This document provides an overview of a presentation on automated negotiation given by Takayuki Ito from Nagoya Institute of Technology in Japan. The presentation covered four parts: an introduction to automated negotiation, bargaining approaches using game theory, multi-issue negotiation using heuristic approaches, and an automated negotiation agent competition. Ito discussed key concepts in negotiation like negotiation protocols, strategies, preferences, and Pareto optimality. For bargaining approaches, he summarized Nash's cooperative bargaining solution and Rubinstein's alternating offers model in non-cooperative games.
Difference Between Search & Browse Methods in Odoo 17
Automated Negotiation
1. AgentSchool 2013 at Dunedin, New Zealand. (Dec. 1, 2013)
Automated Negotiation
Takayuki Ito
Dept. of Computer Science / School of Techno-Business Administration
Nagoya Institute of Technology, Japan
ito@nitech.ac.jp
2. Takayuki Ito
Associate Professor, Nagoya Institute of Technology
E-mail : ito@nitech.ac.jp
http:/
/www.itolab.nitech.ac.jp/~ito/
2000 Dr. of Engineering, Nagoya Institute of Technology
1999 Research fellow, Japan Society for the Promotion of Science (JSPS).
2000 Visiting Researcher, USC/ISI.
2001 Assoc. Professor, Japan Advanced Insti. of Sci. & Tech. (JAIST).
2003 Assoc. Professor, Dept. of CSE, Nagoya Institute of Technology.
2005 Visiting Scholar, Computer Science, Harvard University.
2005 Visiting Researcher, MIT Sloan School of Management.
2006-Now Assoc. Prof., School of Techno-Business Admin., Nagoya Institute of Technology.
2008 Visiting Scientist, Center for Collective Intelligence, MIT Sloan School of Management.
2010 JST PREST researcher (super challenging type)
2011 Japanese Cabinet Office’ NEXT fund Principal Investigator
s
2013 AAMAS 2013 Program Chair
3. Area :
Computer Science, Artificial Intelligence, Multi-Agent Systems,
Automated Negotiations, Auction Theory, Mechanism Design, Smart
City, Smart Grid, etc.
Main achievements:
•2013 JSPS Award
•Prizes for Science and Technology, The Commendation for Science and
Technology by the Minister of Education, Culture, Sports, Science and
Technology, 2013.
•The NEXT Funding Program from the Japanese Cabinet Office
•The Young Scientists' Prize, The Commendation for Science and Technology
by the Minister of Education, Culture, Sports, Science and Technology, 2007.
•Nagao Special Researcher Award, IPSJ 2007
•Best Paper Award, The Fifth International Joint Conference on Autonomous
Agents and Multi-Agent Systems (AAMAS2006, 1/553), 2006.
•2005 Best Paper Award, Japan Society for Softoware Science and Technoglogy.
•2004 IPA Exploratory Software Creation Project, Super Creator Award.
4. Today’s schedule
•
Part 1: Introduction to Automated Negotiation
•
Part 2: Bargaining : Game Theoretic Approaches
•
Part 3: Multi-issue Negotiation : Heuristic Approaches
•
Part 4: Automated Negotiating Agent Competition
6. What is Negotiation?
•
Negotiation is a form of interaction in which a group of agents
with conflicting interests try to come to a mutually acceptable
agreement over some outcome.
•
The outcome is typically represented in terms of the allocation
of resources (commodities, services, time, money, CPU cycles,
etc.)
•
Agents’ interests are conflicting in the sense that they cannot be
simultaneously satisfied, either partially or fully (= trade-off)
•
Automated negotiation would be negotiation that is automated
with some computation support, e.g., fully automated negotiation
among computational agents, partially automated negotiation with
a computational mediator with human negotiators, etc.
7. What is Negotiation?
“Negotiation can be seem as a distributed search
through a space of potential agreements.” [Jennings
2001]
[Jennings 2001] N. R. Jennings, P. Faratin, A. R. Lomuscio, S. Parsons, C. Sierra and M.
Wooldridge, Automated Negotiation: Prospects, Methods and Challenges, International Journal of
Group Decision and Negotiation, 10(2):199-215, 2001
8. What is Negotiation?
issue (attribute) 2
“Negotiation can be seem as a distributed search
through a space of potential agreements.” [Jennings
2001]
issue (attribute) 1
This negotiation space can be seen as 2 dimensional
[Jennings 2001] N. R. Jennings, P. Faratin, A. R. Lomuscio, S. Parsons, C. Sierra and M.
Wooldridge, Automated Negotiation: Prospects, Methods and Challenges, International Journal of
Group Decision and Negotiation, 10(2):199-215, 2001
9. bilateral negotiation
•
We focus on “bilateral negotiations” , that is,
negotiations involving two agents
•
Multi-party negotiations refers negotiations involving
many of ag ents. In g en e ral, auctio ns an d
m e chan isms can been seen as mu lti-party
negotiations. Also, some researchers now focusing on
heuristic-based one.
10. Main ingredients of Negotiation
1. The negotiation object, which defines the set of
possible outcomes
2. The agents conducting the negotiation
3. The protocol according to which agents search for a
specific agreement
4. The individual strategies that determine the agents’
behavior based on their preferences over the
outcomes
11. Negotiation outcomes
•
There are many ways to define the outcomes.
•
Also called as agreements or deals.
•
Characteristics
•
Continuous or discrete
•
Single issue or multiple issues
12. Example 1
•
1.0 litter milk between Alice and Bob
• The issue is (dividing) milk, that is
single issue & continuous
• The possible outcome can be
represented as a number in interval
[0,1.0].
• One possible outcome is 0.2l for Alice
and 0.8l for Bob.
13. Example 2
•
Parking slot 1 and 2 for Charles and Daniel
• The issue is 3 parking slots, that is single issue &
discrete
• The possible outcome can be represented as
assignment of the parking slot
• One possible outcome is slot 1 for Charles and slot
2 for Bob
14. Example 3
•
Buying a house between Seller and Buyer
• The issues are price, design, and place (3 issues),
that is multiple issue (& discrete and continuous)
• The possible outcome can be represented as a
tuple of values of the issues.
• One possible outcome is ($150,000, modern,
Dunedin)
15. Preferences
•
Two different agents prefer different allocations of the
resources.
•
Preference representation or “ordinal utility”
•
Binary preference relation :
•
means outcome o1 is at least as good as
outcome o2 for agent i
•
means
and it is not a case that
16. Utility function
•
One way to define preference relation for agent i is to
define a utility function
to assign real number to
each possible outcome (cardinal utility <-> ordinal utility).
•
The utility function
have
if
•
In multi-issue negotiation, it is possible to have a multiattribute utility function which maps a vector of attribute
values to a real number.
•
A rational agent attempts to reach a deal that maximizes
his/her utility.
represents the relation
.
when we
17. Example
•
There are many definitions of utility functions
•
Example:
•
A quasi-linear utility function
•
The utility, ui, for an item, x, is defined as, vi,
the value of it minus the cost, ci, to acquire it.
•
ui(x) = vi(x) - ci(x)
18. Protocols
•
Given a set of the agents and their preferences/
utilities, we need a protocol.
•
A protocol is rules of interaction for enabling the
agents to search for an agreement
•
One-shot or repeated
•
There are many protocols proposed so far.
•
Example: Alternative-offer protocol (we will see this
in the later section), auction, mediator, etc.
19. Strategy
•
Given a set of agents, their preferences, and an
agreed protocol, the final ingredient is the agent’s
strategy
•
The strategy may specify what offer to make next
or what information to reveal (truthfully or not).
•
A rational agent’s strategy must aim to achieve the
best possible outcome for him/her.
•
Game-theory is analyzing agents’ strategic behavior.
20. Pareto Optimality
•
At the Pareto optimal situation, without reducing
another agent’s utility, there is no agent who can
increase his/her utility.
•
An outcome d is Pareto efficient (Pareto optimal) if there
is no outcome that is better for at least one agent and
not worse for the other agent
•
There is no game outcome d’ for agents A and B s.t.
[ uA(d’) ≥ uA(d) and uB(d’) ≥ uB(d) ]
and [ uA(d’) > uA(d) or uB(d’) > uB(d) ]
21. Example: Cake division
•
When dividing one cake, it is Pareto optimal if the entire cake is
completely divided and allocated to members, and there is no remaining
pieces
•
Pareto optimal does not mean fairness
Blue or yellow can increase his/
her cake without reducing
opponent’s cake!
Without reducing opponent’s utility, there is no
agent who can increase his/her cake.
22. Approaches
•
•
•
Bargaining: Game Theoretic Approaches : Part 2
• How game theory can be used to analyze negotiation.
• Cooperative game or non-cooperative game
• Assumptions:
• Rules of the game, preferences & beliefs of all players are common
knowledge
• A2: Full rationality on the part of all players (=unlimited
computation)
• Preferences encoded in a (limited) set of player types (utility functions)
• Closed systems, predetermined interaction, small sized games
Heuristic Approaches (AI approach): Part 3
• No common knowledge or perfect rationality assumptions needed
• Agent behaviour is modeled directly
• Suitable for open, dynamic environments
• Space of possibilities is very large
Argumentation Approaches : out of scope in this lecture
• Based on formal logics of dialogue games
23.
24. Cooperative game or Noncooperative game
•
There are two ways to model bilateral negotiations : using
cooperative game or using non-cooperative game
•
•
•
In cooperative games, agreements are enforceable or
binding, and it’s possible for the agents to negotiate
outcomes that are mutually beneficial.
In non-cooperative game, the agents are self-interested
and thus they have incentive to deviate from an agreement
to improve his/her utility
Thus, a same game would have the different outcome between
cooperative games and non-cooperative games
27. Cooperative game based bargaining
•
Most of work on cooperative models of bargaining followed from the
seminal work of Nash [Nash1950, Nash1953]
•
[Nash1950] J.F. Nash, The bargaining problem, Econometrica,
18:155-162, 1950
•
[Nash1953] J.F. Nash, Two-person cooperative game, Econometrica,
21:128-140, 1953
•
Nash analyzed the bargaining problem and defined a solution/outcome for
it using an axiomatic approach
•
Nash defined a solution without the details of negotiation process
•
The solution is called as “Nash solution” for bargaining/negotiation
problems and it is widely used as one of the ideal solutions.
•
Assumption
•
The two agents are perfectly rational: each can accurately compare
its preferences for the possible outcomes, they are equal in bargaining
skill, and each has complete knowledge of the preference of the other.
28. Nash solution
•
Definition: A bargaining problem is defined as a pair (S, d). A
bargaining solution is a function f that maps every barging
problem (S, d) to an outcome in S, i.e.,
f(S,d)-> S
•
S is bargaining set that is the set of all utility pairs result from
an agreement.
•
d is the disagreement point where each agent i gets ui(d) even
if there is no agreement
•
Definition : Nash solution is defined as follows:
!
•
Nash product : (u1(x)-u1(d)) x (u2(x)-u2(d))
31. Nash solution
•
Nash proved that the solution that satisfies the five axioms below is
Nash solution and its unique.
•
Axiom 1 (Individual Rationality) : Each agent can get at least
disagreement point. f(S,d) >= d.
•
Axiom 2 (Symmetry) : The solution is independent form agent’s name,
like A or B.
•
Axiom 3 (Pareto Optimality)
•
Axiom 4 (Invariance from Afine Transformation) : The solution should not
change as a result of linear changes to the utility for either agent
•
Axiom 5 (Independence of Irrelevant Alternatives) : Eliminating feasible
alternatives that are not chosen should not affect the solution. Namely,
32. Bargaining based on non-cooperative game
•
Usually, non-cooperative model of bargaining
specifies a procedure of negotiation.
•
Most influential non-cooperative model is the
wallowing Rubinstein’s work.
[Rubinstein 1982] Perfect equilibrium in a bargaining model. Econometrica,
50(1):97-109, Jan 1982.
[Rubinstein 1985] A bargaining model with incomplete information about
time preference. Econometrica, 53:1151-1172, Jan 1985.
33. A brief overview of Rubinstein’s bargaining
[Rubinstein 1982] Perfect equilibrium in a bargaining model. Econometrica, 50(1):97-109, Jan 1982.
•
There are two agents (players) and a unit of
good, a pie, to be split between them (Issue is
divisible).
•
If agent a gets a share of
gets
.
•
If agents cannot reach an agreement, they do not
get anything.
then agent b
34. Alternating offers protocol
θ
•
This game is played over a series of
discrete time periods t = 1,2,3,...
•
The agents take turns in making
offers.
•
t = 1, player A proposes an offer
(Xa,t=1). If player B accepts A’s offer,
they reach an agreement. If not
(reject), goto t = 2.
θ
s(θ)
•
•
t = 2, player B proposes a counter
offer (Xb,t=2). If player A accepts B’s
offer, they reach an agreement. If
not (reject), goto t = 3.
t = 3, ...
θ
s(θ)
M
θ
s(θ)
M’
Concretely,
•
s(θ)
offer
offer
offer
offer
offer
35. Characteristics of the alternating offer protocol
•
The utility is increasing in the player’s share and
decreasing in time.
•
This decrease in utility with time is modeled with a
discount factor,
and
.
•
If a and b receive a share of xa and xb respectively
where xa + xb =1, then their utilities at time t are as
follows:
36. Equilibrium of the alternating offer protocol
•
If this game is played infinitely overtime, then
Rubinstein showed that there is a unique (subgame
perfect) equilibrium outcome in which the players
immediately reach an agreement on the following
shares:
37. drawbacks
•
Rubinstein’s model does not take “deadlines” into account.
•
There is nothing to prevent the agents from haggling
for as long as they wish.
•
A player’s bargaining power depends on the relative
magnitude of the players’ respective costs of haggling.
!
•
A lot of works on this line have been done.
39. Heuristic approaches
•
The heuristic approach is particularly useful when there are
multiple issues to negotiate, and finding an equilibrium offer is
computationally hard.
•
Of course there are Game theoretic approaches to multiissue negotiations (e.g. [Fatima2006]). However, here,
heuristic approaches are more focusing on computational
hardness, complex utilities, etc.
•
[Fatima2006] S.S.Fatima, M.Wooldridge, and N.R.Jennings,
Multi-issue negotiation with deadlines. Journal of Artificial
Intelligence Research, 27:381-417, 2006.
40. Monotonic concession
[Rosenschein & Zlotkin 94] Rules of Encounter
•
Players are not allowed to make offers which have a lower
utility for their opponent than their last offer. The minimum
concession per round can be fixed above 0
=> It guarantees to terminate. But, anyway, they have to
concede.
•
Question: how to make concessions?
•
•
•
If I do not know the opponents preferences
If there are multiple issues
Note: In multi-issue negotiations with unknown opponent
preferences, it is not always possible to make monotonic
concessions
41. Time dependent concession
•
Suppose we have a buyer (the case of the seller is
symmetrical) which desires to buy a good for an aspiration
price Pmin and reservation price Pmax (highest he is willing
to pay); deadline is a time Tmax
•
Price offered at time t will be:
!
•
P(t ) = Pmin + F (t )( Pmax − Pmin )
F(t) gives the fraction of the distance left between the first
(best) offer and the reservation value
1/ β
' min(t , Tmax ) $
"
F (t ) = k a + (1 − k a )%
%
"
Tmax
&
#
43. Time dependent concession
•
Hard-headed (β->0): No concessions, sticks to the initial offer
throughout (the opponent may concede, though)
•
Linear time-dependent concession (β=1): Concession is linear in the
time remaining until the deadline
•
Boulware (β<1): Concedes very slowly; initial offer is maintained until
just before the deadline
•
Conceder (β>1): Concedes to the reservation value very quickly
!
•
Tit-for-tat : Cooperating on the first move and then mirroring
whatever the other player did in the preceding round
44. Multi-issue negotiations
•
Single issue negotiations
•
•
Example: seller and buyer for a bottle of wine negotiating over a price
Multi-issue negotiations
•
Example: seller and buyer negotiating for a house over multiple issues,
price, place, style, architecture, etc.
•
Trade-off between issues : An agent can make concessions in one or
more issues in order to extract concessions in other issues preferable
to him/her
•
Example
•
Buyer concede about style instead of proposing nicer place.
•
Seller concede about price instead of proposing un-preferred
place.
45. Bidding Based Protocol
for Multiple Interdependent Issue
Negotiations
Takayuki Ito#*
!
Collaborative work with
Mark Klain*, Hiromitsu Hattori+, and Katsuhide Fujita#
!
#Nagoya
Institute of Technology, JAPAN
!
*Sloan
School of Management,
Massachusetts Institute of Technology, USA
!
+Kyoto University, JAPAN
46. Summary
Target : Multi-issue Negotiation Protocol!
Negotiation with multiple interdependent issues!
Non-linearity of agent’s utility functions!
Approach : Biding-based Negotiation Protocol!
An agent bids conditions to obtain better utility
as a bid!
Intractability of bid-generation!
Result : Outperform protocols applied in liner
domains!
Difficulty in scalability
47. [Preliminaries]
Modeling Non-linear Utilities
Bumpy non-linear utility space!
The utility is a summation of satisfied contracts’
values
Many constraints
are satisfied
A few/no constraints
are satisfied
Existing protocols assuming linear utility functions are not effective.
How to obtain a solution with high social welfare for non-linear utility
function ?
48. [Preliminaries]
Non-linear Utilities
Non-linear utility space!
m issues with the domain of integers [0, X]!
Issues are common for agents.!
A contract is a vector of issue values s =
(s1,...,sm).!
Agent’s utility function!
The function is represented
in terms of constraints.!
A constraint represents an
acceptable region
and its value (utility).
49. Bidding-based Negotiation Protocol
Sampling --- Bid-generation --- Winner
determination!
An agent submits bids to an mediator for high
individual utility.
An agent samples its utility space to find
high-utility region.
Trade-off between high-utility and the limit
of # of samples.
Samples do NOT always lie on optimal contracts.
How to detect better contract regions ?
50. Adjusting Samples
An agent adjusts samples based on simulated annealing
method.!
Multiple simulated annealing in the utility space!
All random sampling points are initial solutions.!
!
!
!
!
!
!
Each sampling point may move to each close optimal
contract.
51. Bid-generation
A bid is defined as a set of contracts which can
offer the same utility around an adjusted
sampling point.!
An agent can submit a bid iff its utility is larger
than the threshold.
Collect all constraints
satisfied by this point
Find the intersection
of the constraints
Several contracts could be submitted as one bid.
52. Winner Determination
The mediator identifies the combinations of bids as
the final contract.!
The final contract is a consistent bids with the highest social welfare.!
Only one bid from each agent is included.
1. Find mutually consistent bids
Specifying overlapping contract region
Agent 1
The final
contract
2. Select the best contracts
Comparing the summed bid values
Agent 2
53. Experiments
Setting!
Constraints satisfying many issues could have larger weights.!
The maximum value for a constraint: 100 x # of issues
e.g., the possible value for a binary constraint is 200.!
Agents have the same issues and domain for each issue value.!
Domain for issue value is [0,9]!
Approximate search-based winner determination!
The final contract is searched by the simulated annealing.!
The annealer for sample adjustment does not run too long.!
The purpose of the sample adjustment is to find the peak of
the optimum nearest point.
54. Experiments
Linear utility function case!
Comparison between the optimal result and the result
of Hill Climbing protocol (for each issue)!
!
Issues
!
1
2
3
4
5
6
7
8
9
10
! HC
0.973
0.991
0.998
0.989
0.986
0.987
0.986
0.996
0.988
0.991
!
Optimality with linear utility function (4 agents)
The simple HC protocol can produce nearly optimal
results even for a large space.!
The mediator can find the best value for issue 1, then
issue 2, ...
55. Experiments
Hill Climbing / Bidding-based method for non-linear utility
function!
HC mediator tends to converge to a local optimum. !
AR mediator has more chances to find better contract
because agents can generate bids covering multiple optima.
1.2
Optimality Rate
1
0.8
0.6
0.4
HC
AR
0.2
2
3
4
5
6
7
Number of Issues
8
9
10
56. Definition of Optimality Rate
Optimum contract
Sum of all agents’ utility functions and
use the Simulated Annealing (SA) to find
the contract with the highest possible
social welfare.
!
Optimality Rate = (Optimum solution
in the mechanism) / (Optimum
contract)
57. Experiments
Hill Climbing / Bidding-based method for non-linear utility
function!
HC mediator can quickly obtain the final contract. !
AR mediator takes more time, but the final contract is
calculated within a practical time.
4500
AR
3500
CPU time [ms]
4000
HC
3000
2500
2000
1500
1000
500
0
2
3
4
5
6
7
Number of Issues
8
9
10
58. Scalability
Optimality Rate
The impact of the scaling-up of the problem space!
90%+ optimality for up to 8 issues
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
2
3
4
Number of Agents
5 10
9
8
7
6
5
4
3
Number of Issues
2
60. Sampling Rates v.s. Agreement
The Failure rate, % of negotiations that do not lead to an
agreement, is higher when there are more sampling points
Reason: !
When there are many
sampling points, each agent
has a better chance of finding
good local optima in its
utility space.!
However, the num. of bids is
limited for computation time.!
This increases a risk of not
finding an overlap between
the bids
61. Discussion
The number of bids is ...!
# of
!
issues
2
3
4
5
6
7
8
9
10
11
12
13
14
15
# of!
!
bids
54
200
461
758
1074
1341
1636
1746
1972
2086
2238
2326
2491
2648
The winner determination computation grows
exponentially as (# of bids per agent)# of agents!
If we use an exhaustive search method (with
branch cutting),
the problem size is limited to the small one.!
The trade-off between the computation time and the
optimality
62. Summary
An bidding-based protocol for the negotiation with
multiple interdependent issues.!
Our bidding-based protocol outperforms existing
protocols applied in linear problems.
63.
64. ANAC overview
•
This competition brings together researchers from the negotiation community and
provides a unique benchmark for evaluating practical negotiation strategies in multiissue domains. The three previous competitions have spawned novel research in AI in
the field of autonomous agent design which are available to the wider research
community.
•
The declared goals of the competition are:
•
to encourage the design of practical negotiation agents that can proficiently
negotiate against unknown opponents and in a variety of circumstances,
•
to provide a benchmark for objectively evaluating different negotiation
strategies,
•
to explore different learning and adaptation strategies and opponent models,
and
•
to collect state-of-the-art negotiating agents and negotiation scenarios, and
making them available to the wider research community.
65. ANAC 2010-2013
•
•
Competitions
• ANAC2010 Toronoto, Canada
ANAC2010 ANAC2011
Agents
7
18
• ANAC2011 Taipei, Taiwan
Domains
3
8
• ANAC2012 Valencia, Spain
Rules
Discount1Factor
(
• ANAC2013, Saint Paul, USA
!
Organizers
• Tim Baarslag, Delft University of Technology
• Kobi Gal, Ben-Gurion University
• Enrico Gerding, University of Southampton
• Koen Hindriks, Delft University of Technology
• Takayuki Ito, Nagoya Institute of Technology
• Nicholas R. Jennings, University of Southampton
• Catholijn Jonker, Delft University of Technology
• Sarit Kraus, University of Maryland and Bar-Ilan University
• Raz Lin, Bar-Ilan University
• Valentin Robu, University of Southampton
• Colin R. Williams, University of Southampton
ANAC2012
ANAC2013
17
19
24
24
Reserva8on1Value
Bid1History
66.
67. GENIUS: Tournament Environment
•
A research tool for automated multi-issue negotiation
•
Negotiation tournaments in different scenarios
•
analytical toolbox
•
Simplifies and supports agent development
•
repositories of domains and agents
•
Education : teach students to design negotiation algorithms
!
•
Programming is all in Java.
http://mmi.tudelft.nl/negotiation/index.php/Genius
68. Negotiation Domains
•
Agents negotiate based on negotiation domain which has multiple issues.
Utility spaces are not known
Example of Bid (fashion style)
sweaters
Pants
classic
pants
Shoes
boots
Accessories
hat
Utility(A)
1.0
Utility(B)
Utility(AgentB)
Shirts
?
Utility(AgentA)
•
Large variety of domain characteristics possible, and easy to identify
Laptop
Grocery
Energy
Number of
issues
3 issues
5 issues
8 issues
Size
27
1600
390625
Opposition
Weak
Medium
Strong
69. Creating an Agent
•
Implementation itself is very easy if you know Java
•
extend negotiator.Agent class
•
override the three methods:
•
ReceiveMessage()
•
init()
•
chooseAction()
•
Create a package, compile them, and load the main class.
•
See More details in userguide.pdf
71. Summary
•
Part 1: Introduction of Automated Negotiation
•
Part 2: Bargaining : Game Theoretic Approaches
•
Part 3: Multi-issue Negotiation : Heuristic
Approaches
•
Part 4: Automated Negotiating Agent Competition
72. References (1)
•
I. Rahwan and S. Fatima, “Negotiations”, Chapter 4 in the Book “Multiagent Systems: 2nd
edition” edited by G. Weiss, MIT Press, ISBN 978-0-262-01889-0, 2013. (Recommended)
•
N. R. Jennings, P. Faratin, A. R. Lomuscio, S. Parsons, C. Sierra and M. Wooldridge, Automated
Negotiation: Prospects, Methods and Challenges, International Journal of Group Decision and
Negotiation, 10(2):199-215, 2001
•
J.F. Nash, The bargaining problem, Econometrica, 18:155-162, 1950
•
J.F. Nash, Two-person cooperative game, Econometrica, 21:128-140, 1953
•
A. Rubinstein, Perfect equilibrium in a bargaining model. Econometrica, 50(1):97-109, Jan 1982.
•
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