Resource:
Let X be a set of alternatives that an agent can choose. A preference
ordering is a scheme whereby an agent ranks all possible alternatives in
order of preference. For any two alternatives x and y at most one of the
following three statements holds:
x is preferred to y (π₯β»π¦)
y is preferred to x (π₯βΊπ¦ )
x and y are equally attractive (π₯βΌπ¦)
Also note that π₯β½π¦ means that either π₯β»π¦ or π₯βΌπ¦
The standard model of rational choice assumes the following standard
properties on these preferences:
Completeness : any two bundles can be compared: for all x and y
in X, either π₯β»π¦ or π₯βΌπ¦ or π₯βΊπ¦
Intuitively, if we do not have completeness, then the preference
ordering is silent about some comparisons. For example, if we have
neither π₯β»π¦ nor π₯βΌπ¦ nor π₯βΊπ¦, then the preference ordering βhas not yet
made up its mind about how to rank x and yβ.
Reflexivity : Any bundle is at least as good as itself: π₯βΌπ₯
Transitivity : if x is preferred or indifferent to y and y to z,
then x is preferred or indifferent to z: If π₯βπ¦ & π¦βπ§ then π₯βz .
To get an intuitive understanding of transitivity, consider a set of
three alternative \(\{a,b,c\}\), so that that the preference ordering
makes a comparison between \(a\) and \(b\) (i.e. we have either aβ»b or
aβΌb or aβΊb) and the preference ordering makes a comparison between \(b\) and \(c\) and between \(c\) and \(a\). Then transitivity means that the
three elements can be arranged on a line, with the element further to
the right always being preferred.
When an ordering is complete, then the relation \(\succ\) already
contains all information, since π₯βΌπ¦ is equivalent to having neither π₯β»π¦
nor π₯βΊπ¦.
Moreover, the model of rational choice assumes that when faced with a
subset \(S\) of the set of alternatives \(X\), the agent chooses an
element \(a\in S\) such that there is no other element \(b\in S\) such
that \(b\succ a\).
One can prove that from these properties it follows that the set \(X\) can be written as a union of disjoint sets, \(X=\cup_{i}\ A_{i}\),
where \(\forall i\ \forall a,b\in A_{i}\ a\sim b\) and \(\forall i\ \forall a\in A_{i}\ \text{if}\ b\notin A_{i}\ \text{then}\text{\ \ }a\succ\text{b\ or\ }b\succ a\).
The sets \(A_{i}\) are called the indifferent sets.
The standard model of consumer choice \(X\) is a rational choice model
where the set \(X\) of alternatives is the set of bundles of
consumption. Typically, it is assumed that the consumer can consume of
each good any amount corresponding to a real number, so \(X=R^{n}\).
In the case of two goods the \(A_{i}\) are typically (i.e. when the
preferences are βwell-behavedβ) curves, which are thus called
βindifference curvesβ.
For vectors \(x,y\), it is common to use the notation π₯β₯π¦ to mean that
each component of \(x\) is higher than the corresponding component of \(y\). The following are properties that are commonly assumed for
preferences over consumption bundles:
Monotonicity (more is better) :
Weak Monotonicity : If π₯β₯π¦ then π₯βπ¦
Strong Monotonicity : If π₯β₯π¦ and π₯β π¦ then π₯β»π¦ (remember x and y
are bundles of goods β vectors)
Local nonsatiation : βπ₯βπ and βπ>0 (as small as you
want) β π¦βπ with \(\text{distance}(x,y)\) <π such that π¦β»π₯.
Here \(\text{distan}\text{ce}(x,y)\) can be taken to be the geometric
distance.
The following are also common additional assumptions:
Convexity: The formal definition is often stated as follows:
Given \(x,y,z\in X\) then \(\forall t\in[0,1]\):
if \(x\curlyeqsucc z\) and \(y\curlyeqsucc z\) then \(\text{tx}+(1-t)y\succcurlyeq z\)
To interpret it, consider complete preferences and consider any two
elements \(a,b\in X\). Suppose \(a\succcurlyeq b\). Then the
definition implies that \(\forall t\in[0,1]\) \(\text{ta}+(1-t)b\succcurlyeq b\), since both \(a\) and \(b\) are
at least as good as \(b\). Since \(t\in[0,1]\) we have
that \(\text{ta}+(1-t)b\) can be looked at as a compromise between \(a\) and \(b\). Hence the definition means that any compromise between
any two elements is at least as good as the worse of the two elements.
Or put in different words: If \(a\) is at least as good as \(b\) and we
start with \(b\) and move in the direction of \(a\) (we are mixing b
with \(a\) by choosing a point \(\text{ta}+(1-t)b\) that lies on the
line segment connecting \(b\) to \(a\).), then things can only improve.
More succinctly, we obtain the following consequence of convexity: if
one is indifferent between x and y, a mixture of x and y will always be
preferred.