bargaining in bundle over multiple issues in finite horizon alternating offers protocol
Download
Skip this Video
Download Presentation
Bargaining in-Bundle over Multiple Issues in Finite-Horizon Alternating-Offers Protocol

Loading in 2 Seconds...

play fullscreen
1 / 17

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


  • 67 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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

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

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
  • 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

Thank you for your kind attention

ad