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Suppose we are faced with the problem of making a decision. One approach to the problem might be to determine the desired outcome and then to behave in a way that leads to that result. This leaves open the question of whether it is always possible to achieve the desired outcome. An alternative approach is to list the courses of action that are available and to determine the outcome of each of those behaviours. One of these outcomes is preferred because it is the one that maximises something of value (for example, the amount of money received). The course of action that leads to the preferred outcome is then picked from the available set.We will call the second approach “making an optimal decision”. In this book, we will develop a mathematical framework for studying the problem of making an optimal decision in a variety of circumstances.

A man hears that his young daughter always takes a nickel when an adult relative offers her a choice between a nickel and a dime. He explains to his daughter, “A dime is twice as valuable as a nickel, so you should always choose the dime”. In a rather exasperated tone, his daughter replies “Daddy, but then people will not offer me any money”.

In this chapter, we add an extra layer of complexity to our models of decision making by introducing the idea of a state-dependent decision process. The processes we will consider can either be deterministic or stochastic. To begin with, we will assume that the process must terminate by an a priori fixed time (a “finite horizon” model). In principle, decisions can be made at times = 0, 1, 2,..., — 1, although the actual number of decisions made may be fewer than if the process terminates early as a consequence of the actions taken. Models that have no a priori restriction on the number of decisions to be taken (“infinite horizon” models) will be considered in the next chapter.

An interactive decision problem involves two or more individuals making a decision in a situation where the payoff to each individual depends (at least in principle) on what every individual decides. Borrowing some terminology from recreational games, which form only a subset of examples of interactive decision problems, all such problems are termed “games” and the individuals making the decisions are called “players”. However, recreational games may have restrictive features that are not present in general games: for example, it is not necessarily true that one player “wins” only if the other “loses”. Games that have winners and losers in this sense are called ; these are considered in Section 4.7.3.

So far we have considered static games in which decisions are assumed to be made simultaneously (or, at least, in ignorance of the choices made by the other players). However, there are many situations of interest in which decisions are made at various times with at least some of the earlier choices being public knowledge when the later decisions are being made. These games are called because there is an explicit time-schedule that describes when players make their decisions.

For ease of exposition, most of this book is devoted to models in which players have discrete and finite strategy sets. However, several classic games describe situations in which the players do not choose actions from a discrete set; instead their pure strategy sets are subsets of the real line. In this chapter, we give a few examples to show how the concepts of game theory are easily extended to such cases. Economic models of a duopoly provide examples with pure-strategy Nash equilibria, and the so-called War of Attrition has an equilibrium involving mixed strategies.

Consider the following two (related) questions. In the Prisoners’ Dilemma, uncooperative behaviour was the predicted outcome although cooperative behaviour would lead to greater payoffs for all players if was cooperative. Interpreting the Prisoners’ Dilemma as a generalised social interaction, we can ask the question: Is external (e.g., governmental) force required in order to sustain cooperation or can such behaviour be induced in a liberal, individually rational way? In the Cournot duopoly, cartels were not stable. However, in many countries, substantial effort is expended in making and enforcing anticollusion laws. So it seems that, in reality, there is a risk of cartel formation. How can cartels be stable?

So far we have considered two-player games in the framework of Classical Game Theory, where the outcome depends on the choices made by rational and consciously reasoning individuals. The solution for this type of game (the Nash equilibrium) was based on the idea that each player uses a strategy that is a best response to the strategy chosen by the other, so neither would change what they were doing. For symmetric Nash equilibria, (σ*, σ*), we can give an alternative interpretation of the Nash equilibrium by placing the game in a population context. In a population where everyone uses strategy σ*, the best thing to do is follow the crowd; so if the population starts with everyone using σ*, then it will remain that way — the is in equilibrium. Nash himself introduced this view, calling it the “mass action interpretation”. A natural question to ask is then: What happens if the population is close to, but not actually at, its equilibrium configuration? Does the population tend to evolve towards the equilibrium or does it move away? This question can be investigated using Evolutionary Game Theory, which was invented for biological models but has now been adopted by some economists.

In the previous chapter, we investigated the concept of an evolutionarily stable strategy. Although this concept implicitly assumes the existence of some kind of evolutionary dynamics, it gives an incomplete description. First, an ESS may not exist — in which case the analysis tells us nothing about the evolution of the system described by the game. Second, the definition of an ESS deals only with monomorphic populations in which every individual uses the same strategy. But, if the ESS is a mixed strategy, then all strategies in the support of the ESS obtain the same payoff as the evolutionarily stable strategy itself. So it is pertinent to ask whether a polymorphic population with the same population profile as that generated by the ESS can also be stable. To address these questions, we will look at a specific type of evolutionary dynamics, called replicator dynamics.