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

1. Bargaining in-Bundle over Multiple Issues in Finite-Horizon Alternating-Offers Protocol Francesco Di Giunta and Nicola Gatti Politecnico di Milano Milan, Italy

2. 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

3. 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?

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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.

12. 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

13. 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

14. 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

15. 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

16. 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

17. Finally Thank you for your kind attention