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<article-title>Approximate and Online Multi-Issue Negotiation</article-title>
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<author><a href="mailto:shaheen@csc.liv.ac.uk"><name>Shaheen S. Fatima</name></a></author>
<aff>Department of Computer Science <br/>University of Liverpool Liverpool L69 3BX, UK</aff>

<author><a href="mailto:mjw@csc.liv.ac.uk"><name>Michael Wooldridge</name></a></author>
<aff>Department of Computer Science <br/>University of Liverpool Liverpool L69 3BX, UK</aff>

<author><a href="mailto:nrj@ecs.soton.ac.uk"><name>Nicholas R. Jennings</name></a></author>
<aff>School of Electronics and Computer Science<br/> University of Southampton Southampton SO17 1BJ, UK</aff>

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<abstract>
<title>ABSTRACT</title>
<p>This paper analyzes bilateral multi-issue negotiation between selfinterested
autonomous agents. The agents have time constraints in
the form of both deadlines and discount factors. There arem &#62; 1
issues for negotiation where each issue is viewed as a pie of size
one. The issues are "indivisible" (i.e., individual issues cannot be
split between the parties; each issue must be allocated in its entirety
to either agent). Here different agents value different issues
differently. Thus, the problem is for the agents to decide how to
allocate the issues between themselves so as to maximize their individual
utilities. For such negotiations, we first obtain the equilibrium
strategies for the case where the issues for negotiation are
known a priori to the parties. Then, we analyse their time complexity
and show that finding the equilibrium offers is an NP-hard
problem, even in a complete information setting. In order to overcome
this computational complexity, we then present negotiation
strategies that are <italic>approximately optimal</italic> but computationally efficient,
and show that they form an equilibrium. We also analyze the
relative error (i.e., the difference between the true optimum and the
approximate). The time complexity of the approximate equilibrium
strategies is O(<italic>nm</italic>/&#949;<sup>2</sup>) where n is the negotiation deadline and &#949;
the relative error. Finally, we extend the analysis to <italic>online</italic> negotiation
where different issues become available at different time points
and the agents are uncertain about their valuations for these issues.
Specifically, we show that an approximate equilibrium exists for
online negotiation and show that the expected difference between
the optimum and the approximate is O(&#8730;m) . These approximate
strategies also have polynomial time complexity.</p>
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