Definition 1.  We define a static capital system as a triple (X,\Sigma,\mu) with counting measure \mu where \mu(X)=m and  is called the monetary constant, \Sigma is a collection of subsets of X such that \mu(\cup_{A\in\Sigma}A)=m, elements of which are called owners, and \mu(A) is called the worth of A for an owner A.

Note we are not requiring \Sigma to be closed under any operations (i.e. it is not an algebra of sets).  Suppose we have two structures on X, (X,\Sigma_1,\mu) and (X,\Sigma_2,\mu).   Let f:\Sigma_1\to P(\Sigma_2) (i.e. a multi-valued map into \Sigma_2).  Such a function is called a trade (and may correspondingly be thought of as a change of ownership).  We define the trade utility of a trade f as a map u_f:\Sigma_1\to\mathbb{Z} by

\displaystyle u_f(A)=\mu\left(\bigcup f(A)\right)-\mu(A).

Again, \cup f(A) need not be in \Sigma_2, but we can of course still define the counting measure on it.

Definition 2.  A composite trade is a map g\circ f:\Sigma_1\to P(\Sigma_3) where f:\Sigma_1\to P(\Sigma_2) and g:\Sigma_2\to P(\Sigma_3) are trades.

Note that g\,\circ:P(\Sigma_2)\to P(\Sigma_3) since it is defined on the image of f.  g\,\circ simply evaluates g on all sets in f(A).

Definition 3.  Let (X,\Sigma_t,\mu)_{t\geq 0} be a continuum of static capital systems.  We say (X,\Sigma_t,\mu)_{t\geq 0} is a capital system if

  1. for every t\geq 0 and \varepsilon\geq t there is a unique trade f_{t,\varepsilon}:\Sigma_t\to P(\Sigma_{t+\varepsilon});
  2. f_{t,0}=1 (i.e. f_{t,0}(A)=\{A\});
  3. if f_{t,\varepsilon_1} and f_{\varepsilon_1,\varepsilon_2} are trades such that \varepsilon_1+\varepsilon_2=\varepsilon, then f_{t,\varepsilon}=f_{\varepsilon_1,\varepsilon_2}\circ f_{t,\varepsilon_1} for all t,\varepsilon_1,\varepsilon_2\geq 0.

Example 4.  A capital system is in a socialist state at time t if \mu(A)=\mu(B) for all A,B\in\Sigma_t.  We may further say (X,\Sigma_t,\mu)_{t\geq 0} is socialist during T\subseteq[0,\infty) if (X,\Sigma_t,\mu) is in a socialist state for all t\in T.  A capital system is in a communist state at time t if \Sigma_t=\{X\}. Similarly we have the definition for communist during a set T\subseteq[0,\infty).

Note that by this definition a communist state implies a socialist state.  In the above regards, a communist state can be thought of as having a single owner (say, “the people”), and socialist state has owners with equal worth.

Definition 5.  A dynamic capital system is a capital system (X_t,\Sigma_t,\mu)_{t\geq 0} where (X_t,\Sigma_t,\mu) is a static capital system for all t where \mu(X_t)=m_t and X_t,X_s are comparable (in the inclusion sense) for all s,t\geq 0.  In particular the function m:[0,\infty)\to\mathbb{N} defined by m(t)=m_t is called the monetary policy.  If m_t is strictly increasing during an interval, we say (X_t,\Sigma_t,\mu) is expansionary during that interval.  Similarly it is  contractionary if it is strictly decreasing on some interval.

Definition 6.  A dynamic capital system (X_t,\Sigma_t,\mu)_{t\geq 0} is rational if u_{f_{t,\varepsilon}}\geq 0 for all t,\varepsilon\geq 0.

Of course if \varepsilon is 0 we have f_{t,0}=1 and thus the condition is satisfied for this case:

\displaystyle u_{f_{t,0}}(A)=\mu\left(\bigcup \{A\}\right)-\mu(A)=0.

So in a rational dynamic capital system we have the inequality

\displaystyle\mu(A)\leq\mu\left(\bigcup f_{t,\varepsilon}(A)\right)\leq m_{t+\varepsilon}

with A\in\Sigma_t.  If \lim_{t\to\infty}m_t exists and is finite, then the rational dynamic capital system (X_t,\Sigma_t,\mu) is said to have an end game.


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