Strategic Negotiation and Cooperation Among Autonomous Agents

1
Automated Negotiation
Sarit Kraus Bar-Ilan, Israel UMD,USA
2
Plan of the course

• Introduction
• Rules of Encounters
• Strategic Negotiation
• Auctions
• protocols
• strategies
• Argumentation

3
Machines Controlling and Sharing Resources

• Electrical grids (load balancing)
• Telecommunications networks (routing)
• PDAs (schedulers)
• Shared databases (intelligent access)
• Traffic control (coordination)

4
Broad Working Assumption

• Designers (from different companies, countries,
  etc.) come together to agree on standards for how
  their automated agents will interact (in a given
  domain)
• Discuss various possibilities and their
  tradeoffs, and agree on protocols, strategies,
  and social laws to be implemented in their
  machines

5
Attributes of Standards

• Efficient Pareto Optimal
• Stable No incentive to deviate
• Simple Low computational and communication
  cost
• Distributed No central decision-maker
• Symmetric Agents play equivalent roles

Designing protocols for specific classes of
domains that satisfy some or all of these
attributes
6
Distributed Artificial Intelligence (DAI)

• Distributed Problem Solving (DPS) Centrally
  designed systems, built-in cooperation, have
  global problem to solve
• Multi-Agent Systems (MAS) Group of
  utility-maximizing heterogeneous agents
  co-existing in same environment, possibly
  competitive

7
Phone Call Competition Example

• Customer wishes to place long-distance call
• Carriers simultaneously bid, sending proposed
  prices
• Phone automatically chooses the carrier
  (dynamically)

ATT
Sprint
MCI
0.20
0.23
0.18
8
Best Bid Wins

• Phone chooses carrier with lowest bid
• Carrier gets amount that it bid

MCI
Sprint
ATT
0.20
0.23
0.18
9
Attributes of the Mechanism

• Distributed
• Symmetric
• Stable
• Simple
• Efficient

Carriers have an incentive to invest effort in
strategic behavior
ATT
MCI
Sprint
0.20
0.23
0.18
10
Best Bid Wins, Gets Second Price

• Phone chooses carrier with lowest bid
• Carrier gets amount of second-best price

MCI
Sprint
ATT
0.20
0.23
0.18
11
Attributes of the Mechanism

• Distributed
• Symmetric
• Stable
• Simple
• Efficient

Carriers have no incentive to invest effort in
strategic behavior
ATT
MCI
Sprint
0.20
0.23
0.18
12
Database Domain
Common Database
TOD
13
Negotiation
A discussion in which interested parties
exchange information and come to an agreement.
Davis and Smith, 1977

• Two-way exchange of information
• Each party evaluates information from its own
  perspective
• Final agreement is reached by mutual selection

14
Game Theory--Short Introduction

• Game theory is the study of decision making in
  multi-person situations where the outcome depends
  on everyones choice.
• In Decision Theory and the theory of competitive
  equilibrium from economics the other participants
  actions are considered as an environmental
  parameter. The effect of the of the
  decision-makers actions on the other
  participants is not taken into consideration.

15
Describing a Game

• Essential elements players, actions,
  information, strategies, payoffs, outcome, and
  equilibria.
• Ways to present social interactions as a game
• Extensive formthe most complete description.
• Strategic form many details are omitted.
• Coalitional form binding agreements exist.

16
Example of two players game
dindia
op
deal
0 2-
1 2
deal
Dsikh
3- 0
2- 1-

blow
17
Nash Equilibrium

• An action profile is an order set a(a1,,aN) of
  one action for each of the N players in the game.
• An action profile a is a Nash Equilibrium (Nash
  53) of a strategic game, if each agent j does
  not have a different action yielding an outcome
  that it prefers to that generated when chooses
  aj, given that every other player I chooses ai.

18
2,1-
blow
3-,5
op
sik
2,5
yes
Ind
2,1-
3,4
op
blow
yes
0.4
sik
Ind
dealH
dealH
c
1,4
0.6
Ind
sik
dealH
Ind
dealH
dealH
1,4
dealH
sik
op
op
4- ,4
-3,0-
19
Rules of Encounter
Jeffrey S. Rosenschein Gilad Zlotkin
20
Domain Theory

• Task Oriented Domains
• Agents have tasks to achieve
• Task redistribution
• State Oriented Domains
• Goals specify acceptable final states
• Side effects
• Joint plan and schedules
• Worth Oriented Domains
• Function rating states acceptability
• Joint plan, schedules, and goal relaxation

21
Postmen Domain
Post Office
TOD
a
?
?
c
b
?
?
f
?
e
d
22
Database Domain
Common Database
TOD
23
Fax Domain
faxes to send
TOD
a
c
b
Cost is only to establish connection
f
e
d
24
Slotted Blocks World
SOD
3
1
2
3
1
2
25
The Multi-Agent Tileworld
WOD
hole
agents
tile
B
A
2
2
5
5
2
obstacle
4
3
2
26
Task Oriented Domain (TOD)

• A tuple lt T, A, c gt where
• T is the set of all possible tasks
• A A1 , , An is a list of agents
• c is a monotonic function c 2T ? ?

An encounter is a list T1 ,, Tn of finite sets
of tasks from T such that agent Ak needs to
achieve all the tasks in Tk (also called agent
Aks goal).
27
Building Blocks

• Domain
• A precise definition of what a goal is
• Agent operations
• Negotiation Protocol
• A definition of a deal
• A definition of utility
• A definition of the conflict deal
• Negotiation Strategy
• In Equilibrium
• Incentive-compatible

28
Deal and Utility in two-agent TOD

• Deal ? is a pair (D1, D2) D1 ? D2 T1 ? T2
• Conflict deal ? (T1, T2)
• Utilityi(?) Cost(Ti) Cost(Di)

29
Negotiation Protocols

• Agents use a product-maximizing negotiation
  protocol (as in Nash bargaining theory)
• It should be a symmetric PMM (product maximizing
  mechanism)
• Examples 1-step protocol, monotonic concession
  protocol

30
Building Blocks

• Domain
• A precise definition of what a goal is
• Agent operations
• Negotiation Protocol
• A definition of a deal
• A definition of utility
• A definition of the conflict deal
• Negotiation Strategy
• In Equilibrium
• Incentive-compatible

31
Negotiation with Incomplete Information
Post Office
?
a
b
h
1
g
c

• What if the agents dont know each others
  letters?

f
e
d
?
?
2
1
32
1 Phase Game Broadcast Tasks
Post Office
?
a
b
h
1
g
c

• Agents will flip a coin to decide who delivers
  all the letters.

e
f
d
?
?
2
1
33
Hiding Letters
Post Office
?
a
b
h
(1)
(hidden)
g
c
e
f
d
They then agree that agent 2 delivers to f and e.
?
?
2
1
34
Another Possibility for Deception
Post Office
a
c
b
?

• They will agree to flip a coin to decide who goes
  to b and who goes to c.

?
1, 2
1, 2
35
Phantom Letter
Post Office
b, c, d
a
b, c
c
?
b
1, 2

• They agree that agent 1 goes to c.

?
1, 2
?
d
1 (phantom)
36
Negotiation over Mixed Deals

• Mixed deal (D1, D2) p
• The agents will perform (D1, D2) with probability
  p, and the symmetric deal (D2, D1) with
  probability 1 p

Theorem With mixed deals, agents can always
agree on the all-or-nothing deal
37
Hiding Letters with MixedAll-or-Nothing Deals
Post Office
?
a
b
h
(1)
(hidden)
g
c

• They will agree on the mixed deal where agent 1
  has a 3/8 chance of delivering to f and e.

e
f
d
?
?
2
1
38
Phantom Letters with Mixed Deals
Post Office
b, c, d
a
b, c
c
?
b

• They will agree on the mixed deal where A has 3/4
  chance of delivering all letters, lowering his
  expected utility.

1, 2
?
1, 2
?
d
1 (phantom)
39
Sub-Additive TODs

• TOD lt T, A, c gt is sub-additive if for all finite
  sets of tasks X, Y in T we have
• c(X ? Y) ? c(X) c(Y)

40
Sub-Additivity
X
Y
c(X ? Y) ? c(X) c(Y)
41
Sub-Additive TODs

• The Postmen Domain, Database Domain, and Fax
  Domain are sub-additive.

The Delivery Domain (where postmen dont have
to return to the Post Office) is not sub-additive.
?
?
42
Incentive Compatible Mechanisms
a
?
a
b
h
(1)
(hidden)
?
g
c
Sub-Additive
1, 2
?
1, 2
e
f
d
Hidden
Phantom
?
?
?
1
(phantom)
Pure
L
L
2
1
A/N
T/P
T
Mix
L
T/P
Theorem For all encounters in all sub-additive
TODs, when using a PMM over all-or-nothing deals,
no agent has an incentive to hide a task.
43
Decoy Tasks
Decoy tasks, however, can be beneficial even with
all-or-nothing deals
Sub-Additive
Hidden
Phantom
Decoy
Pure
L
L
L
A/N
T
T/P
L
Mix
L
T/P
L
44
Concave TODs

• TOD lt T, A, c gt is concave if for all finite sets
  of tasks Y and Z in T , and X ? Y, we have
• c(Y ? Z) c(Y) ? c(X ? Z) c(X)

Concavity implies sub-additivity.
45
Concavity
Z
X
Y

• The cost Z adds to X is more than the cost it
  adds to Y.(Z - X is a superset of Z - Y)

46
Concave TODs

• The Database Domain and Fax Domain are concave
  (not the Postmen Domain, unless restricted to
  trees).

Z
1
?
This example was not concave Z adds 0 to X, but
adds 2 to its superset Y (all blue nodes).
?
2
?
1
X
1
2
?
?
?
1
1
47
Three-Dimensional Incentive Compatible Mechanism
Table
Theorem For all encounters in all concave TODs,
when using a PMM over all-or-nothing deals, no
agent has any incentive to lie.
Concave
Hidden
Phantom
Decoy
Pure
L
L
L
A/N
T
T
T
Mix
L
T
T
Sub-Additive
Hidden
Phantom
Decoy
Pure
L
L
L
A/N
T
T/P
L
Mix
L
T/P
L
48
Modular TODs

• TOD lt T, A, c gt is modular if for all finite sets
  of tasks X, Y in T we have
• c(X ? Y) c(X) c(Y) c(X ? Y)

Modularity implies concavity.
49
Modularity
X
Y

• c(X ? Y) c(X) c(Y) c(X ? Y)

50
Modular TODs

• The Fax Domain is modular (not the Database
  Domain nor the Postmen Domain, unless restricted
  to a star topology).

Even in modular TODs, hiding tasks can be
beneficial in general mixed deals.
51
Three-Dimensional Incentive Compatible Mechanism
Table
Modular
H
P
D
Pure
L
T
T
Concave
A/N
T
T
T
H
P
D
Mix
L
T
T
Pure
L
L
L
A/N
T
T
T
Sub-Additive
H
P
D
Mix
L
T
T
Pure
L
L
L
A/N
T
T/P
L
Mix
L
T/P
L
52
Related Work

• Coalitions Formations Shehory, Sandholm
• Mechanism designEphrati, Kraus, Tennenholtz
• Other models of negotiation Sycara, Durfee,
  Lesser, Gasser, Gmytrasiewicz, Jennings
• Consensus mechanisms, voting techniques, economic
  models Ephrati, Wellman, Sandholm

53
Conclusions

• By appropriately adjusting the rules of encounter
  by which agents must interact, we can influence
  the private strategies that designers build into
  their machines
• The interaction mechanism should ensure the
  efficiency of multi-agent systems

Rules of Encounter
Efficiency
54
Conclusions

• To maintain efficiency over time of dynamic
  multi-agent systems, the rules must also be
  stable
• The use of formal tools enables the design of
  efficient and stable mechanisms, and the precise
  characterization of their properties

Stability
Formal Tools
55
Strategic Negotiation

• Collaborators Jon Wilkenfeld, Rina
  Schwartz-Azoulay, Orna Shechter, Esti Freitsis

56
DAI Overview

• AI
• DAI
• DPS MA
• strategic
  negotiation

57
Strategic Negotiation Model

• Model of alternative offers (Rubinstein) which
  takes negotiation time into consideration
  reduces negotiation time.
• During the strategic-negotiations agents
  communicate their respective desires to reach
  mutually beneficial agreement.
• The model provides a unified to many problems.

58
Structure of the Negotiation

• There are N self motivated agents, randomly
  designated 1,2,...
• All the agents negotiate to reach an agreement.
• The negotiation process may include several
  equidistant iterations 0,1,2 ?Time and can
  continue forever. In each time period t, agent
  j(t) t mod N makes an offer.

59
Structure of the Negotiation - cont.

• The other agents respond simultaneously YES4
  or NO8 or OPTM.
• If the offer was accepted4 by all the agentsthe
  last offer is implemented.
• If at least one agent opts outM a conflict
  occurs.
• Otherwise (the offer was rejected8 by at least
  one agent), the negotiation proceeds to period
  t1. ???

60
Applications

• Information servers (large databases).
• Resources sharing.
• Tasks distribution.
• Computer assisted negotiation.
• Union/management negotiation.

61
Negotiation on data allocation in multi-server
environment
62
Environment Description

• There are several information servers. Each
  server is located at a different geographical
  area.
• Each server receives queries from the clients in
  its area, and sends documents as responses to
  queries. These documents can be stored locally,
  or in another server.

63
Environment Description
the query
document/s
distance
serverj
serveri
a query
the document/s
a client
area j
area i
64
Environment Description - cont.

• The information is clustered in datasets
  (corresponding to file, fragment, etc.)
• Each new dataset has to be allocated to one of
  the servers by mutual agreement among the
  servers.
• Each server wants to store the datasets in a
  location which reduces its communication and
  storage costs.
• A negotiation session is initiated when a set of
  new datasets arrive.

65
Motivation

• Cooperation among servers with similar areas of
  interest (e.g., Web servers).
• The Data and Information System component of the
  Earth Observing System (EOSDIS) of NASAA
  distributed knowledge system which supports
  archival and distribution of data at multiple and
  independent servers.

66
Motivation - cont.

• Each data collection, or file, is called a
  dataset. The datasets are huge, so each dataset
  has only one copy.
• The current policy for data allocation in NASA is
  static old datasets are not reallocated each
  new dataset is located by the server with the
  nearest topics (defined according to the topics
  of the datasets stored by this server).

67
Related Work -File Allocation Problem

• The original problemHow to distribute files
  among computers, in order to optimize the system
  performance.
• Our problemHow can self-motivated servers
  decide about distribution of files, when each
  server has its own objectives.

68
Basic Definitions

• SERVERS the set of the servers.
• DATASETS the set of datasets (files) to be
  allocated.
• Allocationa mapping of each dataset to one of
  theservers. The set of all possible allocation
  is denoted by Allocs.
• U the utility function of each server.

69
The Conflict Allocation

• If at least one server opts outM of the
  negotiation, then the conflict allocation
  conflict_alloc is implemented.
• We consider the conflict allocation to be the
  static allocation. (each dataset is stored in the
  server with closest topics).

70
Utility Function

• Userver(alloc,t) specifies the utility of server
  from alloc?Allocs at time t.
• It consists of
• The utility from the assignment of each dataset.
• The cost of negotiation delay.
• Userver(alloc,0) S Vserver(x,alloc(x)).
• x?DATASETS

71
Parameters of utility

• query price payment for retrieved docoments.
• usage(ds,s) the expected number of documents of
  dataset ds from clients in the area of server
  s.
• storage costs, retrieve costs, answer costs.

72
Cost over time

• Cost of communication and computation time of the
  negotiation.
• Loss of unused information new documents can not
  be used until the negotiation ends.
• Datasets usage and storage cost are assumed to
  decrease over time, with the same discount ratio
  (p-1).
• Thus, there is a constant discount ratio of the
  utility from an allocation Userver(alloc,t)d
  tUserver(alloc,0) - tC.

73
Assumptions

• Each server prefers any agreement over
  continuation of the negotiation indefinitely.
• The utility of each server from the conflict
  allocation is always greater or equal to 0.
• OFFERS - the set of allocations that are
  preferred by all the agents over opting out.

74
Equilibrium

• Nash equilibriumA strategy profile p is a Nash
  Equilibriumif no player has a different strategy
  yielding an outcome that he prefers to that
  generated when it chooses pi.
• Subgame Perfect EquilibriumIf the strategy
  profile induced in every subgame is a Nash
  Equilibrium of this subgame.

75
Negotiation Analysis - Simultaneous Responses

• Simultaneous responsesA server, when
  responding, is not informed of the other
  responses.
• TheoremFor each offer x ? OFFERS, there is a
  subgame-perfect equilibrium of the bargaining
  game, with the outcome x offered and unanimously
  accepted in period 0.

76
Choosing the Allocation

• The designers of the servers can agree in advance
  on a joint technique for choosing x
• giving each server its conflict utility.
• maximizing a social welfare criterion
• the sum of the servers utilities.
• or the generalized Nash product of the servers
  utilities P (Us(x)-Us(conflict)).

77
Choosing the Allocation - cont.

• The problem of finding an optimal allocation is
  NP-complete (a reduction from the multiprocessors
  scheduling).
• When finding x is intractable, we suggest the
  following protocol
• each server will search for an allocation
• the allocation which maximizes the predefined
  social welfare criterion will be chosen.

78
Search Methods

• We have implemented the following algorithms
• A backtracking algorithmSearching the search
  space of the allocation problem.
• A random restart hill-climbing algorithmStarts
  with a random allocation and tries to improve it.
• A genetic algorithmSearching by simulating an
  evolution process. Each individual represents an
  allocation. The algorithm involves reproduction,
  crossover and mutation of individuals.

79
Experimental Evaluation

• How do the parameters influence the results of
  the negotiation?
• vcost(alloc) the variable costs due to an
  allocation (excludes storage_cost and the gains
  due to queries).
• vcost_ratio the ratio of vcosts when using
  negotiation, and vcosts of the static allocation.

80
Effect of Parameters on The Results

• As the number of servers grows, vcost_ratio
  increases (more complex computations) L.
• As the number of datasets grows, vcost_ratio
  decreases (negotiation is more beneficial) J.
• Changing the mean usage did not influence
  vcost_ratio significantlyK, but vcost_ratio
  decreases as the standard deviation of the usage
  increasesJ.

81
Influence of Parameters - cont.

• When the standard deviation of the distances
  between servers increases, vcost_ratio
  decreasesJ.
• When the distance between servers increases,
  vcost_ratio decreasesJ.
• In the domains tested,
• answer_cost ? vcost_ratio ? L.
• storage_cost ? vcost_ratio ? L.
• retrieve_cost ? vcost_ratio ? J.
• query_price ? vcost_ratio ? J.

82
Social Criteria

• We studied the effect of the choice of the social
  welfare criterion on the results.
• We compare the following criteria
• Sum of agents utilities.
• Product of agents utilities.
• Maximizing the sum achieves lower vcost_ratio.
• Maximizing the product achieves lower dispersion
  of the agents utilities.

83
Incomplete Information

• Each server knows
• The usage frequency of all datasets, by clients
  from its area.
• The usage frequency of datasets stored in it, by
  all clients.

84
Incomplete Information - cont.

• A revelation mechanism
• First, all the servers report simultaneously all
  their private information
• for each dataset, the past usage of the dataset
  by this server.
• for each server, the past usage of each local
  dataset by this server.
• Then, the negotiation proceeds as in the complete
  information case.

85
Incomplete Information - cont.

• LemmaThere is a Nash equilibrium where each
  server tells the truth about its past usage of
  remote datasets, and the other servers usage of
  its local datasets.
• Lies concerning details about local usage of
  local datasets are intractable.

86
Summary negotiation on data allocation

• We have considered the data allocation problem in
  a distributed environment.
• We have presented the utility function of the
  servers, which expresses their preferences.
• We have proposed using a negotiation protocol for
  solving the problem.
• For incomplete information situations, a
  revelation process was added to the protocol.

87
Negotiations in the pollution sharing problem

• Collaborator Esti Freitsis

88
Environment Description

• There are some closely grouped plants in an
  industrial region.
• Each plant can produce several types of products.
• Each plant has a utility function (profit).
• There are several types of pollution substances.
• Each plant has norms, restricting maximal
  emission of each polluting substance that it
  emits. The pollution always has to be below these
  norms. We refer to the situation when only these
  norms have to be carried out as usual
  circumstances.

89
Special circumstances

• Sometimes there is a need to reduce pollution for
  some period because of external factors such as
  weather (high humidity, wind towards residential
  area). In this case plants receive new norms. We
  refer to this situation as special circumstances.

90
Current solution

• Current solution each plant reduce pollution
  according to the new norms.
• Disadvantage for one plant it is less costly to
  reduce one substance while for another it is less
  costly to reduce another substance.

91
Negotiations

• Our solution plants negotiate to reach
  beneficial agreements about the emission of what
  substances and by which percent each of them must
  be reduced.
• The conflict solution following the new norms.
• We consider complete information situations.

92
Negotiations Protocols

• Simultaneous responsesan agent responding to an
  offer is not informed of the other responses.
• Sequential responses an agent responding to an
  offer is informed of the responses of the
  preceding agents (assuming that the agents are
  ordered).

93
Negotiations strategies for simultaneous responses

• As in the data allocation case
• For each possible agreement x that is better to
  all the plants than the conflict solution there
  is a subgame-perfect equilibrium of the
  bargaining game, with the outcome x offered and
  unanimously accepted in period 0.

94
Negotiations strategies for sequential responses

• Assumption there is a time period, T where
  negotiation cannot continue anymore. In T the
  conflict allocation is implemented.
• Perfect equilibrium by backward induction
• At T-1 if negotiations hasnt ended, AT-1
  suggests the best agreement to itself which is
  better to all agents than the conflict solution
  (denoted by OT-1 ) the other agents accept.
• At T-2, AT-2 suggests the best agreement to
  itself which is better to all agents than the
  conflict solution and OT-1 (denoted by OT-2).
  The other agents accept.
• By induction, at the first time period A0 O0 the
  others accept.

95
Assumptions about the environment

• Profit is a linear function of the number of
  items of each product produced by the plant
• Pollution is a linear function of the number of
  items of each product produced.

96
Techniques which were checked

• Strategic negotiations
• Sequential responses backtracking
• Simultaneous response Maximization of the sum
  with guaranties of default profit
• Simplex method - method for linear optimization
• Nash Product
• Praxis - method for multi-variable nonlinear
  function minimization.
• Hill Climbing

97
Simulation Parameters

• Number of plants is varied from 5 to 20.
• Number of pollution types is varied from 5 to 20.
  For each product pollution of some type is
  produced with probability 1/2.
• Each plant produces Max_prod different types of
  products. Max_prod is varied from 5 to 20.
  Pollution and profit per item of product and
  pollution constraints are set randomly.
• Results Average of 25 simulation runs.

98
Plants utility as the function of the number of
plants
99
Standard Deviation as the function of the number
of plants
100
Computation time as a function of number of plants
101
Plants utility as the function of the number of
pollution substances
102
Standard deviation as the function of the number
of pollution substances
103
Computation time as a function of the number of
pollution substances
104
Plants utility as a function of the number of
products
105
Standard deviation as a function of the number
of products
106
Computation time as the function of the number of
products
107
Computation time as a function of the number of
products
108
Conclusions

• Maximizing the sum yields the highest average
  utility, but also the highest standard deviation
  requires agreement between the designers on
  selecting a solution.
• Backward induction yields a reasonable average
  utility with low standard deviations and no need
  for designers agreement on detailed protocol.
• On going work incomplete information.

109
Sharing Resources Through Negotiation

• Joint resource public communication system
  satellite
• Agents self motivated.
• Environment no central controller.

110
Environment Description

• Two agents must share a joint resource the
  resource can only be used by one agent at a time.
  No central controller.
• One agent (A) is using the resource, and the
  second (W) wants to use it too.
• The agents negotiate to reach an agreement a
  schedule that divides the usage of the resource
  lts,tgt.

111
Environment Description -cont

• A continues to use the resource as the
  negotiation proceeds A gains over time.
• W is not able to use the resource W loses over
  time.
• Opting out causes damage to the resourceboth
  agents wait q time steps.
• Additional option an agent can leave the
  negotiation.

112
Applying the strategic model

• We developed a detailed utility function for the
  agents (U_A U_W). Parameters type of goal,
  dead-lines, costs of negotiation, gains from
  goal, etc.
• Main factor in the negotiation the best
  agreement for A, which is still better for W than
  Opting out (O_n).

113
Perfect equilibrium strategies

• O_n depends on the specific situation we proved
  lemmas which specify the value of O_n as a
  function of the utility function parameters.
• Complete information Negotiation ends at most
  after one step with an agreement, or W leaves.
• The strategies are simple.

114
Experiments Using MINUET
Agent 2
Agent 1
Send request lt5,3gt
Working on goal 102

Receive request lt5,3gt
Resources
1001 - free 1002 - busy
115
Experiments Results
Nego.
EDF
Metric
Utility score
91
91
Abandon goals
9.6
8.4
21.2
Nego./Alter.
15.5
116
Summary

• A strategic model of negotiation, taking the
  passage of time into account.
• We consider wide range of situationscomplete
  /incomplete informationNgt2 agentsagents lose
  over time/some lose and some gain over time

117
Summary--cont.

• The model was applied to different domains.
• We found simple and stable strategies.
• Negotiation ends without delay.