Approximate and online multi-issue negotiation

S.S. Fatima

Loughborough University, UKS.S.Fatima@lboro.ac.uk

M. Wooldridge N.R. Jennings

University of Liverpool, UK University of Southampton, UK mjw@csc.liv.ac.uk nrj@ecs.soton.ac.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): 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