Editing Ordinal utility (section)

== Additivity with three or more goods{{Anchor|PI}} {{Anchor|assessment}} ==
{{anchor|additivity3}}
When there are three or more commodities, the condition for the additivity of the utility function is surprisingly ''simpler'' than for two commodities. This is an outcome of [[Debreu theorems|Theorem 3 of Debreu (1960)]]. The condition required for additivity is '''preferential independence'''.<ref name=KeeneyRaiffa1993/>{{rp|104}}

A subset A of commodities is said to be ''preferentially independent'' of a subset B of commodities, if the preference relation in subset A, given constant values for subset B, is independent of these constant values. For example, suppose there are three commodities: ''x'' ''y'' and ''z''. The subset {''x'',''y''} is preferentially-independent of the subset {''z''}, if for all <math>x_i,y_i,z,z'</math>:
:<math>(x_1,y_1, z)\preceq (x_2,y_2, z) \iff (x_1,y_1, z')\preceq (x_2,y_2, z')</math>.
In this case, we can simply say that:
:<math>(x_1,y_1)\preceq (x_2,y_2)</math> for constant ''z''.

Preferential independence makes sense in case of [[independent goods]]. For example, the preferences between bundles of apples and bananas are probably independent of the number of shoes and socks that an agent has, and vice versa.

By Debreu's theorem, if all subsets of commodities are preferentially independent of their complements, then the preference relation can be represented by an additive value function. Here we provide an intuitive explanation of this result by showing how such an additive value function can be constructed.<ref name=KeeneyRaiffa1993/> The proof assumes three commodities: ''x'', ''y'', ''z''. We show how to define three points for each of the three value functions <math>v_x, v_y, v_z</math>: the 0 point, the 1 point and the 2 point. Other points can be calculated in a similar way, and then continuity can be used to conclude that the functions are well-defined in their entire range.

'''0 point''': choose arbitrary <math>x_0,y_0,z_0</math> and assign them as the zero of the value function, i.e.:
:<math>v_x(x_0)=v_y(y_0)=v_z(z_0)=0</math>

'''1 point''': choose arbitrary <math>x_1>x_0</math> such that <math>(x_1,y_0,z_0)\succ(x_0,y_0,z_0)</math>. Set it as the unit of value, i.e.:
:<math>v_x(x_1)=1</math>
Choose <math>y_1</math> and <math>z_1</math> such that the following indifference relations hold: 
:<math>(x_1,y_0,z_0)\sim(x_0,y_1,z_0)\sim(x_0,y_0,z_1)</math>. 
This indifference serves to scale the units of ''y'' and ''z'' to match those of ''x''. The value in these three points should be 1, so we assign
:<math>v_y(y_1)=v_z(z_1)=1</math>

'''2 point''': Now we use the preferential-independence assumption. The relation between <math>(x_1,y_0)</math> and <math>(x_0,y_1)</math> is independent of ''z'', and similarly the relation between <math>(y_1,z_0)</math> and <math>(y_0,z_1)</math> is independent of ''x'' and the relation between <math>(z_1,x_0)</math> and <math>(z_0,x_1)</math> is independent of ''y''. Hence
:<math>(x_1,y_0,z_1)\sim(x_0,y_1,z_1)\sim(x_1,y_1,z_0).</math>
This is useful because it means that the function ''v'' can have the same value – 2 – in these three points. Select <math>x_2, y_2, z_2</math> such that
:<math>(x_2,y_0,z_0)\sim(x_0,y_2,z_0)\sim(x_0,y_0,z_2)\sim(x_1,y_1,z_0)</math>
and assign
:<math>v_x(x_2)=v_y(y_2)=v_z(z_2)=2.</math>

'''3 point''': To show that our assignments so far are consistent, we must show that all points that receive a total value of 3 are indifference points. Here, again, the preferential independence assumption is used, since the relation between <math>(x_2,y_0)</math> and <math>(x_1,y_1)</math> is independent of ''z'' (and similarly for the other pairs); hence
:<math>(x_2,y_0,z_1)\sim(x_1,y_1,z_1)</math>
and similarly for the other pairs. Hence, the 3 point is defined consistently.

We can continue like this by induction and define the per-commodity functions in all integer points, then use continuity to define it in all real points.

An implicit assumption in point 1 of the above proof is that all three commodities are ''essential'' or ''preference relevant''.<ref name=Ted/>{{rp|7}} This means that there exists a bundle such that, if the amount of a certain commodity is increased, the new bundle is strictly better.

The proof for more than 3 commodities is similar. In fact, we do not have to check that all subsets of points are preferentially independent; it is sufficient to check a linear number of pairs of commodities. E.g., if there are <math>m</math> different commodities, <math>j=1,...,m</math>, then it is sufficient to check that for all <math>j=1,...,m-1</math>, the two commodities <math>\{x_j,x_{j+1}\}</math> are preferentially independent of the other <math>m-2</math> commodities.<ref name=KeeneyRaiffa1993/>{{rp|115}}

=== Uniqueness of additive representation ===
An additive preference relation can be represented by many different additive utility functions. However, all these functions are similar: they are not only increasing monotone transformations of each other ([[#Uniqueness|as are all utility functions representing the same relation]]); they are increasing [[linear transformation]]s of each other.<ref name=Ted/>{{rp|9}} In short,
::An additive ordinal utility function is ''unique up to increasing linear transformation''.