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.