Bargaining in bundle over multiple issues in finite horizon alternating offers protocol
Download
1 / 17

Bargaining in-Bundle over Multiple Issues in Finite-Horizon Alternating-Offers Protocol - PowerPoint PPT Presentation


  • 68 Views
  • Uploaded on

Bargaining in-Bundle over Multiple Issues in Finite-Horizon Alternating-Offers Protocol. Francesco Di Giunta and Nicola Gatti Politecnico di Milano Milan, Italy. Summary. Introduction to alternating-offers bargaining, open problems, and topic of the paper Review of the single-issue solution

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Bargaining in-Bundle over Multiple Issues in Finite-Horizon Alternating-Offers Protocol' - vladimir-reyes


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Bargaining in bundle over multiple issues in finite horizon alternating offers protocol

Bargaining in-Bundle over Multiple Issues in Finite-Horizon Alternating-Offers Protocol

Francesco Di Giunta and Nicola Gatti

Politecnico di Milano

Milan, Italy


Summary
Summary Alternating-Offers Protocol

  • Introduction to alternating-offers bargaining, open problems, and topic of the paper

  • Review of the single-issue solution

  • Basic ideas for our multi-issue solution

  • Development of the multi-issue solution

  • Conclusions and further work


Alternating offers bargaining
Alternating-offers bargaining Alternating-Offers Protocol

  • Two rational agents - a buyer b and a seller s – make offers and counteroffers in order to reach an agreement (e.g., on price, quality, quantity,… of a good to be sold)

  • They have opposite interests and they both lose utility as time passes by

  • Different settings:

    • finite-horizon vs infinite-horizon

    • single-issue vs multi-issue

    • complete information vs incomplete information

  • The problem is: how should the two rational agents behave? Which should be their strategies?


Alternating offers bargaining1
Alternating-offers bargaining Alternating-Offers Protocol

  • Game-theoretical analysis pioneered by [Stahl, 1972] and [Rubinstein, 1982]

  • Long time interest in the game theory and in the artificial intelligence community

  • The single issue problem with complete information is solved

  • Slow further developments towards the solution of realistic models

  • Main open problems:

    • Incomplete information

    • Multiple issues


Multi issue problem
Multi-issue problem Alternating-Offers Protocol

  • Multi-issue bargaining protocols:

    • Sequential: the issues are negotiated one by one

    • In-bundle: all the issues are negotiated together

  • Sequential bargaining does not assure Pareto-efficiency

  • In-bundle bargaining is said to involve too much computations


Focus of our paper
Focus of our paper Alternating-Offers Protocol

  • We focus on finite-horizon in-bundle alternating-offers bargaining with complete information

  • We show that, for the most common kind of utility functions, the problem is indeed tractable

  • We merge game-theoretical and linear/convex programming techniques


Review of the one issue model
Review of the one-issue model Alternating-Offers Protocol

  • The buyer b and the seller s act alternately at integer times

  • Possible actions at time t are

    • Make an offer (a real number, typically a price)

    • Accept the opponent’s previous offer x: the outcome is (x,t)

    • Exit the negotiation: the outcome is NoAgreement

  • The utility function Ub (Us) of b (s) depends on her

    • Reservation price RPb (RPs)

    • Deadline Tb (Ts)

    • Time discount factor δb (δs)

  • Ub(x,t) = (RPb-x)(δb)t if t ≤ Tb

  • Ub(x,t) = -1 if t > Tb

  • Us(x,t) = (x-RPs)(δs)t if t ≤ Ts

  • Us(x,t) = -1 if t > T

  • Ub(NoAgreement) = Us(NoAgreement) = 0


Review of the one issue solution
Review of the one-issue solution Alternating-Offers Protocol

  • The appropriate notion of solution is subgame perfect Nash equilibrium

  • The protocol is essentially a finite game, so the equilibrium can be found by backward induction:

    • Call T = min {Tb,Ts}

    • At time T the acting agent (say, s) would accept any offer with positive utility

    • At time T-1 agent b would offer x*T-1=RPs or accept any offer x such that Ub(x,T-1) ≥ Ub(x*T-1,T)

    • At time T-2 agent s would offer x*T-2 such that Ub(x*T-2,T-1) = Ub(RPs,T) or accept any offer x such that Us(x,T-2) ≥ Us(x*T-2,T-1)

  • I.e., at each time point t, from T back, it is possible to recursively find the offer x*t that the acting rational agent would do if she would make an offer; such offers x*t (or possible irrational higher ones) are always accepted by the rational opponent.

  • Therefore the agreement is achieved at the very beginning of the bargaining on the value x*0


Towards the multi issue solution
Towards the multi-issue solution Alternating-Offers Protocol

  • The core of the single-issue solution is the calculation of the values x*t that one agent should offer at time t and the other should accept at time t+1

  • In the one-issue situation this is very easy

  • Are there, in the multi-issue situation, tuples x*t of values that act somehow like these values x*t? The answer, for a wide class of multi-issue utility functions, turns out to be yes

  • Is the calculation of these values computationally tractable? Again, the answer is yes

  • Is the attained agreement Pareto-efficient? Yes


Towards the multi issue solution1
Towards the multi-issue solution Alternating-Offers Protocol

  • In single-issue bargaining, value x*t-1 is calculated from x*t as the value such that

    Ui(x*t-1,t) = Ui(x*t,t+1)

    where i is the agent that acts at time t

  • I.e., x*t-1 is obtained as the one step “backward propagation” of x*t along the level curves of the utility function of agent i

  • In multi-issue bargaining, instead, there is no unique “backward propagated” tuple x*t-1=<x*1t-1,…,x*nt-1> but an entire set of tuples X*t-1 which at time t are worth for agent i the same as x*t at time t+1


Basic idea for multi issue bargaining
Basic idea for multi-issue bargaining Alternating-Offers Protocol

  • We take as x*t-1 the tuple in X*t-1 that maximizes the utility of the agent acting at time t-1

  • For a wide range of utility functions, this can be done efficiently with linear/convex programming.


Multi issue bargaining assumptions
Multi-issue bargaining assumptions Alternating-Offers Protocol

  • Linear multi-issue utility function of agent i:

    • Ui(x1,…, xn,t) = ∑jUji(xj,t) if for each j Uji(xj,t) ≥ 0

    • Ui(x1,…, xn,t) = -1 otherwise

      where

    • Uji(xj,t) = uji(xj)(δjb)t if t ≤ Tji

    • Uji(xj,t) = -1 otherwise

      where

    • uji are continuous, concave and strictly monotonic

    • uji are such that the agents have opposite preferences over each issue

    • uji are such that there are feasible agreements


Multi issue bargaining solution
Multi-issue bargaining solution Alternating-Offers Protocol

  • T = minji{Tji} is the global deadline of the bargaining

  • Tuple x*T-1 = <x*1T-1,…,x*nT-1> = <RP1i,…,RPni> where i is the agent that acts at time T

  • To calculate x*t-1 from x*t(be s the agent that acts at time t)

    • Calculate the set X*t-1 of tuples which at time t are worth the same as x*t at time t+1 for agent s

    • Use linear/convex programming to calculate x*t-1 as the value in X*t-1 that maximizes the utility of agent b


Multi issue bargaining solution1
Multi-issue bargaining solution Alternating-Offers Protocol

Be σ* the following strategy profile:

  • At time T accept any offer that has nonnegative value

  • At time t<T accept any offer x such that agreement (x,t) has utility greater or equal to (x*t-1 ,t+1) and otherwise counteroffer x*t


Main results
Main results Alternating-Offers Protocol

It can be shown that

  • Strategy σ* is the unique subgame perfect equilibrium of the protocol

  • The calculation of σ* is linear with T and polynomial with the number of issues

  • With strategy profile σ*, the agreement is achieved immediately and is Pareto-efficient


Conclusions
Conclusions Alternating-Offers Protocol

  • In this paper we have shown that complete information multi-issue bargaining is tractable, despite what is usually believed, for a wide (and the most common) range of utility functions and for the best known bargaining protocol

  • Further work will deal with the incomplete information problem


Finally
Finally Alternating-Offers Protocol

Thank you for your kind attention