THE TWO-ENVELOPE PARADOX AND THE FOUNDATIONS OF RATIONAL DECISION THEORY

Terry Horgan

You are given a choice between two envelopes. You are told, reliably, that each envelope has some money in it—some whole number of dollars, say—and that one envelope contains twice as much money as the other. You don’t know which has the higher amount and which has the lower. You choose one, but are given the opportunity to switch to the other. Here is an argument that it is rationally preferable to switch: Let x be the quantity of money in your chosen envelope. Then the quantity in the other is either 1/2x or 2x, and these possibilities are equally likely. So the expected utility of switching is 1/2(1/2x) + 1/2(2x) = 1.25x, whereas that for sticking is only x. So it is rationally preferable to switch.

There is clearly something wrong with this argument. For one thing, it is obvious that neither choice is rationally preferable to the other: it’s a tossup. For another, if you switched on the basis of this reasoning, then the same argument could immediately be given for switching back; and so on, indefinitely. For another, there is a parallel argument for the rational preferability of sticking, in terms of the quantity y in the other envelope. But the problem is to provide an adequate account of how the argument goes wrong. This is the two-envelope paradox.

In an
earlier paper (Horgan 2000) I offered a diagnosis of
the paradox. I argued that the flaw in the argument is considerably more subtle
and interesting than is usually believed, and that an adequate diagnosis
reveals important morals about both probability and the foundations of decision
theory. One moral is that there is a kind of expected utility, not previously
noticed as far as I know, that I call *nonstandard* expected utility. I proposed
a general normative principle governing the proper application of nonstandard
expected utility in rational decisionmaking. But this
principle is inadequate in several respects, some of which I acknowledged in note
added in press and some of which I have meanwhile discovered. The present paper
undertakes the task of formulating a more adequate general normative principle
for nonstandard expected utility. After preliminary remarks in section 1, and a
summary in section 2 of the principal claims and ideas in Horgan
2000, I take up the business at hand in sections 3-6.

**1. Preliminaries.**

To begin
with, the paradoxical argument is an expected-utility argument. In decision
theory, the notion of expected utility is commonly articulated in something
like the following way (e.g., Jeffrey 1983). Let acts A_{1},…,A_{m} be open to the agent, and let the agent know
this. Let states S_{1},…,S_{n}
be mutually exclusive and jointly exhaustive possible states of the world, and
let the agent know this. For each act A_{i} and each state S_{j},_{ }let the agent know that if A_{i}
were performed and S_{j} obtained, then the
outcome would be O_{ij} and let the agent
assign to each outcome O_{ij} a desirability *D*O_{ij}. These conditions define a *matrix formulation* of a decision
problem. If the states are independent of the acts—probabilistically,
counterfactually, and causally—then the *expected
utility* of each act A_{i} is this:

U(A_{i}) = å_{j} pr(S_{j})×*D*O_{ij}

I.e., the expected utility of A_{i} is the weighted
sum of the desirablities of the respective possible
outcomes of A_{i}, as weighted by the probabilities of the respective
possible states S_{1},…,S_{n}.

Second, the conditions characterizing a matrix formulation of a decision problem are apparently satisfied in the two-envelope situation, in such a way that the paradoxical argument results by applying the definition of expected utility to the relevant matrix. The states are characterized in terms of x, the quantity (whatever it is) in the agent’s chosen envelope. Letting the chosen envelope be M (for ‘mine’) and the non-chosen one be O (for ‘other’), we have two possible states of nature, two available acts, and outcomes for each act under each state, expressible this way:

O contains 1/2x O contains 2x

Stick Get x Get x

Switch Get 1/2x Get 2x

**Matrix 1**

Each of the two states of nature evidently has probability 1/2. So, letting the desirability of the respective outcomes be identical to their numerical values, we can plug into our definition of expected utility:

U(Stick) = [pr(O contains 1/2x)×D(Get x)] + [pr(O contains 2x)×D(Get x)]

= 1/2×D(Get x) + 1/2×D(Get x)

= 1/2x + 1/2x

= x

U(Switch) = [pr(O contains 1/2x)×D(Get 1/2x)] + [pr(O contains 2x)×D(Get 2x)]

= 1/2×D(Get 1/2x) + 1/2×D(Get 2x)

= 1/2×1/2x + 1/2×2x

= 1/4x + x

= 5/4x

Third, the
operative notion of probability, in the paradoxical argument and in decision
theory generally, is *epistemic* in the
following important sense: it is tied to the agent’s total available
information. So I will henceforth call it ‘epistemic probability’. Although I
remain neutral about the philosophically important question of the nature of
epistemic probability, lessons that emerge from the two-envelope paradox yield
some important constraints on an adequate answer to that question. [1]

Fourth,
below it will be useful to illustrate various points by reference to the
following special case of the two-envelope decision situation, which I will
call the *urn case*. Here we stipulate
that the agent knows that the dollar-amounts of money in the two envelopes were
determined by randomly choosing a slip of paper from an urn full of such slips;
that on each slip of paper in the urn was written an ordered pair of successive
numbers from the set {1,2,4,8,16,32}; that there was an equal number of slips
in the urn containing each of these ordered pairs; and that the first number on
the randomly chosen slip went into the envelope the agent chose and the second
went into the other one. Under these conditions, the acts, states, and outcomes
are represented by the following matrix:

Stick Switch

M contains 1 and O contains 2 Get 1 Get 2

M contains 2 and O contains 1 Get 2 Get 1

M contains 2 and O contains 4 Get 2 Get 4

M contains 4 and O contains 2 Get 4 Get 2

M contains 4 and O contains 8 Get 4 Get 8

M contains 8 and O contains 4 Get 8 Get 4

M contains 8 and O contains 16 Get 8 Get 16

M contains 16 and O contains 8 Get 16 Get 8

M contains 16 and O contains 32 Get 16 Get 32

M contains 32 and O contains 16 Get 32 Get 16

**Matrix 2**

Since each of the 10 state-specifications in Matrix 2 has epistemic probability 1/10,

U(Stick) = 1/10(1+2+2+4+4+8+8+16+16+32) = 9.3

U(Switch) = 1/10(2+1+4+2+8+4+16+8+32+16) = 9.3

Fifth,
below I will occasionally refer to the following variant of the original
two-envelope decision situation. You are given an envelope M, and there is
another envelope O in front of you. You are reliably informed that M has a
whole-dollar amount of money in it that was chosen by a random process; that
thereafter a fair coin was flipped; and that if the coin came up heads then
twice the quantity in M was put into O, whereas if the coin came up tails then
half the quantity in M was put into O. I will call this the *coin-flipping situation*, in contrast to
the *original situation* that generates
the two envelope paradox. In this coin-flipping situation, you ought rationally
to switch—as has been correctly observed by those who have discussed it (e.g., Cargile 1992, 212-13, *et. al.* 1994, 44-45, and McGrew *et. al.* 1997, 29).

Finally, it
also will be useful to have before us the following special case of the
coin-flipping situation, which I will call the *coin-flipping urn case*. Here we stipulate that the agent knows that
the whole-dollar amount in his own envelope M was determined by randomly
choosing a slip of paper from an urn full of such slips; that on each slip in
the urn was written one of the numbers in the set {2,4,8,16,32};
and that there was an equal number of slips in the urn containing each of these
numbers. The agent also knows that after the quantity in M was thus determined,
the quantity in O was then determined a fair coin-flip, with twice the quantity
in M going into O if the coin turned up heads, and half the quantity in M going
into O if the coin turned up tails. Under these conditions, the expected
utilities are calculated on the basis of the following matrix:

Stick Switch

M contains 2 and O contains 1 Get 2 Get 1

M contains 2 and O contains 4 Get 2 Get 4

M contains 4 and O contains 2 Get 4 Get 2

M contains 4 and O contains 8 Get 4 Get 8

M contains 8 and O contains 4 Get 8 Get 4

M contains 8 and O contains 16 Get 8 Get 16

M contains 16 and O contains 8 Get 16 Get 8

M contains 16 and O contains 32 Get 16 Get 32

M contains 32 and O contains 16 Get 32 Get 16

M contains 32 and O contains 64 Get 32 Get 64

**Matrix 3**

Since the probability is 1/10 for each of the states in Matrix 3, the expected utilities are

(Stick) = 1/10(2+2+4+4+8+8+16+16+32+32) = 1/10(124) = 12.4

(Switch) = 1/10(1+4+2+8+4+16+8+32+16+64) = 1/10(155) = 15.5

**2. Diagnosis and Theoretical Implications.**

Discussions
of the two-envelope paradox (e.g., Nalebuff 1989, Cargile 1992, Castell and Batens 1994, Jackson *et.
al.* 1994, Broome 1995, Arntzenius and McCarthy
1997, Scott and Scott 1997, Chalmers unpublished) typically claim that there is
something wrong with the probability assignments in the paradoxical
argument—although there are differences of opinion about exactly how the
probabilities are supposed to be mistaken. I disagree. Consider the urn case,
for example. On my construal of the paradoxical reasoning, the symbol ‘x’ goes
proxy for a rigid definite description, which we can render as ‘the actual
quantity in M’ (where ‘actual’ is a construed as rigidifying operator). With
respect to the urn case, the following list of statements constitutes a
fine-grained specification—expressed in terms of the rigid singular term ‘the
actual quantity in M’—of
the epistemic possibilities concerning the contents of envelopes
M and O:

1. The actual quantity in M = 1 & O contains 2

2. The actual quantity in M = 2 & O contains 1

3. The actual quantity in M = 2 & O contains 4

4. The actual quantity in M = 4 & O contains 2

5. The actual quantity in M = 4 & O contains 8

6. The actual quantity in
M = 8 & O contains 4

7. The actual quantity in
M = 8 & O contains 16

8. The actual quantity in
M = 16 & O contains 8

9. The actual quantity in M = 16 & O contains 32

10. The actual quantity in M = 32 & O contains 16

Each
statement on this list has epistemic probability 1/10. Hence, since all the
statements are probabilistically independent of one another, the disjunction of
the five even-numbered statements on the list has probability 1/2, and the
disjunction of the five odd-numbered ones also has one half. But the epistemic
probability of the statement

O contains 1/2(the actual quantity
in M)

is just the epistemic probability of the disjunction of the
even-numbered statements on the list, since each even-numbered disjunct specifies one of the epistemically
possible ways that this statement could be true. Likewise, the epistemic
probability of the statement

O contains 2(the actual
quantity in M)

is just the epistemic probability of the disjunction of the *odd*-numbered statements on the list,
since each of the odd-numbered statements specifies one of the epistemically possible ways that *this* statement could be true. Therefore, in the urn case, the
statements

pr(O contains 1/2(the actual quantity in M)) = 1/2

pr(O
contains 2(the actual quantity in M)) = 1/2

are true. In both , the
constituent statement within the scope of ‘pr’ expresses a *coarse-grained epistemic possibility*, a possibility subsuming
exactly half of the ten equally probable fine-grained epistemic possibilities
corresponding to the statements on the above list. Each of these two
coarse-grained epistemic possibilities does indeed have probability 1/2, since
each possibility is just the disjunction of half of the ten equally probable
fine-grained epistemic possibilities. Moreover, these points about the urn case
generalize straightforwardly to the original two-envelope situation. So, since
the symbol ‘x’ in the paradoxical argument goes proxy for ‘the actual quantity
in M’, the probability assignments employed in the argument are correct.

How then *does* the paradoxical argument go wrong? To come to grips with this
question, we need to appreciate several crucial facts about epistemic
probability and about the concept of expected utility—facts that the argument
helps bring into focus.

First, epistemic probability is *intensional*, in
the sense that the sentential contexts created by the epistemic-probability
operator do not permit unrestricted substitution *salva** veritate* of co-referring singular terms.
Consider the urn case, for example, and suppose that (unbeknownst to the agent,
of course) the actual quantity in M is 16. Then the first of the following two
statements is true and the second is false, even though the second is obtained
from the first by substitution of a co-referring singular term:

pr(M contains the actual quantity in M) = 1

pr(M contains 16) = 1.

Likewise, the first of the following two statements is true and the second false, even though the second is obtained from the first by substitution of a co-referring singular term:

pr(O contains 1/2(the actual quantity in M)) = 1/2

pr(O contains 8) = 1/2.

It should not be terribly surprising, upon reflection, that epistemic probability is intensional in the way belief is, since epistemic probability is tied to available information in much the same way as is rational belief. (This certainly should not be surprising to those who think that epistemic probability is just rational degree of belief.)

Second, it
is important to distinguish between two ways of specifying states, outcomes,
and desirabilities in matrix formulations of decision
problems. On one hand are *canonical*
specifications: the items, as so specified, are epistemically
determinate for the agent, given the total available information—i.e., the
agent knows what item the specification refers to. On the other hand are *noncanonical*
specifications of states, outcomes, and desirabilities:
the items, as so specified, are epistemically
indeterminate for the agent. The paradoxical two-envelope argument employs noncanonical specifications of states and of outcomes/desirabilies; for, the specifications employ the symbol ‘x’
which goes proxy for the noncanonical referring
expression ‘the actual quantity in M’, and the quantity in referred to is epistemically indeterminate (as so specified) for the
agent. (The canonical/noncanonical distinction is discussed
at greater length in Horgan 2000.)

Third, it
needs to be recognized that because expected utility involves epistemic
probabilities, and because epistemic-probability contexts are intensional, the available acts in a given decision problem
can have several different kinds of expected utility. On one hand is *standard* expected utility, calculated by
applying the definition of expected utility to a matrix employing canonical
specifications of states, outcomes, and desirabilities.
On the other hand are various kinds of *nonstandard*
expected utility, calculated by applying the definition to matrices involving
various kinds of noncanonical specifications.

Take the urn version of the two-envelope problem, for instance, and suppose that (unbeknownst to the agent, of course) M contains 16 and O contains 32. The standard expected utilities, for sticking and for switching, are calculated on the basis of a matrix employing canonical state-specifications, like Matrix 2 (in section 1). As mentioned above, since each of the 10 state-specifications in Matrix 2 has epistemic probability 1/10,

U(Stick) = 1/10(1+2+2+4+4+8+8+16+16+32) = 9.3

U(Switch) = 1/10(2+1+4+2+8+4+16+8+32+16) = 9.3

On the other hand, one nonstandard kind of expected utility
for the acts of sticking and switching, which I will call *x-based* nonstandard utility and I will denote by ‘U^{x}’, is calculated by letting ‘x’ go proxy for
‘the actual quantity in M’ and then applying the definition of expected utility
to a matrix with noncanonical state-specifications
formulated in terms of x, viz., Matrix 1 (in section 1). Since each of the two
state-specifications in Matrix 1 has epistemic probability 1/2, and since
(unbeknownst to the agent) M contains 16,

U^{x}(Stick)
= x = 16

U^{x}(Switch)
= 1.25x = 20

Another nonstandard kind of expected utility for the acts of
sticking and switching, which I will call *y-based*
nonstandard utility and I will denote by ‘U^{y}’,
is calculated by letting ‘y’ go proxy ‘the actual quantity in O’ and then
applying the definition of expected utility to a matrix with noncanonical state-specifications formulated in terms of y,
viz.,

M contains 1/2y M contains 2y

Stick Get 1/2y Get 2y

Switch Get y Get y

**Matrix 4**

Since each of the two state-specifications in Matrix 4 has epistemic probability 1/2, and since (unbeknownst to the agent) O contains 32,

U^{y}(Stick)
= 1.25y = 40

U^{y}(Switch)
= y = 32

There is nothing contradictory about these various
incompatible expected-utility values for sticking and switching in this
decision problem, since they involve three different kind*s* of expected utility—the standard kind U, and the two nonstandard
kinds U^{x} and U^{y}.

Fourth,
since a distinction has emerged between standard expected utility and various
types of nonstandard expected utility, it now becomes crucial to give a new,
more specific, articulation of the basic normative principle in decision
theory—the principle of *expected-utility
maximization*, prescribing the selection of an action with maximum expected
utility. This principle needs to be understood as asserting that rationality
requires choosing an action with maximum *standard*
expected utility. Properly interpreted, therefore, the expected-utility
maximization principle says nothing whatever about the various kinds of
nonstandard expected utility that an agent’s available acts might also happen
to possess.

Having extracted these important
morals about epistemic probability and expected utility from consideration of
the paradoxical argument, we are now in a position to diagnose how the argument
goes wrong. Since the kind of expected utility to which the argument appeals is
U^{x}—i.e., x-based nonstandard expected
utility—the principal flaw in the argument is its implicit reliance on a
mistaken normative assumption, viz., that in the two-envelope decision problem,
rationality requires U^{x}-maximization.
Thus, given that U^{x} is the operative
notion of expected utility in the paradoxical argument, the reasoning is
actually correct up through the penultimate conclusion that the expected
utilities of sticking and switching, respectively, are x and 1.25x. But the
mistake is to infer from this that one ought to switch.

Equivocation
is surely at work too. Since the unvarnished expression ‘the expected utility’
is employed throughout, the paradoxical argument effectively trades on the
presumption that the kind of expected utility being described is *standard* expected utility. This
presumption makes it appear that the normative basis for the final conclusion
is just the usual principle that one ought rationally to perform an action with
maximal expected utility. But since that principle applies to standard expected
utility, whereas the argument is really employing a nonstandard kind, the
argument effectively equivocates on the expression ‘the expected utility’.

In light of this diagnosis of the paradoxical argument, and the distinction that has emerged between standard expected utility and various kinds of nonstandard expected utility, important new questions emerge for the foundations of rational decision theory: Is it sometimes normatively appropriate to require the maximization of certain kinds of nonstandard expected utility? If so, then under what circumstances?

Such questions are of interest for at least two reasons. First, the use of an appropriate kind of nonstandard expected utility sometimes provides a suitable shortcut-method for deciding on a rationally appropriate action in a given decision situation. A correctly applicable kind of nonstandard expected utility typically employs a much more coarse-grained set of states, thereby simplifying calculation.

Second (and
more important), maximizing a certain kind of nonstandard expected utility
sometimes is rationally appropriate in a given decision situation even though
the available acts *lack* standard
expected utilities—i.e., even though the total available information does not
determine a uniquely rationally eligible standard probability distribution over
a suitable set of exclusive and exhaustive states of the world. (By a *standard* distribution of epistemic
probabilities, I mean a probability distribution over states *as canonically specified*.) In such
decision situations it is rationally appropriate to maximize a certain kind of
nonstandard expected utility even though the agent’s total available
information makes it rationally *inappropriate*
to adopt any standard probability distribution, because numerous
candidate-distributions all are equally rationally eligible.[2]

The two-envelope situation itself is a case in point. The official description of the situation does not provide enough information to uniquely fix a standard probability distribution that generates standard expected utilities for sticking and switching. (In this respect, the original problem differs from our special case, the urn version.) Nevertheless, the following is a perfectly sound expected-utility argument for the conclusion that sticking and switching are rationally on a par. Let z be the lower of the two quantities in the envelopes, so that 2z is the higher of the two. Then the epistemic possibilities for states and outcomes are described by the following matrix:

M contains z and O contains 2z M contains 2z and O contains z

Stick Get z Get 2z

Switch Get 2z Get z

**Matrix
5**

The two state-specifications in Matrix 5 both have probability 1/2. Hence the expected utility of sticking is 1/2z + 1/2(2z) = 3/2z, whereas the expected utility of switching is 1/2(2z) + 1/2z = 3/2z. So, since these two acts have the same expected utility, they are rationally on a par.

The
soundness of this argument is commonly acknowledged in the literature. What is
not commonly acknowledged or noticed, however, is that the notion of expected
utility employed here is a nonstandard kind. I will call it *z-based* nonstandard expected utility,
and I will denote it by U^{z}. In order to
illustrate the fact that U^{z} differs from
standard expected utility U, return to the urn case, and suppose that
(unbeknownst to the agent, of course) the actual lower quantity z in the
envelopes is 16 (and hence the actual higher quantity in the envelopes, 2z, is
32). Then, as calculated on the basis of Matrix 2 (in section 1), U(Stick) = U(Switch) = 9.3. However,

U^{z}(Stick)
= 1/2z + 1/2(2z) = 1/2×16 + 1/2×32 = 24

U^{z}(Switch)
= 1/2(2z) + 1/2z = 1/2×32 +1/2×16 = 24

And with respect to the original two-envelope situation (as
opposed to the urn case), there are *no*
such quantities as U(Stick) and U(Switch), since there is not any single,
uniquely correct, distribution of standard probabilities over
canonically-specified epistemic possibilities.

The
coin-flipping version of the original decision problem also illustrates the
rational applicability of a suitable kind of nonstandard expected utility—in
this case, E^{x}. The form of reasoning
employed in the original two-envelope paradox not only yields the correct
conclusion, but in *this* situation
also appears to be a perfectly legitimate way to reason one’s way to that
conclusion. This fact is acknowledged in the literature; but once again, it is
not commonly acknowledged or noticed that the notion of expected utility
employed here is nonstandard. In order to illustrate the fact that U^{x} does indeed differ from standard expected
utility, consider the coin-flipping urn case, and suppose that (unbeknownst to
the agent, of course) the actual quantity in M is 16. Then although U(Stick) = 12.4 and U(Switch) = 15.5, as explained in
section 1 above (with reference to Matrix 3), the x-based nonstandard expected
utilities are:

U^{x}(Stick)
= x = 16

U^{x}(Switch)
= 1/2(1/2x) + 1/2(2x) = 1/2×8 + 1/2×32 = 20

And with respect to the original coin-flipping
two-envelope situation (as opposed to our urn version of it), there are *no* such quantities as U(Stick) and
U(Switch), since there is not any single, uniquely correct, distribution of
standard probabilities over canonically-specified epistemic possibilities.

In light of
these observations, in Horgan 2000 I proposed the
following general normative principle for the maximization of various kinds of
nonstandard expected utility in various decision situations. For a given
decision problem, let d be a singular referring expression that is epistemically indeterminate given the total available
information, and hence is noncanonical. Let U^{d}
be a form of nonstandard expected utility, applicable to the available acts in
the decision situation, that is calculated on the basis of a matrix employing noncanonical state-specifications, outcome-specifications,
and desirablity-specifications formulated in terms of d. Suppose that for the given decision situation, the
following *existence condition*
obtains:

(E.C.) There is at least one rationally eligible standard probability distribution over epistemically possible states of nature.

Under these circumstances,

(A) Rationality
requires choosing an act that maximizes U^{d} just in case there
is a unique ratio-scale ordering O of available acts such that (i) for
every rationally eligible standard probability distribution *D* to epistemically
possible states of nature for the given decision situation, U* _{D}* ranks the available acts
according to O, and (ii) U

Here, U* _{D}*
is the standard expected utility as calculated on the basis of

The
proposed normative principle (A) dictates U^{z}-maximization
in the original two-envelope situation, but not U^{x}-maximization
or U^{y}-maximization. It dictates U^{x}-maximization in the coin-flipping version of
the two-envelope situation, but not U^{y}-maximization
or U^{z}-maximization. It has applications
not only as an occasional short-cut method for rational decision-making that is
simpler than calculating standard expected utility, but also (and much more
importantly) as a method for rational decision-making in certain situations
where the available acts have no standard expected utilities at all.

**3. A Residual Theoretical Issue.**

I now think that the proposed principle (A) is inadequate, in four specific ways. I will explain the first problem in this section, and the second in section 4. In section 5 I will propose a new normative principle in place of (A), one that overcomes these two problems. Then in section 6 I will introduce the third and fourth problems, and I will address them by proposing yet another principle, a generalization of the one proposed in section 5.

The first problem is that condition (A) applies only to decision situations for which there is at least one rationally eligible standard probability distribution over epistemically possible states of nature (i.e., situations where (E.C.), the existence condition, holds). Yet there are decision situations for which (i) rationality requires choosing an act that maximizes a given kind of nonstandard expected utility, but (ii) there is no rationally eligible standard probability distribution over epistemically possible states of nature—i.e., no probability distribution over canonically specified states that satisfies all the conditions of the given decision situation. Hence, there is a need to generalize principle (A) in order to cover such decision situations.

We obtain a
case in point by elaborating the original two-envelope decision situation in
the following way. You are told, reliably, that the actual quantity in M has
this feature: if you were to learn what it is, then you would consider it
equally likely that O contains either twice that amount of half that amount;
and likewise, the actual quantity in O is such that if you were to learn what
it is, then you would consider it equally that M contains either twice that
amount or half that amount. I will call this the *expanded version* of the two-envelope situation.

The
expanded version remains a coherent decision problem. For this case too, no
less that the original version, rationality requires choosing an act that
maximizes U^{z}—which means that sticking and
switching are rationally on a par. And, as in the original version, neither act
has a standard expected utility. However, the reason why not is different than
before. In the original version, the lack of standard expected utility was due
to the fact that there were numerous rationally eligible standard probability
distributions to epistemically possible states of
nature—so that there is no rational reason to adopt any one of them, over
against the others. In the expanded version, however, there is *no* rationally eligible standard
probability distribution over the relevant states of nature. Why not? Because
the following argument looms:

1. If I were to learn that M contained the minimum amount 1, then I would not consider it equally likely that O contains either twice that amount of half that amount (because I would know that O contains 2). Hence, M does not contain 1. By parallel reasoning, O does not contain 1.

2. Since neither M nor O contains 1, if I were to learn that M contained 2, then I would

not consider it equally likely that O contains either twice that amount or half that amount

(because I would know that O does not contain 1). Hence, M does not contain 2. By

parallel reasoning, O does not contain 2.

.

.

.

n. Since neither M nor O contains 2n-1, if I were to learn that M contained 2n, then I would not

consider it equally likely that O contains either twice that amount or half that amount

(because I would know that O does not contain 2n-1). Hence, M does not contain 2n. By

parallel reasoning, O does not contain 2n.

Etc.

This argument has a familiar structure: it is a version of the so-called “surprise examination paradox.” Presumably it is flawed in some way—in whatever way constitutes the proper diagnosis of the surprise examination paradox. However, be that as it may, the fact that such a paradox arises from the conditions specified in the expanded two-envelope decision situation has this consequence: no standard probability distribution—i.e., no probability distribution over canonically specified potential states of envelopes M and O—is fully consistent with these specified conditions. For, the canonical state-specifications

M contains 1 and O contains 2

O contains 1 and M contains 2

each would have to be assigned probability zero, and hence the canonical state-specifications

M contains 2 and O contains 4

O contains 2 and M contains 4

each would have to be assigned probability zero, and so forth for all potential quantities in M and in O—whereas the sum of the probabilities constituting a probability distribution must be 1. So in the case of the expanded two-envelope situation, there are no rationally eligible standard probability distributions to epistemically possible states of nature.

Similar
remarks apply, mutatis mutandis, to an expanded version of the coin-flipping
situation that includes this additional condition: you are told, reliably, that
the actual quantity in O has this feature: if you were to learn what it is,
then you would consider it equally likely that M contains either twice that
amount or half that amount. (Presumably it is already true, even for the
earlier-described coin-flipping situation, that the actual quantity *in M* has the corresponding feature
vis-à-vis O.) In this informationally enriched
decision situation, as in the official coin-flipping situation, rationality
requires choosing the act that maximizes U^{x},
viz., switching. But once again there is no rationally eligible standard
probability distribution over canonically specified potential states of nature,
because the conditions of the decision situation collectively have a
surprise-examination structure.

I will not
propose a solution to the surprise-examination paradox, nor is it one required
for present purposes. The crucial points are these. First, the expanded
versions of the original two-envelope situation and the coin-flipping situation
are coherent decision problems, despite the fact that they have a
surprise-examination structure that precludes any rationally eligible standard
probability distribution over canonically-specified potential states of nature.
Second, in each of these situations, rationality requires choosing an act that
maximizes a certain kind of nonstandard expected utility—viz., U^{z} in the expanded version of the original
situation, and U^{x} in the expanded version
of the coin-flipping situation. Third, the general normative principle (A) in Horgan 2000, stating when rationality requires choosing an
act that maximizes a given kind of nonstandard expected utility in a given
decision situation, does not apply to the cases lately described, because principle
(A) applies only when (E.C.) is satisfied—i.e., only when there is at least one
rationally eligible standard probability distribution over epistemically
possible states of nature. Thus arises the following theoretical issue for the
foundations of rational decision theory: articulating a normative principle, to
govern the application of nonstandard expected utility, that is more general
than (A)—a principle that does subsume decision situations like those I have
described in this section.[4]

**4. A Second Residual Theoretical Issue.**

A second
problem arises from the fact that principle (A) is supposed to specify the
conditions under which rationality *requires*
the maximization of a given kind of nonstandard expected utility U^{d}.
In a footnote to (A) in Horgan 2000, I remarked:

Saying that rationality “requires
the maximization” of U^{d} means more than saying that rationality requires
choosing an available act that happens to have a maximal U^{d}-value.
It also means that having a maximal U^{d}-value is itself a *reason* why U^{d}-maximization is
rationally obligatory. The idea is that U^{d} accurately reflects
the comparative rational worth (given the agent’s available information) of the
available acts.

Suppose, however, that U^{d} turns out to
generate the right ratio-scale rankings of the actions, but for purely
accidental and coincidental reasons. Then U^{d} will not
“accurately reflect” those rankings in the sense intended; it will not be a
guaranteed, non-accidental, indicator of them. And having maximal U^{d}-value
will not be a “reason for rational obligatoriness,”
in the sense intended, to choose a U^{d}-maximizing act.
What is wanted, then, is something stronger that clause (ii) of principle (A).
U^{d}
should have some feature *guaranteeing*
that it generates the appropriate ranking of the available acts.

**5. Ratio-Scale
Comparative Rational Worth and a New Normative Principle.**

One
important pre-theoretic idea about rationality is that for some decision
problems, the agent’s total information (including desirabilities
of various potential outcomes of available acts) confers upon each of the
available acts some epistemically determinate, quantitively measurable, *absolute rational worth*. This idea gets explicated in decision
theory in terms of the familiar notion of expected utility—i.e., what I have
here called *standard* expected
utility. The available acts in a given decision problem have absolute rational
worth, for the agent, just in case they have standard expected utilities; and
the absolute worth of each act just *is*
its standard expected utility. Thus, having absolute rational worth requires
that there be a set of epistemically determinate
state-specifications such that (a) the agent has an epistemically
determinate probability distribution over these state-specifications and (b)
for each available act A_{i} and each state-specification, A_{i}
has an epistemically determinate outcome and epistemically determinate desirability under the state as
so specified.

Another
important pre-theoretic idea about rationality is that for some decision problems,
the agent’s total information determines, for the set of available acts, an epistemically determinate, ratio-scale, ranking of *comparative rational worth*. When the
acts each have an absolute rational worth (i.e., a standard expected utility),
this will automatically confer comparative rational worth as well: the standard
expected utilities fix a corresponding ratio-scale ranking of the acts. For
some decision problems, however, the agent’s total information determines a
specific ratio-scale ranking of comparative rational worth for the available
acts, *independently* of any specific
probability distribution over epistemically
determinate states of nature. Sometimes this happens even though there is also
a uniquely correct standard probability distribution, so that the acts have
standard expected utilities too (e.g., the urn case, and the coin-flipping urn
case). Sometimes it happens when there is *not*
a uniquely correct standard probability distribution, so that the acts do not
have standard expected utilities—either (a) because the total information is
consistent with more than one rationally eligible standard probability
distribution over the relevant canonically specified states (e.g., the original
two-envelope situation, and the coin-flipping situation), or (b) because the
total available information has a “surprise examination” structure that
actually *precludes* any rationally
eligible probability distribution over the relevant canonically specified
states (e.g., the extended versions of the original two-envelope situation and
the coin-flipping situation).

Although
nonstandard expected utilities are specific numerical quantities, they are epistemically indeterminate for the agent. Thus, they are
not a measure of absolute rational worth. Nevertheless, in decision situations
like those discussed above, nonstandard expected utility does generate an epistemically determinate ratio-scale ranking of the
available acts (even though the nonstandard expected utilities themselves are epistemically indeterminate). Moreover, for each of these
decision situations, the available acts stand in a unique ratio-scale ranking
of comparative rational worth, independently of any specific probability
distribution over canonically specified states of nature. As I will put it, the
acts stand in a unique ratio-scale ranking of *SPD-independent* comparative rational worth (i.e., comparative
rational worth that is independent of any specific *standard probability distribution*). So in such decision situations,
the normatively appropriate kind of nonstandard expected utility is a kind that
is guaranteed to rank the available acts in accordance with their
SPD-independent ratio-scale comparative rational worth. In the original
two-envelope situation and its urn variant and its extended variant, U^{z} does this (but U^{x}
and U^{y} do not), whereas in the
coin-flipping situation and its urn variant and its extended variant, U^{x} does this (but U^{y}
and U^{z} do not).

In
effect, clause (i) of principle (A) is an attempt to
characterize the relevant kind of SPD-independent ratio-scale comparative
rational worth, and clause (ii) is an attempt to specify how a given type of
nonstandard expected utility U^{d} must be linked to this feature in order for U^{d}-maximization
to be rationally required. But clause (i) is unsatisfactory,
because it fails to apply to relevant situations with a “surprise examination”
structure. And clause (ii) is unsatisfactory too, because it does not preclude
the possibility that U^{d} happens to rank the acts in accordance with their
SPD-independent ratio-scale comparative rational worth for purely fortuitous
and accidental reasons. What we need, then, is a normative principle that (1)
is applicable to decision situations for which there are no rationally eligible
standard probability distribution over the epistemically
possible states of nature (e.g., the extended two-envelope situation, and the
extended coin-flipping situation), and (2) articulates the conditions under
which a specific kind of nonstandard expected utility *non-accidentally* ranks the available acts in a given decision
problem by SPD-independent ratio-scale comparative rational worth.

Consider
the original two-envelope situation, the extended version of the original
situation, and the urn case. Why is it mistaken to use U^{x}
in these decision situations? The fundamental problem is the following. On one
hand, the state-specifications employed in calculating U^{x},
viz.,

O contains 1/2x

O contains 2x

hold constant
the epistemically indeterminate quantity x in
envelope M, while allowing the content of O to vary between the two epistemically indeterminate quantities 1/2x and 2x. But on
the other hand, this asymmetry, with respect to the fixity or variability of epistemically indeterminate features of the actual
situation, does not reflect any corresponding asymmetry in the agent’s total
available information. Yet the effect of the asymmetry is that U^{x}(Switch)
= 5/4U^{x}(Stick). Thus, since switching and sticking are rationally on
a par, U^{x} fails to order these acts by
their ratio-scale comparative rational worth.

By
contrast, why is it correct to use U^{z} in
the original two-envelope situation, in the extended version of it, and in the
urn case? Because on one hand, the two state-specifications employed in
calculating U^{z}, viz.,

M contains z and O contains 2z

M contains 2z and O contains z

are symmetric
with respect to matters of fixity variability concerning the two epistemically indeterminate quantities z and 2z. The
quantities themselves (viz., the lower and the higher of the two actual
quantities in the two envelopes) are both held fixed; and the locations of
these two quantities vary in a symmetrical way, across the two epistemically indeterminate states. On the other hand, this
symmetry with respect to fixity and variability reflects the symmetry of the
agent’s available information concerning the contents of envelopes M and O. The
result is that U^{z}(Switch) = U^{z}(Stick),
so that U^{z} accurately ranks the acts in
accordance with their ratio-scale comparative rational worth.

Consider
now the coin-flipping version of the two-envelope situation, the extended
coin-flipping version, and the coin-flipping urn case. Why is it correct to use
U^{x} in these cases? Because on one hand,
the two state-specifications employed in calculating U^{x},
viz.,

O contains 1/2x

O contains 2x

hold constant
the epistemically indeterminate quantity x in
envelope M, while allowing the content of O to vary between the two epistemically indeterminate quantities 1/2x and 2x. On the
other hand, this asymmetry, with respect to the fixity and variability of epistemically indeterminate features of the actual
situation, directly reflects a corresponding asymmetry in the agent’s total
available information: the agent knows that the quantity x in envelope M was
selected first, and then either 1/2x or 2x was placed in envelope O, depending
on the outcome of a fair coin-toss. That informational asymmetry renders
switching 5/4 as rationally valuable as sticking. So, since the asymmetry is
reflected in the fact that the state-specifications hold fixed the quantity x
in envelope M while allowing the quantity in envelope O to vary between 1/x and
2x, U^{x} accurately ranks switching and
sticking by their ratio-scale comparative rational worth: U^{x}(Switch)
= 5/4U^{x}(Stick).

By
contrast, why is it incorrect to use U^{z} in the coin-flipping
version of the two-envelope situation, the extended coin-flipping version, and
the coin-flipping urn case? Because on one hand, the two state-specifications
employed in calculating U^{z}, viz.,

M contains z and O contains 2z

M contains 2z and O contains z

are symmetric
with respect to matters of fixity and variability concerning the two epistemically indeterminate quantities z and 2z. On the
other hand, these state-specifications thereby fail to reflect the crucial
asymmetry in the agent’s information about the contents of envelopes M and O,
with the result that U^{z} fails to
accurately rank switching and sticking by their ratio-scale comparative ratio
worth of

These
observations point the way toward the general normative principle we are
seeking, concerning the rational appropriateness or inappropriateness of using
a specific kind of nonstandard expected utility in a given decision situation.
For a given decision problem, let d be a singular referring expression that denotes some
numerical quantity and is epistemically indeterminate
given the total available information, and hence is noncanonical.
Let U^{d}
be a form of nonstandard expected utility, applicable to the available acts in
the decision situation, that is calculated on the basis of a matrix employing noncanonical state-specifications, outcome-specifications,
and desirability-specifications formulated in terms of d. We
will say that the set of state-specifications employed to calculate U^{d}
is *symmetry and asymmetry reflecting,
with respect to fixity and variability of features of the decision situation*
(for short, SAR_{f}_{/v}) just in
case any symmetries or asymmetries in these state-specifications reflect
corresponding symmetries and asymmetries in the agent’s total available
information. Then

(B) Rationality requires choosing an act that
maximizes U^{d}
if (i) U^{d} employs
state-specifications that are SAR_{f}_{/v},
and (ii) U^{d}
generates an epistemically determinate ratio-scale
ranking of the available acts.[5]

When the conditions in (B) are met,
the available acts do indeed possess SPD-independent ratio-scale comparative
rational worth, and U^{d} is guaranteed to rank the acts in a way that
accurately reflects their comparative worth. For, the very symmetries and
asymmetries in the agent’s total information that fix determinate ratio-scale
comparative worth for the acts, independently of any specific probabilities for
canonical state-specifications, are directly reflected in the
fixity/variability structure of the noncanonical
state-specifications employed by U^{d}.

**6. Ordinal-Scale Rational Worth and a More
General Normative Principle.**

Although
the two problems with principle (A) described in sections 3 and 4 have now been
dealt with, two further problems need to be addressed; both also arise for
principle (B) and hence will prompt modifications of (B) in turn. The third
problem is that there are decision problems for which (i)
the available acts stand in an *ordinal-scale*,
but not a *ratio-scale*, ordering of SPD-independent
comparative rational worth, and (ii) there is a suitable kind of nonstandard
expected utility that rationally ought to be maximized (because it is
guaranteed to reflect the ordinal-scale comparative rational worth of the
acts).

Here
is a simple example. You are given a choice between two envelopes E1 and E2,
after being reliably informed that first some whole-dollar quantity of money of
$2 or more was chosen by some random process and placed in E1, and then the
square of that quantity was placed into E2. Assuming that the desirability of
an outcome is just the dollar-amount obtained, in this decision situation there
is a kind of nonstandard expected utility definable for this situation that
ought rationally to be maximized, viz. U^{w},
where w = the actual quantity in E1. Since

U^{w}(Choose E1) = w

U^{w}(Choose E2) = w^{2}

and since w^{2}
> w for all potential values of w, rationality requires the U^{w}-maximizing act, viz., choosing E2. However,
since the epistemically possible quantities in E2 are
a non-linear function of the corresponding epistemically
possible quantities in E1, the two acts do not stand not in an SPD-independent *ratio-scale* ranking of comparative
rational worth, but only in an SPD-independent *ordinal-scale* SPD-independent ranking of comparative worth.
(Accordingly, U^{w}
generates only an epistemically determinate
ordinal-scale ranking of the acts.)

The
fourth problem is that rationality sometimes requires maximizing a more general
version of nonstandard expected utility than has so far been discussed, a
version involving several noncanonical
number-denoting terms rather than just one. Consider the following decision
situation, for example. You are given a choice of two envelopes E1 and E2.
Envelope E1 has two slots S1_{E1} and S2_{E1}, and envelope E2
has two slots S1_{E2} and S2_{E2}. Each slot in E1 contains
some dollar-quantity of money, selected by some random process. (The two
selections were independent of one another.) Slot S1_{E2} of E2
contains either half or twice the quantity in slot S1_{E1} of E1, depending
on the outcome of a fair coin-flip. Slot S2_{E2} of E2 contains either
one fourth of, or four times, the quantity in slot S1_{E2}, depending
on the outcome of an independent fair coin-flip.

Letting
x be the actual quantity in S1_{E1} and y be the actual quantity in S2_{E1},
there is a nonstandard expected utility U^{x,y}
definable for this decision problem that yields epistemically
indeterminate expected utilities expressed as mathematical functions of x and
y. Assuming that the desirabilities of the potential outcomes
are just their dollar amounts,

U^{x,y}(Choose E1) = 1/4[(x+y)
+ (x+y) + (x+y) + (x+y)] = x+y

U^{x,y}(Choose E2) =
1/4[(1/2x+1/4y) + (2x+1/4y) + (1/2x+4y) + (2x+4y)] = 5/4x + 17/8y.

U^{x,y}
is guaranteed to reflect the acts’ SPD-independent comparative ordinal-scale
rational worth, because (5/4x + 17/8y) > (x + y) for any permissible values
of x and y. Thus, rationality dictates the maximization of U^{x,y} in this decision situation. (Notice that the
third problem too is illustrated by this case. The stated conditions fix an
SPD-independent *ordinal-scale*
comparative rational worth for the two acts, without fixing any unique
ratio-scale ordering: choosing E2 is rationally preferable to choosing E1, but
not by any specific, probability-independent, ratio.)

So
for some decision problems, a certain kind of nonstandard expected utility
reflects SPD-independent ordinal-scale comparative rational worth of the
available acts, even when they lack PDP-independent *ratio-scale* comparative rational worth. Moreover, for some decision
problems, SPD-independent comparative rational worth is reflected by a kind of
nonstandard expected utility based on several noncanonical
number-denoting terms rather than one. Thus a normative principle more general
than (B) is needed, to govern the rationally appropriate use of nonstandard
expected utility in such cases.

The needed principle can be articulated by generalizing (B) in the
following way. For a given decision problem, let d1,...,dm be
singular referring expressions that denote numerical quantities and are epistemically indeterminate given the total available
information, and hence are noncanonical. Let U^{d1,...,}^{dm}
be a form of nonstandard expected utility, applicable to the available acts in
the decision situation, that is calculated on the basis of a matrix employing noncanonical state-specifications, outcome-specifications,
and desirability-specifications formulated in terms of d1,...,dm.
Then

(C) Rationality requires choosing an act that
maximizes U^{d1,...,}^{dm}_{
}just in case (i) U^{d1,...,}^{dm}
employs state-specifications that are SAR_{f}_{/v},
and (ii) U^{d1,...,}^{dm}
generates an epistemically determinate ordinal-scale
ranking of the available acts.[6]

When these conditions are met, the
available acts do indeed possess SPD-independent ordinal-scale comparative
rational worth, and U^{d1,...,}^{dm}
is guaranteed to rank the available acts in a way that accurately reflects
their comparative rational worth. For, the very symmetries and asymmetries in
the agent’s total information that fix determinate ordinal-scale comparative
worth for the acts, independently of any specific probabilities for canonical
state-specifications, are directly reflected in the fixity/variability
structure of the noncanonical state-specifications
employed by U^{d1,...,}^{dm}.
So we have arrived at a general normative principle governing the maximization
of nonstandard expected utility, a principle that overcomes all four problems
faced by principle (A).

Principle (B), which states only a sufficient condition for the
rationality of maximizing a given kind of nonstandard expected utility (rather
than a sufficient *and necessary*
condition), remains in force. In effect, it is a special case of our
more general normative principle (C).

Let
me make several final observations about principles (C) and (B) and the key
notion they employ, viz., the feature SAR_{f}_{/v}.
First, I take it that the failure to be SAR_{f}_{/v}
is a feature that can be exhibited only by state-specifications of the kind
that figure in *nonstandard* expected
utility, viz., *epistemically** indeterminate* state-specifications.
Only when relevant features of the actual situation are specified in epistemically indeterminate ways does it become possible to
fix or vary them in ways not reflective of one’s total information, within a
set of state-specifications that are mutually exclusive and jointly exhaustive.

Second,
the feature of being SAR_{f}_{/v} is
evidently clear enough to be useful and applicable in concrete decision
situations like those I have described in this paper. Often in such situations,
one can tell by inspection whether or not the state-specifications employed by
a given kind of nonstandard expected utility are SAR_{f}_{/v}.
Indeed, it is evidently very common in practice—in betting decisions, for
example—to rely on calculations of nonstandard expected utilities that are SAR_{f}_{/v}.

But
third, being SAR_{f}_{/v} also has
been characterized somewhat vaguely, in terms of several vague ideas: (1)
symmetries and asymmetries in one’s total information, (2) symmetries and
asymmetries in a set of noncanonical state-specifications,
and (3) a relation of “reflection” between the latter and the former kinds of
symmetries and asymmetries. It would be theoretically desirable to explicate
these notions further, and to employ the explicated versions to articulate a
sharpened normative principle that would replace and explicate the vague
normative principles (C) and (B).

Fourth, the notion of SPD-independent comparative rational worth is also somewhat vague, as so far characterized. It would be theoretically desirable to provide a direct explication of it too, and to explicitly articulate its connection to explicated versions of principles (C) and (B). These tasks of further explication and articulation I leave for a future occasion.[7]

**REFERENCES**

Arntzenius, F.
and McCarthy, D. 1997 “The Two envelope Paradox and Infinite Expectations,” *Analysis*, 57, 42-50.

Broome, J. 1995 “The two-envelope paradox,” *Analysis*, 55, 6-11.

Cargile, J. 1992 “On a Problem about Probability and Decision,” *Analysis*, 54, 211-16.

Castell, P. and Batens, D. 1994
“The Two-Envelope Paradox: The Infinite Case,” *Analysis*, 54, 46-49.

Chalmers, D. Unpublished “The Two-Envelope Paradox: A Complete Analysis?”

Horgan, T. 2000
“The Two-Envelope Paradox, Nonstandard Expected Utility, and the Intensionality of Probability,” *Nous*, in press.

Jeffrey, R. 1983 *The** Logic of Decision*, Second Edition,

Jackson, F., Menzies, P., and Oppy, G. 1994
“The Two Envelope ‘Paradox’,” *Analysis*,
54, 43-45.

McGrew, T.,
Shier, D. and Silverstein, H. 1997 “The Two-Envelope Paradox Resolved,” *Analysis*, 57, 28-33.

Nalebuff, B. 1989 “The Other Person’s Envelope is Always Greener,” *Journal
of Economic Perspectives*, 3, 171-81.

Scott, A. and
Scott, M. 1997 “What’s in the Two Envelope Paradox?” *Analysis*, 57, 34-41.

[1]
Epistemic probability, as understood here, must conform to the axioms of
probability theory. Although the term ‘epistemic probability’ has sometimes
been used for subjective degrees of belief that can collectively fail to
conform to these axioms, I think it is important to reclaim the term from those
who have employed it that way. I would maintain that there are *objective* facts about the kind of
probability that is tied to the agent’s available information—i.e., about what
I am calling *epistemic* probability.
One objective fact is that epistemic probability obeys the axioms of
probability theory.

[2] According to some construals of epistemic probability, rationality permits the initial adoption of virtually any standard probability distribution that obeys the axioms of probability and also is consistent with the agent’s total available information—provided that that one then updates one’s prior standard probabilities, on the basis of new evidence, in accordance with Bayes’ theorem. In my view this tolerant attitude toward prior standard probabilities is mistaken, precisely because rationality prohibits the adoption of any single specific standard probability distribution when numerous candidate-distributions are all equally eligible. But even those who take the tolerant approach to prior standard probabilities can agree about the theoretical importance of nonstandard expected utilities, vis-à-vis decision situations in which the comparative rational worth of the available acts is independent of any specific standard probability distribution over epistemically determinate states of nature.

[3]
This formulation improves upon the version in Horgan
2000 by explicitly building into clause (ii) a feature that the earlier version
effectively took for granted, but should have articulated: viz., that the
ratio-scale ranking of available acts generated by U^{d}
is *epistemically** determinate* for the agent (even though
the U^{d}-quantities
themselves are epistemically indeterminate).

[4]
Note that it would not suffice merely to drop (E.C.) from the specification of
the circumstances under which principle (A) applies, and leave (A) otherwise
intact. For, clause (i) of principle (A) would then
be *vacously*
satisfied in the expanded two-envelope situation by each of U^{x},
U^{y}, and U^{z}.
Principle (A) would thus require the maximization of *all three* of these kinds of nonstandard expected utility, in the
expanded two-envelope situation—a requirement that is not only normatively
inappropriate, but is impossible to fulfill.

[5]
Condition (B) is stated merely as a sufficient condition for the rationality of
U^{d}-maximization,
rather than a sufficient *and necessary*
condition, because it is still not general enough to cover all cases. See
section 6.

[6]
Clause (ii) is non-redundant, because there are decision situations in which
clause (i) is satisfied but clause (ii) is not. Here
is an example. You are given a choice between two envelopes E1 and E2, each of
which contains some whole-dollar quantity of money. You are told that some
quantity n, evenly divisible by 3, was first selected by a random process and
placed into E1, and that the quantity (n/3)^{2} was then placed into
E2. Letting w = the actual quantity in E1, U^{w}(Choose E1) =
w, whereas U^{w}(Choose E2) = (w/3)^{2}.
In this situation U^{w}
is a form of nonstandard expected utility that satisfies clause (i) of principle (C). However, U^{w} does not generate an epistemically determinate ordinal-scale ranking of the
available acts, and hence does not satisfy clause (ii) of (C). For, U^{w}(Choose E1) > U^{w}(Choose
E2) if w < 9, whereas U^{w}(Choose E1) = U^{w}(Choose E2) if w = 9, whereas U^{w}(Choose E1) < U^{w}(Choose
E2) if w > 9.

[7] I dedicate this paper to my wife Dianne, who has patiently endured my envelope obsession. She plans to put my ashes into two envelopes, and then put one envelope on the mantel and sprinkle the other’s contents into the wind at the U.S. Continental Divide.