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[[!meta title="Game Theory: a Critical Introduction"]]

## Index

* Difference between meteorological and traffic-jam-style predictions: 18.
* Frequent assumption (but not always) on game theory that individuals knows
  the rules of the game; that they even known their inner motives: 28.
* Individualism, separation of structure and choice, 31.

# Excerpts

## Intro

What is game theory:

    In many respects this enthusiasm is not difficult to understand. Game theory
    was probably born with the publication of The Theory of Games and Economic
    Behaviour by John von Neumann and Oskar Morgenstern (first published in
    1944 with second and third editions in 1947 and 1953). They defined a game
    as any interaction between agents that is governed by a set of rules
    specifying the possible moves for each participant and a set of outcomes for
    each possible combination of moves.

How it can help:

    If game theory does make a further substantial contribution, then we
    believe that it is a negative one. The contribution comes through
    demonstrating the limits of a particular form of individualism in social
    science: one based exclusively on the model of persons as preference satisfiers.
    This model is often regarded as the direct heir of David Hume’s (the 18th
    century philosopher) conceptualisation of human reasoning and motivation. It
    is principally associated with what is known today as rational choice theory, or
    with the (neoclassical) economic approach to social life (see Downs, 1957, and
    Becker, 1976). Our main conclusion on this theme (which we will develop
    through the book) can be rephrased accordingly: we believe that game theory
    reveals the limits of ‘rational choice’ and of the (neoclassical) economic
    approach to life. In other words, game theory does not actually deliver Jon
    Elster’s ‘solid microfoundations’ for all social science; and this tells us
    something about the inadequacy of its chosen ‘microfoundations’.

Assumptions:

    three key assumptions: agents are instrumentally
    rational (section 1.2.1); they have common knowledge of this rationality
    (section 1.2.2); and they know the rules of the game (section 1.2.3).
    These assumptions set out where game theory stands on the big questions of
    the sort ‘who am I, what am I doing here and how can I know about either?’.
    The first and third are ontological. 1 They establish what game theory takes as
    the material of social science: in particular, what it takes to be the essence of
    individuals and their relation in society. The second raises epistemological
    issues 2 (and in some games it is not essential for the analysis). It is concerned
    with what can be inferred about the beliefs which people will hold about how
    games will be played when they have common knowledge of their rationality.

Instrumental rationality (_Homo economicus_):

    We spend more time discussing these assumptions than is perhaps usual in
    texts on game theory because we believe that the assumptions are both
    controversial and problematic, in their own terms, when cast as general
    propositions concerning interactions between individuals. This is one respect
    in which this is a critical introduction. The discussions of instrumental
    rationality and common knowledge of instrumental rationality (sections 1.2.1
    and 1.2.2), in particular, are indispensable for anyone interested in game
    theory. In comparison section 1.2.3 will appeal more to those who are
    concerned with where game theory fits in to the wider debates within social

    [...]

    Individuals who are instrumentally rational have preferences over various
    ‘things’, e.g. bread over toast, toast and honey over bread and butter, rock
    over classical music, etc., and they are deemed rational because they select
    actions which will best satisfy those preferences. One of the virtues of this
    model is that very little needs to be assumed about a person’s preferences.
    Rationality is cast in a means-end framework with the task of selecting the
    most appropriate means for achieving certain ends (i.e. preference
    satisfaction); and for this purpose, preferences (or ‘ends’) must be coherent
    in only a weak sense that we must be able to talk about satisfying them more
    or less. Technically we must have a ‘preference ordering’ because it is only
    when preferences are ordered that we will be able to begin to make
    judgements about how different actions satisfy our preferences in different
    degrees.

    [...]

    Thus it appears a promisingly general model of action. For instance, it could
    apply to any type of player of games and not just individuals. So long as the
    State or the working class or the police have a consistent set of objectives/
    preferences, then we could assume that it (or they) too act instrumentally so
    as to achieve those ends. Likewise it does not matter what ends a person
    pursues: they can be selfish, weird, altruistic or whatever; so long as they
    consistently motivate then people can still act so as to satisfy them best.

An agent is "rational" in this conext when they have preference ordering" and
if "they select the action that maximizes those preferences:

    Readers familiar with neoclassical Homo economicus will need no further
    introduction. This is the model found in standard introductory texts, where
    preferences are represented by indifference curves (or utility functions) and
    agents are assumed rational because they select the action which attains the
    highest feasible indifference curve (maximises utility). For readers who have
    not come across these standard texts or who have forgotten them, it is worth
    explaining that preferences are sometimes represented mathematically by a
    utility function. As a result, acting instrumentally to satisfy best one’s
    preferences becomes the equivalent of utility maximising behaviour.

Reason and slavery:

    Even when we accept the Kantian argument, it is plain that reason’s
    guidance is liable to depend on characteristics of time and place. For
    example, consider the objective of ‘owning another person’. This obviously
    does not pass the test of the categorical imperative since all persons could
    not all own a person. Does this mean then we should reject slave-holding? At
    first glance, the answer seems to be obvious: of course, it does! But notice it
    will only do this if slaves are considered people. Of course we consider
    slaves people and this is in part why we abhor slavery, but ancient Greece
    did not consider slaves as people and so ancient Greeks would not have been
    disturbed in their practice of slavery by an application of the categorical
    imperative.

Reason dependent on culture:

    Wittgenstein suggests that if you want to know why people act in the way that
    they do, then ultimately you are often forced in a somewhat circular fashion to
    say that such actions are part of the practices of the society in which those
    persons find themselves. In other words, it is the fact that people behave in a
    particular way in society which supplies the reason for the individual person to
    act: or, if you like, actions often supply their own reasons. This is shorthand
    description rather than explanation of Wittgenstein’s argument, but it serves to
    make the connection to an influential body of psychological theory which
    makes a rather similar point.

Cognitive dissonance and free market proponents:

    Festinger’s (1957) cognitive dissonance theory proposes a model where
    reason works to ‘rationalise’ action rather than guide it. The point is that we
    often seem to have no reason for acting the way that we do. For instance, we
    may recognise one reason for acting in a particular way, but we can equally
    recognise the pull of a reason for acting in a contrary fashion. Alternatively,
    we may simply see no reason for acting one way rather than another. In such
    circumstances, Festinger suggests that we experience psychological distress. It
    comes from the dissonance between our self-image as individuals who are
    authors of our own action and our manifest lack of reason for acting. It is like
    a crisis of self-respect and we seek to remove it by creating reasons. In short
    we often rationalise our actions ex post rather than reason ex ante to take them
    as the instrumental model suggests.

    [...]

    Research has shown that people seek out and read advertisements for
    the brand of car they have just bought. Indeed, to return us to economics, it is
    precisely this insight which has been at the heart of one of the Austrian and
    other critiques of the central planning system when it is argued that planning
    can never substitute for the market because it presupposes information
    regarding preferences which is in part created in markets when consumers
    choose.

Infinite regress of the economics of information acquisiton (i.e learning, eg.
from a secret service):

    Actually most game theorists seem to agree on one aspect of the problem
    of belief formation in the social world: how to update beliefs in the presence
    of new information. They assume agents will use Bayes’s rule. This is explained
    in Box 1.6. We note there some difficulties with transplanting a technique from
    the natural sciences to the social world which are related to the observation we
    have just made. We focus here on a slightly different problem. Bayes provides
    a rule for updating, but where do the original (prior) expectations come from?
    Or to put the question in a different way: in the absence of evidence, how do
    agents form probability assessments governing events like the behaviour of
    others?

    There are two approaches in the economics literature. One responds by
    suggesting that people do not just passively have expectations. They do not
    just wait for information to fall from trees. Instead they make a conscious
    decision over how much information to look for. Of course, one must have
    started from somewhere, but this is less important than the fact that the
    acquisition of information will have transformed these original ‘prejudices’.
    The crucial question, on this account, then becomes: what determines the
    amount of effort agents put into looking for information? This is deceptively
    easy to answer in a manner consistent with instrumental rationality. The
    instrumentally rational agent will keep on acquiring information to the point
    where the last bit of search effort costs her or him in utility terms the same
    amount as the amount of utility he or she expects to get from the
    information gained by this last bit of effort. The reason is simple. As long as
    a little bit more effort is likely to give the agent more utility than it costs,
    then it will be adding to the sum of utilities which the agent is seeking to
    maximise.

    [...]

    This looks promising and entirely consistent with the definition of
    instrumentally rational behaviour. But it begs the question of how the agent
    knows how to evaluate the potential utility gains from a bit more information
    _prior to gaining that information_. Perhaps he or she has formulated expectations of
    the value of a little bit more information and can act on that. But then the
    problem has been elevated to a higher level rather than solved. How did he or
    she acquire that expectation about the value of information? ‘By acquiring
    information about the value of information up to the point where the
    marginal benefits of this (second-order) information were equal to the costs’,
    is the obvious answer. But the moment it is offered, we have the beginnings of
    an infinite regress as we ask the same question of how the agent knows the
    value of this second-order information. To prevent this infinite regress, we
    must be guided by something _in addition_ to instrumental calculation. But this
    means that the paradigm of instrumentally rational choices is incomplete. The
    only alternative would be to assume that the individual _knows_ the benefits that
    he or she can expect on average from a little more search (i.e. the expected
    marginal benefits) because he or she knows the full information set. But then
    there is no problem of how much information to acquire because the person
    knows everything!

Infinite recursion of the common knowledge (CKR):

    If you want to form an expectation about what somebody does, what
    could be more natural than to model what determines their behaviour and
    then use the model to predict what they will do in the circumstances that
    interest you? You could assume the person is an idiot or a robot or whatever,
    but most of the time you will be playing games with people who are
    instrumentally rational like yourself and so it will make sense to model your
    opponent as instrumentally rational. This is the idea that is built into the
    analysis of games to cover how players form expectations. We assume that
    there is common knowledge of rationality held by the players. It is at once
    both a simple and complex approach to the problem of expectation
    formation. The complication arises because with common knowledge of
    rationality I know that you are instrumentally rational and since you are
    rational and know that I am rational you will also know that I know that you
    are rational and since I know that you are rational and that you know that I
    am rational I will also know that you know that I know that you are rational
    and so on…. This is what common knowledge of rationality means.

    [...]

    It is difficult to pin down because common knowledge of X
    (whatever X may be) cannot be converted into a finite phrase beginning with ‘I
    know…’. The best one can do is to say that if Jack and Jill have common
    knowledge of X then ‘Jack knows that Jill knows that Jack knows …that Jill
    knows that Jack knows…X’—an infinite sentence. The idea reminds one of
    what happens when a camera is pointing to a television screen that conveys the
    image recorded by the very same camera: an infinite self-reflection. Put in this
    way, what looked a promising assumption suddenly actually seems capable of
    leading you anywhere.

    [...]

    The problem of expectation formation spins hopelessly out of control.

    Nevertheless game theorists typically assume CKR and many of them, and
    certainly most people who apply game theory in economics and other
    disciplines

Uniformity: Consistent Alignment of Beliefs (CAB), another weak assumption
based on Harsanyi doctrine requiring equal information; followed by a
comparison with Socract dialectics:

    Put informally, the notion of _consistent alignment of beliefs_ (CAB) means that
    no instrumentally rational person can expect another similarly rational
    person who has the same infor mation to develop different thought
    processes. Or, alternatively, that no rational person expects to be surprised
    by another rational person. The point is that if the other person’s thought is
    genuinely moving along rational lines, then since you know the person is
    rational and you are also rational then your thoughts about what your
    rational opponent might be doing will take you on the same lines as his or
    her own thoughts. The same thing applies to others provided they respect
    _your_ thoughts. So your beliefs about what your opponents will do are
    consistently aligned in the sense that if you actually knew their plans, you
    would not want to change your beliefs; and if they knew your plans they
    would not want to change the beliefs they hold about you and which support
    their own planned actions.

    Note that this does not mean that everything can be deterministically
    predicted.

Reason reflecting on itself:

    These observations are only designed to signal possible trouble ahead
    and we shall examine this issue in greater detail in Chapters 2 and 3. We
    conclude the discussion now with a pointer to wider philosophical currents.
    Many decades before the appearance of game theor y, the Ger man
    philosophers G.F.W.Hegel and Immanuel Kant had already considered the
    notion of the self-conscious reflection of human reasoning on itself. Their
    main question was: can our reasoning faculty turn on itself and, if it can,
    what can it infer? Reason can certainly help persons develop ways of
    cultivating the land and, therefore, escape the tyranny of hunger. But can it
    understand how it, itself, works? In game theory we are not exactly
    concerned with this issue but the question of what follows from common
    knowledge of rationality has a similar sort of reflexive structure. When
    reason knowingly encounters itself in a game, does this tell us anything
    about what reason should expect of itself?

    What is revealing about the comparison between game theory and
    thinkers like Kant and Hegel is that, unlike them, game theory offers
    something settled in the form of CAB. What is a source of delight,
    puzzlement and uncertainty for the German philosophers is treated as a
    problem solved by game theory. For instance, Hegel sees reason reflecting
    on reason as it reflects on itself as part of the restlessness which drives
    human history. This means that for him there are no answers to the
    question of what reason demands of reason in other people outside of
    human history. Instead history offers a changing set of answers. Likewise
    Kant supplies a weak answer to the question. Rather than giving substantial
    advice, reason supplies a negative constraint which any principle of
    knowledge must satisfy if it is to be shared by a community of rational
    people: any rational principle of thought must be capable of being followed
    by all. O’Neill (1989) puts the point in the following way:

        [Kant] denies not only that we have access to transcendent meta-
        physical truths, such as the claims of rational theology, but also that
        reason has intrinsic or transcendent vindication, or is given in
        consciousness. He does not deify reason. The only route by which we
        can vindicate certain ways of thinking and acting, and claim that those
        ways have authority, is by considering how we must discipline our
        thinking if we are to think or act at all. This disciplining leads us not to
        algorithms of reason, but to certain constraints on all thinking,
        communication and interaction among any plurality. In particular we are
        led to the principle of rejecting thought, act or communication that is
        guided by principles that others cannot adopt.
        (O’Neill p. 27)

Summary:

    To summarise, game theory is avowedly Humean in orientation.  [...]
    The second [aspect] is that game theorists seem to assume _too much_ on behalf
    of reason [even more than Hume did].

Giddens, Wittgenstein language games and the "organic or holistic view of the relation between action and structure" (pages 30-31):

    The question is ontological and it connects directly with the earlier
    discussion of instrumental rationality. Just as instrumental rationality is not
    the only ontological view of what is the essence of human rationality, there is
    more than one ontological view regarding the essence of social interaction.
    Game theory works with one view of social interaction, which meshes well
    with the instrumental account of human rationality; but equally there are
    other views (inspired by Kant, Hegel, Marx, Wittgenstein) which in turn
    require different models of (rational) action.

State (pages 32-33):

    Perhaps the most famous example of this type of
    institutional creation comes from the early English philosopher Thomas
    Hobbes who suggested in Leviathan that, out of fear of each other,
    individuals would contract with each other to form a State. In short, they
    would accept the absolute power of a sovereign because the sovereign’s
    ability to enforce contracts enables each individual to transcend the dog-
    eat-dog world of the state of nature, where no one could trust anyone and
    life was ‘short, nasty and brutish’.

    Thus, the key individualist move is to draw attention to the way that
    structures not only constrain; they also enable (at least those who are in a
    position to create them). It is the fact that they enable which persuades
    individuals consciously (as in State formation) or unconsciously (in the case
    of those which are generated spontaneously) to build them. To bring out
    this point and see how it connects with the earlier discussion of the
    relation between action and structure it may be helpful to contrast Hobbes
    with Rousseau. Hobbes has the State emerging from a contract between
    individuals because it serves the interests of those individuals. Rousseau
    also talked of a social contract between individuals, but he did not speak
    this individualist language. For him, the political (democratic) process was
    not a mere means of ser ving persons’ interests by satisfying their
    preferences. It was also a process which changed people’s preferences. People
    were socialised, if you like, and democracy helped to create a new human
    being, more tolerant, less selfish, better educated and capable of cherishing
    the new values of the era of Enlightenment. By contrast, Hobbes’ men and
    women were the same people before and after the contract which created
    the State. 4

Game theory as justification of individualism (pages 32-33), which reminds the
discussion made by Dany-Robert Dufour in La Cite Perverse; it is also noted
that the State is considered a "collective action agency":

    Where do structures come from when they are separate from actions?  An
    ambitious response which distinguishes methodological individualists of all
    types is that the structures are merely the deposits of previous interactions
    (potentially understood, of course, as games). This answer may seem to threaten
    an infinite regress in the sense that the structures of the previous
    interaction must also be explained and so on. But, the individualist will want
    to claim that ultimately all social str uctures spring from interactions
    between some set of asocial individuals; this is why it is ‘individualist’.

    [...]

    Returning to game theory’s potential contribution, we can see that, in so
    far as individuals are modelled as Humean agents, game theory is well placed
    to help assess the claims of methodological individualists. After all, game
    theory purports to analyse social interaction between individuals who, as
    Hume argued, have passions and a reason to serve them. Thus game theory
    should enable us to examine the claim that, beginning from a situation with
    no institutions (or structures), the self-interested behaviour of these
    instrumentally rational agents will either bring about institutions or fuel their
    evolution. An examination of the explanatory power of game theory in such
    settings is one way of testing the individualist claims.
    In fact, as we shall see in subsequent chapters, the recurring difficulty

    [...]

    Suppose we take the methodological individualist route and see
    institutions as the deposits of previous interactions between individuals.
    Individualists are not bound to find that the institutions which emerge in
    this way are fair or just. Indeed, in practice, many institutions reflect the
    fact that they were created by one group of people and then imposed on
    other groups. All that any methodological individualist is committed to is
    being able to find the origin of institutions in the acts of individuals qua
    individuals. The political theory of liberal individualism goes a stage
    further and tries to pass judgement on the legitimacy of particular
    institutions. Institutions in this view are to be regarded as legitimate in so
    far as all individuals who are governed by them would have broadly
    ‘agreed’ to their creation.

    Naturally, much will turn on how ‘agreement’ is to be judged because
    people in desperate situations will often ‘agree’ to the most desperate of
    outcomes. Thus there are disputes over what constitutes the appropriate
    reference point (the equivalent to Hobbes’s state of nature) for judging
    whether people would have agreed to such and such an arrangement. We set
    aside a host of further problems which emerge the moment one steps outside
    liberal individualist premises and casts doubt over whether people’s
    preferences have been autonomously chosen. Game theory has little to
    contribute to this aspect of the dispute. However, it does make two
    significant contributions to the discussions in liberal individualism with
    respect to how we might judge ‘agreement’.

Prisioner's dilemma and the hobbesian argument for the creation of a State (pages 36-37):
resolution would require a higher State in the next upper level of recursion:

    Finally there is the prisoners’ dilemma game (to which we have dedicated the
    whole of Chapter 5 and much of Chapter 6). Recall the time when there were
    still two superpowers each of which would like to dominate the other, if
    possible. They each faced a choice between arming and disarming. When both
    arm or both disarm, neither is able to dominate the other. Since arming is
    costly, when both decide to arm this is plainly worse than when both decide to
    disarm. However, since we have assumed each would like to dominate the
    other, it is possible that the best outcome for each party is when that party
    arms and the other disarms since although this is costly it allows the arming
    side to dominate the other. These preferences are reflected in the ‘arbitrary’
    utility pay-offs depicted in Figure 1.4.
    
    Game theory makes a rather stark prediction in this game: both players will
    arm (the reasons will be given later). It is a paradoxical result because each
    does what is in their own interest and yet their actions are collectively self-
    defeating in the sense that mutual armament is plainly worse than the
    alternative of mutual disarmament which was available to them (pay-off 1 for
    utility pay-offs depicted in Figure 1.4.
    
    Game theory makes a rather stark prediction in this game: both players will
    arm (the reasons will be given later). It is a paradoxical result because each
    does what is in their own interest and yet their actions are collectively self-
    defeating in the sense that mutual armament is plainly worse than the
    alternative of mutual disarmament which was available to them (pay-off 1 for
    each rather than 2). The existence of this type of interaction together with the
    inference that both will arm has provided one of the strongest arguments for
    the creation of a State. This is, in effect, Thomas Hobbes’s argument in
    Leviathan. And since our players here are themselves States, both countries
    should agree to submit to the authority of a higher State which will enforce an
    agreement to disar m (an argument for a strong, independent, United
    Nations?).