than the private bid. After
the job is submitted the bid cannot be changed. Through-
out this article the notions of jobs and users are used inter-
The following Highest-Bid (HB) on-line algorithm is
based on the Online Greedy Migration (GM) algorithm ex-
tensively studied in , with the addition of a payment cal-
culation. The algorithm is used to determine whether to as-
sign a job to a node or to place it in a queue of waiting jobs.
It supports various payment schemes by assuming that p?
which is the current payment per unit of run-time for job
i, was already calculated somehow by one of the payment
Algorithm 1 Highest-Bid (HB)
Upon job arrival, job termination or bid update do:
1. If it is a new job j set its total payment: pj= 0.
2. For each running or newly terminated job i, set: pi=
assignment of the job and p?
per unit of run-time.
i, where t?
iis the elapsed time since the last
iis the current payment
3. Sort the set of new and currently queued and uncom-
pleted jobs in a descending order according to their
bid. Break ties according to the submission time.
4. Assign the first n jobs from the sorted list to the n
nodes, possibly preempting jobs with lower bids.
5. For each assigned job i, determine its current p?
6. Queue unassigned jobs until the next run of the algo-
In terms of algorithmic mechanism design, each user has
a well defined utility function (see below) that represents
the user’s preference over various outcomes of the HB algo-
rithm. It is assumed that users will try to game the system,
for example by reporting false information to optimize their
respective utilities .
For each job i, let:
• wiand ˜ wibe the private and the reported bids respec-
• ϕibe the flow time (in seconds), defined as the total
time between the submission time and the time when
the job was finished. Note that ϕidepends on ˜ wi.
• BSDibe the bounded slowdown factor of the job, de-
where τiis the actual run-time  in seconds and 60
is a threshold value which prevents getting excessive
BSD values in extreme conditions.
The private valuation viof job i which represents the
user’s preference for a shorter completion time, was defined
in  as: −wiϕi. In this paper we use the same definition:
Definition: Quasi-Linear Utility
For job i define: ui= vi− pi.
By using Eq. 2 the utility function of job i becomes:
ui= −wiϕi− pi
Ideally users truth-reveal their private bids, which can be
formally stated as:
Full-Truthfulness (FT) Model: For each job i: wi =
However, assuming that users are self-interested, ra-
tional and strategic, they may report untruthful bids (i.e.
wi?= ˜ wi) if it maximizes their utility.
2.2. The Untruthful Bids Model
To the best of our knowledge, there is no sufficiently
clear definition of untruthful behavior that fits the clus-
ter environment we are modelling.
model which can be used for this purpose. The reported
bid range can be parameterized by 0 ≤ β < 1, such that
˜ wi∈ [(1 − β)wi,wi]. Assume that users strongly prefer to
avoid higher payments. Assume that ˜ wiis chosen randomly
and uniformly from the above range defined by the private
bid. We now define the users’ behavior model implemented
in this paper:
Small Risk-aggressive Group (SRG) Model: Divide
all of the jobs randomly and uniformly into two groups: the
first group contains 90% of the jobs with β = 10% and is
called the “risk-conservative group”; and the second group
contains the remaining 10%, with β = 90%, and is called
the “risk-aggressive group”.
In words, our first assumption in the SRG model is that
users facing uncertainties have simple strategies, and there-
fore can only underbid their values.
would never overbid and would never delay their arrival
time to the system. Our second assumption in this model
is that the majority of users are conservative, and therefore
their willingness to underbid is very small. Only a small
group of aggressive users can severely underbid their val-
Although this model assumes that all of the jobs are
known in advance, it also can be applied without knowing
the total amount of jobs. Each time a new job is submitted
a biased 10%-90% coin is flipped so that the job has a 10%
chance of being “aggressive” and a 90% chance of being
“conservative”. More elaborate models of user untruthful-
ness could make the market more complicated to analyze
and alter the original FT bid distribution, which is assumed
Below we define a
to be bi-modal. For example, increasing the percentage of
the risk-aggressive users to 80% and lowering the percent-
age of the risk-conservative users to 20% (the β values stay
unchanged) gives a distribution which is more exponential
rather than bi-modal. In case of the SRG the distribution is
nearly identical to FT.
It is not reasonable to assume that every job is highly
important and consequently has a higher bid; a much more
realistic assumption is that only a fraction of jobs are really
important (highly prioritized). The majority of the jobs are
regular jobs with normal priorities. The aforementioned bi-
modal distribution for FT is generated by uniformly com-
bining two normal distributions such that the first one is
used for 80% of the jobs and the second one is used for
the other 20%. The first distribution models jobs with low
bids (normally prioritized) with a mean of 30, whereas the
second one models jobs with high bids (highly prioritized)
whose mean is 150. For both distributions the standard
deviation is 15. Throughout the article, low-bid jobs re-
fer to jobs with wi∈ [0,60], middle-bid jobs refer to jobs
with wi ∈ (60,120) and high-bid jobs refer to jobs with
In the next section we present the performance of Al-
gorithm HB with preemptions and other non-preemptive
3. Performance of HB for the SRG model
In this section we compare the performance of Algo-
rithm HB on the FT and the SRG models. We also compare
Algorithm HB with preemptions to other, non-preemptive
3.1. SRG vs. FT
As was stated earlier, we have taken an empirical ap-
proach in this study and carried out simulations using
real-world workloads from three homogeneous clusters :
DAS2 , LPC  and REQUIN . Each workload
consists of records for each submitted job that includes the
job submission time, its run-time, the user’s estimate of its
run-time, and other non-relevant parameters. Since bidding
was not used in these workloads we used randomly gener-
ated bidding values. For each workload, Table 1 presents
the number of jobs, the actual number of nodes used (in the
simulations) and the mean (average) run-time of all the jobs
(in seconds). Note that since our model does not deal with
parallel jobs, we converted such jobs to sequences of se-
rial jobs . Also note that due to under-utilization of both
the DAS2 and the LPC clusters, we artificially reduced the
number of nodes in these clusters by 25%, in order to create
a competitive environment. Although the simulations were
performed for all of the aforementioned workloads only the
results of the LPC workload are presented since the results
for the other workloads were conceptually the same.
Table 1: Workloads
No. serial jobs
Average BSD vs. Private bid
Fig. 1 presents the average BSD for different values of pri-
vatebids(wi). Inthefigure, theredlinepresentstheaverage
BSD of the risk-aggressive users in the SRG model, while
the blue line presents the average BSD of all the users in the
Figure 1: BSD comparison: SRG vs. FT
From the figure it can be seen that the average BSD of
the low-bid jobs is higher in the SRG model. The number of
middle-bid jobs is very small which leads to a noisy results
for this range of bids. In the case of high-bid jobs, there is
model have a greater ˜ withan the majority of other jobs.
Since the HB algorithm prefers jobs with higher bids over
jobs with lower bids, lowering a job’s bid leads to increased
slowdown of this job. As a result, the risk-aggressive jobs
are subjected to a greater slowdown.
Measure of SSJs
Based on the analysis in , it is assumed that users show
tolerance to the growing slowdown up to a certain point
(threshold). We now define this threshold value for exces-
sive BSD values:
Definition: Severely Slowdowned Job (SSJ)
A job i such that, BSDi≥ 5.
The value of 5 was chosen based on the analysis in .
of private bids (wi). In the figure, the red line presents the
percentage of SSJs of the risk-aggressive users in the SRG
model, while the blue line presents the percentage of SSJs
of all the users in the FT model.
Figure 2: SSJ comparison: SRG vs. FT
From the figure it can be seen that the percentage of low-
bid SSJs is higher in the SRG model than in the FT model
(as before, the results for middle-bid jobs are noisy). This
means that a non-negligible percentage of risk-aggressive
users’ jobs are subject to excessive slowdown. It is as-
sumed that a job’s bid correlates with its importance and
that the user wants more important jobs to have less slow-
down. Therefore, good performance means that jobs with
higher bids do not become SSJs and have the lowest slow-
down possible. The above results show that risk-aggressive
users are penalized by degraded performance compared to
3.2. HB vs. Non-Preemptive Algorithms
In this section we compare the performance of the HB
preemptive algorithm to several non-preemptive algorithms
using the adjusted workloads described in the previous sec-
tion. Specifically, we extend the analysis presented in 
by incorporating the SRG model.
We used the following three algorithms:
1. A non-preemptive version of the HB (NPHB) algo-
rithm that sorts the jobs in the queue according to their
reported bid and then assigns jobs to available nodes.
2. WSPT - similar to NPHB, except that the queue is
sorted according to the ratio between the reported bid
and the run-time: ˜ wi/τi, known as the Smith promi-
nent ratio rule . Note that this algorithm assumes
that the run-times are known.
3. WSPT-EST - a version of WSPT, in which the actual
run-time τi, is replaced by a user estimation of the
run-time ˜ τi. Note that the run-time estimation either
appears in the workload records (DAS2, LPC) or was
generated (REQUIN) according to .
Average BSD vs. Private bid
Fig. 3 presents the average BSD of the risk-aggressive users
for different values of private bids (wi) and for the afore-
mentioned algorithms. In the figure, the green line presents
the average BSD for the HB algorithm, while the orange,
blue and red lines present the average BSD for NPHB,
WSPT and WSPT-EST algorithms respectively.
Figure 3: BSD comparison: HB vs.
From the figure it can be seen that the WSPT-EST per-
forms less well than WSPT, but both these algorithms per-
form better than HB for low-bid jobs. In case of the high-
bid jobs, the performance of HB improves and the average
BSD values become smaller. As for the middle-bid jobs, the
results are noisy as before. The HB algorithm also outper-
forms the NPHB algorithm for the whole spectrum of bids.
These results correlate with the performance results in ;
the difference, however, is the new perspective of the SRG
Measure of SSJs
Fig. 4 presents the percentage of SSJs of the risk-aggressive
users for different values of private bids (wi) and for the
aforementioned algorithms. In the figure, the green line
presents the percentage of SSJs for the HB algorithm, while
the orange, blue and red lines present the percentage of SSJs
for NPHB, WSPT and WSPT-EST algorithms respectively.
From the figure it can be seen that the HB algorithm has
consistently fewer SSJs for almost the whole spectrum of
bids, which means that its performance is better. As in the
previous case, the results correlate with .
The next section analyzes the utility values produced
by the HB algorithm combined with various payment
4. Analysis of Different Payment Schemes
The previous section presented the performance of the
HB algorithm without considering market aspects of pay-
Figure 4: SSJ comparison: HB vs.
ments and utilities. It was concluded that risk-aggressive
users are penalized in terms of BSD for reporting untruthful
bids. This section analyzes users’ utilities under different
payment schemes and under different truthfulness models
(FT and SRG). As in the previous section only the results
of the LPC workload are presented since the results for the
other workloads were conceptually the same. For our anal-
ysis we used the following two payment schemes:
1. First price: each running job pays exactly the bid it re-
ported. Whenever the number of jobs drops below the
number of nodes, the payment of each job becomes the
reservation price of the node. Obviously, it is econom-
ically unreasonable to request a full price if there is no
demand. Note that in order to simplify the model, the
reservation price of each node is set to 1.
2. k-th price: each running job pays the bid of the job
with the highest bid in the queue; if the waiting queue
is empty the payment of each job becomes the reserva-
tion price of the node. Note that this scheme is the nat-
ural dynamic adaptation of the classical second-price
scheme. Essentially each job pays the minimal bid it
could have reported and still be running.
The utility function (Eq. 3) becomes linearly lower as
a job’s run-time grows; i.e., comparing the utility value
of a short job with the utility value a long job almost al-
ways shows that the short job has a higher utility. This ef-
fect is negated by dividing the jobs according to their run-
times and comparing them within the same run-time group.
Therefore all of the jobs were divided into three equal sized
groups, SHORT, MEDIUM and LONG according to their
respective run-times and the utility values were analyzed
within these groups. Naturally, the utility values of the
LONG group were generally much smaller than the values
of the SHORT and MEDIUM. Fig. 5 presents the average
utility values for different values of private bids (wi) and for
the First price scheme. In the figure, the red line presents
the average utility values of the risk-aggressive users in the
SRG model, while the blue line presents the average utility
(a) SHORT group(b) MEDIUM group(c) LONG group
Figure 5: Utility comparison of HB using First price between SRG and FT
values of all the users in the FT model. From the figure it
can be seen that for the SHORT and the MEDIUM run-time
groups the utility values of both low-bid and high-bid jobs
are higher for the FT model. In the case of the LONG group
(Fig. 5(c)) the utility values of the low-bid jobs are lower in
the SRG model than in the FT model; however for the high-
bid jobs it is exactly the opposite. To get a better grasp of
this situation Fig. 6(a) and Fig. 6(b) present a detailed view
of the low-bid and the high-bid jobs for the LONG group.
By referring to Fig. 1 and Fig. 2 it can be seen that in
the SRG model the BSD of high-bid jobs is very small and
virtually coincides with the BSD in the FT model. Thus by
paying less, the risk-aggressive users benefit in terms of the
overall utility. This result is not suprising since the First
price payment scheme inherently lacks incentive compati-
bility. However, from the mechanism designer perspective
encouraged to game the system whereas risk-conservative
users are left at the mercy of the aggressive users. Creating
a formally proven, fully incentive on-line compatible mech-
anism for our model is a hard task. Moreover, only a small
amount of empirical work has been done with on-line mod-
els in mind (cf. Section 6). Therefore, a first step towards
incentive compatibility can be made by using the k-th price
payment scheme, which is known to be incentive compat-
ible in auctions with k winners . The job assignment
step in the HB algorithm that assigns n jobs from the sorted
list to n nodes can be viewed as a momentary auction with
n winners. Thus the k-th price scheme is an approxima-
tion of a fully incentive compatible mechanism in our case.
Fig. 6(c) presents the utility values of high-bid jobs using
k-th price and is analogous to Fig. 6(b). From the figure
it can be seen that the utility values of the risk-aggressive
users in the SRG model coincide with the utility values of
theusersintheFTmodel. Clearly, inthiscasethereisnoin-
centive for risk-aggressive users to report smaller bids since
the utility would remain essentially the same as though they
had reported truthful bids.
In order to understand the differences between First price
and k-th price we compared the utility values in both cases
for the SRG and the FT models. It turns out that the util-
ity values are the same for the SHORT group and almost
the same for the MEDIUM group with k-th price having
slightly higher values. The most significant differences are
presented in Fig. 7 that compares the utility values of high-
bid jobs from the LONG group. In the figure the green
line presents the values for the k-th price scheme, while the
orange line presents the values for the First price scheme.
The figure shows that the utility values in the case of k-th
price are higher for both the SRG (Fig. 7(b)) and the FT
(Fig. 7(a)) models. In the k-th price scheme each payment
is no greater than a job’s reported bid which is exactly the
payment in case of First price. By Eq. 3 the utility value for
the k-th price scheme is no lower than the utility value for
the same job in the case of the First price scheme. This also
explains Fig. 6(c): by using k-th price the utility values of
the users in the FT model become higher and virtually equal
to the utility values of the risk-aggressive users.
risk-aggressive users can benefit from reporting untruthful
bids under the First price payment scheme. However, the
k-th price scheme is more robust and using it instead of the
First price scheme results not only in higher utility values
for all types of users, but also prevents risk-aggressive users
from deriving any benefit from reporting untruthful bids.
Although the k-th price might seem harder to compute, it is
actually simpler than other VCG-like mechanisms .
5. Market Stability
This section initiates the empirical study of market sta-
bility in the context of on-line computational markets.
Specifically, we analyze market stability in terms of risk-
aggressive users’ utility and the mechanism designer’s rev-
enue. As usual, stability is associated with various formal
theoretical equilibrium concepts. In this section we for-
mulate a simple one-shot game to model the interaction
between the mechanism designer and the risk-aggressive
users. We then look for a pure Nash-equilibrium in this
(a) Low-bid jobs (First price)(b) High-bid jobs (First price)(c) High-bid jobs (k-th price)
Figure 6: Detailed utility comparison of HB between SRG and FT (LONG jobs group)
(a) High-bid jobs (FT)(b) High-bid jobs (SRG)
Figure 7: Detailed utility comparison of LONG group jobs between First price and k-th price
We first identify two major players: the mechanism de-
signer and an aggregate consumer playing on behalf of the
risk-aggressive users, where the former essentially wishes
to maximize revenue and the latter strives to maximize per-
sonal (aggregated) utility. More formally, the strategy space
of the mechanism designer is any combination of the pos-
sible allocation algorithms (HB, NPHB or WSPT) and pay-
ment schemes (First price or k-th price). Ideally, the mech-
anism designer wants to maximize revenue. However, since
in real-life consumers have several optional competing sup-
pliers, the mechanism designer also needs to consider the
quality of the service. Therefore, in our game we model
the payoff of the mechanism designer in terms of the To-
tal Weighted Flow Time (TWFT) as well. This metric was
used in  to evaluate the aforementioned algorithms un-
der the same basic on-line model. The strategy space of the
risk-aggressive users is any of the following β-parameter
values: 5%,25%,50%,60%,70%,80%,90%. Note that the
specific value of β is chosen only once before the submis-
sion of any job and cannot be changed later.
In order to use the k-th price scheme with the NPHB
and the WSPT algorithms we had to adapt it since these
algorithms lack preemptive ability and in the case of WSPT
the queue has a different order. In case of the NPHB the
k-th price was reduced to First price whenever the bid of a
running job was smaller than the maximal bid of a queued
job. In case of the WSPT the price for a running job i was
set to be: ( ˜ wj/τj)τi, where ˜ wj/τjis the largest ratio among
the queued jobs. This is the minimal bid that job i could
have reported and would still be scheduled to run.
Table 2 and Table 3 present the payoff matrices of the
game for the LPC workload. The results for the DAS2 and
REQUIN workloads were nearly the same and therefore are
omitted. The first table corresponds to the First price pay-
ment scheme and the second one to the k-th price payment
scheme. Both tables show the results for the LONG run-
time group since it was the focus of the previous section
and it provides the most interesting insights. The results
for the SHORT and MEDIUM run-time groups are almost
the same as for the LONG group. As in the standard pay-
off matrix, the rows in both Table 2 and Table 3 represent
the strategies of the first player, namely the mechanism de-
signer, and are colored in blue, whereas the columns repre-
sent the risk-aggressive users and are colored in red. Each
cell in the matrix contains the payoff values, where the first
one is the payoff of the mechanism designer and the second
one is the payoff of the risk-aggressive users.
The payoffs of the mechanism designer are values of
the revenue and quality of service metric defined by:
inator is TWFT of all the jobs. Each value is normalized
by the maximal payoff (across both tables) which occurs in
case of the First price scheme for the combination of HB
with β = 5%. As was mentioned earlier, the idea of using
jwjϕj(i,j go over all the jobs). The numerator
is the sum of the payments of all the jobs, while the denom-
Table 2: Payoff matrix for the First price scheme (LONG jobs group)
1.000, 0.8330.990, 0.8420.976, 0.845
0.862, 0.8010.853, 0.8100.841, 0.813
0.904, 0.8270.895, 0.8420.883, 0.860
Table 3: Payoff matrix for the k-th price scheme (LONG jobs group)
0.306, 1.0000.303, 0.9880.296, 0.961
0.636, 0.8540.629, 0.8580.619, 0.847
0.774, 0.8550.765, 0.8690.755, 0.879
TWFT to evaluate algorithms is not new. Adding weight to
the traditional total flow time metric reflects the nature of
the bids in our model which can be interpreted as weights
or priorities in non-economic parlance. The payoffs of the
risk-aggressive users are their average utility values normal-
ized by the maximal utility value (across both tables) which
occurs in case of the k-th price scheme for the combination
of HB with β = 5%. Blue numbers in the matrix cells are
the maximal payoffs of the mechanism designer for each β
value, while the red numbers are the maximal payoffs of the
risk-aggressive users for each possible strategy of the mech-
anism designer. Clearly the Nash equilibrium occurs either
for the combination of HB with β = 50% in case of the
First price scheme, or for the combination of WSPT with
β = 70% in case of the k-th price scheme.
The HB algorithm (for both payment schemes) is very
sensitive to changes in the β parameter in terms of the ra-
tio between the minimal and the maximal risk-aggressive
users payoff values. This can be accounted by the preemp-
tive capability of the HB algorithm which responds on-line
to newly submitted jobs with high bids. On the other hand,
the WSPT algorithm (for both payment schemes) is particu-
larly non-responsive to changes in β since it sorts the queue
of the waiting jobs according to the ratio ˜ wi/τiand not ac-
cording to ˜ wi. The total revenue in the case of the First
price payment scheme is almost independent of the actual
algorithm and the revenue values differ only by a few per-
cent. However, the maximal payoff values of the mecha-
nism designer are generated exclusively for the HB algo-
rithm. Effectively the payoff values are reduced into TWFT
values and since the HB algorithm has the best performance
in terms of TWFT it consequently has the highest payoff.
Naturally, the mechanism designer would prefer to choose
the HB algorithm with the First price scheme since it pro-
However, this choice is problematic from the users’ per-
spective because it does not give them the highest possible
utility and motivates users to game the mechanism. Had our
mechanism been a monopoly and if users had not had any
alternative, the aforementioned problems could have been
ignored. However the prevalence of other clusters and allo-
cation mechanisms drives users to seek other mechanisms
that give them better utility and better stability.
Table 3 helps analyze the payoffs under the k-th price
scheme in greater detail. As can be seen in this table the
Nash equilibrium occurs for the combination of WSPT with
β = 70%. The WSPT algorithm has an informational ad-
vantage that allows the mechanism designer to get better
payoffs but at the same time it violates our model by requir-
ing job run-times. Examining the results without the WSPT
line gives an equilibrium for the combination of NPHB with
β = 25%. Although in terms of TWFT the HB algorithm
is better, the payoffs from the NPHB algorithm are higher
because of its higher revenues. Since a higher TWFT means
that more jobs with high values are delayed in the waiting
queue, the k-th price of each running job increases, thus
resulting in higher revenue.
There is no clear recommendation as to which combi-
nation of algorithm and payment scheme to use. By mod-
elling the interaction between the mechanism designer and
analyze the market stability in our model. The results indi-
cate that there are Nash equilibria for both the First price
and the k-th price schemes. However, in both cases the
equilibria occur when risk-aggressive users report untruth-
ful bids. Clearly, the combination of the HB algorithm with
k-th price discourages these users from reporting untruthful
bids since it reduces their utility. Further study of market
stability is necessary to understand how it affects user be-
havior and to determine whether an equilibrium in which
users are truthful can be achieved in our model or in some
closely related model.
6. Related Work
The field of market-based resource allocation in grids as
well as in clusters has been extensively studied [6, 17, 12,
4, 10, 3]. It is largely divided into two approaches, the first
of which is the on-line auction with an unknown number of
future users that constantly submit jobs with auctions per-
formed on-line (sometimes known as the spot-market). The
second approach is a periodic auction that first gathers bids
and then clears the market by determining the winners and
the payments, sometimes known as the reservation-market
since until the next auctioning round the resources are re-
served for the winners of the previous auction. This ap-
proach is typically incorporated into batch schedulers and
requires some additional information, e.g. job run-time.
Tycoon  is an example of a deployed resource al-
locator that consists of a number of components, namely
resources, users, agents, a resource discovery service and a
trusted bank. The users employ the resource discovery ser-
vice to get a picture of the current load on every node and
bid separately on each node they need. The nodes are inde-
pendent and each one runs multiple jobs simultaneously in
proportion to the job bid. The agent’s role is to optimize, on
behalf of the user, the task of finding the nodes in which the
job could get a larger share for a smaller price. Each user
pays the bid of the job directly unless there is no load on
the node, in which case only the reservation price is paid.
The work in  is a theoretical study of a job scheduler
in a cluster environment. The authors formulate an on-line
model with k resources and continuously arriving agents
that submit requests for resources. Each request consists
of the bid, the length of the job, the arrival and the depar-
ture time of the agent. An auction takes place each time a
new agent arrives or a running job terminates. As in many
simple auctions (“single-parameter”), each agent pays the
amount of the smallest bid that it could have reported and
still get an allocation. If a job does not finish running before
the departure time, the utility is set to be zero and the agent
pays nothing. By assuming restricted misreporting of the
arrival and departure times and that all job lengths are in an
[Lmin,Lmax] interval, this allocation mechanism is incen-
tive compatible and is O(log(Lmax/Lmin))-competitive
compared to the offline case. As in our case this mecha-
nism also uses a (slightly different) greedy approach to the
allocation step and the same payment scheme (k-th price).
The differences, however, are that we do not require the de-
parture times of the agents and the job lengths; secondly,
instead of traditional competitive analysis we compared the
performance of various algorithms using real data.
Incentive compatibility is an important and desired prop-
erty in auctions since it encourages users to truthfully reveal
incentive compatibility in combinatorial auctions is compu-
tationally infeasible (an NP-complete problem ). A sig-
vise various approximate algorithms that provide restricted
incentive compatibility.In  the authors introduce a
greedy allocation scheme that ensures incentive compati-
bility for a restricted type of users. The authors of  im-
plemented a reservation-market mechanism in which they
restrict the expressiveness of users’ requests for resources.
As a result, by using dynamic programming they designed a
computationally tractable incentive compatible mechanism.
It was evaluated by modelling the untruthful behavior of a
single user as randomly varying his or her bid. Obviously
the basic idea is the same as in our case, however we pur-
pose a more elaborate model in which there is a randomly
selected group of untruthful users.
To the best of our knowledge, only the work of Ng et
al.  analyzes user behavior in a real working market-
based resource allocator system, called Mirage , which
allocates testbed resources in repeated first-price combina-
torial auctions. Because it is inherently not incentive com-
patible, Mirage makes it possible to study users’ behavior
and the strategies employed to game the mechanism. Four
strategies are described, the first of which is identical to the
strategy of lowering the reported bid in our SRG model.
7. Conclusions and Future Work
In this work we studied an on-line model of a computing
marketthateffectively sellscomputingnodesina cluster (or
multi-clusters) to constantly arriving new jobs. In addition
we used a simple model of user behavior (the SRG model)
which assumes that the majority of the users are risk-
conservative while a small minority are risk-aggressive. By
using the greedy-preemptive HB algorithm it was shown
that although untruthful risk-aggressive users are penal-
ized for their behavior in terms of the slowdown, they are
still able to increase their utility if the First price payment
scheme is used. Switching to the k-th price scheme resulted
in lower utility for the risk-aggressive users; in fact, the util-
ity values were effectively the same as in the case of full
truthfulness. By looking at our model from the point of
view of a two-player game we were able to analyze market
stability. Although strategy combinations that yield equi-
libria exist, no clear recommendation can be derived from
them. Results indicate that stability does not necessarily
mean full truthfulness. Further work is necessary to study
this type of game between the users and the mechanism de-
signer in greater detail.
Our work points to various paths for future work. We
avoided the theoretical treatment of our problem; however
this approach could lead to more fundamental results. An-
other approach would be to continue the study of the sta-
bility issue by looking at more than two players. It was
assumed that there is only one mechanism designer, but in
market in its own right. The sellers are the mechanism de-
signers who have many choices in terms of the actual mech-
anism they use, and the buyers are the users who want to use
the computing resources the designers offer.
In our model users submit the maximal amount of pay-
ment per unit of time they are willing to pay, but at the same
time it is assumed that the run-time of jobs is unknown.
Thus the resulting total payment that a user might be re-
quired to pay is effectively unbounded. In reality budgets
are limited, so the question is how to decide what is more
important to the user - staying within the limits of the bud-
get or getting a better performance and a better utility? How
would our model and allocation mechanism work if we also
required users to submit a budget ceiling along with the bid?
Clearly, these are interesting and unexplored questions.
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