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.

Modern data-mining applications, often called “big-data” analysis, require us to manage immense amounts of data quickly. In many of these applications, the data is extremely regular, and there is ample opportunity to exploit parallelism. Important examples are:

1. (1) The ranking of Web pages by importance, which involves an iterated matrix-vector multiplication where the dimension is many billions.

2. (2) Searches in “friends” networks at social-networking sites, which involve graphs with hundreds of millions of nodes and many billions of edges.

To deal with applications such as these, a new software stack has evolved. These programming systems are designed to get their parallelism not from a “super-computer,” but from “computing clusters” – large collections of commodity hardware, including conventional processors (“compute nodes”) connected by Ethernet cables or inexpensive switches. The software stack begins with a new form of file system, called a “distributed file system,” which features much larger units than the disk blocks in a conventional operating system. Distributed file systems also provide replication of data or redundancy to protect against the frequent media failures that occur when data is distributed over thousands of low-cost compute nodes.

On top of these file systems, many different higher-level programming systems have been developed. Central to the new software stack is a programming system called MapReduce. Implementations of MapReduce enable many of the most common calculations on large-scale data to be performed on computing clusters efficiently and in a way that is tolerant of hardware failures during the computation.

MapReduce systems are evolving and extending rapidly. Today, it is common for MapReduce programs to be created from still higher-level programming systems, often an implementation of SQL. Further, MapReduce turns out to be a useful, but simple, case of more general and powerful ideas. We include in this chapter a discussion of generalizations of MapReduce, first to systems that support acyclic workflows and then to systems that implement recursive algorithms.

Our last topic for this chapter is the design of good MapReduce algorithms, a subject that often differs significantly from the matter of designing good parallel algorithms to be run on a supercomputer.

There are many sources of data that can be viewed as a large matrix. We saw in Chapter 5 how the Web can be represented as a transition matrix. In Chapter 9, the utility matrix was a point of focus. And in Chapter 10 we examined matrices that represent social networks. In many of these matrix applications, the matrix can be summarized by finding “narrower” matrices that in some sense are close to the original. These narrow matrices have only a small number of rows or a small number of columns, and therefore can be used much more efficiently than can the original large matrix. The process of finding these narrow matrices is called dimensionality reduction.

We saw a preliminary example of dimensionality reduction in Section 9.4. There, we discussed UV-decomposition of a matrix and gave a simple algorithm for finding this decomposition. Recall that a large matrix M was decomposed into two matrices U and V whose product UV was approximately M. The matrix U had a small number of columns whereas V had a small number of rows, so each was significantly smaller than M, and yet together they represented most of the information in M that was useful in predicting ratings of items by individuals.

In this chapter we shall explore the idea of dimensionality reduction in more detail. We begin with a discussion of eigenvalues and their use in “principal component analysis” (PCA). We cover singular-value decomposition, a more powerful version of UV-decomposition. Finally, because we are always interested in the largest data sizes we can handle, we look at another form of decomposition, called CUR-decomposition, which is a variant of singular-value decomposition that keeps the matrices of the decomposition sparse if the original matrix is sparse.

Eigenvalues and Eigenvectors

We shall assume that you are familiar with the basics of matrix algebra: multiplication, transpose, determinants, and solving linear equations for example. In this section, we shall define eigenvalues and eigenvectors of a symmetric matrix and show how to find them.

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.