Approximate and online multi-issue negotiation S.S. Fatima Loughborough University, UK M. Wooldridge N.R. Jennings University of Liverpool, UK University of Southampton, UK

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)

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)

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

Overview 1. Single issue negotiation 2. Extension to multiple issues 3. Complexity of negotiating multiple issues 4. Approximately optimal strategies 5. Summary

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

Alternating offers protocol Time Agent Offer 1 a  b x (accept/reject) 2 b  a y (accept/reject) - n

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 Optimal Offers

Equilibrium strategies (n = 2) δ = 1/4 first mover: a Offer: (x, y) x: a’s share; y: b’s share TimeSize of pieOffering agentOffer 11a → b(3/4, 1/4) (not symmetric) 21/4b → a(0, 1/4) Backward Induction Agreement

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: δ c for issue c (1 ≤ c ≤ m) Utility: U a (x, t) = ∑ c k a c U(x c, t)

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

Example Divisible issues: Complete information m = 2 n = 2 δ 1 = δ 2 = 1/2 UTILITIES: U a = x 1 + 2x 2 ; U b = 2y 1 + y 2 TimeSize of pieOffering agent Package Offer 11, 1a → b[(1/4, 3/4); (1, 0)] OR [(3/4, 1/4); (0, 1)] 21/2, 1/2b → a[(0, 1/2); (0, 1/2)] U b = 1.5 Agreement

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) OFFER [x, y] s.t. U b (y, t) = U b (y t+1, t+1) If more then one such [x, y] perform trade-offs across issues to find best offer RECEIVE [x, y] If U a (x, t) ≥ U a (x t+1, t+1) ACCEPT else REJECT

Making trade-offs Agent a’s trade-off problem at time t: Find a package [x t, y t ] to m Maximize ∑ k a c x t c c=1 m such that ∑ k b c y t c = U b (x t+1, t+1) 0 ≤ x t c ≤ 1, 0 ≤ y t c ≤ 1 c=1 This is the fractional knapsack problem The optimal solution to the fractional knapsack problem can be found using a Greedy method

Making trade-offs Agent a’s perspective (time t) Agent a considers the m issues in the increasing order of k a /k b and assigns to b the maximum possible share for each of them until b’s cumulative utility equals U b (y t+1, t+1)

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)

Indivisible issues Agent a’s trade-off problem: To find a package [x t, y t ] that m Maximize ∑ k a c x t c c=1 m such that ∑ k b c y t c = U b (y t+1, t+1) x t c = 0 or 1; y t c = 0 or 1 c=1 This is the integer knapsack problem which is NP-hard The problem of finding the optimal offers for indivisible issues is also NP hard

Knapsack problem: Approximate solution An approximate solution to integer knapsack problem can found using dynamic programming Fully polynomial time approximation; time complexity: O(m/ε 2 ) z: approximate solution z*: optimal solution Relative error of approximation: (z - z*) / z* ≤ ε

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/ε 2 )

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

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)

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

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