Median-based Rules for Decision-making under Complete Ignorance

Abstract

This article characterizes a class of rules for decision-making when an agent knows the possible states of the world and the outcome of each of his/her actions for each state, but does not have any information about the probabilities of the states. The existing literature in this framework has mainly considered ‘max’-based or ‘min’-based rules and their variants. Such rules reflect rather extreme forms of optimism or pessimism on the part of an agent. In contrast, this paper focuses on the median outcome(s) and characterizes a class of decision-making rules which reflects a more ‘balanced’ attitude towards uncertainty. We also discuss a possible interpretation of our result in terms of the ranking of alternative social states that an individual may have when he/she is under the Rawlsian ‘veil of ignorance’.

JEL Classifications: D81

Keywords

Complete ignorance median-based rules median outcome non-probabilistic uncertainty

Introduction

This article characterizes a class of rules for decision-making under the type of non-probabilistic uncertainty considered first by Arrow and Hurwicz (1972). Under this type of uncertainty, the agent knows different possible states of the world and the outcome of each of her actions for each state, but does not have any probabilistic information, such as exact probabilities, the likelihood ranking,¹

See Kelsey (1993) for a discussion of decision-making when the agent has only the likelihood ranking of the states, but not their exact probabilities.

or probability intervals²

See Gilboa and Schmeidler (1989) for a model of decision-making where the agent has a probability interval for each state of the world.

for these states. Following Arrow and Hurwicz (1972), several writers (see, e.g., Barbera & Jackson, 1988; Barrett & Pattanaik, 1994; Maskin, 1979) have discussed different rules of decision-making under uncertainty of the Arrow–Hurwicz type. All these contributions, however, focus on ‘max’-based or ‘min’-based rules and variants of such rules. In light of the agent’s usually limited capacity for processing information, it seems intuitively plausible to assume that an agent, when confronted with the problem of choice under uncertainty, may concentrate on some ‘focal’ outcomes³

The idea that the agent may consider only some focal outcomes of each available action goes back to Milnor (1954) and Shackle (1954). It may be worth recalling that the paper of Arrow and Hurwicz (1972) was published in a volume in honour of Shackle.

for each action. It is, however, not clear why the agent will necessarily look only at the extreme outcomes, that is, the best or worst outcomes, of each action. An alternative focal point for each action may be its median outcome(s).⁴

For a precise definition of the median outcome(s) of an action, see ‘Notation and Assumptions’ section.

The ranking of actions on the basis of their extreme outcomes involves excessive optimism or pessimism on the part of the agent. In contrast, the focus on the median outcome(s) in ranking alternative actions can be interpreted as a characteristic of more balanced behaviour. Though decision rules based on the median outcome(s) seem to have considerable intuitive plausibility, the structure of these rules in the Arrow–Hurwicz framework has not been explored so far. The purpose of this article is to fill this gap in the literature by providing an axiomatic characterization of a class of median-based decision rules for choice under non-probabilistic uncertainty of the Arrow–Hurwicz type.⁵

Nitzan and Pattanaik (1984) characterize a class of median-based decision rules in a framework which was first introduced by Kannai and Peleg (1984), and which is very different from that of Arrow and Hurwicz (1972). Nitzan and Pattanaik (1984), as well as Kannai and Peleg (1984), do not introduce the states of the world into their model, they assume that the agent knows only the set of outcomes for each action.

Further, the class of median-based rules characterized in this paper, although formulated as individual decision-making problems, has implications for social welfare judgements as well. When individuals form their ranking of social states under the ‘veil of ignorance’ as conceptualized by Rawls (1971), focusing on the individual(s) with median level(s) of well-being for each social state may seem to be a more balanced approach than focusing on the worst off individual. We briefly comment on how our analysis can be applied to this specific issue.

The structure of the article is as follows. In the next section, we introduce the basic notation and assumptions. The section after that presents the axioms with illustrative examples. The main result and its proof are given in ‘The Main Result’ section. The next section contains an example of a median-based rule. In the section after that, we provide an interpretation of median-based rules, characterized in this paper, in terms of an individual’s ‘impersonal’⁶

Though my intuitive interpretation is closer to the intuition of Rawls’ (1971) ‘veil of ignorance’, I have used Harsanyi’s (1955) term ‘impersonal’ because it is convenient for indicating absence of bias while decision-making (the reader will recall that despite the considerable intuitive similarity between the approaches of Harsanyi (1955) and Rawls (1971), the concept of impersonality in Harsanyi (1955) is somewhat different from the corresponding notion in Rawls (1971)).

or impartial evaluation of alternative social states. The final section concludes this article.

Notation and Assumptions

Assumption 1. The universal set of outcomes, X, is a non-empty and convex subset of $ℜ^{n}$ , where n is some fixed positive integer.

Assumption 2. The agent has a convex ordering $\underset{˜}{≻}$ over X, such that for some $x, y \in X, x \underset{˜}{≻} y$ and not $(y \underset{˜}{≻} x)$ .

The asymmetric and symmetric factors of $\underset{˜}{≻}$ are given by $≻$ and ∼, respectively. Let $d (x, y)$ denote the Euclidean distance between $x, y \in X$ .

Let Ψ be a non-empty set of states of the world and let a generic element of Ψ be denoted by s. A ‘decision problem’ is defined as a (finite) non-empty subset of Ψ . Let Z be the class of all decision problems. Let the elements of Z be denoted by $S, S^{'}, S^{″}$ , etc. Given $S \in Z$ , let A(S) denote the set of all possible functions $a : S \to X$ . The elements of A(S) are called ‘actions’. For a decision problem $S = {s_{1}, \dots, s_{m}}$ , where m is a positive integer, an action specifies exactly one n-tuple of real numbers for each of the m states of the world, and hence, can be thought of as an m × n vector of real numbers. It may be worth noting here that this representation of actions as m × n vectors of real numbers allows us later to introduce the property of continuity of the agent’s ordering over actions for a given decision problem (see Assumption 3).

A typical decision problem with two states of the world $S = {s_{1}, s_{2}}$ , two actions $a, b \in A (S)$ and outcomes $a (s_{1}), a (s_{2}), b (s_{1}), b (s_{2}) \in X$ is described as follows:

\begin{matrix} S \\ s_{1} & s_{2} \\ a & a (s_{1}) & a (s_{2}) \\ b & b (s_{1}) & b (s_{2}) \end{matrix}

An action $a \in A (S)$ is trivial, if for all $s, s^{'} \in S, a (s) = a (s^{'})$ . A trivial action $a \in A (S)$ , such that for all $s \in S, a (s) = x$ , is denoted by $〚 x 〛$ .

Let $ℜ (S)$ be the set of all weak preference orderings defined over A(S). Let $\cup_{S \in Z} ℜ (S)$ be denoted by $ℜ$ . We define a rule to be a non-empty valued correspondence C from Z to $ℜ$ such that for all $S \in Z, C (S) \subset ℜ (S)$ .

Assumption 3. For all $S \in Z$ , the agent has a weak preference ordering $R_{S}$ defined over A(S), such that,

(i) $R_{S}$ is continuous⁷

As noted earlier, actions for any given decision problem with m states of the world can be thought of as m × n vectors of real numbers. Therefore, continuity of the agent’s ordering over A(S) can be defined in the usual fashion.

over A(S),

and

(ii) for all $x, y \in X, x \underset{˜}{≻} y$ iff $〚 x 〛 R_{s} 〚 y 〛$ .

$I_{S}$ and $P_{S}$ are the symmetric and asymmetric factors, respectively, corresponding to $R_{S}$ .

Remark: Given Assumption 3, the ordering $\underset{˜}{≻}$ over X is continuous.

For all decision problems $S \in Z$ , and, for all $a, b \in A (S)$ , we write $a * b$ , iff the outcomes of action a corresponding to the different states of the world in S constitute a permutation of the outcomes of action b.

Let $S = {s_{1}, s_{2}, \dots, s_{m}} \in Z$ be a decision problem and let $a \in A (S)$ . Let the outcomes in the set A(S) be indexed as $x_{1}, x_{2}, \dots, x_{m}$ such that for all $k (m > k \geq 1), x_{k} \underset{˜}{≻} x_{k + 1} .$ ⁸

If there are more than one way of indexing the outcomes in this fashion, we choose one of them and keep it fixed.

Then the set of median outcome(s) of action a is denoted by med(a) and is defined to be:

{x_{\frac{_{m - 1}}{2} + 1}}

if m is odd, and

{x_{\frac{m}{2}}, x_{\frac{m}{2} + 1}}

if m is even.

For all $S \in Z$ and for all $a, b \in A (S)$ , we say that med(a) and med(b) are ‘similar’, iff $| m e d (a) | = | m e d (b) |$ ⁹

$| m e d (a) |$ represents cardinality of the set of median outcome(s) from action a.

and there exists a one-to-one function h from med(a) to med(b) such that for all

x \in m e d (a), x \sim h (x)

. We say that the agent follows a ‘median-based rule’, iff for all

S \in Z

, for all

R_{S} \in C (S)

and for all

a, b \in A (S)

, [if med(a) and med(b) are similar, then

a I_{s} b

For example, consider the following decision problem $S = {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}}$ , actions $a, b \in A (S)$ and outcomes $x_{1}, x_{2}, {x^{'}}_{2}, x_{3}, {x^{'}}_{3}, {x^{″}}_{3}, y_{3}, y_{4}, {y^{'}}_{4}, y_{5}, y_{6}, {y^{'}}_{6} \in X$ such that ${x^{'}}_{2} \sim {y^{'}}_{4}, x_{3} \sim y_{5}, x_{1} \underset{˜}{≻} x_{2} \sim {x^{'}}_{2} \underset{˜}{≻} x_{3} \sim {x^{'}}_{3} \sim {x^{″}}_{3}$ and $y_{3} \underset{˜}{≻} y_{4} \sim {y^{'}}_{4} \underset{˜}{≻} y_{5} \underset{˜}{≻} y_{6} \sim {y^{'}}_{6}$

\begin{matrix} S \\ S_{1} & S_{2} & S_{3} & S_{4} & S_{5} & S_{6} \\ a & {x^{'}}_{2} & x_{3} & {x^{'}}_{3} & x_{1} & x_{2} & {x^{″}}_{3} \\ b & y_{5} & y_{3} & y_{4} & y_{6} & {y^{'}}_{4} & {y^{'}}_{6} \end{matrix}

We choose the indexing of outcomes, such that, $x_{1} \underset{˜}{≻} x_{2} \underset{˜}{≻} {x^{'}}_{2} \underset{˜}{≻} x_{3} \underset{˜}{≻} {x^{'}}_{3} \underset{˜}{≻} {x^{″}}_{3}$ and $y_{3} \underset{˜}{≻} y_{4} \underset{˜}{≻} {y^{'}}_{4} \underset{˜}{≻} y_{5} \underset{˜}{≻} y_{6} \underset{˜}{≻} {y^{'}}_{6}$ . Then, $m e d (a) = {{x^{'}}_{2}, x_{3}}, m e d (b) = {{y^{'}}_{4}, y_{5}}$ and a median-based rule will yield $a I_{s} b$ .

Note that the class of median-based rules is not necessarily a singleton. Consider the following decision problem $S = {s_{1}, s_{2}}$ and two actions $a, b \in A (S)$ , such that $b (s_{1}) ≻ a (s_{1})$ and $a (s_{2}) ≻ b (s_{2})$ . It is easy to see that both $a P_{s} b$ and $b P_{s} a$ are consistent with a median-based rule as we have defined it. This shows that a median-based rule need not be singleton valued.

\begin{matrix} S \\ s_{1} & s_{2} \\ a & a (s_{1}) & a (s_{2}) \\ b & b (s_{1}) & b (s_{2}) \end{matrix}

The Axioms

We shall now introduce several plausible properties that the agent may satisfy. The properties are also illustrated with examples. We shall later characterize median-based rules in terms of these properties.

Axiom 1. Neutrality: Suppose $S, S^{'} \in Z, | S | = | S^{'} |, a, b \in A (S), a^{'}, b^{'} \in A (S^{'})$ . Further, suppose there exists a one-to-one function f from S to $S^{'}$ such that, for all $s, \hat{s} \in S, [a (s) \underset{˜}{≻} b (\hat{s}) iff a^{'} (f (s)) \underset{˜}{≻} b^{'} (f (\hat{s}))],$ $[b (\hat{s}) \underset{˜}{≻} a (s) iff b^{'} (f (\hat{s})) \underset{˜}{≻} a^{'} (f (s))],$ $[a (s) \underset{˜}{≻} a (\hat{s}) iff a^{'} (f (s)) \underset{˜}{≻} a^{'} (f (\hat{s}))],$ $[a (\hat{s}) \underset{˜}{≻} a (s) iff a^{'} (f (\hat{s})) \underset{˜}{≻} a^{'} (f (s))],$ $[b (s) \underset{˜}{≻} b (\hat{s}) iff b^{'} (f (s)) \underset{˜}{≻} b^{'} (f (\hat{s}))],$ and $[b (\hat{s}) \underset{˜}{≻} b (s) iff b^{'} (f (\hat{s})) \underset{˜}{≻} b^{'} (f (s))]$ . Then, for all $R_{s} \in C (S)$ and all $R_{s^{'}} \in C (S^{'}), [a R_{S} b iff a^{'} R_{s^{'}} b^{'}]$ and $[b R_{S} a iff b^{'} R_{s^{'}} a^{'}]$ .

Suppose two decision problems S and $S^{'}$ have equal number of the states of the world. Neutrality then requires that, if the ranking of outcomes from two actions a and b, in decision problem S is ‘analogous’ to the ranking of the outcomes of two actions $a^{'}$ and $b^{'}$ , in the decision problem $S^{'}$ , then the ranking of a and b will be similar to the ranking of $a^{'}$ and $b^{'}$ .

For example, neutrality implies $[a R_{S} b iff a^{'} R_{s^{'}} b^{'}]$ and $[b R_{S} a iff b^{'} R_{s^{'}} a^{'}]$ in the following two decision problems S and $S^{'}$ , where ${s^{'}}_{1} = f (s_{1}), {s^{'}}_{2} = f (s_{2}),$ and $p, q, r, x, y, z \in X$ , such that $p ≻ q ≻ r ≻ x ≻ y ≻ z$ .

\begin{matrix} S & S^{'} \\ s_{1} & s_{2} & {s^{'}}_{1} & {s^{'}}_{2} \\ a & r & y & a^{'} & q & x \\ b & z & p & b^{'} & y & p \end{matrix}

Neutrality has several plausible implications that have been discussed in the literature for decision-making under complete uncertainty. First, neutrality implies that the identities of the states of a decision problem do not matter while ranking actions in a decision problem; only order of the outcomes under different states matters. Thus, neutrality is similar to the well-known ‘symmetry’ axiom introduced by Arrow and Hurwicz (1972), but it is stronger than the ‘symmetry’ axiom. The ‘symmetry’ axiom as discussed in Arrow and Hurwicz (1972) requires the image set of the mapping from one decision problem to the other to be identical with the domain set, whereas the image set can be different than the domain set under neutrality.

Second, neutrality implies that, while ranking two actions, only the ranking of outcomes from these two actions are relevant. The ranking of outcomes, at least one of which does not occur in the two actions under consideration, is of no importance. This may be noted as ‘independence of the irrelevant outcomes’.

Thus, in the presence of neutrality, only the ordering of the relevant outcomes under the different states is considered. At first sight, this may seem implausible. Consider the following example with two states and two actions where the outcomes are assumed to be monetary magnitudes.

\begin{matrix} S & S^{'} \\ s_{1} & s_{2} & {s^{'}}_{1} & {s^{'}}_{2} \\ a & 7 & 1 & a^{'} & 7000 & 4.99 \\ b & 5 & 5 & b^{'} & 5 & 5 \end{matrix}

Suppose, an outcome x is at least as good as y iff $x \geq y$ . It is possible for an agent to have $a I_{s} b$ and $a^{'} P_{s^{'}} b^{'}$ , violating neutrality. However, the Arrow–Hurwicz (1972) framework of complete ignorance provides only ordinal information about an agent’s preference over the outcomes. Since the ordering of outcomes from $a, b \in A (S)$ is the same across states as the ordering of outcomes from $a^{'}, b^{'} \in A (S^{'})$ neutrality seems to be a plausible axiom in this framework.

Lemma 1: Suppose the agent satisfies neutrality. Then, for every decision problem $S = {s_{1}, s_{2}, \dots, s_{m}} \in Z$ and for all actions $\bar{a}, a \in A (S)$ , such that, $a * \bar{a}$ and $a (s_{1}) \underset{˜}{≻} a (s_{2}) \underset{˜}{≻} \dots \underset{˜}{≻} a (s_{m})$ , we must have $a I_{s} \bar{a}$ .

Proof: Let $S = {s_{1}, s_{2}, \dots, s_{m}} \in Z$ and let $\bar{a}, a \in A (S)$ such that $a * \bar{a}$ and $a (s_{1}) \underset{˜}{≻} a (s_{2}) \underset{˜}{≻} \dots \underset{˜}{≻} a (s_{m})$ . Since $a * \bar{a}$ , there exists a one-to-one function f from S to S such that for all $s \in S, a (s) = \bar{a} (f (s))$ and hence $a (s) \sim \bar{a} (f (s))$ . Therefore, by neutrality $[a R_{s} \bar{a} iff \bar{a} R_{s} a]$ and $[\bar{a} R_{s} a iff a R_{s} \bar{a}]$ . Since, by connectedness of R_s, we have $(a R_{s} \bar{a} or \bar{a} R_{s} a)$ , it follows that $a I_{s} \bar{a}$ .

Axiom 2. Duality: Suppose $S, S^{'} \in Z, | S | = | S^{'} |, a, b \in A (S), a^{'}, b^{'} \in A (S^{'})$ , and $s, \hat{s} \in S$ . Further, suppose there exists a one-to-one function g from S to $S^{'}$ such that, for all $s, \hat{s} \in S, [a (s) \underset{˜}{≻} b (\hat{s}) iff b^{'} (g (\hat{s})) \underset{˜}{≻} a^{'} (g (s))]$ , $[b (\hat{s}) \underset{˜}{≻} a (s) iff a^{'} (g (s)) \underset{˜}{≻} b^{'} (g (\hat{s}))]$ , $[a (s) \underset{˜}{≻} a (\hat{s}) iff a^{'} (g (\hat{s})) \underset{˜}{≻} a^{'} (g (s))]$ , $[a (\hat{s}) \underset{˜}{≻} a (s) iff a^{'} (g (s)) \underset{˜}{≻} a^{'} (g (\hat{s}))]$ , $[b (s) \underset{˜}{≻} b (\hat{s}) iff b^{'} (g (\hat{s})) \underset{˜}{≻} b^{'} (g (s))]$ , and $[b (\hat{s}) \underset{˜}{≻} b (s) iff b^{'} (g (s)) \underset{˜}{≻} b^{'} (g (\hat{s}))]$ . Then, for all $R_{s} \in C (S)$ and all $R_{s^{'}} \in C (S^{'}), [a R_{s} b iff b^{'} R_{s^{'}} a^{'}]$ and $[b R_{s} a iff a^{'} R_{s^{'}} b^{'}]$ .

Suppose two decision problems S and $S^{'}$ have the same number of states of the world. Duality then requires that, if the ranking of outcomes from two actions a and b in decision problem S is the ‘reverse’ of the ranking of the outcomes of two actions $a^{'}$ and $b^{'}$ in the decision problem $S^{'}$ , then the ranking of a and b must be the ‘reverse’ of the ranking of $a^{'}$ and $b^{'}$ .

In the following two decision problems S and $S^{'}$ such that ${s^{'}}_{1} = g (s_{1}), {s^{'}}_{2} = g (s_{2}), {s^{'}}_{3} = g (s_{3}), p, q, r \in X$ and $p ≻ q ≻ r$ , duality implies $[a R_{s} b iff b^{'} R_{s^{'}} a^{'}]$ and $[b R_{s} a iff a^{'} R_{s^{'}} b^{'}]$ .

\begin{matrix} S & S^{'} \\ s_{1} & s_{2} & s_{3} & {s^{'}}_{1} & {s^{'}}_{2} & {s^{'}}_{3} \\ a & p & q & r & a^{'} & r & q & p \\ b & r & r & P & b^{'} & p & p & r \end{matrix}

Axiom 3. Weak Dominance: For all decision problems, $S \in Z$ , and for all $a, b \in A (S)$ , if $a (s) ≻ b (s)$ for all $s \in S$ , then $a R_{s} b$ for all $R_{s} \in C (S)$ .

Thus, if, for every state of the world, an action yields an outcome that is better than the outcome from another action, then the former action is at least as good as the latter one. For example, in the following decision problem S, where $a, b \in A (S)$ , $p \underset{˜}{≻} q ≻ r \in X$ , weak dominance requires $a R_{s} b$ .

\begin{matrix} S \\ s_{1} & s_{2} \\ a & p & q \\ b & r & r \end{matrix}

Next we consider an axiom, which will be referred to in our discussion though we shall not use it in our characterization result.

Axiom 4. Strict Dominance: For all decision problems $S \in Z$ , and for all $a, b \in A (S)$ , if [ $a (s) \underset{˜}{≻} b (s)$ for all $s \in S$ ] and [ $a (s) ≻ b (s)$ for some $s \in S$ ], then $a P_{s} b$ for all $R_{s} \in C (S)$ .

It is clear that Axiom 4 is much stronger than Axiom 3. But Axiom 4 still has considerable plausibility.

The Main Result

Proposition 1. Suppose, Assumptions 1 through 3 hold. Then the agent follows a median-based rule if he/she satisfies neutrality, duality and weak dominance.

We proceed to the proof of Proposition 1 via a series of lemmas. Throughout the proof, it is to be understood that Assumptions 1–3 hold, and the agent satisfies neutrality, duality and weak dominance.

Lemma 2: For all $S \in Z$ such that $| S | \leq 2$ , and for all $a, b \in A (S)$ such that $b (s) \sim v$ for all $s \in S$ , and ${v} = m e d (a)$ , we must have $a I_{s} b$ .

Proof: If $| S | = 1$ , then, $a I_{s} b$ follows immediately by reflexivity of $R_{s}$ . If $| S | = 2$ , then, $a I_{s} b$ follows from reflexivity of $R_{s}$ and neutrality.

Lemma 3: Let $S = {s_{1}, \dots, s_{2 m + 1}} \in Z$ be such that m is a positive integer. Let $a, b \in A (S)$ be such that $a (s_{1}) ≻ a (s_{2}) ≻ \dots ≻ a (s_{2 m + 1})$ , and $b (s) \sim a (s_{m + 1})$ for all $s \in S$ . Then, we must have $a I_{s} b$ .

Proof: Consider S and $a, b \in A (S)$ as specified in the statement of Lemma 3. For the sake of convenience, we represent $a, b, a^{'}, b^{'} \in A (S)$ as follows:

\begin{matrix} S \\ s_{1} & s_{2} & \dots \dots \dots \dots \dots \dots .. & s_{2 m} & s_{2 m + 1} \\ a & a (s_{1}) & a (s_{2}) & \dots \dots \dots \dots \dots \dots .. & a (s_{2 m}) & a (s_{2 m + 1}) \\ b & b (s_{1}) & b (s_{2}) & \dots \dots \dots \dots \dots \dots .. & b (s_{2 m}) & b (s_{2 m + 1}) \\ a^{'} & a (s_{2 m + 1}) & a (s_{2 m}) & \dots \dots \dots \dots \dots \dots .. & a (s_{2}) & a (s_{1}) \\ b^{'} & b (s_{2 m + 1}) & b (s_{2 m}) & \dots \dots \dots \dots \dots \dots .. & b (s_{2}) & b (s_{1}) \end{matrix}

Recall that $b (s) \sim a (s_{m + 1})$ for all $s \in S$ and $a (s_{1}) ≻ a (s_{2}) ≻ \dots ≻ a (s_{2 m + 1})$ . Hence, by neutrality, $a I_{s} a^{'}$ and $b I_{s} b^{'}$ . By transitivity of $R_{s}$ , we then have:

(a R_{s} b iff a^{'} R_{s} b^{'}) and (b R_{s} a iff b^{'} R_{s} a^{'})

(1)

By duality, we have:

(a R_{s} b iff b^{'} R_{s} a^{'}) and (b R_{s} a iff a^{'} R_{s} b^{'})

(2)

By connectedness of $R_{s}$ , either $a P_{s} b$ , or $b P_{s} a$ , or $a I_{s} b$ . If $a P_{s} b$ , then, by (1) and (2), $a^{'} P_{s} b^{'}$ and $b^{'} P_{s} a^{'}$ , which is a contradiction. Similarly $b P_{s} a$ yields a contradiction. Thus, we must have $a I_{s} b$ .

Lemma 4: Let $S = {s_{1}, \dots, s_{2 m}} \in Z$ be such that m is a positive integer. Let $a, b \in A (S)$ be such that $a (s_{1}) ≻ a (s_{2}) ≻ \dots ≻ a (s_{m}) \underset{˜}{≻} a (s_{m + 1}) ≻ a (s_{m + 2}) ≻ \dots ≻ a (s_{2 m})$ , [ $b (s_{i}) \sim a (s_{m})$ for $i = 1, \dots, m$ ], and [ $b (s_{i}) \sim a (s_{m + 1})$ for $i = m + 1, \dots, 2 m$ ]. Then we must have $a I_{s} b$ .

Proof: The proof follows exactly similar logic as described in the proof of Lemma 3.

Lemma 5: Let $x, y \in Z$ be such that $x ≻ y$ and let $\tilde{ε}$ be such that $d (x, y) > \tilde{ε} > 0.$ Then, for every positive integer m, there exists $w_{1}, w_{2}, \dots, w_{m} \in] x, y [$ such that $x ≻ w_{1} ≻ w_{2} ≻ \dots ≻ w_{m} ≻ y$ , and $\tilde{ε} > d (w_{1}, y) > d (w_{2}, y) > \dots > d (w_{m}, y)$ .

Proof: We first show that, for all $q, r \in X$ such that $q ≻ r$ , and all $\tilde{ε}$ such that $d (q, r) > \tilde{ε} > 0$ , there exists $w \in] q, r [$ such that,

\tilde{ε} > d (w, r) > 0, and q ≻ w ≻ r .

(3)

Let $q, r \in X$ be such that, $q ≻ r$ . Let $\tilde{ε}$ be such that, $d (q, r) > \tilde{ε} > 0$ .

Since $q ≻ r$ , by convexity of $\underset{˜}{≻},$

w ≻ r for all w \in] q, r [.

(4)

Noting $q ≻ r$ , by the continuity of $\underset{˜}{≻},$ there exists $w \in] q, r [$ such that

\tilde{ε} > d (w, r) > 0 and q ≻ w .

(5)

Equation (3) follows from (4) and (5).

Now, let $x, y \in X$ such that $x ≻ y$ , let $\tilde{ε}$ be such that $d (x, y) > \tilde{ε} > 0$ , and let m be any positive integer.

Since $x, y \in X, x ≻ y$ , and $d (x, y) > \tilde{ε} > 0$ , by (3), there exists $w_{1} \in] x, y [$ , such that

\tilde{ε} > d (w_{1}, y) > 0, and x ≻ w_{1} ≻ y .

(6)

Since $w_{1}, y \in X, w_{1} ≻ y$ , and $d (w_{1}, y) > ε^{'} > 0$ , for some positive $ε^{'}$ , by (3) again, there exists $w_{2} \in] w_{1}, y [$ such that,

ε^{'} > d (w_{2}, y) > 0, and w_{1} ≻ w_{2} ≻ y .

(7)

Thus, we have $w_{1}, w_{2} \in] x, y [$ such that, $d (x, y) > \tilde{ε} > d (w_{1}, y) > ε^{'} > d (w_{2}, y)$ , and $x ≻ w_{1} ≻ w_{2} ≻ y$ . Continuing in this fashion, for all $x, y \in X$ such that, $x ≻ y$ , all $\tilde{ε}$ such that $d (x, y) > \tilde{ε} > 0$ , and every positive integers m, there exist $w_{1}, w_{2}, \dots, w_{m} \in] x, y [$ such that $x ≻ w_{1} ≻ w_{2} ≻ \dots ≻ w_{m} ≻ y$ , and $\tilde{ε} > d (w_{1}, y) > d (w_{2}, y) > \dots > d (w_{m}, y)$ .

Lemma 3 showed that, in a decision problem with an odd number of states of the world, if two actions a and b are such that b always yields outcomes that are indifferent to the median outcome of a, and if a does not yield indifferent outcomes for any two distinct states of the world, then a and b must be indifferent. Our next lemma, Lemma 6, extends Lemma 3 by relaxing the requirement that a does not yield indifferent outcomes for any two distinct states of the world. Lemma 7 extends Lemma 4 in an analogous fashion.

Lemma 6: Let $S = {s_{1}, \dots, s_{2 m + 1}} \in Z$ be such that $m (m \geq 1)$ is a positive integer. Let $a, b \in A (S)$ be such that, $a (s_{1}) \underset{˜}{≻} a (s_{2}) \underset{˜}{≻} \dots \underset{˜}{≻} a (s_{2 m + 1})$ , and $b (s) \sim a (s_{m + 1})$ for all $s \in S$ . Then, we must have $a I_{s} b$ .

Proof: Consider $S = {s_{1}, \dots, s_{2 m + 1}} \in Z$ . Let $a, b \in A (S)$ be such that $a (s_{1}) \underset{˜}{≻} a (s_{2}) \underset{˜}{≻} \dots \underset{˜}{≻} a (s_{2 m + 1})$ , and $b (s) \sim a (s_{m + 1})$ for all $s \in S$ . Now, partition S into $S_{1}, S_{2}, \dots, S_{t}$ such that, [for all $j \in {1, \dots, t}$ and all $s, s^{'} \in S_{j}, a (s) \sim a (s^{'})$ ] and [for all $j \in {1, \dots, t - 1}$ , all $s \in S_{j}$ , and all $s^{'} \in S_{j + 1}, a (s) ≻ a (s^{'})$ ].¹⁰

If t = 1, then $a, b \in A (S)$ are trivial actions such that, Lemma 6 follows immediately, by reflexivity of $R_{s}$ .

For all

j \in {1, \dots, t}

, let

m (j)

be the cardinality of

S_{j}

By our assumption, there exist $y_{0}, y \in X$ such that $y_{0} ≻ y$ . Then, by Lemma 5, there exists $y_{1}, \dots, y_{t} \in] y_{0}, y [$ such that, $y_{0} ≻ y_{1} ≻ y_{2} ≻ \dots ≻ y_{t} ≻ y$ and $d (y_{0}, y) > d (y_{1}, y) > d (y_{2}, y) > \dots > d (y_{t}, y)$ . Let $ε$ be a positive number such that $ε = \min (d (y_{0}, y_{1}), d (y_{1}, y_{2}), \dots, d (y_{t - 1}, y_{t}))$ . Consider $(ε / k)$ , where $k (k \geq 2)$ is any positive integer. Then, for every $j \in {1, \dots, t}$ , by Lemma 5, there exist $w_{j, 1}, w_{j, 2}, \dots, w_{j, m (j)}$ such that, $y_{j - 1} ≻ w_{j, 1} ≻ w_{j, 2} ≻ \dots ≻ w_{j, m (j)} ≻ y_{j}$ , and $(ε / k) > d (w_{j, 1}, y_{j}) > d (w_{j, 2}, y_{j}) > \dots > d (w_{j, m (j)}, y_{j})$ .

Let $a_{ε / k}$ be an action such that for every $j \in {1, \dots, t}$ , and $\begin{array}{l} n \in {1, 2, \dots, m (j)}, a (s_{m (1) + m (2) + \dots + m (j - 1) + n}) = w_{j, n} \end{array}$ . It is clear by Lemma 5 that for all $s \in S, a_{ε / k} (s) ≻ a (s)$ , and $(ε / k) > d (a_{ε / k} (s), a (s)) > 0$ .

Hence, as k \to \infty, a_{ε / k} converges to action a .

(8)

Further, note that, for every k, $V_{k} ≻ V$ , where ${V_{k}} = m e d (a_{ε / k})$ and ${V} = m e d (a)$ . Then, by Lemma 2, $a_{ε / k} I_{s} 〚 V_{k} 〛$ for every k.¹¹

Note that as $k \to \infty, 〚 V_{k} 〛$ converges to $〚 V 〛$

Again, by Assumption 3,

〚 V_{k} 〛 P_{s} 〚 V 〛

. Hence, by transitivity of

R_{s}, a_{ε / k} P_{s} 〚 V 〛

for all k.

Finally, noting (8) and $a_{ε / k} P_{s} 〚 V 〛$ , we have

a R_{s} 〚 V 〛, by continuity of R_{s} .

(9)

All that remains to be shown is that [not $a P_{s} 〚 V 〛$ ], which given (9), will give us $a I_{s} 〚 V 〛$ . Suppose $a P_{s} 〚 V 〛$ . Then, by continuity of $R_{s}$ , there exists large enough $k^{'}$ , such that, $a P_{s} 〚 V_{k^{'}} 〛$ . But, given $a_{ε / k} I_{s} 〚 V_{k} 〛$ for every k, we must have $a_{ε / k^{'}} I_{s} 〚 V_{k^{'}} 〛$ . Therefore, by transitivity of $R_{s}$ , it follows that, for some $k^{'}, a P_{s} a_{ε / k^{'}}$ . This contradicts weak dominance, since, as we noted earlier, $a_{ε / k} (s) ≻ a (s)$ for all $s \in S$ . This completes the proof of Lemma 6.

Lemma 7: Let $S = {s_{1}, \dots, s_{2 m}} \in Z$ be such that $m (m > 1)$ is a positive integer. Let $a, b \in A (S)$ be such that, $a (s_{1}) \underset{˜}{≻} a (s_{2}) \underset{˜}{≻} \dots \underset{˜}{≻} a (s_{m / 2}) \underset{˜}{≻} a (s_{(m / 2) + 1}) \underset{˜}{≻} a (s_{(m / 2) + 2}) \underset{˜}{≻} \dots \underset{˜}{≻} a (s_{2 m})$ , $[b (s_{i}) \sim a (s_{m / 2}) for i = 1, \dots, m / 2]$ , and $[b (s_{i}) \sim a (s_{(m / 2) + 1}) for i = (m / 2) + 1, \dots, 2 m]$ . Then, we must have $a I_{s} b$ .

Proof: Consider $S = {s_{1}, \dots, s_{2 m}} \in Z$ such that $m (m > 1)$ is a positive integer. Let $a, b \in A (S)$ be such that, $a (s_{1}) \underset{˜}{≻} a (s_{2}) \underset{˜}{≻} \dots \underset{˜}{≻} a (s_{m / 2}) \underset{˜}{≻} a (s_{(m / 2) + 1}) \underset{˜}{≻} a (s_{(m / 2) + 2}) \underset{˜}{≻} \dots \underset{˜}{≻} a (s_{2 m})$ , $[b (s_{i}) \sim a (s_{m / 2}) for i = 1, \dots, m / 2]$ , and $[b (s_{i}) \sim a (s_{(m / 2) + 1}) for i = (m / 2) + 1, \dots, 2 m]$ . Now, partition S into $S_{1}, S_{2}, \dots, S_{t}$ such that, [for all $j \in {1, \dots, t}$ , and all $s, s^{'} \in S_{j}, a (s) \sim a (s^{'})$ ], and [for all $j \in {1, \dots, t - 1}$ , all $s \in S_{j}$ , and all $s^{'} \in S_{j + 1}, a (s) ≻ a (s^{'})$ ]. The rest of the proof is similar to the proof of Lemma 6.

Proof of Proposition 1: Let $S = {s_{1}, \dots, s_{m}} \in Z$ be any decision problem such that $a, b \in A (S)$ , $| m e d (a) | = | m e d (b) |$ , and there exists a one-to-one function h from med(a) to med(b) such that for all $x \in m e d (a), x \sim h (x)$ . Given $a \in A (S)$ , it can be checked that there exists $\bar{a} \in A (S)$ such that $a * \bar{a}$ , and $\bar{a} (s_{1}) \underset{˜}{≻} \dots \underset{˜}{≻} \bar{a} (s_{m})$ . Then by Lemma 1, we have $a I_{s} \bar{a}$ . By Lemmas 2, 3, 6 and 7, we have $R_{s}$ . By transitivity of , we get $a I_{s} b$ .

An Example

We have shown that, given Assumptions 1–3, if the agent satisfies neutrality, duality and weak dominance, he/she must follow a median-based rule. We now give an example where Assumptions 1–3 as well as neutrality, duality and weak dominance are all satisfied.

Let X be any non-empty and convex subset of $ℜ^{n}$ and let $\underset{˜}{≻}$ be any convex and continuous ordering over X, such that, for some $x, y \in X, x ≻ y$ . Thus, by our specification, Assumptions 1–3 (i) are satisfied. Let U be a real valued and continuous utility function representing $\underset{˜}{≻}$ . Clearly, such a utility function U exists (see Debreu, 1987). For every decision problem $S \in Z$ , let $R_{s}$ defined over A(S) be such that, for all $a, b \in A (S), a R_{s} b, iff \underset{x \in m e d (a)}{Σ} U (x) \geq \underset{y \in m e d (b)}{Σ} U (y)$ . Clearly, for every $S \in Z, R_{s}$ is continuous and for all $x, y \in X, 〚 x 〛 R_{s} 〚 y 〛 iff x \underset{˜}{≻} y$ and hence Assumption 3 (ii) is satisfied. Further, for all $S \in Z$ , and for all $a, b \in A (S)$ , if $| m e d (a) | = | m e d (b) |$ and there exists a one-to-one function h from med(a) to med(b) such that for all $x \in m e d (a), x \sim h (x)$ , then $\underset{x \in m e d (a)}{Σ} U (x) = \underset{y \in m e d (b)}{Σ} U (y)$ and hence $a I_{s} b$ .

Median-based Rules and an Individual’s Ranking of Alternative Social States under the Veil of Ignorance

Let $N = {1, 2, \dots, n}$ be a given society. A social state can then be thought of as an n-tuple of well-being levels (denoted by real numbers), with exactly one well-being level for each individual. The real numbers representing the well-being levels of the different individuals are assumed to be interpersonally comparable. An individual’s ranking, in terms of social welfare, of two social states, say a and b can be thought of as the individual’s impersonal or impartial ranking of a and b. Following Rawls (1971), an individual’s impersonal ranking of social states a and b can, in turn, be identified with the individual’s ranking of a and b when he/she does not know which of the n individuals he/she will be in any given social state. Thus, what we called actions in the preceding sections can now be interpreted as the different social states. The different states of the world can now be interpreted as ‘being individual 1’, ‘being individual 2’, … and ‘being individual n’. For any given social state, say, a and any given state of the world, say ‘being individual i’, the outcome for the agent is simply the real number representing the well-being of individual i, in the social state a. When the problem of decision-making under uncertainty, which we discussed in the preceding sections, is interpreted in this fashion, it is clear that, if the agent satisfies our axioms, then two social states with the same median individual well-being levels must be indifferent to each other in the agent’s impersonal ranking of social states which has been identified with the agents ranking of social sates in terms of social welfare. This, of course, is different from the Rawlsian ranking of social states, which is determined by the well-being ranking of the worst off individuals in the two social sates. But the procedure of ranking social states on the basis of the median individual well-being level(s) in those social states seems to have some intuitive appeal.

Concluding Remarks

Most of the articles, which discuss non-probabilistic uncertainty of the Arrow–Hurwicz type, focus on what may be called positional decision rules. The positional rules characterized in this literature mainly consider the best or the worst outcomes. Lexicographic variants of such rules have also been discussed. It is, however, surprising that none of the articles in this area have dealt with the case when the agent makes decision on the basis of the median outcome(s) of her actions. In this paper, we have sought to fill this gap by providing an axiomatic characterization of median-based rules.

Three points may be noted about the class of decision rules discussed in this article. First, the rules in this class make use of only ordinal properties of preferences. Second, we characterize a class of decision rules rather than a single decision rule. We characterize the decision-making process of an agent who is indifferent between two actions when the median outcomes from those two actions are similar. The decision rules characterized in this paper are, however, silent about the ranking of two actions when the sets of their median outcomes are not similar. Finally, it may be noted that, if there are at least three different indifference classes defined by the agent’s ordering over X, then, for some decision problem S, the agent’s ranking over the actions will violate strict dominance, which, as we have noted earlier, is stronger than the property of weak dominance, but is still fairly plausible. Consider outcomes $x, x^{'}, y, z, z^{'} \in X$ such that, $x ≻ x^{'} ≻ y ≻ z ≻ z^{'}$ . Let $S = {s_{1}, s_{2}, s_{3}} \in Z$ and let $a, b \in A (S)$ be such that:

\begin{matrix} S \\ s_{1} & s_{2} & s_{3} \\ a & x & y & z \\ b & x^{'} & y & z^{'} \end{matrix}

Under every median-based rule, as defined in this paper, the agent will be indifferent between actions a and b, and, therefore, her ranking of actions will violate the plausible property of strict dominance. This limitation arises because, in our framework, the agent focuses only on the sets of median outcomes of different actions, ignoring information about other possible outcomes. All decision rules, which incorporate such a ‘focal-point’ approach, including the decision rules considered by Arrow and Hurwicz (1972),¹²

In the Arrow–Hurwicz (1972) decision rules, the agent focuses only on the best and worst outcomes of each action, ignoring all intermediate outcomes.

suffer from this limitation.

Before concluding, I would like to mention the literature on ‘regret theory’ (see, among others, Bell [1982] and Loomes and Sudgen [1982, 1987] for details), which also tends to favour median outcome(s) while ranking actions. The fundamental idea underlying regret theory involves measuring regret by comparing utility from the realized outcome of a chosen action with that from another outcome which might have been possible, had one chosen differently. Regret theory essentially assumes cardinal measurability of such ‘regret’ and thus uses a framework that is very different from the one that I use in this paper. To the best of my knowledge, this paper is the first one to provide characterization of a class of median-based rules in the Arrow–Hurwicz framework, where the agent’s preferences are ordinal. Like the other basic rules in this framework, the class of median based rules characterized in this paper can be broadened further. Characterizations of more complete classes of median-based rules when the sets of median outcomes of actions are dissimilar, and also a lexicographic variation of the class of rules discussed in this paper are areas of future research.

Acknowledgements

I am thankful to Dr Prasanta K. Pattanaik for his suggestions and encouragement, and to the referee(s) for valuable comments.

References

Arrow

K. J.

Hurwicz

(1972). An optimality criterion for decision-making under ignorance. In Carter

C. F.

Ford

J. L.

(Eds.), Uncertainty and expectations in economics: Essays in honor of G.L.S. Shackle (pp. 1–11). Oxford: Basil Blackwell.

Barbera

Jackson

(1988). Maximin, leximin, and the protective criterion: Characterizations and comparisons. Journal of Economic Theory , 46(1), 34–44.

Barrett

C. R.

Pattanaik

P. K.

(1994). Decision-making under complete uncertainty. In Dickinson

Driscoll

Sen

(Eds.), Risk and uncertainty in economics (pp. 20–36). Aldershot: Edward Elgar.

Bell

(1982). Regret in decision making under uncertainty. Operations Research , 30(5), 961–981.

Debreu

(1987). Theory of value: An axiomatic analysis of economic equilibrium (Vol. 17). New Haven and London: Yale University Press.

Gilboa

Schmeidler

(1989). Maximin expected utility with non-unique prior. Journal of Mathematical Economics , 18(2), 141–153.

Harsanyi

J. C.

(1955). Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility. Journal of Political Economy , 63(4), 309–321.

Kelsey

(1993). Choice under partial uncertainty. International Economic Review , May 1, 297–308.

Kannai

Peleg

(1984). A note on the extension of an order on a set to the power set. Journal of Economic Theory , 32(1), 172–175.

10.

Loomes

Sugden

(1982). Regret theory: An alternative theory of rational choice under uncertainty. Economic Journal , 92(368), 805–824.

11.

Loomes

Sugden

(1987). Some implications of a more general form of regret theory. Journal of Economic Theory , 41(2), 270–287.

12.

Maskin

(1979). Decision-making under ignorance with implications for social choice. Theory and Decision , 11(3), 319–337.

13.

Milnor

(1954). Games against nature. In Thrall

Coombs

Davis.

(Eds.), Decision processes (pp. 49–59). New York, NY: Wiley.

14.

Nitzan

Pattanaik

P. K.

(1984). Median-based extensions of an ordering over a set to the power set: An axiomatic characterization. Journal of Economic Theory , 34(2), 252–261.

15.

Rawls

(1971). A theory of justice . Cambridge, MA: Harvard University Press.

16.

Shackle

G. L. S.

(1954). Expectation in economics. In Carter

C. F.

Meredith

G. P.

Shackle.

G. L. S.

(Eds.), Uncertainty and business decisions (pp. 90–97). Liverpool: Liverpool University Press.