Representation of Preference Orders by Hyperreal Numbers

Kiyoshi Ikeda

                          ky-ikeda@jaist.ac.jp

School of Information Science, Japan Advanced Institute of Science and Technology

Game theory is now a basic framework of microeconomics in analyzing economic behavior of consumers and producers in a market. One of the assumptions in this theory is that each agent behaves so as to maximize his or her own payoff , and this assumption is used to explain a lot of competitive or cooperative behavior of the agents in a market. Such payoffs are usually given in the analysis of a game as integers, sometimes as real numbers, in any case they are given by a numerical function, which is to be interpreted concretely as an agent’s preference for goods or social welfare in the real world. Our aim is to study the validity of such numerical representation, in other words, we shall examine what kind of conditions are needed to represent an agent’s preference by means of numerical values.

To put it more precisely, suppose we are in a situation to play a game, and let S be the set of items, one of which we will gain after playing the game. Because some of the items may be preferable rather than others, we shall have to calculate the best strategy to obtain as desirable item as possible. According to the methodology of game theory, this calculation will generally be carried out as follows: we assign a real-valued function U : S → R which somehow reflect our preference,^{†} and choose the best strategy that maximize the expected value of the function. Now, our question is: How can we assign such a numerical function U : S → R ?

von Neumann–Morgenstern Utility Functions

One of the answer is the axiomatic formulation given by von Neumann and Morgenstern in their pioneering work [5], which is the most generally accepted formulation ensuring the existence of numerical utilities. Let us review and summarize some basic results of expected utility theory from the work of von Neumann and Morgenstern. The following description is based on Fishburn [2].

A binary relation ⪯ on a set D is a weak order if it satisfies the following conditions:

(i) for all x, y, z ∈ D, x ⪯ y and y ⪯ z imply x ⪯ z,

(ii) for all x, y ∈ D, x ⪯ y or y ⪯ x.

The strict part ≺ and indifference part ∼ are defined from ⪯ by

x ≺ y if and only if x ⪯ y and not (y ⪯ x),

x ∼ y if and only if x ⪯ y and y ⪯ x.

Let S denote a set of items. A simple lottery is a mapping λ : S → [0, 1] with the following properties:

(i) ∑_{s∈S} λ(s) = 1, (ii) there is a finite S′ ⊆ S such that λ(s) = 0 for all s ∈ S \ S′.

Let Δ(S) denote the set of all simple lotteries. Given λ, μ ∈ Δ(S) and θ ∈ [0, 1], define the convex combination θλ + (1 − θ)μ by

[ θλ + (1 − θ)μ](s) = θλ(s) + (1 − θ)μ(s) for all s ∈ S.

It can easily be verified that the convex combination θλ + (1 − θ)μ also belongs to Δ(S).

†Such a function U : S → R is referred to as a utility function, and the value U(s) for s ∈ S is called the utility of the item s.

Now we consider the following axioms for a preference relation ⪯ on Δ(S) : For all x, y, z in Δ(S),

B1 ⪯ is a weak order on Δ(S),

B2 if x ≺ y then θx + (1 − θ)z ≺ θy + (1 − θ)z for all θ ∈ (0, 1),

B3 if x ≺ y and y ≺ z, then there are θ, ξ ∈ (0, 1) such that

θx + (1 − θ)z ≺ y and y ≺ ξx + (1 − ξ)z.

Theorem 1 (see Fishburn [2])

A preference order ⪯ on Δ(S) satisfies B1–B3 if and only if there is a utility function U : S → R such that

λ ≺ μ ⟺ ∑_{s∈S} λ(s)U(s) < ∑_{s∈S} μ(s)U(s).

Also, U is unique up to a positive affine transformation, i.e. if V : S → R is another function with the same property, then there exist real constants a > 0 and b such that

V (s) = a U(s) + b for all s ∈ S.

Additive Utilities

The above formulation by von Neumann and Morgenstern is based on the abstract notion of ‘preference on the lotteries’, so that the meaning of B1–B3 is not quite clear. To avoid such abstraction, we propose an alternative system of axioms which is based on more concrete notion of ‘preference on the consumption plans’. The advantage of our formulation is that a utility function is ensured to be additive, i.e. utilities can be added together.

Let S = {α_{1}, α_{2}, ⋯, α_{n}} be a set of n-types of indivisible items for some n ≥ 2 , which may include a diamond, a car, a grain of sesame, and so on. The set Ω(S) of consumption plans of these items can be defined as

Ω(S) = { k_{1}α_{1} + k_{2}α_{2} + ⋯ + k_{n}α_{n} | k_{1}, k_{2} ⋯, k_{n} ∈ N }^{†},

where k_{1}α_{1} + k_{2}α_{2} + ⋯ + k_{n}α_{n} simply denotes “k_{1} of α_{1}, k_{2} of α_{2}, ⋯, and k_{n} of α_{n}”. The additive operation + is naturally applied to consumption plans:

(k_{1}α_{1} + ⋯ + k_{n}α_{n}) + (l_{1}α_{1} + ⋯ + l_{n}α_{n}) = (k_{1} + l_{1})α_{1} + ⋯ + (k_{n} + l_{n})α_{n}.

Let us assume that there exists a preference relation ⪯ on Ω(S) satisfying the following conditions:

A1 ⪯ is a weak order on Ω(S).

A2 For all x, y, z ∈ Ω(S), x ≺ y implies x + z ≺ y + z.

A3 For all x, y, z ∈ Ω(S), x ∼ y implies x + z ∼ y + z.

The ordering condition A1 is an assumption that a preference relation of a rational person should be at least a weak order. The monotonicity condition A2 is an assertion that the strict part of a preference relation should be preserved by adding the same items to both sides of an inequality. For example, consider the case of two items (α_{1}, α_{2}) = ( cookie , cake ) and suppose, in the first place, we have a preference

2 × cookie ≺ 1 × cake ,

which means that we like a cake better than two cookies. Then, by A2, the following preferences are inferred:

⇒  3 × cookie ≺ 1 × cake + 1 × cookie ,
⇒  1 × cake + 3 × cookie ≺ 2 × cake + 1 × cookie ,
⇒  ⋯

by adding a cookie to both sides, then adding a cake, and so on. The meaning of A3 can be described in a similar way.

†N denotes the set of all natural numbers including zero.

The following theorem states that it is enough for us to assume these simple scheme of conditions A1–A3 to represent a preference relation in terms of hyperreal numbers. Let ε be an infinitely small number, i.e. 0 < ε and ε < 1/k for k = 1, 2, 3, · · ·.

Theorem 2 (Ikeda [4]) Let S = {α_{1}, α_{2}, ⋯, α_{n}} be a set of n-types of items for some n ≥ 2 . A preference relation ⪯ on Ω(S) satisfies A1–A3 if and only if there are hyperreal numbers u_{1}, ⋯, u_{n} such that for all k_{1}, ⋯, k_{n}, l_{1}, ⋯, l_{n} ∈ N,

∑_{i=1}^{n} k_{i}α_{i} ⪯ ∑_{i=1}^{n} l_{i}α_{i} ⟺ ∑_{i=1}^{n} k_{i}u_{i} ≤ ∑_{i=1}^{n} l_{i}u_{i}

in which u_{1}, ⋯, u_{n} are of the form:

( u_{1} ; u_{2} ; ⋮ ; u_{n} ) = ( \begin{matrix}r_{11} & ⋯ & r_{1n} \\ r_{21} & ⋯ & r_{2n} \\ ⋮ & ... & ⋮ \\ r_{n1} & ⋯ & r_{nn} \end{matrix} ) ( 1 ; ε ; ⋮ ; ε^{n−1} )

where {r_{ij}}_{1≤i,j≤n} is an n × n matrix of real numbers satisfying either r_{ij} = 0 for all i, j, or that the first column vector is nonzero, i.e. (r_{11}, r_{12}, ⋯, r_{n1}) ≠ (0, 0, ⋯, 0) . Also, the first column vector is unique up to a positive real multiple, i.e. if {s_{ij}}_{1≤i,j≤n} is another matrix with the same property, then there exists a positive real number a > 0 such that

( s_{11} ; ⋮ ; s_{n1} ) = a ( r_{11} ; ⋮ ; r_{n1} )

In other words, the ‘real parts’ of the hyperreal numbers u_{1}, ⋯, u_{n} are uniquely determined up to a positive real multiple.

Example 3 Let us give an example illustrating why hyperreal numbers get into our consideration. Let (α_{1}, α_{2}) = ( sesame , diamond ), and suppose we have a preference

0 ≺ α_{1} and kα_{1} ≺ α_{2} for all k > 0.

That is to say, a grain of sesame is better than nothing, and a diamond is better than any finite grains of sesame. This is consistent with the axioms A1–A3, but cannot be represented additively by real numbers because there is no pair of real numbers a_{1}, a_{2} such that 0 < ka_{1} < a_{2} for all k > 0. Thus the utilities must be expressed as (u_{1}, u_{2}) = (ε, 1). In other words, a grain of sesame has infinitely small utility in comparison with a diamond.

References

[1] Brams, S. J. and Fishburn, P. C. (2000). Fair Division of Indivisible Items between Two People with Identical Preferences: Envy-freeness, Pareto-optimality, and Equity. Social Choice and Welfare, 17, 247–267.

[2] Fishburn, P. C. (1970). Utility Theory for Decision Making. Wiley, New York.

[3] Goldblatt, R. (1998). Lectures on the Hyperreals: An Introduction to Nonstandard Analysis. Springer, New York.

[4] Ikeda, K. (2001). Non-Archimedean Additive Utilities for Discrete Items. manuscript.

[5] von Neumann, J. and Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton University Press. second edition, 1947; third edition, 1953.