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Approximate and online multi-issue negotiation
S.S. Fatima
Loughborough University, [email protected]
M. Wooldridge N.R. Jennings
University of Liverpool, UK University of Southampton, UK [email protected] [email protected]
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): 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
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
Time Size of pie Offering agent Offer
1 1 a → b (3/4, 1/4)(not symmetric)
2 1/4 b → 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: δc for issue c (1 ≤ c ≤ m)
Utility: Ua(x, t) = ∑c kacU(xc, 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 δ1= δ2 = 1/2 UTILITIES: Ua = x1 + 2x2; Ub = 2y1 + y2
Time Size of pie Offering agent
Package Offer
1 1, 1 a → b [(1/4, 3/4); (1, 0)]OR[(3/4, 1/4); (0, 1)]
2 1/2, 1/2 b → a [(0, 1/2); (0, 1/2)]Ub = 1.5
Agreement
Optimal strategies
For t = nThe offering agent takes 100 percent of all the issuesThe receiving agent acceptsFor t < n (Agent a’s perspective)
OFFER [x, y]
s.t. Ub(y, t) = Ub(yt+1, t+1)If more then one such [x, y]perform trade-offs across issues to find best offer
RECEIVE [x, y]
If Ua(x, t) ≥ Ua(xt+1, t+1) ACCEPTelse REJECT
Making trade-offs
Agent a’s trade-off problem at time t: Find a package [xt, yt] to m
Maximize ∑ kac xt
c
c=1
m
such that ∑ kbc yt
c = Ub(xt+1, t+1) 0 ≤ xtc ≤ 1, 0 ≤ yt
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 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)
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 [xt, yt] that m
Maximize ∑ kac xt
c
c=1
m
such that ∑ kbc yt
c = Ub(yt+1, t+1) xtc = 0 or 1; yt
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/ε2)
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/ε2)
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