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A Recursive Protocol for Negotiating Contracts Under Non-monotonic Preference Structures

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Abstract

Negotiating contracts with multiple interdependent issues may yield non-monotonic preference spaces for the participating agents. These negotiations are specially challenging because of the complexity and dimension of the search space. Automated negotiation mechanisms designed and proven useful for monotonic utility spaces may fail in these negotiation scenarios. This paper presents a novel solution to the problem of automated multi-issue negotiations in the context of complex utility spaces. We seek to address the challenge of intractably large contract spaces and utility functions with multiple local optima in automated negotiation scenarios. A protocol for automated bilateral multi-attribute negotiation processes is proposed, in which the individual agents’ preferences can be non-monotonic and discontinuous. The protocol is based on a recursive non-mediated bargaining mechanism, which involves two agents who simultaneously exchange proposals defined as regions within the negotiation space. An agreement on a region implies a new bargaining which is restricted to that region. This recursive process is governed by a set of rules which modulate the joint exploration of the negotiation space until an agreement is found or a deadline expires. The protocol is experimentally evaluated under monotonic and non-monotonic preference scenarios, confirming that the protocol is able to produce outcomes close to the Pareto frontier in acceptable negotiation time, outperforming previous approaches.

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Abbreviations

A i :

Agent A i

ATH i :

Set of acceptance thresholds for agent A i

\({{ath_{r_{im}}^{i}}}\) :

Acceptance threshold for agent A i and region size r i

\({{b_{r_{im}}^{t_{n}}}}\) :

Bargaining thread of size r im starting at t n

BELL 1,2:

Bell utility functions

BTH:

Bargaining thread

CES1,2,3,4:

Constant elasticity of substitution utility functions

fbell :

Bell function

F RS (x):

Exponential function to generate the distribution of region sizes

\({{F_{ATH}^{i}(x)}}\) :

Exponential function to generate the distribution of acceptance thresholds

\({{F_{QTH}^{i}(x)}}\) :

Exponential function to generate the distribution of quality thresholds

lnsro :

Upper bound on the number of trials when searching for root offers in a BTH

lnro :

Upper bound on the number of generated root offers in a BTH

lndro :

Upper bound on the number of root offer descendants

lnfc :

Upper bound on the number of unaccepted child regions

lnco :

Upper bound on the number of child offers

lnrco :

Upper bound on the number of each offer’s rejected children

LNBT :

Set of upper bounds \({lnbt_{r_{im}}}\) for the number of child threads

MaxIter :

Upper bound of negotiation rounds

\({{\mathcal{N}_{d}}}\) :

Negotiation dialogue

nsc :

Number of sampled contracts

\({{O_{i}^{t}}}\) :

Agent i’s offer at instant t

OSD:

Overall satisfaction degree

QTH i :

Set of quality thresholds for agent A i

\({{qth_{r_{im}}^{i}}}\) :

Quality threshold for agent A i and region size r i

QR :

Set of radii for concentric regions in the evaluation of offer quality

\({{q_{r_{im}}}}\) :

For offers of size r im , radius of the concentric region to compute quality

R = < c, r > :

Region of size r and center c

r :

Size of a region

r 1 :

Maximum region size

r m :

Minimum region size

RNP:

Region based negotiation protocol

s :

contract

\({{(R_{b},R_{s})_{r_{im}}^{t_{n}+a}}}\) :

Exchange of offers within a BTH \({b_{r_{im}}^{t_{n}}}\) at instant t n  + a

\({{(res_{b},res_{s})_{r_{im}}^{t_{n}+a}}}\) :

Exchange of responses within a BTH \({b_{r_{im}}^{t_{n}}}\) at instant t n  + a

RS :

Set of region sizes r i

SNP:

Similarity based negotiation protocol

\({{S_{rs}^{R}}}\) :

Acceptable contracts in S R

S R :

Set of nsc contracts in R

τ r :

Decay factor for\({F_{RS}^{i}}\)

\({{\tau_{a}^{i}}}\) :

Decay factor for \({F_{ATH}^{i}}\)

\({{\tau_{q}^{i}}}\) :

Decay factor for \({F_{QTH}^{i}}\)

U i :

Agent i’s utility function

\({{U_{i}^{obj}}}\) :

Aspirational or objective utility

\({{U_{i}^{rs}}}\) :

Reservation utility

\({{vq_{(R_{s})_{r_{im}}^{t_{n}+a}}}}\) :

Request for the movement of offer \({\mathbf{(R_{s})_{r_{im}}^{t_{n}+a}}}\)

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Authors and Affiliations

  1. Computer Engineering Department, Escuela Politecnica Superior, Universidad de Alcala, Alcala de Henares, 28805, Madrid, Spain

    Ivan Marsa-Maestre, Miguel A. Lopez-Carmona, Juan A. Carral & Guillermo Ibanez

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  1. Ivan Marsa-Maestre
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  2. Miguel A. Lopez-Carmona
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  3. Juan A. Carral
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  4. Guillermo Ibanez
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Correspondence to Miguel A. Lopez-Carmona.

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Marsa-Maestre, I., Lopez-Carmona, M.A., Carral, J.A. et al. A Recursive Protocol for Negotiating Contracts Under Non-monotonic Preference Structures. Group Decis Negot 22, 1–43 (2013). https://doi.org/10.1007/s10726-011-9254-6

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