Recourse Allocation In P2P Framework

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Recourse AllocationIn P2P Framework

• Yoni Peleg

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Outline
Resource Allocation
Cryptographic
Graph Theory
Open subjects

• Previous and current work on resource allocation
  in MAS.
• P2P Framework, Multi-Hop Cellular Networks
• Overview Motivation
• Problems in these models
• Resource Allocation in these models
• Solving the difficulties with
• Cryptographic methods
• Random graph theory

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Resource Allocation Overview
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Allowing an agent to use other agents resources
  in order to complete its task.
• Works in resource allocation can be divided into
  2 main areas
• Economical (Marketplace)
• Matching

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Economics Resource Allocation
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Assumes a central marketplace (usually doesnt
  refer to its implementation).
• Sellers and buyers meet in the marketplace in
  order to trade resources.
• Solves questions about the price of the
  resources
• Will an equilibrium be reached?
• How long does it take to reach the equilibrium?
• What is the price of the resources in the
  equilibrium point?

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Economics Resource Allocation (cont.)
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• A lot of sellers (easy to find an agent with an
  available resource).
• All the available resources are identical.

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Matching Resources
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Resources are rare
• Only a few agents have resources to share.
• Resources are not necessarily identical
• For example, using the resource of a closer agent
  might be preferable due to communication costs.
• Cooperative / Non-Cooperative Environments
• Agents might not share their resource for free.
• Task Match a supplier to each consumer.

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The Contact Net
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• A fundamental work in resource allocation
• Each consumer starts an auction for a resource.
• All suppliers participate in the auction.
• The consumer chooses the best possible offer.
• Big disadvantage Using broadcast.

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Centralized Match-Mark
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Well have a special new agent (a matchmark) that
  will
• Hold a list of all consumers / suppliers (will
  get a message from the other agents anytime they
  supply / need a resource).
• Match appropriate couples.
• Disadvantages
• Hot Spot
• Inefficient (not scalable for systems with a big
  number of agents)

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Distributed Match-Mark
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• The (suppliers, consumers) list is distributed
  between many agents.
• Each agent can also be a matchmark.

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Distributed Match-Mark lower bounds
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Lower bound is known if the matchmarks cant pass
  information between each other
• One match-mark that is familiar withO( )
  agents exists.
• An agent that is familiar with O( )
  matchmarks exists.
• Can be improved to O( ) if the max distance
  in a graph between a supplier and a consumer is
  k.
• Can we do better by allowing matchmarks to pass
  information between each other?

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P2P Framework properties
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Symmetric
• All the agents have the same contribution to the
  matching process.
• Distributed Control
• The control of the system is distributed between
  all the agents.
• System is more robust, no hot spot.

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P2P Framework properties (cont.)
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Simple agents.
• Each agent has connections only to a constant
  number of other agents.

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Algorithm For A Cooperative Environment
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Parameters
• TTL How deep in the network the message will
  arrive.
• TimeOut How long to wait for a response.
• d To how many neighbors to send a message when
  searching for a resource.

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Algorithm For A Cooperative Environment
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Algorithm (Vanzin 2006)
• When an agent a needs a resource, it sends a
  message to d of its neighbors. a attaches to
  the message the TTL parameter.
• Each agent that receives the message, reduces 1
  from the TTL parameter, and passes the message
  forward (with the new TTL).

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Algorithm For A Cooperative Environment
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Algorithm (Vanzin 2006)
• An agent that can supply the resource will
  contact a directly.
• Stop passing the message when TTL reach 0.
• If a didnt receive a resource by TimeOut it
  sends another request with a bigger TTL / d.

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Algorithm For A Cooperative Environment
Cryptographic
Graph Theory
Open subjects
Resource Allocation
TTL 3 d 2
s
s
t
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Algorithm For A Cooperative Environment
Cryptographic
Graph Theory
Open subjects
Resource Allocation
TTL 32 d 2
s
t
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Algorithm For A Cooperative Environment
Cryptographic
Graph Theory
Open subjects
Resource Allocation
TTL 321 d 2
s
t
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Algorithm For A Cooperative Environment
Cryptographic
Graph Theory
Open subjects
Resource Allocation
TTL 3210 d 2
Algorithm Failed!
s
t
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Algorithm For A Cooperative Environment
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Analysis of the algorithm (Vanzin 2006)
• If d1
• Random walk
• Will find the resource in O(log(n)) (average
  case)
• Long time to find a resource
• d is large
• Find the resource in shorter time
• Big message complexity

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Non-Cooperative Environment
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Well assume that agents only care about
  increasing their personal utility
• Well ignore attacks like denial of service
  that only try to hurt the system.
• Well have to pay an agent in order to use its
  resource
• Auctions (like in the CNET).
• Well have to pay other agents if we want them to
  pass our seek for resource message.

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Difficulties In Non-Cooperative Environment
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• An agent may try to
• Get paid without passing a message.
• Avoid paying other agents that pass its message.
• Block bids of distant-agents
• Man in the middle attack
• Get paid for passing messages to a DECOY agent.
• Register many copies of itself to the system.
• Whitewash

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Multi Hop Cellular Phone Network
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Regular cellular phone
• All calls are transferred through a central
  location.

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Multi Hop Cellular Phone Network
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• A possible problem in case of multiple calls in a
  small area.
• Also relevant to sensors communications.

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Multi Hop Cellular Phone Network
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• An alternative the calls will be routed by the
  cell-phones themselves.

T
S
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Difficulties
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Problem keeping the cell-phone on and passing
  other users messages costs money (battery).
• The greedy user may keep his cell-phone off, and
  just take advantage of other users for passing
  his messages (free ride).

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Non-Economic Solutions
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Hardware Solution
• Add hardware (like an OTP) to the cell-phones
  that will transfer the messages.
• Home users cant change the hardware (criminal
  organizations might be able to do so).
• Task the hardware should be as simple as
  possible.
• Wont work in a general P2P framework.

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Non-Economic Solutions
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Repetitive solution
• Assume that any cell-phone can listen to the
  outgoing messages of the cell-phones close to it
• Cant understand the content of the message
• Can tell if a message was sent
• No user has incentive to send fake messages
• They also cost buttery power
• If a user didnt forward your message punish it
  in the future.
• Doesnt fit a general P2P model.

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Economic Solutions
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Central Bank Solution (CBS)
• A trusted (centralized) third party
• All the agents have its public key
• All the agents can verify that the bank has
  signed a message

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Economic Solutions (CBS)
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Naïve algorithm
• The consumer connects to the bank, and sends it
  the search message.
• The bank signs this message and charges the
  consumer.
• Any agent in the path
• Checks that the message is legitimate (the
  signature on the message is valid).
• Send the bank a receipt a proof that it got
  the message (cryptographic hash signature), and
  the names of the previous and the next agents in
  the search path.

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Economic Solutions (CBS)
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Naïve algorithm (cont.)
• The bank will pay both the agent that sends the
  confirmation and the agents previous and after it
  in the search path.
• This way, each agent will have incentive both to
  pass the message forward and to send the receipt.
• Also, the bank can track the path and check that
  its legitimate.

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Economic Solutions (CBS)
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Problem of the naïve algorithm the entire
  control of the message distribution process
  passes through the bank
• Back to centralized recourse allocation

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Economic Solutions (Sprite, Zhong 2003)
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Possible distributed algorithm?
• We can let the last agent on each search path
  serve as the bank.
• It has no incentive to lie (itll only take money
  from the consumer and give it to the agents in
  the path).
• Big problems
• Coalitions
• Fake identities

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Micro-payment Scheme (Jakobsson 2003)
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Use lottery tickets
• When entering the system, each agent gets from
  the bank a secret personal random-number.
• There exists a global function F N X N - 0,1
  (known to all the agents in the system) that
  represents the lottery ticket.
• The function returns 1 (a win) with probability
  p (parameter we can choose).
• Cryptographic hash function

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Micro-payment Scheme (Jakobsson 2003)
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Use lottery tickets (cont.)
• The consumer sends the central bank the message
  it wants to distributes.
• The bank charges the consumer for the
  distribution, adds to the message a random number
  and signs it.
• Each middle-agent checks F with the random
  number and its personal key to see if it has won.

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Micro-payment Scheme (Jakobsson 2003)
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Use lottery tickets (cont.)
• An agent connects to the central bank only if it
  won
• In that case, also the previous agent and the
  next agents in the search path get paid.
• Winning is rare (probability p)
• The central bank is connected only at the
  beginning of a lookup for a resource and in a
  winning case.

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Micro-payment Scheme (Jakobsson 2003)
Cryptographic
Graph Theory
Open subjects
Resource Allocation

• Use lottery tickets (cont.)
• Agent has incentive to pass forward messages even
  if it didnt win
• The next agent might win and then itll get paid.
• Possible attack lie about the previous / next
  agents identity can be solved with statistical
  methods.

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The middle man attack problem
Graph Theory
Open subjects
Resource Allocation
Cryptographic

• Two different scenarios
• Regular P2P Framework
• An agent that wants to participate in the
  auction, may avoid forwarding the auction message
  to other agents in order to increase its
  probability of winning.
• itll loose the payment for forwarding the bid.
• An agent might publish a bid of its own in
  order to check what the other agents offer
• cost money.

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The middle man attack problem
Graph Theory
Open subjects
Resource Allocation
Cryptographic

• Two different scenarios (cont.)
• Multi-hop cellular phones framework
• A distant agent may not be able to connect to the
  manager of the auction directly.
• A middle agent can decide to avoid passing a bid
  only if its a better offer than its own bid.
• itll loose the payment for passing the bid.
• The value of the bid can be sealed from the
  middle-agent using cryptographic methods.

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The middle man attack problem
Graph Theory
Open subjects
Resource Allocation
Cryptographic

• Regular P2P framework

Hi, my nameis 32.13.1.6 and Im looking for a
resource X
56.91.0.1
32.13.1.6
22.2.8.3
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The middle man attack problem
Graph Theory
Open subjects
Resource Allocation
Cryptographic
Hi, my nameis 56.91.0.1 and Im looking for a
resource X
56.91.0.1
32.13.1.6
22.2.8.3
42
The middle man attack problem
Graph Theory
Open subjects
Resource Allocation
Cryptographic
I better bid bellow 900NIS
56.91.0.1
Resource X For 900 NISFrom 22.2.8.3
32.13.1.6
22.2.8.3
43
The middle man attack problem
Graph Theory
Open subjects
Resource Allocation
Cryptographic
Resource X For 899 NISFrom 56.91.0.1
56.91.0.1
32.13.1.6
22.2.8.3
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The middle man attack problem
Graph Theory
Open subjects
Resource Allocation
Cryptographic

• The solution
• Look at the following network
• All the agents want toparticipate in the
  auction.
• Bob and Cain needto choose (independently)whethe
  r to follow the protocoland pass David messages.

Bob
Avi
David
Cain
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The middle man attack problem
Graph Theory
Open subjects
Resource Allocation
Cryptographic

• The solution
• Can be viewed as the following game table

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The middle man attack problem
Graph Theory
Open subjects
Resource Allocation
Cryptographic

• The game has two Nash equilibriums
  Cooperate-Cooperate and Defect-Defect.
• If there are k paths from David to Avi, its
  enough that the agents in one path cooperate to
  cause all the agents that defected to loose the
  game.
• If an agent doesnt participate in the auction,
  it has a dominate strategy to cooperate.

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The middle man attack problem
Graph Theory
Open subjects
Resource Allocation
Cryptographic

• A reminder d is the parameter that represents
  the number of neighbors each agent forward the
  auction message to
• We would like to analyze the influence of the
  parameter d on the probability that there wont
  be a single agent that appears in all the paths
  between two random agents.

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Uniform Model
Graph Theory
Open subjects
Resource Allocation
Cryptographic

• The probability space
• A graph with n nodes, each node representsan
  agent.
• Each vertex has d outgoing-edges (Gd-out graph).
• The edges are drawn uniformly.
• Reasonable model for P2P frameworks (for example
  BitTorrent).
• The following calculations will be done for d3.

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Analysis Of The Uniform Model
Graph Theory
Open subjects
Resource Allocation
Cryptographic

• What is the expected number of edges that enter
  the target agent?

Probability that a random edge will enter the
target vertex
Probability that all the d edges dont enter the
target vertex
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Analysis Of The Uniform Model
Graph Theory
Open subjects
Resource Allocation
Cryptographic

• What is the expected number of edges that enter
  the target agent?

Probability that a vertex has an outgoing edge to
the target vertex
Expected number of edges that will enter the
target vertex
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Analysis Of The Uniform Model
Graph Theory
Open subjects
Resource Allocation
Cryptographic

• Assume that we know the number of vertexes with
  distance of up do j from t
• What is the expected number of vertexes of
  distance j1 from t?

t
s
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Analysis Of The Uniform Model
Graph Theory
Open subjects
Resource Allocation
Cryptographic

• Assume that we know the number of vertexes with
  distance of up do j from t, What is the expected
  number of vertexes of distance j1 from t?

X Distance j1
s
t
Y Distance j
Z - Distance less than j
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Analysis Of The Uniform Model
Graph Theory
Open subjects
Resource Allocation
Cryptographic
Probability that a vertex has an outgoing edge to
a vertex in Y
Expected value of X
X dist j1, Y dist j, Z dist
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Analysis Of The Uniform Model
Graph Theory
Open subjects
Resource Allocation
Cryptographic

• We would like to see until what stage we can
  expect the number of vertexes of distance j to
  be at least twice the number of vertexes of
  distance j-1.

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Analysis Of The Uniform Model
Graph Theory
Open subjects
Resource Allocation
Cryptographic
Lemma Let T1,T2,,Tn be a sequence of numbers
such that for each i, Ti1.5Ti-1. Then
Proof By induction.
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Analysis Of The Uniform Model
Graph Theory
Open subjects
Resource Allocation
Cryptographic
If we assume that in each step the number of
vertexes of distance j is at least 1.5 times the
number of vertexes of distance j1 well get that
Thus by our assumption the expected value of X
X dist j1, Y dist j, Z dist
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Analysis Of The Uniform Model
Graph Theory
Open subjects
Resource Allocation
Cryptographic

• What is the probability that the number of
  vertexes doesnt multiply by 1.5 (at least)?

Chernoff bound (for YO(n))
For small Y the situation is less promising 5
of the vertexes in the graph dont have incoming
edges!
Still we can get reasonable results using the
fact that Y is an integer.
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Analysis Of The Uniform Model
Graph Theory
Open subjects
Resource Allocation
Cryptographic

• Conclusions
• if the number of vertexes of distance j from t is
  smaller than 0.1n then the number of vertexes of
  distance j1 from t is at least 1.5 times the
  previous distance (with very high probability).
• Best strategy for d3 is TTLlog(0.1n).

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Analysis Of The Uniform Model
Graph Theory
Open subjects
Resource Allocation
Cryptographic

• Conclusions (cont.)
• Choosing TTLlog(0.1n)2 will promise (with good
  probability) that no agent will be placed on all
  the paths between s to a random vertex t.
• We can repeat the calculations for other values
  of d and their appropriate TTL.
• Solution is robust against coalitions
  withconstant size.
• Results are supported by simulations

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Non Uniform Model
Graph Theory
Open subjects
Resource Allocation
Cryptographic

• The probability space
• A graph with n nodes, each node representsan
  agent.
• Agents are located uniformly on L X L square
• L a parameter.
• Each vertex has d outgoing-edges (Gd-out graph).

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Non Uniform Model
Graph Theory
Open subjects
Resource Allocation
Cryptographic

• The probability space (cont.)
• Edge between vertexes u and v is drawn with
  probability
• Where
• Reasonable model for cellular phones network (in
  reality using different parameters).

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Non Uniform Model
Graph Theory
Open subjects
Resource Allocation
Cryptographic

• In order to reach all the vertexes in a search we
  need to set TTLO(n) in worst case.
• Turns out that this also holds for our uniform
  model

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Non Uniform Model
Graph Theory
Open subjects
Resource Allocation
Cryptographic

• First, bound E(D)
• Get maximum when an agent is located in the
  middle of the square, and minimum on the edge.
• With high probability
• Next, we would like to calculate the probability
  of a connection between vertexes s and t that
  goes through TTL vertexes.

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Non Uniform Model
Graph Theory
Open subjects
Resource Allocation
Cryptographic

• Draw rectangles around the straight line between
  s and t.

s

• Since the probability for an edge drops
  exponentially, consider only paths from s
  through vertexes inside the rectangles to t.

t
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Non Uniform Model
Graph Theory
Open subjects
Resource Allocation
Cryptographic

• Calculate the probability by
• Calculate the expected number of vertexes in each
  rectangle.
• Calculate the probability for a path to any of
  the vertexes in the rectangle.
• Problem miss paths

s
t
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Open Subjects
Open subjects
Resource Allocation
Cryptographic
Graph Theory

• Correct the analysis of the non-uniform model
• Try to use circles that represent the error rate
  instead of rectangles.
• Formalize the robustness of the model against
  random coalitions.
• Gather formal simulation results.
• Try to deal with the ability of an agent to
  conduct an auction for predicting the current
  best offer.

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Thanks!!!