Approximate and online multi-issue negotiation

1. Approximate and online multi-issue negotiation S.S. Fatima Loughborough University, UK S.S.Fatima@lboro.ac.uk M. Wooldridge N.R. Jennings University of Liverpool, UK University of Southampton, UK mjw@csc.liv.ac.uknrj@ecs.soton.ac.uk

2. The Problem To study the strategic behaviour of agents for bilateral multi-issue negotiation and determine optimal strategies Optimal strategies depend on • Protocol • Deadline • Utility functions • Whether all the issues are known to the agents at the beginning of negotiation • Type of issues (divisible or indivisible)

3. Setting • Deadline • An agent’s cumulative utility is the sum of utilities from individual issues • Divisible and indivisible issues • All the issues are known to the agents at the beginning • The issues become known one by one (online negotiation)

4. Objective • To identify those scenarios for which optimal strategies are • easy to compute • hard to compute • To develop a fast algorithm for finding approximately optimal strategies

5. Overview • Single issue negotiation • Extension to multiple issues • Complexity of negotiating multiple issues • Approximately optimal strategies • Summary

6. Single issue negotiation • Agents a and bnegotiate over an issue - a pie of size 1 • Deadline: n and Discount factor: δ • Utility from (x,y): Ua(x, t) = x δt-1 if t ≤ n 0 otherwise Ub(y, t) = y δt-1 if t ≤ n 0 otherwise • The agents negotiate using Rubinstein’s alternating offer’s protocol

7. Alternating offers protocol TimeAgentOffer 1 a b x (accept/reject) 2 b  a y (accept/reject) - - n

8. Optimal Offers How much should an agent offer in the first time period? Let n=1 and a be the first mover Agent a proposes to keep the whole pie; agent b accepts

9. Equilibrium strategies (n = 2) δ = 1/4 first mover: a Offer: (x, y) x: a’s share; y: b’s share Agreement Backward Induction

10. Multiple issues • Set of issues: S = {1, 2, …, m} • Each issue is a pie of size 1 • Deadline: n (for all the issues) • Discount factor: δc for issue c (1 ≤ c ≤ m) • Utility: Ua(x, t) = ∑c kacU(xc, t)

11. Package deal procedure • Issues negotiated using alternating offer’s protocol • An offer specifies a division for each of the m issue • The agents are allowed to accept/reject a complete offer An agent reason backwards and makes tradeoffs across the issues to maximize its cumulative utility

12. Example Divisible issues: Complete information m = 2 n = 2 δ1= δ2 = 1/2 UTILITIES: Ua = x1 + 2x2; Ub = 2y1 + y2 Agreement

13. Optimal strategies For t = n The offering agent takes 100 percent of all the issues The receiving agent accepts For t < n(Agent a’s perspective)

14. Making trade-offs Agent a’s trade-off problem at time t: Find a package [xt, yt] to m Maximize ∑kac xtc c=1 m such that ∑ kbc ytc = Ub(xt+1, t+1) 0 ≤ xtc ≤ 1, 0 ≤ ytc ≤ 1 c=1 This is the fractional knapsack problem The optimal solution to the fractional knapsack problem can be found using a Greedy method

15. Making trade-offs • Agent a’s perspective (time t) Agent a considers the m issues in the increasing order of ka/kb and assigns to b the maximum possible share for each of them until b’s cumulative utility equals Ub(yt+1, t+1)

16. Equilibrium solution • An agreement on all the m issues occurs in the first time period • The equilibrium solution is Pareto-optimal • The equilibrium solution is not unique • Time to compute the equilibrium offer for the first time period is O(mn)

17. Indivisible issues Agent a’s trade-off problem: To find a package [xt, yt] that m Maximize ∑kac xtc c=1 m such that ∑ kbc ytc = Ub(yt+1, t+1) xtc = 0 or 1; ytc = 0 or 1 c=1 • This is the integer knapsack problemwhich is NP-hard • The problem of finding the optimal offers for indivisible issues is also NP hard

18. Knapsack problem:Approximate solution An approximate solution to integer knapsack problem can found using dynamic programming Fully polynomial time approximation; time complexity: O(m/ε2) z: approximate solution z*: optimal solution Relative error of approximation: (z - z*) / z* ≤ ε

19. Equilibrium for indivisible issues • At every time step, the above offers form an ε-approximate equilibrium • Time complexity of finding approximate equilibrium offer for time period t is O(m/ε2)

20. Online negotiation • The agents know that they will negotiate more issues in the future but are uncertain about their valuations for those issues • The issues become known at different time points • The agents must settle an issue as soon as it is made known (i.e., prior to having information about the future issues - the agents have a probability distribution over the possible future issues) • Once an issue is settled it cannot be renegotiated

21. Online integer knapsack problem • The weights and profits for items are made known one at a time • An algorithm must decide whether or not to include an item as soon as its weights and profits are known without knowing the details of future items • For uniformly distributed weights and profits, an approximate solution can be found using a greedy algorithm • Time complexity: O(m) • Expected error E[z* - z] = O(√m)

22. Equilibrium for online negotiation Time complexity of finding equilibrium offer for time period t: O(m) Expected approximation error: E[z* - z] = O(√m)

23. Future Work • To find optimal strategies for online negotiation where the coefficients of utility functions have distributions other than uniform • To find optimal strategies for the case of interdependent issues • To find optimal strategies for non-linear utility functions