This book contains some fundamental results and insights relating to the automation of negotiation. The focus is on competitive negotiations where the outcome (i.e., whether or not an agreement is reached and if so what agreement is reached) depends on the strategies of all parties. We studied how to model and analyse such strategic negotiations using game theory.

From a strategic point of view, there are four key determinants of the outcome of a negotiation:

1. the negotiation deadline,

2. the discount factor,

3. the information that the players have about the negotiation parameters,

4. the protocol/procedure used for negotiation, and

5. the negotiation agenda.

We explored a number of procedures and compared them in terms of the properties (i.e., time of agreement, Pareto optimality, uniqueness, symmetry) of their equilibrium outcomes. But this book is not just about strategic negotiations, it is about their automation. In this vein, we examined the negotiation procedures and their equilibria from a computational perspective. The following are the main sources of complexity that can make the automation of negotiation difficult:

1. The type of issues. In terms of the computational complexity of calculating offers, divisible issues are easier to deal with than indivisible or discretely divisible issues.

2. The form of utility functions. In terms of the computational complexity of calculating offers, linear utility functions are easier to deal with than nonlinear ones.

While normal-form games capture the strategic structure of decision-making settings, they abstract away one key aspect of playing games that seems quite central to their character. Specifically, they assume that players make choices and act simultaneously, with no knowledge of the choices of their counterparts. But in many strategic situations, the players do not move simultaneously: they take turns in making their moves. An obvious example of such a situation is the game of chess. Another example is a bargaining scenario where a buyer and a seller take turns in making offers to each other in order to reach an agreement on the price of a commodity. Such situations may be modelled using sequential-moves games. In this chapter, we will study how these games are represented, and what notions of equilibria are used to analyse and solve them.

In a purely sequential-moves game, the players not only take turns in making their moves, but they typically know what the players did in all the previous moves. In contrast, when a player makes a move in a simultaneous-moves game, he does not know the other players' moves.

Purely sequential-moves and purely simultaneous-moves situations rarely arise in practice; many interesting real-world situations involve both simultaneous and sequential moves. Thus, we will learn how to model such situations with games and how to analyse and solve those games.

So far in this book, we have been focusing on bilateral (i.e., one-to-one) negotiations. One can easily imagine situations where more than two agents might need to negotiate with one another. For example, in a typical trading scenario, a seller might want to negotiate with multiple potential buyers. For such scenarios, we need multilateral negotiation protocols, that is protocols allowing one-to-many and many-to-many negotiations. In this chapter, we introduce a number of such protocols.

8.1 Alternating offers protocol with multiple bargainers

Consider the following many-to-many negotiation scenario. A group of two or more agents are together able to jointly produce a surplus, that is some joint gains. However, no subset of the group is able to do this. For example, say we have a group of five musicians that come together and create a piece of music that no subset could produce. By selling their music, they make a joint profit of £500,000. The key question here is “how should the joint gains be split between the individuals?”. We can think of the joint gains as a pie of unit size. The individuals must decide how they will divide the pie between themselves. An agreement requires the approval of all the players; no subset is allowed to reach an agreement.

Since there are more than two agents, we cannot use the alternating offers protocol described in Chapter 5. The protocol must be extended to deal with multiple players.

So far in this book, we have studied negotiation using two approaches: strategic and heuristic. The strategic approach is a non-cooperative game-theoretic approach. There is, however, another possibility: an axiomatic approach. The axiomatic approach will be the main focus of this chapter. Among the three approaches, the strategic and heuristic ones are better suited to the design of negotiating agents. Nevertheless, we include this chapter in order to explain the key concepts that underlie an axiomatic approach and, in the spirit of the Nash program (Nash, 1953; Binmore, 1985; Binmore and Dasgupta, 1987), show that the outcomes of some of the strategic models of bargaining can be very close to the outcomes of some axiomatic models.

We begin by understanding the key differences between cooperative and non-cooperative games. Following this, we introduce some of the prominent axiomatic models for single and multi-issue negotiation and show how some of their outcomes relate to those of strategic models. In the end, we give a comparative account of the axiomatic and strategic approaches in terms of their similarities and differences.

11.1 Background

Game-theoretic analysis of negotiation can be done using one of two possible approaches: axiomatic or strategic. In the former approach, negotiation is modelled as a cooperative game, while in the latter, it is modelled as a non-cooperative game.

In order for us to apply mathematical techniques to the analysis of negotiation, we must be able to express the domain of interest to us in appropriate mathematical terms. Such a formal representation can then be used as the basis upon which to construct computer programs that can negotiate in this domain. For example, suppose you are aiming to build a computer program that can negotiate on your behalf in the purchase of a used car. Then your formalisation should capture at least the following two things:

1. All attributes of the car and the associated purchase that might play a part in determining how desirable (or otherwise) the car is, both for you and for your negotiation counterparts. Relating to the car itself, these attributes might include make, model, colour, age, mileage, condition of bodywork and so on, and relating to the purchase would include price (of course!), length of insurance, and potentially others such as the vendor of the car.

2. A utility function, which defines, for every possible combination of values of the attributes characterising the negotiation domain, the utility that you would obtain from a deal on the purchase of a car.

Our aim in this chapter is twofold. First, we present a three-point classification scheme for negotiation domains, which was introduced by Rosenschein and Zlotkin (1994). This scheme provides a useful point of reference for understanding negotiation domains.

In the previous chapter, we took the agenda as given and studied the strategic behaviour of agents for the different multi-issue negotiation procedures. This study showed that, for a given agenda, the procedure is a key determinant of the outcome of a negotiation. In this chapter, we will learn the importance of the agenda: we will treat the procedure as given and see how the outcome of a negotiation can be changed by changing the agenda.

Given this influence, an economic agent will clearly prefer an agenda that maximises her individual utility. Such an agenda is called the agent's optimal agenda. But it may not always be computationally easy to find such an agenda. Thus, we will focus on some specific negotiation settings and study polynomial-time methods for finding an optimal agenda. These methods have both economic and computational significance: by using them, a player will be able to maximise her utility, and the methods have computational feasibility. In other words, they facilitate the design of software agents that can not only negotiate optimally over a given set of issues, but also choose the right agenda before actual negotiation begins.

Let us begin by looking at some example scenarios where the parties must choose an agenda for negotiation.

We hope that by now you will have a good understanding of the scope and applicability of negotiation techniques, as well as a feel for the kinds of techniques used to analyse negotiation settings and build negotiating systems. Our aim in this chapter is to describe briefly some other research areas that are related closely to negotiation. Specifically, we discuss the domain of social choice theory (which is concerned with the general problem of group decision making using techniques such as voting), the area known as argumentation (which is about trying to make sense of domains when there are conflicting arguments about the domain), and fair division (which is concerned specifically with the problem of how to divide goods/resources among a group of agents).

13.1 Social choice

We begin by looking at the domain of social choice theory. Social choice theory addresses itself to the problem of how a group of agents can make a group decision when they have conflicting preferences. The mechanisms that social choice theory studies for this problem are typically voting procedures, in the sense that we are familiar with voting procedures in everyday life, where they are used for political decision making in democratic societies.

The basic setting considered in social choice theory is as follows. As usual, we have a set P = {1,…,|P|} of agents, who in this chapter we will often refer to as voters.

As we saw in Chapter 1, the purpose of negotiation is to resolve a conflict between individuals and reach a mutually acceptable agreement with respect to some domain. A classic example of such conflict occurs when there are some scarce resources, and the individuals in question must reach a mutually acceptable agreement over how to divide/allocate the resources between themselves. The resources could be anything the individuals want: commodities, services, money, food, and so on. Typically, the negotiating individuals will have conflicting preferences over the possible divisions. Negotiation can be used to find a mutually acceptable way of dividing the resources. In this chapter, we will show how game theory can be used to analyse situations in which participants have conflicting preferences. We will later apply these techniques to the analysis of negotiation settings.

Before we start our analysis of such situations, we must identify and extract the information that is required to understand a situation of conflicting preferences. This information is represented in a game-theoretic model. In general, the models used in game theory represent abstractions of the underlying domain, which aim to capture all and only those aspects of the domain that are relevant for the decisions that must be made by those participating in it. The mathematical models of such conflicts of interests are called games.

In the previous chapters, we saw that finding optimal offers can be computationally expensive, especially in the context of solving the trade-off problem for the PDP and for finding optimal agendas. Because software agents have limited computational resources at their disposal, heuristic approaches can be useful for the design of negotiating agents in such situations. In such approaches, a heuristic is used to generate counter offers that are “good enough” (i.e., close to optimum) but not necessarily optimal.

Dealing with the bounded rationality of agents is just one of the reasons for using a heuristic approach. Another reason is that an opponent might not always play equilibrium strategies. In such cases, a strategy that is a best reply to the opponent's strategy must be played rather than an equilibrium one.

Irrespective of the reason for using this approach, heuristics can be designed for a number of purposes. For example, they can be designed to generate optimal counter offers, to predict information about an opponent, or to find optimal agendas. Research into the design of heuristics has been growing continuously and several different approaches – including local search methods, approximation methods, and machine learning methods – have been adopted. Below, we introduce a number of representative works in terms of their key underlying ideas, goals, and main results.