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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?).
Too much trust in that type of instrumental rationality might lead to lower
outcomes in some games:
The term rationalisable has been used to describe such strategies because a
player can defend his or her choice (i.e. rationalise it) on the basis of beliefs
about the beliefs of the opponent which are not inconsistent with the game’s
data. However, to pull this off, we need ‘more’ commonly known rationality
than in the simpler games in Figures 2.1 and 2.3. Looking at Figure 2.4 we see
that outcome (100, 90) is much more inviting than the rationalisable outcome
(1, 1). It is the deepening confidence in each other’s instrumental rationality
(fifth-order CKR, to be precise) which leads our players to (1, 1). In summary
notation, the rationalisable strategies R2, C2 are supported by the following
train of thinking (which reflects the six steps described earlier):
-- 48
Nash-equilibrium: self-confirming strategy:
A set of rationalisable strategies (one for each player) are in a Nash
equilibrium if their implementation confirms the expectations of each player
about the other’s choice. Put differently, Nash strategies are the only
rationalisable ones which, if implemented, confirm the expectations on which
they were based. This is why they are often referred to as self-confirming
strategies or why it can be said that this equilibrium concept requires that
players’ beliefs are consistently aligned (CAB).
-- 53
Arguments against CAB:
In the same spirit, it is sometimes argued (borrowing a line from John von
Neumann and Oskar Morgenstern) that the objective of any analysis of games is
the equivalent of writing a book on how to play games; and the minimum
condition which any piece of advice on how to play a game must satisfy is
simple: the advice must remain good advice once the book has been published.
In other words, it could not really be good advice if people would not want to
follow it once the advice was widely known. On this test, only (R2, C2) pass,
since when the R player follows the book’s advice, the C player would want to
follow it as well, and vice versa. The same cannot be said of the other
rationalisable strategies. For instance, suppose (R1, C1) was recommended: then
R would not want to follow the advice when C is expected to follow it by
selecting C1 and likewise, if R was expected to follow the advice, C would not
want to.
Both versions of the argument with respect to what mutual rationality entails
seem plausible. Yet, there is something odd here. Does respect for each other’s
rationality lead each person to believe that neither will make a mistake in a
game? Anyone who has talked to good chess players (perhaps the masters of
strategic thinking) will testify that rational persons pitted against equally
rational opponents (whose rationality they respect) do not immediately assume
that their opposition will never make errors. On the contrary, the point in
chess is to engender such errors! Are chess players irrational then? One is
inclined to answer no, but why? And what is the difference as
-- 57
Limits conceptualizing reason as an algorithm ("Humean approach to reason
is algorithmic"):
Harsanyi doctrine seems to depend on a powerfully algorithmic and controversial
view of reason. Reason on this account (at least in an important part) is akin
to a set of rules of inference which can be used in moving from evidence to
expectations. That is why people using reason (because they are using the same
algorithms) should come to the same conclusion. However, there is genuine
puzzlement over whether such an algorithmic view of reason can apply to all
circumstances. Can any finite set of rules contain rules for their own
application to all possible circumstances? The answer seems to be no, since
under some sufficiently detailed level of description there will be a question of
whether the rule applies to this event and so we shall need rules for applying
the rules for applying the rules. And as there is no limit to the detail of the
description of events, we shall need rules for applying the rules for applying
the rules, and so on to infinity. In other words, every set of rules will require
creative interpretation in some circumstances and so in these cases it is
perfectly possible for two individuals who share the same rules to hold
divergent expectations.
This puts a familiar observation from John Maynard Keynes and Frank
Knight regarding genuine uncertainty in a slightly different way, but
nevertheless it yields the same conclusion. There will be circumstances under
which individuals are unable to decide rationally what probability assessment
to attach to events because the events are uncertain and so it should not be
surprising to find that they disagree. Likewise, the admiration for
entrepreneurship found among economists of the Austrian school depends on
the existence of uncertainty. Entrepreneurship is highly valued precisely
because, as a result of uncertainty, people can hold different expectations
regarding the future. In this context, the entrepreneurs are those who back
their judgement against that of others and succeed. In other words, there
would be no job for entrepreneurs if we all held common expectations in a
world ruled by CAB!
A similar conclusion regarding ineliminable uncertainty is shared by social
theorists who have been influenced by the philosophy of Kant. They deny that
reason should be understood algorithmically or that it always supplies answers
as to what to do. For Kantians reason supplies a critique of itself which is the
source of negative restraints on what we can believe rather than positive
instructions as to what we should believe. Thus the categorical imperative (see
section 1.2.1), which according to Kant ought to determine many of our
significant choices, is a sieve for beliefs and it rarely singles out one belief.
Instead, there are often many which pass the test and so there is plenty of
room for disagreement over what beliefs to hold.
Perhaps somewhat surprisingly though, a part of Kant’s argument might
lend support to the Nash equilibrium concept. In particular Kant thought that
rational agents should only hold beliefs which are capable of being
universalised. This idea, taken by itself, might prove a powerful ally of Nash.
[...] Of course, a full Kantian perspective is
likely to demand rather more than this and it is not typically adopted by game
theorists. Indeed such a defence of Nash would undo much of the
foundations of game theory: for the categorical imperative would even
recommend choosing dominated strategies if this is the type of behaviour that
each wished everyone adopted. Such thoughts sit uncomfortably with the
Humean foundations of game theory and we will not dwell on them for now.
Instead, since the spirit of the Humean approach to reason is algorithmic, we
shall continue discussing the difficulties with the Harsanyi—Aumann defence
of Nash.
-- 58-60
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