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Phillips, P.J. (2007) Statistical Economic Equilibrium. Available at SSRN: https://ssrn.com/abstract=957187
STATISTICAL ECONOMIC EQUILIBRIUM
Peter J Phillips
A ‘philosophical’ solution is presented for the equilibrium selection problem or the question regarding which
equilibrium state the economic system ‘chooses’ from among the many possible equilibrium states. An economic
system is constructed. At a given time, the system occupies an equilibrium macro-state. Consistent with each possible
macro-state are a large number of micro-states where the economic elements are in motion. Each micro-state has an
equal a priori probability and the system shows no preference for any particular micro-state. This permits the
conclusion that the equilibrium macro-state in which the economic system is most likely to be found is that macro-
state with which the largest number of micro-states is consistent. The economic system is found in a particular
equilibrium state with a higher probability than it is found in any other equilibrium state. In a sense, it is this
equilibrium state that the system ‘chooses’.
Author Details: Peter J Phillips, Associate Professor of Finance, School of Commerce, University of Southern
Queensland, West Street, Toowoomba, Queensland, Australia, 4350; Email: phillips@usq.edu.au. Website:
https://www.growkudos.com/profiles/42744/.
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THE STATISTICAL NATURE OF THE ECONOMIC SYSTEM
One of the most important unresolved problems in modern economic theory is the equilibrium selection
problem. Specifically, which equilibrium state does the economic system ‘choose’ from among the
multiplicity of equilibrium states that may exist? The equilibrium existence theorems of Arrow and Debreu
prove that there is at least one equilibrium point that exists to satisfy the data of a fully specified theoretical
economic system in commodity space . However, there may exist more than one equilibrium state and
the existence theorems provide no theoretical indication regarding which equilibrium state will be ‘chosen’
by the economic system. There is, however, a philosophical answer to this philosophical question: the
economic system does not choose at all. Rather, the economic system may be found in a particular
equilibrium state with a greater probability than any other equilibrium state.
The economic system is statistical in nature. The economist cannot hope to predict with perfect certainty
(i.e. probability = 1) events that occur in the economic system. The best that can be done is to construct a
theoretical framework for the economic system that permits the statistical analysis of economic events. In
this manner, we assign probabilities to particular events that permit conclusions regarding the likelihood
that such events might occur. In the physical sciences too, the statistical nature of the physical world has
long been recognised and the old program of developing theoretical frameworks in which events occur
deterministically was swept away a long time ago. The statistical nature of the economic system effectively
rules out the possibility of a ‘deterministic’ theory of equilibrium that would permit the economist to state
with certainty that the economic system will ‘choose’ a particular equilibrium state.
Statistical mechanics is relevant to the development of an answer to the question regarding which
equilibrium state the economic system ‘chooses’ because a similar or analogous problem was solved by the
statistical mechanics. Specifically, a physical system such as a cylinder containing gas or a beaker containing
water may be extremely complex. For example, such a system may contain molecules in constant
motion and continuously varying positions and momentums and interacting with other particles. The
complexity is so great that the scientist has incomplete knowledge of the system. In order to draw
conclusions about the macro-state of the physical system, the physicist must utilise statistical averages over
the system’s micro-states, replacing precise determinism with probabilities of molecule motion such that
the probability of deriving a particular measurement at the macro level can be computed.
This is analogous to the situation that economists find themselves in when it comes to describing
characteristics of the macro-state of the economic system. There are an extremely large number of
economic agents in constant ‘economic motion’, continuously changing their consuming and producing
behaviours and their interactions with each other. As such, the economist’s knowledge of the economic
system—tastes, technology and wealth distribution—is inevitably incomplete. If economists are to draw
conclusions about the macro-state of an economic system, the equilibrium position it ‘chooses’ to occupy,
then it would stand to reason that a statistical analysis of the economic system that permits the
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computation of the probability of observing a particular macro-state is required and the hope of a
deterministic solution to the equilibrium selection problem must be given up.
PART ONE
The context is a mathematical-economic system in Hilbert space . This diverges somewhat from the
standard mathematical-economic systems that are set in real vector space . Fortunately, Chichilnisky and
Heal (1998) have utilised Hilbert space formalism to construct a system
with traders indexed by , a commodity space , endowments and preferences of the trader
. Chichilnisky and Heal’s (1998) mathematical-economic system has the following features or
characteristics:
(1) A trader may have a zero endowment of some goods. That is, may have some zero coordinates;
(2) A trade is a real valued function on R;
(3) The trading space is contained in H;
(4) H is a weighted space of real valued measurable functions with the inner product
, where is a finite measure on R;
(5) Both the consumption set for an agent and the price space are H;
(6) When no short sales are permitted the consumption set is H +;
(7) An allocation is a vector which is feasible if the sum of all does not exceed the sum
of all individual agents’ endowments;
(8) The trader’s preferences are represented by increasing quasi-concave functions satisfying and
In addition to this, Chichilnisky and Heal (1998, p.166) make the following assumptions:
(1) Each trader has a preference represented by a function such as that outlined in point (8) above and
the preference is uniformly non-satiated;
(2) If an indifference surface of positive utility intersects a coordinate axis, those of superior utility do so
as well; and
(3) An agent exists whose preferences are smooth.
The most important feature of this economic system is that just one property, limited arbitrage, is sufficient
and necessary for the existence of competitive economic equilibrium in this system. Within the context of
the economic system, , Chichilnisky and Heal (1998) prove that
limited arbitrage is both necessary and sufficient for the existence of competitive economic equilibrium.
This result is summarised below.
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Definition (1): A feasible allocation is a competitive equilibrium when (1) there is a price
with and for any with ; and (2)
(Chichilnisky and Heal, 1998, p.167).
A problem that has attracted a great deal of attention is the problem of the existence of competitive
equilibrium and the conditions necessary and sufficient to ensure its existence. Chichilnisky and Heal
(1998) prove that within the system designed by them, and, specifically, for there is a
single condition that is both necessary and sufficient for the existence of competitive equilibrium. This
condition is limited arbitrage. Limited arbitrage requires that there exists a price at which no trader can
increase utility by increasing the scale of zero-cost trades (Chichilnisky and Heal, 1998, p.168). In order to
develop limited arbitrage, Chichilnisky and Heal introduce the mathematical objects called cones. Trader-
specific cones, global cones and market cones are defined as follows
1
,
2
:
Definition (2): A trader-specific cone, , (the cone of trader ), consists of all directions of net trades in the
trading space along which the trader’s utility exceeds that of any other vector in the space. Formally,
Definition (3): A global cone is the set of directions of net trades along which utility never ceases to
increase:
Definition (4): Lastly, the market cone for a trader i is (with short sales allowed):
Limited arbitrage is the condition necessary and sufficient for the existence of economic equilibrium.
Specifically, Chichilnisky and Heal defined limited arbitrage for (i.e. short sales permitted) as follows:
Definition (5): There exists a vector
Hp
assigning a strictly positive value to all vectors in ,
or,
1
Chichilnisky and Heal’s (1998) definitions.
2
More discussion of these particular mathematical objects is contained in Chichilnisky and Heal (1998) and the
references listed therein.
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Theorem (1): Limited arbitrage is necessary and sufficient for the existence of competitive economic
equilibrium.
3
The proofs of necessity and sufficiency are presented in Chichilnisky and Heal (1998) and the
references contained therein and are therefore omitted.
Before proceeding, let us first elaborate upon the mathematical-economic system constructed by
Chichilnisky and Heal (1998) described above. This elaboration shall involve the statement of a number of
postulates and definitions with associated explanation.
Postulate (1): The economic system is statistical (non-
deterministic) in nature.
Postulate (2): At a given time, the economic system is in one of its possible macro-states , , ..
Postulate (3): This macro-state is consistent with a number of micro-states, , .
Postulate (4): The economic system is found to be in each of its micro-states with equal a priori
probability.
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The system has no preference for a particular micro-state.
Postulate (5): At a given time, the economic system is in macroscopic equilibrium.
Definition (6): The macro-state is the (equilibrium) state occupied by the economic system at a given
time. It is the equilibrium ‘chosen’ by the system. A macro-state is therefore characterised by the property
of limited arbitrage. In addition, the properties of the economic system are macroscopically at rest
(unchanging).
Definition (7): The micro-states associated with the macro-state are finite in number and each
represents a different configuration of the properties of the economic system or different configurations of
the data of the economic model. Microscopically, the system is not at rest but is in continuous motion from
micro-state to micro-state.
Definition (8): The properties of the economic system are economic elements that take on particular
values. For example, takes a particular value as do the other elements: , , , and
so on.
Definition (9): The economic elements of the economic system are the items without which it is not
possible to talk about our system. The economic elements of this system include trader preferences,
traders, endowments etc. .
3
Also see Chichilnisky and Heal (1998, p.172)
4
This is essentially the fundamental postulate of statistical mechanics (see, for example, Wilde and Singh (1998)). This
postulate is, in principle, empirically verifiable.
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At a given time, the economic system occupies a particular equilibrium macro-state. This is the equilibrium
state that is ‘chosen’ by the system. By postulate (5), the economic system is, at a given time, in macroscopic
equilibrium. Every macro-state is therefore an equilibrium state. Consistent with this macro-state are a
large number of micro-states. That is to say, the equilibrium macro-state is consistent with a number of
configurations of the data of the model. In an equilibrium macro-state the values of the economic elements
(properties) are constant or ‘at rest’ whilst microscopically the properties are in a state of constant change.
Each micro-state has a different configuration or, equivalently, is defined by different combinations of
values of its economic elements. That is to say, each micro-state has different properties. By postulate (4),
each micro-state, or combination of values of economic elements, has an equal a priori probability.
In Figure 1 the large circles represent possible macro-states for an economic system in equilibrium.
Associated with each macro-state are a finite number of micro-states, shown here as small black circles.
The diagram displays a small finite number of both macro-states and micro-states. However, the numbers
may be very large. That is, there may be many possible equilibrium macro-states and, associated with each
of these, there may be very many micro-states. The number of micro-states associated with a particular
macro-state may be different from the number of micro-states associated with an alternative macro-state.
Hence, the number of small black circles differs across macro-states in the diagram below. The small black
circles representing micro-states are also shown in different patterns within each macro-state. This
highlights the fact that the micro-states consistent with each macro-state need not be identical across
macro-states.
Figure 1: Macro and Micro States
According to the fourth postulate stated above, the economic system is found in each of its micro-states
with equal a priori probability. The economic system does not ‘prefer’ one micro-state over another. This
being the case, the expectation value associated with each particular microstate is simply:
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So we have an economic system in Hilbert space H that is statistical (non-deterministic) in nature. The
economic system is found at a given time in a particular equilibrium macro-state which has consistent
with it a number of micro-states each with equal a priori probability. From the various (equilibrium)
macro-states the economic system ‘chooses’ a particular macro-state . It remains to be explained which
(equilibrium) macro-state the economic system will ‘choose’ from among all the macro-states .
PART TWO
It is in answering this question that the fourth postulate becomes as important as its statistical mechanical
analogue (see footnote four). The economic system will ‘choose’ that equilibrium state that has the highest
probability. The fourth postulate permits the conclusion that the equilibrium state with the highest
probability is that macro-state with which the largest number of micro-states is consistent. That is, the
equilibrium state with which the largest number of configurations of the data of the system is consistent.
We have therefore, two tasks: (1) determine the number of micro-states for a given macro-state; and (2)
present an explanation for why the system will ‘choose’ the macro-state with the highest number.
The macro-state of the economic system is defined by particular constraints on the various economic
elements. Specifically, in an equilibrium macro-state the values of the economic elements (properties) are
constant or ‘at rest’
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whilst microscopically the properties are in a state of constant change.
Macroscopically, therefore, the observer sees a calm economic system. This being said, only particular
values of the economic elements, that is, only particular properties of the economic system (particular
configurations of the data of the model) will be consistent with the constrained values prescribed by the
particular equilibrium macro-state under consideration. The economic system in an equilibrium macro-
state is constrained to a constant number of traders with constant preferences and constant
endowments. This constrained economic system is . Remembering that the economic system is
constantly in motion microscopically, the number of micro-states consistent with the constrained
macro-state is:
The integral is a sum over a region of H satisfying the constraints of constant number of traders, constant
preferences and constant endowments.
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The result permits the computation of the number of micro-states
consistent with an equilibrium macro-state. Once the number of micro-states consistent with each possible
macro-state has been computed, the economist is in a position to determine which of the possible macro -
5
This is in keeping with the standard economic interpretation of equilibrium.
6
Similar integrals are found in most texts on statistical mechanics.
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states is most likely to prevail. The equilibrium macro-state that is most likely to prevail is that with which
the largest number of micro-states is consistent. That is, the macro-state for which the integral is greatest.
Let denote the probability of a particular micro-state . This probability is equal for all i. Then
the probability of a macro-state which has, say, four consistent micro-states is greater than the
probability of another macro-state with only two consistent micro-states:
Within the context of the economic system , therefore, the
equilibrium macro-state—where the properties of the system are constant or at rest—with the largest
number of consistent micro-states is also the equilibrium macro-state with the highest probability. At a
given time, the observer is more likely to observe the economic system in this macro-state than any other
macro-state. Which macro-state is ‘chosen’ by the system? The answer is that the system does not ‘choose’
anything at all. At a given time there is a chance that the economic system will occupy any one of its possible
equilibrium states (macro-states). However, also at a given time, one of these equilibrium states will have
a higher probability than the others. This is the equilibrium state with the highest number of associated
micro-states. It is in this equilibrium state that the economic system is most likely to be found.
A PARTICULAR SITUATION
Consider a particular situation: an n-person game. In standard game theoretical fashion, let be an n-
person game where S is the set of strategies and f is the set of payoff profiles. Each player
chooses a strategy that is associated with a strategy profile that results in a payoff
to player i. Mixed strategies are allowed and equilibrium for a particular game is one of the possible
Nash Equilibria.
Definition (9): The micro-states , of which there are , correspond to each of the n-tuples of an n-
dimensional payoff matrix.
Definition (10): Each macro-state, , is an equilibrium macro-state and is therefore a particular n-tuple
that is one of the possible Nash equilibria.
The probability that a particular micro-state (n-tuple) will be verified as being present in the system is
. It is postulated (postulate 4) that the system, or game, has no a priori preference for any
particular micro-state. This does not mean that the particular players of the game would not prefer a
particular micro-state. Their preferences are reflected in their choices of strategy and it is no contradiction
to say that the system or game has no preference for a particular micro-state. Hence, each micro-state or n-
tuple has an equal a priori probability equal to .
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At least one of the n-tuples is a Nash equilibrium. In a dynamically unfolding game the number of players,
their choice of strategy and their interactions with other players are all in continuous motion. That is, the
game is microscopically in a state of flux, moving constantly from n-tuple to n-tuple. Macroscopically, the
game is in a state of rest and is characterised by one of its possible Nash equilibria. The question that must
be answered is: which of the macro-states (which Nash equilibrium) does the game choose? Which of the
Nash equilibria has the highest probability? In order to answer this question we would need to picture the
game in continuous microscopic motion. In order to visualise, we constrain the analysis to a 2-person game
to simplify the payoff matrix.
Assume that the 2-person game has the payoff matrix presented in Figure 2 below. A matrix can have
0 to N Nash equilibria. The matrix below can have up to three Nash Equilibria. Each of the Nash
Equilibria is represented by an N in the particular cell of the matrix. Now, in macroscopic equilibrium we
will observe the game in one of these particular cells. However, in a continuously unfolding game, the game
is microscopically in motion moving continuously from cell to cell as players’ choices and interactions with
other players change. Of course, the number of players is also changing continuously so the payoff matrix
is of continuously varying dimensions. Microscopically, the system is continuously moving from n-tuple to
n-tuple. If we observe the game in macroscopic equilibrium we must, by definition, observe the game in
Nash equilibrium.
Figure 2: Payoff Matrix with N Nash Equilibria
There are three Nash equilibria. Microscopically the game is moving continuously from n-tuple to n-tuple.
This is depicted in the diagram by the arrows. Remember, we have restricted the game to 2 players to
constrain the number of dimensions. One can imagine, therefore, that when there are more than two
players and the number of players continuously changes, the payoff matrix continuously changes
dimensions and each successive n-tuple may have more elements or fewer elements than its predecessor.
Which of the Nash equilibria, N, does the game ‘choose’? Our answer is that the game will ‘choose’ the N
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that has the highest probability. The N with the highest probability is the N with which the largest number
of microstates (configurations of the game) is consistent.
There are three possible equilibrium macro-states (three N’s). Microscopically the game is in continuous
motion ‘hitting’ cell after cell. For example, an unfolding dynamic game might follow the path
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:
over a given time.
8
Each of the micro-
states (or n-tuples) in a particular path or series of configurations has equal a priori probability. Since there
are eleven configurations in this particular path, each has an a priori probability of 0.090909. In equilibrium
the game is constrained to a particular n-tuple. The constrained game is and the number of micro-
states consistent with the constrained macro-state is:
Because strategies are mapped into the n-tuples of the payoff matrix via a payoff function, the changes in
and are reflected in changes in position of the game in the payoff matrix. In the example above, where
an unfolding dynamic game was said to be assumed to follow the path:
, the high probability Nash
equilibrium is (a, b) since it is the equilibrium macro-state with which the largest number of micro-states
in this path or series of configurations in consistent. This Nash equilibrium has the highest probability
(0.3636 = 4(1/11)).
REMARKS
The equilibrium selection problem is usually framed as a question regarding which equilibrium state the
economic system ‘chooses’ from among the many possible equilibrium states. An economic system can be
constructed, supported by a number of postulates, assumptions and definitions. At a given time, the system
occupies an equilibrium macro-state. It is an equilibrium state because the system is macroscopically ‘at
rest’ with its economic elements exhibiting particular constant values. Consistent with each possible
macro-state are a large number of micro-states where the economic elements are ‘in motion’ exhibiting
constantly changing values. Each micro-state has, by the stated postulates, an equal a priori probability and
the system shows no preference for any particular micro-state. This postulate permits the conclusion that
the equilibrium macro-state in which the economic system is most likely to be found is that macro-state
with which the largest number of micro-states is consistent. The economic system does not ‘choose’ an
equilibrium state but is found in a particular equilibrium state with a higher probability than it is found in
any other equilibrium state.
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We use the word ‘path’ loosely. The system is continuously changing configurations. Strictly speaking (a, b), for
example, is not recurring but emerging at subsequent configurations. There is not one configuration of the game
consistent with the cell (a, b) but, in this case, four configurations each emerging over time. That they occur at different
times differentiates the configurations.
8
When player numbers can change, the n-tuples will not all be ordered pairs (but triples etc).
11
References
Chichilnisky, G. and Heal, G. 1998, “A Unified Treatment of Finite and Infinite Economies: Limited Arbitrage
is Necessary and Sufficient for the Existence of Equilibrium and the Core,” Economic Theory, 12,
pp.163-176.
Cox, J.C., Ingersoll, J.E. and Ross, S.A. 1985, “An Intertemporal General Equilibrium Model of Asset Prices,”
Econometrica, Vol.53, No.2, March.
Wilde, R. and Singh, S. 1998, Statistical Mechanics: Fundamentals and Modern Applications, John Wiley and
Sons, New York, New York.
