Introduction

In this chapter, we consider weighted voting games: a form of social choice system that seems much simpler than most schemes considered in this handbook, but which is, nevertheless, widely used for many important real-world social choice problems. There are at least two very good reasons for studying weighted voting systems: first, as we have already mentioned, they are widely used in practice; and second, for all their apparent simplicity, they possess interesting mathematical and computational properties, making them interesting objects from the point of view of scientific study.

Weighted voting games originated in the domain of cooperative game theory (Chalkiadakis et al., 2011). They model decision-making situations in which a set of voters must make a binary (yes/no) decision on some particular issue; each voter is allocated a numeric weight, and the decision is carried if the sum of weights of voters in favour of it meets or exceeds some specific given threshold, called the quota. Weighted voting games have many applications beyond social choice theory. For example, they can be used to model settings where each player has a certain amount of a given resource (say, time, money, or manpower), and there is a goal that can be reached by any coalition that possesses a sufficient amount of this resource.

The remainder of this chapter is structured as follows.

1. • In Section 16.2, we present the basic models and solution concepts that will be used throughout the remainder of the chapter. We start by defining cooperative games in characteristic function form, and then introduce weighted voting games. We go on to describe some key solution concepts for cooperative games: the core, the Shapley value, and the Banzhaf index.

2. • In Section 16.3, we consider computational properties of weighted voting games: in particular, the complexity of computing the core, the Shapley value, and the Banzhaf index.

3. • In Section 16.4, we consider the (sometimes unintuitive) relationship between the weight that a voter is assigned in a weighted voting game and the power that this voter then wields.

4. • In Section 16.5, we consider the extent to which weighted voting games can be considered as a representation scheme for yes/no voting systems (i.e., simple cooperative games). We give a condition on yes/no voting systems that is both necessary and sufficient for such a system to be representable as a weighted voting game.

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.

In our everyday lives, negotiation is ubiquitous. At work, we bargain with clients about the terms of a contract; we bargain with our boss about a pay rise when the contract is signed. At home, we bargain with our partners about who will tidy the house; we bargain with our children about how many stories they can read before bed. And politicians, of course, routinely bargain in situations that have life or death consequences. The purpose of negotiation is to reach an agreement, and in particular, agreement in the presence of conflicting goals and preferences. If your preferences, goals, and aspirations were completely aligned with mine, then there would be no conflict, and hence there would be no need for negotiation. In this case, the best outcome for me would also be the best outcome for you, and so we could simply determine such an outcome, and then implement it. Neither of us would have any incentive other than to find such a mutually optimal best outcome – our joint problem is nothing more than an optimisation problem. Unfortunately, of course, the real world is not like that. More often than not, people have very different goals and preferences, and when this occurs, some method is required to find an outcome that will be acceptable to all concerned. Negotiation provides such a mechanism.

So far in the book, we have been focusing on situations in which all the negotiating participants are software agents. However, there are many applications where automated negotiators should be able to interact proficiently and collaborate with people. Automated negotiators can be used with humans in the loop or without them. When used with humans, they can alleviate some of the effort required of people during negotiations and can assist people less qualified in the negotiation process (Kersten and Lai, 2007). Also, there may be situations in which automated negotiators can even replace human negotiators (Durenard, 2013). Another possibility is for people embarking on important negotiation tasks to use these agents as a training tool, prior to actually performing the task (Lin et al., 2014). For example, automated negotiators in e-commerce applications can bargain over a price with their site's buyers (Kauppi et al., 2013). Personalised agents that support their users can negotiate with humans with conflicting preferences towards deciding on a meeting time (Tambe, 2008). They can also negotiate the formation of a “care plan” for patients requiring a course of medical treatment (Amir et al., 2013). A person who is preparing for a job interview can train herself with an agent that plays the role of the employer (Lin et al., 2009).

In this chapter we discuss a number of applications where agents can be used to negotiate on behalf of their human counterparts. In general, agents can be used for any kind of negotiation over the allocation of limited resources. For our discussion, however, we focus on the following representative cases. While some of the applications will involve direct negotiations between the parties, others will go through a trusted mediator. Also, while some will only involve software agents, others will deal with both software agents and human negotiators. Then, some applications will be for bilateral negotiations and others for multilateral ones. Finally, some will be industrial applications and others commercial ones.

12.1 Business process management

A business process is composed of a number of interdependent tasks that must be executed in a controlled and ordered way. This execution involves the consumption of resources. In most organisations, these resources are grouped into business units that control the way in which they are deployed. This is also the case with the British Telecom (BT) business process of providing a quote to a customer for installing a network to deliver a specific type of telecommunications service. For the management of this process, Jennings et al. (2000) and Norman et al. (1997) developed an agent-based negotiation framework called Adept (Advanced Decision Environment for Process Tasks).

In this chapter, we turn our attention to negotiation mechanisms – the protocols that will be used by agents when negotiating over domains of the type discussed in the preceding chapter. The first protocol we look at is concerned with negotiation over a single issue. Many real-world situations involve negotiation over a single issue. Typical examples include a buyer and a seller negotiating over the sale price of a commodity, a landlord and a tenant negotiating over the rental price of some accommodation, and an employer and an employee negotiating over the employee's salary. We will refer to the issue over which negotiation takes place as a “pie”, with the terminology arising from the analogy of a group of people negotiating about how to share a pie amongst themselves.

We know that, in order to negotiate, the parties must follow a protocol. There are many different protocols in the literature on negotiation, but arguably the most influential of these is the alternating offers protocol. This protocol was first analysed by Ingolf Stahl (1972) in the context of the division of a discretely divisible pie (i.e., a pie that can only be divided in a finite number of ways), and later by Ariel Rubinstein (1982) in the context of the division of a continuously divisible pie (i.e., any division of the pie is possible).