Alternating-offers bargaining with one-sided uncertain deadlines: an efficient algorithm

In the arena of automated negotiations we focus on the principal negotiation protocol in bilateral settings, i.e. the alternating-offers protocol. In the scientific community it is common the idea that bargaining in the alternating-offers protocol will play a crucial role in the automation of electronic transactions. Notwithstanding its prominence, literature does not present a satisfactory solution to the alternating-offers protocol in real-world settings, e.g. in presence of uncertainty. In this paper we game theoretically analyze this negotiation problem with one-sided uncertain deadlines and we provide an efficient solving algorithm. Specifically, we analyze the situation where the values of the parameters of the buyer are uncertain to the seller, whereas the parameters of the seller are common knowledge (the analysis of the reverse situation is analogous). In this particular situation the results present in literature are not satisfactory, since they do not assure the existence of an equilibrium for every value of the parameters. From our game theoretical analysis we find two choice rules that apply an action and a probability distribution over the actions, respectively, to every time point and we find the conditions on the parameters such that each choice rule can be singularly employed to produce an equilibrium. These conditions are mutually exclusive. We show that it is always possible to produce an equilibrium where the actions, at any single time point, are those prescribed either by the first choice rule or by the second one. We exploit this result for developing a solving algorithm. The proposed algorithm works backward by computing the equilibrium from the last possible deadline of the bargaining to the initial time point and by applying at each time point the actions prescribed by the choice rule whose conditions are satisfied. The computational complexity of the proposed algorithm is asymptotically independent of the number of types of the player whose deadline is uncertain. With linear utility functions, it is O([email protected][email protected]?) where m is the number of the issues and [email protected]? is the length of the bargaining.

[1]  Nicola Gatti,et al.  Bargaining in Bundle over Multiple Issues in Finite-Horizon Alternating-Offers Protocol , 2006, AI&M.

[2]  Avi Pfeffer,et al.  Representations and Solutions for Game-Theoretic Problems , 1997, Artif. Intell..

[3]  Nicholas R. Jennings,et al.  A Software Framework for Automated Negotiation , 2004, SELMAS.

[4]  Nicholas R. Jennings,et al.  Multi-issue negotiation under time constraints , 2002, AAMAS '02.

[5]  Barry O'Neill Handbook of Game Theory, Vol. 3 , 2004 .

[6]  Barry O'Neill Handbook of Game Theory, Volume 3: Edited by Robert Aumann and Sergiu Hart, Elsevier, New York, 2002 , 2004, Games Econ. Behav..

[7]  Michael Wooldridge,et al.  A Classification Scheme for Negotiation in Electronic Commerce , 2001 .

[8]  J. Nash NON-COOPERATIVE GAMES , 1951, Classics in Game Theory.

[9]  Tuomas Sandholm,et al.  Bargaining with Deadlines , 1999, AAAI/IAAI.

[10]  N. R. Jennings,et al.  Multi-Issue Negotiation with Deadlines , 2006, J. Artif. Intell. Res..

[11]  Zizhuo Wang,et al.  A unified framework for dynamic pari-mutuel information market design , 2009, EC '09.

[12]  Nicholas R. Jennings,et al.  On Efficient Procedures for Multi-issue Negotiation , 2006, TADA/AMEC.

[13]  Tuomas Sandholm,et al.  Finding equilibria in large sequential games of imperfect information , 2006, EC '06.

[14]  Jeffrey S. Rosenschein,et al.  Rules of Encounter - Designing Conventions for Automated Negotiation among Computers , 1994 .

[15]  Catholijn M. Jonker,et al.  An agent architecture for multi-attribute negotiation using incomplete preference information , 2007, Autonomous Agents and Multi-Agent Systems.

[16]  J. K. Hunter,et al.  Measure Theory , 2007 .

[17]  Jeffrey S. Rosenschein and Gilad Zlotkin Rules of Encounter , 1994 .

[18]  Alvin E. Roth,et al.  Bargaining under a deadline: evidence from the reverse ultimatum game , 2003, Games Econ. Behav..

[19]  Michael Manove,et al.  Bargaining with Deadlines and Imperfect Player Control , 1993 .

[20]  Pattie Maes,et al.  Agents that buy and sell , 1999, CACM.

[21]  Drew Fudenberg,et al.  Game theory (3. pr.) , 1991 .

[22]  Peter Bro Miltersen,et al.  Computing sequential equilibria for two-player games , 2006, SODA '06.

[23]  Nicola Gatti,et al.  Bargaining over multiple issues in finite horizon alternating-offers protocol , 2006, Annals of Mathematics and Artificial Intelligence.


[25]  David M. Kreps,et al.  Sequential Equilibria Author ( s ) : , 1982 .

[26]  Daniel J. Seidmann,et al.  Deadline Effects and Inefficient Delay in Bargaining with Endogenous Commitment , 1993 .

[27]  Tuomas Sandholm Agents in Electronic Commerce: Component Technologies for Automated Negotiation and Coalition Formation , 2004, Autonomous Agents and Multi-Agent Systems.

[28]  Francesco Amigoni,et al.  A formal framework for connective stability of highly decentralized cooperative negotiations , 2007, Autonomous Agents and Multi-Agent Systems.

[29]  Nicholas R. Jennings,et al.  Negotiation decision functions for autonomous agents , 1998, Robotics Auton. Syst..

[30]  Stefan Napel,et al.  Bilateral Bargaining - Theory and Applications , 2002, Lecture notes in economics and mathematical systems.

[31]  Sarit Kraus,et al.  Strategic Negotiation in Multiagent Environments , 2001, Intelligent robots and autonomous agents.

[32]  P. Bahr,et al.  Sampling: Theory and Applications , 2020, Applied and Numerical Harmonic Analysis.

[33]  Alessandro Lazaric,et al.  Reinforcement learning in extensive form games with incomplete information: the bargaining case study , 2007, AAMAS '07.

[34]  Bernhard von Stengel,et al.  Fast algorithms for finding randomized strategies in game trees , 1994, STOC '94.

[35]  Larry Samuelson,et al.  Bargaining Under Two-Sided Incomplete Information: The Unrestricted Offers Case , 1988, Oper. Res..

[36]  A. Rubinstein Perfect Equilibrium in a Bargaining Model , 1982 .

[37]  A. Rubinstein,et al.  Bargaining and Markets. , 1991 .

[38]  Nicola Gatti,et al.  Alternating-Offers Bargaining Under One-Sided Uncertainty on Deadlines , 2006, ECAI.

[39]  E. Rowland Theory of Games and Economic Behavior , 1946, Nature.

[40]  R. Selten,et al.  A Generalized Nash Solution for Two-Person Bargaining Games with Incomplete Information , 1972 .

[41]  Nicholas R. Jennings,et al.  An agenda-based framework for multi-issue negotiation , 2004, Artif. Intell..