1

AN OVERVIEW

1.1INTRODUCTION

1.1.1Why study game theory?
Game theory is everywhere these days.

After thrilling a whole generationof post-1970 economists, it is spreading like a bushfire throughthe social sciences.

Two prominent game theorists,Robert Aumann and Oliver Hart, explain the attraction in thefollowing way:

Game Theory may be viewed as asort of umbrella or ‘unified field’ theory for the rational side ofsocial science…[it] does not use different, ad hoc constructs…itdevelops methodologies that apply in principle to all interactivesituations.

(Aumann and Hart, 1992)

Of course, you might say, twopractitioners would say that, wouldn’t they. But the view is widelyheld, even among apparently disinterested parties.

Jon Elster, for instance, awell-known social theorist with very diverse interests, remarks ina similar fashion:

if one accepts that interaction isthe essence of social life, then… game theory provides solidmicrofoundations for the study of social structure and socialchange.

(Elster, 1982)

In many respects this enthusiasmis not difficult to understand. Game theory was probably born withthe publication of The Theory of Games and Economic Behaviour byJohn von Neumann and Oskar Morgenstern (first published in 1944with second and third editions in 1947 and 1953).

They defined a game as anyinteraction between agents that is governed by a set of rulesspecifying the possible moves for each participant and a set ofoutcomes for each possible combination of moves. One is hard put tofind an example of social phenomenon that cannot be so described.Thus a theory of games promises to apply to almost any socialinteraction where

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2

THE ELEMENTS OF GAME THEORY

2.1INTRODUCTION
This chapter introduces the central ideas in game theory. It beginsby showing how rational players can logically weed out strategieswhich are strategically inferior (sections 2.3 and 2.4).

Such elimination of strategies relies on what game theory refersto as dominance reasoning and it sometimes requires the assumptionof common knowledge of rationality (CKR).

It is important because it yields clear predictions of whatinstrumentally rational players will do in some games by means of astep-by-step logic.

However, in many games dominance reasoning offers no clear (oruseful) predictions of what might happen.

In these circumstances, game theorists commonly turn to the Nashequilibrium solution concept, named after its creator John Nash(section 2.5). The basic idea behind this concept is that rationalplayers should not want to change their strategies if they knewwhat each of them had chosen to do.

This solution concept helps to refine the predictions of gametheory. However, there is a cost in terms of generality.

The step to Nash seems to require rather more than theassumptions of rationality and CKR.

In section 1.2.2 of the previous chapter we described theessence of the extra requirement: the assumption that players’beliefs will be consistently aligned (CAB).

In some games even this move does not generate predictionsadequately because there are some games in which no specific set ofstrategies is recommended by the Nash equilibrium.

In the jargon, there are games in which there is either no Nashequilibrium in pure strategies, or there are many. 1

Thus predictions made using the Nash equilibrium concept can beeither non-existent or indeterminate.

As a result game theorists have attempted to refine the Nashequilibrium concept.

We present two such refinements: the Bayesian Nash equilibriumconcept for games of incomplete information (section 2.6) and theidea of

trembling hand perfect equilibria (section 2.7).

They embody two of the central ideas which have been at play inthe project of refining the Nash equilibrium to overcome theproblems it encounters in many games.

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3

DYNAMIC GAMES

Backward induction and some extensive form refinements of the Nashequilibrium

3.1INTRODUCTION

This chapter looks at games which have a dynamic structure; thatis, games in which one player makes a move after the other (ratherthan choosing strategies simultaneously).

In these situations, game theory needs to specify the preciseprotocol of moves.

Diagrammatically dynamic games resemble tree diagrams (recallsection 2.2) which are known formally as the extensive form.

So the extensive form representation is appropriate for adynamic game, whereas the matrix representation (or, more formally,the normal form) which we have been using so far is suitable forinteractions in which players choose simultaneously.

The next section begins with an illustration of the advantagesof the extensive form as compared with the normal form for suchgames.

It continues by showing that in some games the extensive formcan help pinpoint a solution which proved elusive while the gamewas viewed in its matrix, or normal, form.

In terms of the discussion of the previous chapter, the study ofa game’s dynamic structure potentially helps by reducing the numberof Nash equilibria.

The hallmark of the analysis of extensive form games is the useof a type of reasoning called backward induction.

Section 3.3 focuses on a particular refinement of the Nashequilibrium which results from a marriage of the original ideabehind the Nash equilibrium and backward induction: the famous(within game theoretical circles) subgame perfect Nashequilibrium.

Section 3.4 is devoted to some (important) controversies whichresult from the application of the common knowledge of rationality(CKR) axiom in dynamic (i.e. extensive form) games.

Sections 3.5 and 3.6 sketch three other major refinements of theNash equilibrium: sequential equilibria, proper equilibria andthose which depend on the use of forward induction. The chapterconcludes with a critical assessment of the Nash equilibriumconcept as well the attempts to refine it.

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4

BARGAINING GAMES

4.1INTRODUCTION

Liberal theorists often explain the State with reference to stateof nature.

For instance, within the Hobbesian tradition there is a starkchoice between a state of nature in which a war of all against allprevails and a peaceful society where the peace is enforced by aState which acts in the interest of all.

The legitimacy of the State derives from the fact that peoplewho would otherwise live in Hobbes’s state of nature (in which lifeis ‘brutish, nasty and short’) can clearly see the advantages ofcreating a State.

Even if a State had not surfaced historically for all sorts ofother reasons, it would have to be invented.

Such a hypothesised ‘invention’ would require a cooperative actof coming together to create a State whose purpose will be tosecure rights over life and property.

Nevertheless, even if all this were common knowledge, it wouldnot guarantee that the State will be created.

There is a tricky further issue which must be resolved.

The people must agree to the precise property rights which theState will defend and this is tricky because there are typically avariety of possible property rights and the manner in which thebenefits of peace will be distributed depends on the preciseproperty rights which are selected (see Box 4.1).

In other words, the common interest in peace cannot be the onlyelement in the liberal explanation of the State, as anywell-defined and policed property rights will secure the peace.

The missing element is an account of how a particular set ofproperty rights are selected and this would seem to require ananalysis of how people resolve conflicts of interest. This is wherebargaining theory promises to make an important contribution to theliberal theory of the State because it is concerned precisely withinteractions of this sort.

To be specific, the bargaining problem is the simplest, mostabstract, ingredient of any situation in which two (or more) agentsare able to produce some benefit throughcooperating with one another, provided they agree in advanceon a division between them. If they fail to agree, the

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5

THE PRISONERS’ DILEMMA

5.1INTRODUCTION: THE DILEMMA AND THE STATE

The prisoners’ dilemma fascinates social scientists because it isan interaction where the individual pursuit of what seems rationalproduces a collectively self-defeating result.

Each person does what appears best to them and yet the outcomeis painfully inferior for all.

Even though there is nothing obviously faulty with their logic,their attempt to improve their prospects makes everyone worseoff.

The paradoxical quality of this result helps explain part of thefascination. But the major reason for the interest is purelypractical.

Outcomes in social life are often lessthan we might hope and the prisoners’ dilemma provides onepossible key to their understanding.

The name comes from a particular illustration of the interactionwhich is credited to Albert Tucker in the 1950s.

In this example, two people are picked up by the police for arobbery and placed in separate cells. They both have the option toconfess to the crime or not, and the district attorney tells eachof them what is likely to happen and makes each an offer.

Figure 5.1 sets out the likely consequences presented by thedistrict attorney in terms of years in prison.

Player B

Player A

Not confess
Confess

Not confess
1, 1
5, 0

Confess
0, 5
3, 3

Figure 5.1

The rationale behind these (negative) pay-offs is something likethis. If both ‘confess’ then the judge, being in no doubt overtheir guilt, will give them 3 years each in prison. Whereas if theyboth ‘don’t confess’ then

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6

REPEATED GAMES AND REPUTATIONS

6.1INTRODUCTION

Many social interactions are repeated either with the same personor with people who are drawn from the same social group.

Indeed, since one-off encounters typically occur only betweenstrangers, the analysis of repeated games promises to extend thescope of game theory considerably.

In addition, when the same game is played repeatedly thestrategic options for players expand significantly, becoming in theprocess more life-like in a number of respects.

For instance, it becomes possible to condition what you do onwhat your opponent has done in previous rounds.

Thus you can punish or reward your opponent depending on whatthey have done in the past.

By definition, this cannot be done when the game is played onlyonce.

Likewise players learn things about theiropponents from the way they have behaved in the past.

Such learning can be exploited by players behaving in particularways to develop reputations for playing the game in particularways.

Therefore the analysis of repeated games also promises furtherinsights regarding the types of behaviour which we might expectfrom instrumentally rational players.

This chapter considers repeated games in various settings.

We begin with the finitely repeated prisoners’ dilemma game insection 6.2 and make use of backward induction and the subgameperfect Nash equilibrium concept from section 3.3.

Perhaps somewhat surprisingly, mutual defection remains the onlyNash equilibrium.

The following two sections discuss, respectively, indefinitelyrepeated prisoners’ dilemma and the related free rider games.

We show (section 6.4) that mutual cooperation is a possible Nashequilibrium outcome in these games provided there is a ‘sufficient’degree of uncertainty over when the repetition will cease.

There are some significant implications here both for liberalpolitical theory and for the explanatory power of game theory.

We notice that this result means that mutual cooperation mightbe achieved without the intervention of a collective agency likethe State and/or without appealing to some expanded notion ofrational agency (see Chapter 5). In other words cooperationcould

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