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‘₯≻𝑦)
y is preferred to x (π‘₯≺𝑦 )
x and y are equally attractive (π‘₯βˆΌπ‘¦)
Also note that π‘₯≽𝑦 means that either π‘₯≻𝑦 or π‘₯βˆΌπ‘¦
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‘₯≻𝑦 or π‘₯βˆΌπ‘¦ or π‘₯≺𝑦
Intuitively, if we do not have completeness, then the preference ordering is silent about some comparisons. For example, if we have neither π‘₯≻𝑦 nor π‘₯βˆΌπ‘¦ nor π‘₯≺𝑦, then the preference ordering β€˜has not yet made up its mind about how to rank x and y’.
Reflexivity : Any bundle is at least as good as itself: π‘₯∼π‘₯
Transitivity : if x is preferred or indifferent to y and y to z, then x is preferred or indifferent to z: If π‘₯β‹Ÿπ‘¦ & π‘¦β‹Ÿπ‘§ then π‘₯β‹Ÿz .
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 a≻b or a∼b or aβ‰Ίb) 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‘₯βˆΌπ‘¦ is equivalent to having neither π‘₯≻𝑦 nor π‘₯≺𝑦.
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 β€˜well-behaved’) curves, which are thus called β€˜indifference curves’.
For vectors \(x,y\), it is common to use the notation π‘₯β‰₯𝑦 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‰₯𝑦 then π‘₯β‹Ÿπ‘¦
Strong Monotonicity : If π‘₯β‰₯𝑦 and π‘₯≠𝑦 then π‘₯≻𝑦 (remember x and y are bundles of goods β†’ vectors)
Local nonsatiation : βˆ€π‘₯βˆˆπ‘‹ and βˆ€πœ€>0 (as small as you want) βˆƒ π‘¦βˆˆπ‘‹ with \(\text{distance}(x,y)\) <πœ€ such that 𝑦≻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.