2.1 Three-Step Acyclicity

Consider having a cup of coffee with one, two, or three lumps of sugar. Suppose that you can’t taste the difference between a cup with one lump and a cup with two lumps. Nor can you taste the difference between a cup with two lumps and a cup with three lumps. And, when you can’t taste any difference, you prefer having less sugar (to keep your intake down). Still, you can taste the difference between a cup with one lump and a cup with three lumps – and, due to your sweet tooth, you prefer the latter.Footnote ¹⁰

Let A, B, and C be the sure prospects of having a cup with one, two, and three lumps of sugar respectively. You prefer A to B, B to C, and C to A. Let ‘ $X\succ Y$’ denote that X is (strictly) preferred to Y.Footnote ¹¹ Then we can state your preferences as follows:Footnote ¹²

1. (1) $A\succ B\succ C\succ A$.

Your preferences are cyclic. More specifically, your preferences violate the following requirement:Footnote ¹³

All violations of Three-Step Acyclicity have the same form as the preferences in (1). So, to show that Three-Step Acyclicity is a requirement of rationality, all we need to show is that preferences of the kind in (1) are irrational.

The standard version of the money-pump argument runs as follows.Footnote ¹⁴ Suppose that you start off with A. An exploiter offers you a trade from A to C. Since you prefer C to A, you accept this offer. Then, after this first trade, you are offered a second trade from C to B. Since you prefer B to C, you also accept this second trade. Finally, after the second trade, you are offered a third trade from B to  $A^{-}$, where $A^{-}$ is just like A except that you have less money and

1. (2) $A\succ A^{-}\succ B$.

We can let the payment be monetary to fit the exploitation framing, but the main point is that $A^{-}$ is less preferred than A by being certainly inferior with respect to some dimension you care about and the same in other respects.Footnote ¹⁵ Let a souring of a prospect X be a prospect that is just like X except that it is certainly inferior in a dimension the agent cares about. (And let a sweetening of a prospect X be a prospect that is just like X except that it is certainly superior in a dimension the agent cares about.)

That there is an $A^{-}$ like this follows from (1) by the following requirement of rationality:Footnote ¹⁶

The idea is that, if you (strictly) prefer X to Y, you must prefer X with some margin. So there should be some, perhaps minimal, amount you’re willing to pay to get X rather than Y.Footnote ¹⁷ This is what blocks trivial responses to the money-pump argument where the agent avoids exploitation by preferring not to pay for anything.Footnote ¹⁸

Now, since you prefer $A^{-}$ to B, you also accept the third trade. So you end up with $A^{-}$ (that is, you pay for A) when you could have kept A for free.

In this example, you followed an approach known as myopic choice – that is, you assessed each choice in isolation, as if it were the only choice you would ever make.Footnote ¹⁹ We distinguish myopic choice from naive choice, which is to (i) consider the prospects of all available plans and assess which of these prospects are choice-worthy in a choice between all of them and (ii) choose in accordance with a plan to end up with one of these choice-worthy prospects – without taking into consideration whether you would later depart from that plan.Footnote ²⁰

Do you avoid exploitation if you follow naive choice rather than myopic choice? To follow naive choice in this case, you need to first consider the prospects of the available plans (that is, A, $A^{-}$, B, and C) and assess which of these prospects are choice-worthy. But how do you choose among three or more prospects if you have cyclic preferences over those prospects? Consider the Maximization Rule:Footnote ²¹

Given your cyclic preferences, you can’t maximize in a choice between A, $A^{-}$, B, and C, since each prospect is less preferred than another prospect. We avoid this problem with the following alternative rule:Footnote ²²

So far, we haven’t made any assumptions about your preference between $A^{-}$ and C. Yet, since you prefer C to A, you may, plausibly, also prefer C to a souring of A.Footnote ²³ So we could plausibly suppose

1. (3) $C\succ A^{-}$.

Then, given the Uncovered-Choice Rule, the only prospects you would be rationally permitted to choose from A, $A^{-}$, B, and C are A, B, and C. Note that $A^{-}$ is ruled out because A is preferred to $A^{-}$ and to every option that $A^{-}$ is preferred to.

Even so, does the naive approach avoid exploitation in this case? It does not. Naive choice combined with the Uncovered-Choice Rule still allows you to accept the first two trades, since you regard B as choice-worthy. So you are rationally permitted to choose in accordance with the plan to accept the first two trades to get B. Then, when you face the third offer of trading from B to  $A^{-}$, the Uncovered-Choice Rule allows you to choose  $A^{-}$ (because A is no longer the prospect of some available plan). So adopting naive choice does not save you from exploitation.

The standard version of the money-pump argument isn’t very convincing, however. In order for a money-pump argument to be compelling, the agent must know in advance what decision problem they face – that is, they must know the whole exploitation set-up in advance.Footnote ²⁴ The exploiter must not rely on any knowledge about what will happen that is unavailable to the agent, because being exploited by someone who knows more than you need not be a sign of irrationality.Footnote ²⁵ But, if you know the whole exploitation set-up in advance, you can use foresight to see that some of the trades aren’t in your interest.Footnote ²⁶ To see how, consider the decision tree of the standard money-pump set-up in Figure 1, which we can call the Standard Money Pump.

Figure 1

The Standard Money Pump

Here, the three squares represent choice nodes corresponding to the three consecutive trade offers. You accept a trade by going up at the corresponding choice node, and you turn the trade down by going down.Footnote ²⁷ The item on the upper left of each square is what you get if you accept the trade, and the item on the lower left is what you keep if you turn the trade down (that is, what you give up if you accept the trade). The preferences stated below the decision tree are the agent’s preferences, which are held constant throughout.Footnote ²⁸ So, in this case and in the other cases we’ll consider, agents do not revise their preferences during the decision problem.Footnote ²⁹

If you have foresight, you can use backward induction. To use backward induction is to predict what you would choose at later choice nodes and to take those predictions into account when you choose at earlier nodes.Footnote ³⁰ First, consider the trade at node 3. At this node, you have a choice between $A^{-}$ and B. And, since you prefer $A^{-}$ to B, you would accept the trade to $A^{-}$ at node 3. (The choices that are prescribed by backward induction are marked by the thicker lines in the decision tree.) This assumes that the only thing that should guide your choice at a node is your preference between the still feasible options; we accept the following principle:Footnote ³¹

In what follows, we’ll take Decision-Tree Separability for granted (until Section 7, where we take on challenges to this kind of separability).

Using backward induction at node 2, we take into account the prediction that $A^{-}$ would be chosen at node 3. Given this prediction, accepting the trade at node 2 effectively results in your final holding being $A^{-}$ whereas turning it down results in your final holding being C. Since you prefer C to  $A^{-}$, you would turn down the trade at node 2.

And, taking this prediction into account at node 1, we find that accepting the trade at node 1 effectively results in your final holding being C whereas turning it down results in your final holding being A. Since you prefer C to A, you accept the trade at node 1. Hence, using backward induction, you will end up with C after accepting the first trade and then turning down the second. So you avoid paying for something you could have kept for free. And so the standard money-pump argument is blocked.Footnote ³²

Nevertheless, we can revise the exploitation set-up so that it works against people who use backward induction.

One way to do so is to repeat the trade offers in case they are rejected but with no more than three trade offers in total, as in the decision problem in Figure 2, the Money Pump with Repeated Offers.Footnote ³³

Figure 2

The Money Pump with Repeated Offers

At any of the final nodes (that is, nodes 3, 4, 6, and 7), you would accept the trades you are offered. In other words, you would go up at each of these nodes.

Using backward induction at nodes 2 and 5, you take into account the prediction that you would accept each of the final trades. So the choice at node 2 is effectively between $A^{-}$ (accepting the trade) and B (turning it down). Since you prefer $A^{-}$ to B, you would accept the trade at node 2 (that is, you would go up to node 3). Likewise, the choice at node 5 is effectively between B (accepting the trade) and C (turning it down). Since you prefer B to C, you would accept the trade at node 5 (that is, you would go to node 6).

Using backward induction at node 1, you take all of these predictions into account. So the choice at node 1 is effectively between $A^{-}$ (accepting the trade) and B (turning it down). Since you prefer $A^{-}$ to B, you accept this first trade. So you go up to node 2 and then up to node 3, where you finally choose  $A^{-}$. And then you end up with $A^{-}$ even though you could have kept A for free.

Still, this argument relies upon a contentious form of backward induction. Specifically, one may challenge our assumption that you would choose rationally and retain your trust in your future rationality even at choice nodes that could only be reached by irrational choices.Footnote ³⁴ To see the problem, suppose that you predict not only rational choices but also some irrational choices in the Money Pump with Repeated Offers. We mark these predicted choices at nodes that would follow irrational choices by thick dashed lines in Figure 3.

Figure 3

The Money Pump with Repeated Offers (blocked by predicted irrationality)

Taking these predictions into account at node 1, the choice at that node is effectively between C (accepting the trade) and B (turning it down). Since you prefer B to C, it’s rationally required that you turn down the trade at node 1. And the move to node 2 is irrational. So the predicted irrational choices would only be made at nodes that follow irrational choices. Hence, in order to rule out these predictions, we must assume that you would choose rationally and retain your trust in your future rationality even at nodes that follow irrational choices. But it’s implausible that you would be rationally required to retain your trust in your future rationality at nodes where you’ve already made irrational choices.Footnote ³⁵

This worry about backward induction does not apply to decision problems that are BI-terminating. A decision problem is BI-terminating if and only if the choices that are prescribed by backward induction are terminal, that is, the prescribed choices are not followed by any potential further choice nodes.Footnote ³⁶ To defend the prescriptions of backward induction in BI-terminating decision problems, we only need to assume that, at nodes that can be reached without making any irrational choices, you retain (i) your rationality and (ii) your trust in your rationality at nodes that can be reached without making any irrational choices. While the money pumps we’ve considered so far are not BI-terminating, the Upfront Money Pump in Figure 4 is BI-terminating.Footnote ³⁷

Figure 4

The Upfront Money Pump

Since you prefer C to A, you would accept the trade at node 3. Using backward induction, you would take this prediction into account at node 2. Given the predicted choice at node 3, the choice at node 2 is effectively between B (accepting the trade) and C (turning it down). Since you prefer B to C, you would accept the trade at node 2. And, taking this prediction into account at node 1, the initial choice is effectively between $A^{-}$ (accepting the trade) and B (turning it down). Since you prefer $A^{-}$ to B, you accept the trade at node 1 and end up with $A^{-}$ even though you could have kept A for free.

This backward-induction argument assumes that you would also make rational choices and retain your trust in your future rationality at nodes that follow irrational choices. But, since the Upfront Money Pump is BI-terminating, the choices that are prescribed by backward induction can be defended by a more compelling argument without this assumption. We only need the weaker assumption that, at nodes that can be reached without making any irrational choices, you retain (i) your rationality and (ii) your trust in your rationality at nodes that can be reached without making any irrational choices. The argument takes the form of a proof by contradiction.Footnote ³⁸

Assume that all three choice nodes can be reached without making any irrational choices. So, at these nodes, you retain your rationality and your trust in your rationality at these nodes. Accordingly, you would accept the trade at node 3, since you prefer C to A. Taking this into account at node 2, the choice at node 2 is effectively between B and C. And, since you prefer B to C, it’s irrational to turn down the trade at node 2. This contradicts our assumption that all three choice nodes can be reached without making any irrational choices.

Next, assume that nodes 1 and 2 can be reached without making any irrational choices. So, at these nodes, you retain your rationality and your trust in your rationality at these nodes. Since we have already shown that it’s irrational to go down at (at least) one of nodes 1 and 2, it must be irrational to go down at node 2. So it must be rationally required to accept the trade at node 2.Footnote ³⁹ Accordingly, you would accept the trade at node 2. Taking this into account at node 1, the choice at node 1 is effectively between $A^{-}$ and B. And, since you prefer $A^{-}$ to B, it’s irrational to turn down the trade at node 1. This contradicts our assumption that nodes 1 and 2 can be reached without making any irrational choices.

Hence it’s irrational to turn down the trade at node 1. So it’s rationally required to accept the first trade from A to  $A^{-}$. And so you end up with $A^{-}$ when you could have kept A for free.

For this argument, we only assumed that, at nodes that can be reached without making any irrational choices, you retain (i) your rationality and (ii) your trust in your rationality at nodes that can be reached without making any irrational choices.

If your preferences are robust under a uniform monetary sweetening (for instance, a penny), we can extend the Upfront Money Pump so that you pay an arbitrarily high amount. Consider the decision problem in Figure 5, the Ruinous Upfront Money Pump.Footnote ⁴⁰

Figure 5

The Ruinous Upfront Money Pump

With the same backward-induction argument as before, we find that it’s rationally required to accept the first trade. So you end up paying over $1,000,000 for A when you could have kept A for free. And, as the Ruinous Upfront Money Pump is still a BI-terminating decision problem, we can defend the prescriptions of backward induction in this case with the same minimal assumptions we relied on for the Upfront Money Pump.

It may be objected that choosing $A^{-}$ at node 1 in the Upfront Money Pump is not a sign of irrationality, since the sequence of choices leading to A isn’t available in the relevant sense at that node. The idea being that the sequence of choices leading to A isn’t securable at node 1, because, at that node, you can’t make your future self make those choices.Footnote ⁴¹

But the target of the money-pump argument isn’t your choice at the first node, which does seem rational given your preferences. The target is the structure of your preferences. And the reason why you can’t secure the sequence of choices that leads to A (even though you can secure the choice of  $A^{-}$) is the cyclic structure of your preferences.Footnote ⁴²

It may next be objected that, even though you prefer $A^{-}$ to B, you could still prefer B to a more specific version of  $A^{-}$, such as $A^{-}$-when-you-could-have-kept-A.Footnote ⁴³ And, if so, you could be rationally required to turn down the trade at node 1 in the Upfront Money Pump even though you prefer $A^{-}$ to B. Hence, by individuating final outcomes (and thereby prospects) finely enough, you may avoid exploitation even though you have the preferences in (1).

This objection assumes that there are no restrictions on how we may individuate final outcomes. But, if there were no such restrictions, requirements of rationality such as Three-Step Acyclicity would be compatible with the rationality of any sequence of choices, because any alleged violation would disappear given some more fine-grained individuation of final outcomes.Footnote ⁴⁴ To get around this problem, we need to adopt a principle of individuation for final outcomes.

For the purposes of our theory of rationality, it seems that we only need to treat final outcomes as distinct if it’s rational to distinguish them preferentially; but, if it is rational to distinguish them preferentially, we need to treat them as distinct:Footnote ⁴⁵

Of course, if it were rationally permitted not to be indifferent between A and $A^{-}$-when-you-could-have-kept-A, these prospects may still be treated as distinct. But, as we shall see in Section 7, it is irrational not to be indifferent between such prospects.Footnote ⁴⁶

Does the Principle of Individuation by Rational Indifference violate the transitivity of identity (the principle that, if $X=Y=Z$, then X = Z)? Consider again the example of having a cup of coffee with one, two, or three lumps of sugar: You can’t tell the difference between a cup with one lump and a cup with two lumps. Nor can you tell the difference between a cup with two lumps and a cup with three lumps. But you can tell the difference between a cup with one lump and a cup with three lumps. As before, let A, B, and C be the sure prospects of having a cup with one, two, and three lumps of sugar respectively. When you can’t tell the difference between two prospects, it’s arguably rationally required to be indifferent between them. So then it’s rationally required to be indifferent between A and B and between B and C. Yet it seems rational to have a preference between A and C. So we find that $A=B=C\ne A$, which violates the transitivity of identity. This objection, however, is blocked if we allow indirect ways of telling the difference between the options. You can tell that a cup with one lump (A) tastes noticeably different from a cup with three lumps (C), whereas a cup with two lumps (B) does not taste noticeably different from a cup with three lumps (C). This difference in how they compare to C is a noticeable difference between A and B. And, in the same manner (changing what needs to be changed), you can distinguish B and C.Footnote ⁴⁷

It may also be objected that you could resist exploitation in the Upfront Money Pump if you adopt self-regulation. Basically, self-regulation forbids, if it can be avoided, choosing options that may be followed by a rationally permitted sequence of choices that has a prospect that you would not have chosen in a direct choice between the prospects of all available plans.Footnote ⁴⁸ Following the Uncovered-Choice Rule, you wouldn’t choose $A^{-}$ in a direct choice between A, $A^{-}$, B, and C. So self-regulation prescribes that you turn down the first trade in the Upfront Money Pump. And then you avoid exploitation.

Nevertheless, cyclic preferrers who adopt self-regulation are still vulnerable to the arguments we used to defend the choices that backward induction prescribes in the Upfront Money Pump. This is the main problem with the self-regulation defence of cyclicity.Footnote ⁴⁹ Moreover, this objection to self-regulation works in all the decision problems we will rely on in our overall money-pump argument for Expected Utility Theory.

For a (potential) second way of exploiting cyclic preferrers who rely on self-regulation, we sour all three options in the cycle. From (1), we have, by Unidimensional Continuity of Preference,

1. (4) $A\succ A^{-}\succ B\succ B^{-}\succ C\succ C^{-}\succ A$,

where $A^{-}$, $B^{-}$, and $C^{-}$ are sourings of A, B, and C respectively.

Consider the decision problem in Figure 6, the Three-Way Money Pump.Footnote ⁵⁰

Figure 6

The Three-Way Money Pump

In the Three-Way Money Pump, you would go up at each of nodes 2, 3, and 4. So, no matter what you choose at node 1, you end up paying for something that you could have had for free. For instance, if you go up at node 1 and up at node 2, then you end up with $A^{-}$ when you could have had A (by going down at node 1 and down at node 4).

This exploitation scheme doesn’t work, however. The exploiter cannot use it without potentially giving up something. The problem is that you can’t be relied on to choose a certain option at node 1, so you might end up paying for any one of A, B, and C no matter what your initial holding were. For instance, if you start with A, the exploiter potentially needs to trade you one of B and C to get your money. And then the scheme looks less profitable for the exploiter.Footnote ⁵¹ It also looks less irrational for you, because you may end up with a final holding that you prefer to your initial holding.Footnote ⁵² If you start off with A and end up with $B^{-}$ or with  $C^{-}$, you do not pay for what you could have kept for free.

It may be objected that whether some behaviour is a sign of irrationality shouldn’t depend on how profitable it is for an exploiter. Isn’t the mere fact that you chose $X^{-}$ when you could have had X a sufficient sign of irrationality? If so, our task constructing money pumps would be much easier. But, if that were a sign of irrationality, then the money-pump argument would prove too much, since clearly rational preferences would be irrational in some cases with an infinite series of trades (as we shall see in Section 8). Being exploited by giving exploiters a free lunch seems worrying in a separate way from merely making a sequence of choices that has a prospect that is less preferred than the prospect of some alternative sequence of choices.

Nevertheless, we can modify the set-up so that cyclic preferrers who rely on self-regulation still end up paying for what they could have kept for free. For this variation, we need to sour each option in the cycle once more. From (4), we have, by Unidimensional Continuity of Preference,

1. (5) $A\succ A^{-}\succ A^{--}\succ B\succ B^{-}\succ B^{--}\succ C\succ C^{-}\succ C^{--}\succ A$,

where $A^{--}$, $B^{--}$, and $C^{--}$ are sourings of $A^{-}$, $B^{-}$, and $C^{-}$ respectively.

Let a state of nature be a description of the world that resolves all of the agent’s uncertainty except that it leaves open what the agent will choose.Footnote ⁵³ Let an event be a set of states of nature.Footnote ⁵⁴ Let $\mathrm{\neg }E$ be the complement of event E, that is, the event that E does not occur. Let $E\mathrm{\&}{E}^{*}$ be the intersection of events E and $E^{*}$, that is, the event that both E and $E^{*}$ occur. Let a partition of states of nature be a set of events such that (i) each event in the set includes at least one state of nature and (ii) each state of nature is a member of exactly one event in the set. And let a gamble be a distribution of prospects over a partition of states of nature.

Suppose then that E₁ and E₂ are two independent chance events such that E₁ occurs with a $1/3$ probability and E₂ occurs with a $1/2$ probability. And consider the gambles G₁, $G_1^{-}$, and G₂ whose outcomes depend on E₁ and E₂:

Here, $\{E_1,\mathrm{\neg }{E}_1\mathrm{\&}{E}_2,\mathrm{\neg }{E}_1\mathrm{\&}\mathrm{\neg }{E}_2\}$ is a partition of states of nature.

We adopt the following requirement of rationality:Footnote ⁵⁵

From (5), we have, by the Weak Principle of Equiprobable Unidimensional Dominance,

1. (6) $G_1\succ G_1^{-}\succ G_2$, and $G_1\succ G_2$.

Now, consider the decision problem in Figure 7, the Self-Regulation Money Pump.

Figure 7

The Self-Regulation Money Pump

Here, the circles represent chance nodes where the way forward depends on a chance event. Chance nodes 2 and 4 go up if and only if E₁ occurs. And chance nodes 3 and 6 go up if and only if E₂ occurs.

You start off with gamble G₁. At choice node 1, you are offered a trade from G₁ to  $G_1^{-}$. If you turn down the trade at node 1, you would get an offer to trade from the outcome of G₁ to the outcome of G₂ after the chance events have resolved. This second trade offer would be offered to you at one of choice nodes 5, 7, and 8.

At each of nodes 5, 7, and 8, you would accept the second trade offer. Using backward induction or self-regulation, you take these predictions into account at node 1. And then the choice at node 1 is effectively between $G_1^{-}$ (accepting the trade) and G₂ (turning it down).

So, taking future choices into account at node 1, neither of the prospects that are effectively available at that node – that is, $G_1^{-}$ and G₂ – would be chosen in a direct choice between the prospects of all available plans given the Uncovered-Choice Rule, because G₁ is preferred to both of them. Hence self-regulation forbids neither $G_1^{-}$ nor G₂, since you can’t avoid choosing an option with a prospect that you would not have chosen in a direct choice between the prospects of all available plans. So it seems that, even if you rely on self-regulation, you should accept the first trade (since you prefer $G_1^{-}$ to G₂). But then you end up with $G_1^{-}$ when you could have kept G₁ for free. Hence this money pump isn’t blocked by self-regulation, and it makes you pay for what you could have kept for free.Footnote ⁵⁶

2.2 Acyclicity

So far, we have only considered arguments that rational preferences conform to Three-Step Acyclicity. We haven’t considered the following, more general, requirement:Footnote ⁵⁷

Yet we can show that Acyclicity is a requirement of rationality in much the same way as Three-Step Acyclicity.

Suppose that you violate Acyclicity by having the following, arbitrarily large, cycle of preferences:

1. (7) $A_1\succ A_2\succ \dots \succ A_n\succ A_1$.

From (7), we have, by Unidimensional Continuity of Preference,

1. (8) $A_1\succ A_1^{-}\succ A_2\succ \dots \succ A_n\succ A_1$,

where $A_1^{-}$ is a souring of A₁. We can then extend the Upfront Money Pump to handle this arbitrarily large cycle. Consider the decision problem in Figure 8, the Upfront Acyclicity Money Pump.Footnote ⁵⁸

Figure 8

The Upfront Acyclicity Money Pump

Here, you are first offered the opportunity to trade from A₁ to $A_1^{-}$. If you were to turn down that offer, you would be offered a trade from A₁ to A₂. Then, if you were to turn down that offer too, you would be offered a trade from A₁ to A₃, and so on until you would be offered a final trade from A₁ to A_n.

By backward induction, we find (in the same manner as in the Upfront Money Pump) that it’s rationally required to accept the first trade. So you end up with $A_1^{-}$ when you could have kept A₁ for free.

To spell out the assumptions of the money-pump arguments, we will rely on the notion of plans being available. Let a plan at a node n be a specification of what to choose at each choice node that can be reached from n while following the specification. Let us say that one follows a plan from a node n if and only if, for each choice node  $n^{*}$ that can be reached from n while choosing in accordance with the plan, one would choose in accordance with that plan if one were to face  $n^{*}$. Moreover, let us say that one intentionally follows a plan from a node n if and only if one follows the plan from n and, for all nodes  $n^{*}$ such that $n^{*}$ can be reached from n by following the plan, if one were to face  $n^{*}$, one would either form or have formed at $n^{*}$ an intention to choose in accordance with the plan at every choice node that can both be reached from n and be reached from $n^{*}$ by following the plan. Finally, let us say that a plan is available at a node n if and only if the plan can be intentionally followed from n.Footnote ⁵⁹

We assume the following requirement of rationality:Footnote ⁶⁰

This is the main assumption of money-pump arguments: that it is irrational to knowingly pay (in some currency you care about) for what you could have kept for free.

We also assume the following principle:

Given this principle, rational preferences must not lead to any conflicts with any requirements of rationality in any possible situation. The underlying idea is that there’s no rational luck.Footnote ⁶¹ Whether you are rational shouldn’t depend on what situation you happen to find yourself in. So whether it’s unlikely that you will ever face a money-pump set-up is irrelevant.Footnote ⁶²

Putting this together, we have a money-pump argument that Acyclicity is a requirement of rationality, and this argument relies on the following requirements of rationality:

• Backward induction at nodes that can be reached without making irrational choices

• The Principle of Unexploitability

• Unidimensional Continuity of Preference

And, in addition, the argument relies on the following principles:

• Decision-Tree Separability

• The possibility of the Upfront Acyclicity Money Pump

• The Principle of Preferential Invulnerability

We need the possibility of the Upfront Acyclicity Money Pump, since the Principle of Preferential Invulnerability only covers possible situations. This assumption is substantial, since it may be rejected if some outcomes cannot occur in the relevant sequential patterns. We will also take it as part of the description of the Upfront Money Pump to handle this arbitrarily large cycle. Consider the decision problem in the Upfront Money Pump (and the other decision problems we will discuss) that, at each node, all plans at that node are available. We do not, however, assume that the money-pump situations we discuss are likely to arise. As conceived here, the money-pump argument against cyclic preferences does not aim to show that having acyclic preferences is useful, or that cyclic preferences are likely to have bad effects.Footnote ⁶³

One worry about the Upfront Acyclicity Money Pump (and the other money pumps we’ll discuss) is that that decision problem is impossible if the violating preferences are non-practical preferences. Consider the following example.Footnote ⁶⁴ Let A be the sure prospect of staying at home. Let B be the sure prospect of going to Rome. And let C be the sure prospect of going mountaineering. Here, your preferences may plausibly be sensitive to what alternatives are available. Suppose that you prefer A-when-the-only-alternative-is-B to B, prefer B to C, and prefer C to A-when-the-only-alternative-is-C. These transitive (and plausible) preferences are practical in the sense that, for each pairwise preference, there is a possible choice between the compared prospects. Accordingly, preferences are non-practical if and only if it is not the case that, for each pairwise preference, there is a possible choice between the compared prospects. But suppose that you also prefer C to A-when-the-only-alternative-is-B. This makes your preferences cyclical. So, to defend Acyclicity, we need to show that these preferences are irrational. The additional preference, however, is non-practical, since there is no possible choice between C and A-when-the-only-alternative-is-B. So, for these cyclic preferences, the Upfront Acyclicity Money Pump is impossible.

To handle this problem, we rely on the Principle of Individuation by Rational Indifference.Footnote ⁶⁵ The fine-grained prospects A-when-the-only-alternative-is-B and A-when-the-only-alternative-is-C differ not only in what the alternative would be but also in that you are a coward in the latter but not in the former. Plausibly, the difference you are rationally permitted to care about is not the difference in alternative but the difference in cowardice. So you are rationally required to be indifferent between A-when-the-only-alternative-is-B and A-and-not-being-a-coward. The Principle of Individuation by Rational Indifference then entails that these prospects should be treated as the same. Crucially, it’s possible to have a choice between A-and-not-being-a-coward and C. So there is a possible instance of the Upfront Acyclicity Money Pump for your following practical preference cycle: A-and-not-being-a-coward is preferred to B, B is preferred to C, and C is preferred to A-and-not-being-a-coward.Footnote ⁶⁶

What about cycles with fewer steps than three? Consider the following irreflexivity requirement:Footnote ⁶⁷

And consider the following asymmetry requirement:Footnote ⁶⁸

Whether it’s even possible to violate these principles depends on how we define strict preference. Let ‘ $X\succsim Y$’ denote that X is at least as preferred as Y.Footnote ⁶⁹ We then adopt the following definition of that X is preferred to Y:Footnote ⁷⁰

Given this definition of strict preference, violations of One- and Two-Step Acyclicity are impossible.Footnote ⁷¹

4.1 Transitivity

Consider, once more, having a cup of coffee with one, two, or three lumps of sugar. Suppose, as before, that you can taste the difference between a cup with one lump and a cup with three lumps but you can neither taste the difference between a cup with one lump and a cup with two lumps nor taste the difference between a cup with two lumps and a cup with three lumps. This time, however, you only care about taste. Accordingly, you’re indifferent between having a cup with one lump and having a cup with two lumps and indifferent between having a cup with two lumps and having a cup with three lumps. And, due to your sweet tooth, you prefer having a cup with three lumps to having a cup with one lump.Footnote ¹⁰²

While your preferences are acyclic, they still violate Transitivity – the second basic axiom of Expected Utility Theory:Footnote ¹⁰³

Since we have already established that Acyclicity and Completeness are requirements of rationality, we have already established the irrationality of the following kinds of violations of Transitivity:

1. (1) $A\succ B\succ C\succ A$.

2. (14) $A\succsim B\succsim C\parallel A$.

Preferences of the kind in (1) violate Acyclicity, and preferences of the kind in (14) violate Completeness. But these aren’t the only kinds of non-transitive preferences. In order to complete the argument that Transitivity is a requirement of rationality, we also need to show that the following complete and acyclic kinds of non-transitive preferences are irrational:Footnote ¹⁰⁴

1. (15) $A\succ B\succ C\sim A$.

2. (16) $A\succ B\sim C\sim A$.

The money pumps we discussed in Section 2 don’t work for agents who have the preferences in (15) or (16). In the Upfront Money Pump, for example, agents with these preferences do not prefer C to A, so they aren’t rationally required to accept the trade from A to C. And then the argument falls apart.

This is easiest to see in case of the preferences in (16). Given the prediction that either of A and C may be chosen at node 3, agents with the preferences in (16) can rely on the following dominance version of backward induction – which, given Completeness, is equivalent to precautionary backward induction:

Dominance backward induction faces much the same worry as Minimal Unidimensional Precaution and precautionary backward induction, namely, that it seems less plausible in case you’re sure that, if you were to choose X, you wouldn’t make a sequence of choices that has a less preferred prospect than any rationally allowed outcome of Y. But much the same response applies here too (see Section 3).

Given dominance backward induction, the agent with the preferences in (16) is rationally required to turn down the trade at node 2, because the rationally allowed outcomes of doing so (that is, A and C) are both at least as good as the rationally allowed outcome of accepting the trade (B) and one of the rationally allowed outcomes of turning the trade down (A) is preferred to the rationally allowed outcome of accepting it. Plausibly, C is not only at least as preferred as A but also at least as preferred as  $A^{-}$. Therefore, taking into account at node 1 what may be chosen at later nodes, we find (given dominance backward induction) that it’s rationally required to turn down the trade at node 1. Hence the earlier money-pump argument is blocked.

One way to extend the money-pump argument for Acyclicity so that it also works for agents who have the preferences in (15) or (16) is the souring approach, which is to convert those acyclic, non-transitive preferences into cyclic preferences of the kind in (1) by breaking the indifferences with the help of sourings.Footnote ¹⁰⁵

Suppose that you have the preferences in (15). From (15), we have, by Unidimensional Continuity of Preference,

1. (17) $A\succ A^{-}\succ B$,

where $A^{-}$ is a souring of A.

Now, consider the following requirement of rationality:Footnote ¹⁰⁶

This requirement is plausible. Since a souring of Y does not improve Y in any dimension the agent cares about, a souring of Y should tip the scale between the two indifferent prospects X and Y in favour of the unsoured X. Nevertheless, Unidimensional IP-Transitivity may seem to assume, to some extent, the point at issue in an argument that Transitivity is a requirement of rationality. We will deal with this worry shortly.

From (15) and (17), we have, by Unidimensional IP-Transitivity,

1. (18) $C\succ A^{-}$.

Then – from (15), (17), and (18) – we have

1. (19) $B\succ C\succ A^{-}\succ B$.

We have derived cyclic preferences of the kind in (1), which can be shown to be irrational with the Upfront Money Pump.

Next, suppose that you have the preferences in (16). The first two steps proceed as before. From (16), we have, by Unidimensional Continuity of Preference,

1. (20) $A\succ A^{-}\succ B$,

where $A^{-}$ is a souring of A. From (16) and (20), we have, by Unidimensional IP-Transitivity,

1. (21) $C\succ A^{-}$.

From (21), we have, by Unidimensional Continuity of Preference,

1. (22) $C\succ C^{-}\succ A^{-}$,

where $C^{-}$ is a souring of C. From (16) and (22), we have, by Unidimensional IP-Transitivity,

1. (23) $B\succ C^{-}$.

Finally – from (20), (22), and (23) – we have

1. (24) $B\succ C^{-}\succ A^{-}\succ B$.

As before, we have derived cyclic preferences of the kind in (1), which can be shown to be irrational with the Upfront Money Pump.

Hence we have a money-pump argument that Transitivity is a requirement of rationality, and this argument relies on the following requirements of rationality:

• Backward induction at nodes that can be reached without making irrational choices

• Completeness (defended in Section 3)

• The Principle of Unexploitability

• Unidimensional Continuity of Preference

• Unidimensional IP-Transitivity

And, in addition, the argument relies on the following principles:

• Decision-Tree Separability

• The possibility of the Upfront Money Pump

• The Principle of Preferential Invulnerability

Still, as mentioned, Unidimensional IP-Transitivity is a special case of Transitivity. So it assumes, at least in part, the point at issue in an argument for Transitivity. Many of the alleged counter-examples to Transitivity would also be counter-examples to Unidimensional IP-Transitivity. For instance, let A be a free trip to Austin, let B be a free trip to Boston, and let $B^{-}$ be the trip to Boston at a cost of $1.Footnote ¹⁰⁷ It may seem rationally permitted to be indifferent between A and B and between A and $B^{-}$ while one prefers B to  $B^{-}$. If we rebut alleged counter-examples of this kind with the help of Unidimensional IP-Transitivity, we assume the point at issue.Footnote ¹⁰⁸

But there is a better response to these alleged counter-examples. If your indifference between two prospects is insensitive to sourings in this manner, then your indifference would have the same problematic insensitivity to sourings as preferential gaps. And then you would be open to a variation of the Precaution Money Pump (from Section 3), replacing preferential gaps with indifference.

Even so, there are other ways to amend the money-pump argument for Transitivity. The following eventwise approach makes use of dominance in terms of events. That is, we assume the following requirement of rationality:Footnote ¹⁰⁹

This principle avoids the earlier worry about IP-Transitivity. Note that the standard, alleged, counter-examples to Transitivity would not be counter-examples to the Strong Principle of Eventwise Dominance. So the Strong Principle of Eventwise Dominance does not assume the point at issue against these alleged counter-examples. So it does not assume the point at issue in an argument for Transitivity.

(Yet, since we will rely on Transitivity for the argument for the strong strict-preference version of Independence in Section 5.3, it may seem that the Strong Principle of Eventwise Dominance assumes, in part, the point at issue in an argument for Independence. In this respect, the souring approach is better. Still, in Section 5.1, we will rebut the commonly claimed counter-examples to the Strong Principle of Eventwise Dominance without relying on Transitivity.)

Suppose that you violate Transitivity by having the preferences in one of (1), (15), and (16). Then you have the following preferences:

1. (25) $A\succ B\succsim C\succsim A$.

Now, consider gambles G₁, G₂, and G₃, which have different outcomes in positive-probability events E₁, E₂, and E₃ that are such that $\{E_1,E_2,E_3\}$ is a partition of states of nature:

From (25), we have, by the Strong Principle of Eventwise Dominance, the following preferences over the gambles:Footnote ¹¹⁰

1. (26) $G_1\succ G_2\succ G_3\succ G_1$.

Once more, we have derived cyclic preferences of the kind in (1), which can be shown to be irrational with the Upfront Money Pump.Footnote ¹¹¹

So we have a money-pump argument that Transitivity is a requirement of rationality, and this argument relies on the following requirements of rationality:

• Backward induction at nodes that can be reached without making irrational choices

• Completeness (defended in Section 3)

• The Principle of Unexploitability

• The Strong Principle of Eventwise Dominance

• Unidimensional Continuity of Preference

And, moreover, the argument relies on the following principles:

• Decision-Tree Separability

• The possibility of the Upfront Money Pump

• The Principle of Preferential Invulnerability

Hence we have two alternative arguments for Transitivity, which rely on notably different assumptions.

4.2 Transitivity of Strict Preference

While we have already argued that Transitivity is a requirement of rationality, it is worth investigating whether we can make do with more compelling assumptions if we merely seek to defend the following (logically weaker) requirement of rationality:Footnote ¹¹²

In order to show that Transitivity of Strict Preference is a requirement of rationality, we need to show that all kinds of violating preferences are irrational. Violations of Transitivity of Strict Preference can be of the following kinds:

1. (1) $A\succ B\succ C\succ A$.

2. (15) $A\succ B\succ C\sim A$.

3. (27) $A\succ B\succ C\parallel A$.

Notably absent from this list of violations are preferences of the following kind:

1. (16) $A\succ B\sim C\sim A$.

The preferences in (16) violate Transitivity (that is, transitivity of at least as preferred as) but not Transitivity of Strict Preference. This, as we shall see, allows us to make do without not only Unidimensional IP-Transitivity but also the Strong Principle of Eventwise Dominance.

The preferences in (1) violate Acyclicity, and the preferences in (27) violate Completeness. So, given Acyclicity and Completeness, we only need to show that preferences of the kind in (15) are irrational.

Suppose that you have the preferences in (15). From (15), we have, by Unidimensional Continuity of Preference,

1. (17) $A\succ A^{-}\succ B$,

where $A^{-}$ is a souring of A.

We also assume, as a requirement of rationality, the mirror image of Unidimensional Continuity of Preference:

The underlying idea behind this principle is the same as for Unidimensional Continuity of Preference: if you (strictly) prefer X to Y, then you must prefer X with some margin. So there should be some, perhaps minimal, amount you are willing to forgo to get X rather than Y.

From (15), we have, by Unidimensional Continuity of Dispreference,

1. (28) $B\succ C^{+}\succ C$,

where $C^{+}$ is a sweetening of C.

Next, instead of Unidimensional IP-Transitivity, we assume the following requirement of rationality:Footnote ¹¹³

Back in Section 4.1, when we assumed Unidimensional IP-Transitivity for the souring approach, we assumed that the souring of one of two indifferent prospects made it less preferred than the other. And we had to rule out that you may still be indifferent between the prospects. Here, we needn’t do so. All we assume is that a sweetening of one of two indifferent prospects does not make it less preferred than the other.

From (15) and (28), we have, by Unidimensional PI-Acyclicity,

1. (29) It is not the case that $A\succ C^{+}$.

From (29), we have, by Completeness,

1. (30) $C^{+}\succsim A$.

Now, consider the decision problem in Figure 12, the Strict-Preference Money Pump.

Figure 12

The Strict-Preference Money Pump

At node 4, you are both rationally permitted to go up and rationally permitted to go down, since you are indifferent between A and C.

Taking this into account at node 3, we find that the prospect of going up ( $C^{+}$) is preferred to one of the rationally allowed outcomes of going down (C) and the prospect of going up is at least as preferred as every rationally allowed outcome of going down. So, by dominance backward induction, it is rationally required that you accept the trade at node 3. Alternatively, we could rely on Minimal Unidimensional Precaution, which also prescribes going up at node 3 – since C is a souring of $C^{+}$ and you do not prefer A to C.

Taking this into account at node 2, it is rationally required to accept the trade at that node, since you prefer B to  $C^{+}$.

Finally, taking this prediction into account at node 1, it is rationally required to accept the initial trade, since you prefer $A^{-}$ to B. So you end up with $A^{-}$ even though you could have kept A for free.

Note that the Strict-Transitivity Money Pump is BI-terminating. So we only need to assume that, at nodes that can be reached without making irrational choices, you retain (i) your rationality and (ii) your trust in your rationality at nodes that can be reached without making irrational choices.

Hence we have a money-pump argument that Transitivity of Strict Preference is a requirement of rationality, and this argument relies on the following requirements of rationality:

• Acyclicity (defended in Section 2.2)

• Backward induction at nodes that can be reached without making irrational choices

• Completeness (defended in Section 3)

• The Principle of Unexploitability

• Unidimensional Continuity of Dispreference

• Unidimensional Continuity of Preference

• Unidimensional PI-Acyclicity

And the argument also relies on the following principles:

• Decision-Tree Separability

• The Irrationality of Single Sourings

• The possibility of the Strict-Preference Money Pump

• The Principle of Future-Choice Independence

• The Principle of Preferential Invulnerability

• The Principle of Rational Decomposition

We only need the Irrationality of Single Sourings, the Principle of Future-Choice Independence, and the Principle of Rational Decomposition to derive Minimal Unidimensional Precaution (see Section 3). So we could drop these assumptions if we assume Minimal Unidimensional Precaution as a requirement of rationality. As a requirement of rationality, Minimal Unidimensional Precaution is equivalent (given standard backward induction) to dominance backward induction in the Strict-Preference Money Pump. So we may also drop Minimal Unidimensional Precaution in this argument if we assume dominance backward induction at nodes that can be reached without making irrational choices.

Note that this approach, which makes do with Unidimensional PI-Acyclicity without Unidimensional IP-Transitivity and the Strong Principle of Eventwise Dominance, does not work for the preferences in (16), where C is indifferent not only to A but also to B. If we tried this approach on those preferences, we would see that, once we have sweetened C, the sweetening of C may be preferred not only to A but also to B. And then the approach is blocked.

4.3 Strong Acyclicity

We can extend both the souring approach and the eventwise approach to cover violations of the following weakening of Transitivity (and strengthening of Acyclicity):Footnote ¹¹⁴

Violations of this weaker requirement can only be of the following general kind:

1. (31) $A_1\succ A_2\succsim \dots \succsim A_n\succsim A_1$.

So suppose that you violate Strong Acyclicity by having the preferences in (31).

For the souring approach, consider first the case where your preferences are, more specifically, of the following cyclical kind:

1. (7) $A_1\succ A_2\succ \dots \succ A_n\succ A_1$.

In that case, we can show that your preferences are irrational with the Upfront Acyclicity Money Pump.

Consider next the remaining case where your preferences over A₁, A₂, …, and A_n do not contain a cycle of strict preference. That is, your preferences are merely weakly cyclical – that is, you have a cycle of strict preference except that some (but not all) strict-preference relations in the cycle have been replaced by indifference. Find three prospects – A_i, A_j, and A_k – that are adjacent in this weak cycle such that

1. (32) $A_i\sim A_j\succ A_k$.

From (32), we have, by Unidimensional Continuity of Preference,

1. (33) $A_j\succ A_j^{-}\succ A_k$,

where $A_j^{-}$ is a souring of A_j. Then, from (32) and (33), we have, by Unidimensional IP-Transitivity,

1. (34) $A_i\succ A_j^{-}\succ A_k$.

Next, we replace the (32) part of the weak cycle with (34). We repeat this procedure if necessary until we end up with a cycle of strict preference. Finally, we use the Upfront Acyclicity Money Pump to show that these cyclic preferences are irrational.

Hence we have a money-pump argument that Strong Acyclicity is a requirement of rationality, and this argument relies on the following requirements of rationality:

• Backward induction at nodes that can be reached without making irrational choices

• The Principle of Unexploitability

• Unidimensional Continuity of Preference

• Unidimensional IP-Transitivity

And, moreover, the argument relies on the following principles:

• Decision-Tree Separability

• The possibility of the Upfront Acyclicity Money Pump

• The Principle of Preferential Invulnerability

For the eventwise approach, consider gambles G₁, G₂, …, and G_n, which have different outcomes in positive-probability events E₁, E₂, …, and E_n that are such that $\{E_1,E_2,\dots ,E_n\}$ is a partition of states of nature:

From (31), we have, by the Strong Principle of Eventwise Dominance,

1. (35) $G_1\succ G_2\succ \dots \succ G_n\succ G_1$.

We have derived cyclic preferences of the kind in (7), which can be shown to be irrational with the Upfront Acyclicity Money Pump.

So we have a money-pump argument that Strong Acyclicity is a requirement of rationality, and this argument relies on the following requirements of rationality:

• Backward induction at nodes that can be reached without making irrational choices

• The Principle of Unexploitability

• The Strong Principle of Eventwise Dominance

• Unidimensional Continuity of Preference

And, in addition, the argument relies on the following principles:

• Decision-Tree Separability

• The possibility of the Upfront Acyclicity Money Pump

• The Principle of Preferential Invulnerability

Notably, unlike the money-pump argument for Transitivity, these money-pump arguments for Strong Acyclicity do not rely on Completeness.Footnote ¹¹⁵

5.1 Allais, Ellsberg, and Independence for Constant Outcomes

The two most prominent objections to Independence are the Allais Paradox (put forward by Maurice Allais) and the Ellsberg Paradox (put forward by Daniel Ellsberg). These paradoxes directly challenge not only the biconditional weak-preference version of Independence but also the following, logically weaker, requirement:Footnote ¹²¹

Violations of this variant of Independence can only be of the following kind:

1. (36) $ApC\succ BpC$, and $BpD\succ ApD$,

where p is a probability such that $0<p<1$. As we shall see, the Allais Paradox and the Ellsberg Paradox both feature seemingly rational preferences of this kind.

The Allais Paradox involves four lotteries. In lottery L₁, one gets $1M for certain; in lottery L₂, there is a $10\mathrm{\%}$ probability of getting $5M, an $89\mathrm{\%}$ probability of getting $1M, and a $1\mathrm{\%}$ probability of getting $0; in lottery L₃, there is an $11\mathrm{\%}$ probability of getting $1M and an $89\mathrm{\%}$ probability of getting $0; and, in lottery L₄, there is a $10\mathrm{\%}$ probability of getting $5M and a $90\mathrm{\%}$ probability of getting $0:Footnote ¹²²

Many people have the following preferences, which we can call Allais Preferences:

1. (37) $L_1\succ L_2$, and $L_4\succ L_3$.

To see that Allais Preferences violate the weak strict-preference version of Independence for Constant Outcomes, let A be a sure prospect of $1M; let B be the prospect of a $10/11$ probability of $5M, otherwise $0; let C be a sure prospect of $1M (just like A); and let D be a sure prospect of $0:

If we let p be $11/100$, then L₁ is equivalent to ApC, L₂ is equivalent to BpC, L₃ is equivalent to ApD, and L₄ is equivalent to BpD. So (37) can be stated as follows:

1. (36) $ApC\succ BpC$, and $BpD\succ ApD$.

Accordingly, Allais Preferences violate the weak strict-preference version of Independence for Constant Outcomes.

The Ellsberg Paradox features an urn that is known to contain 30 red balls and 60 balls that are either black or yellow (and this is all that is known with respect to the proportions in the urn). The proportion of black to yellow balls is unknown. A ball will be drawn at random from the urn. Just like the Allais Paradox, the Ellsberg Paradox involves four lotteries. Lottery L₁ pays $100 if the ball is red, otherwise $0; lottery L₂ pays $100 if the ball is black, otherwise $0; lottery L₃ pays $100 if the ball is red or yellow, otherwise $0; and lottery L₄ pays $100 if the ball is black or yellow, otherwise $0:Footnote ¹²³

Many people have the following preferences, which we can call Ellsberg Preferences:

1. (38) $L_1\succ L_2$, and $L_4\succ L_3$.

Ellsberg Preferences violate the weak strict-preference version of Independence for Constant Outcomes. To see this, let p be the unknown probability of the ball’s being either red or black (which, given the agent’s knowledge, is equivalent to the ball’s not being yellow); let A be the prospect of a $1/(3p)$ probability of $100, otherwise $0; let B be the prospect of a $1-1/(3p)$ probability of $100, otherwise $0; let C be the sure prospect of $0; and let D be the sure prospect of $100:

We see that L₁ is equivalent to ApC, L₂ is equivalent to BpC, L₃ is equivalent to ApD, and L₄ is equivalent to BpD. So (38) can be stated as follows:

1. (36) $ApC\succ BpC$, and $BpD\succ ApD$.

Accordingly – just like Allais Preferences – Ellsberg Preferences violate the weak strict-preference version of Independence for Constant Outcomes.

Hence both Allais and Ellsberg Preferences entail preferences of the kind in (36), so they both violate the weak strict-preference version of Independence for Constant Outcomes. Consequently, Allais and Ellsberg Preferences violate the (logically stronger) biconditional weak-preference version of Independence.Footnote ¹²⁴Footnote ¹²⁵

As we have seen, both Allais and Ellsberg Preferences violate the weak strict-preference version of Independence for Constant Outcomes. Accordingly, Allais and Ellsberg Preferences are both irrational if this principle is a requirement of rationality. Can we show that it is a requirement of rationality? We can.

Suppose that you violate the weak strict-preference version of Independence for Constant Outcomes by having the preferences in (36). From (36), we have, by Unidimensional Continuity of Preference,

1. (39) $ApC\succ A^{-}p{C}^{-}\succ A{}^{--}p{C}^{--}\succ BpC$, and

   $BpD\succ B^{-}p{D}^{-}\succ B{}^{--}p{D}^{--}\succ ApD$,

where $A^{-}p{C}^{-}$ and $B^{-}p{D}^{-}$ are sourings of ApC and BpD respectively and where $A{}^{--}p{C}^{--}$ and $B{}^{--}p{D}^{--}$ are sourings of $A^{-}p{C}^{-}$ and $B^{-}p{D}^{-}$ respectively.

For simplicity, we may assume the following requirement of rationality – even though, strictly, we don’t need it:

We have, by the Souring Principle,

1. (40) $A\succ A^{-}\succ A^{--}$, $B\succ B^{-}\succ B^{--}$, and $C\succ C^{-}\succ C^{--}$.

Now, suppose that E₁ and E₂ are two independent chance events such that E₁ occurs with probability $1/2$ and E₂ occurs with probability p. And consider gambles G₁, $G_1^{-}$, and G₂ whose outcomes depend on these two events:

Like before, we assume that the Weak Principle of Equiprobable Unidimensional Dominance is a requirement of rationality. The Weak Principle of Equiprobable Unidimensional Dominance should be acceptable even if one is risk-averse.Footnote ¹²⁶ In terms of risk, the dominated prospect must be less preferable than the dominating prospect. For every potential undesired final outcome of the dominating prospect, the dominated prospect has a corresponding (soured) final outcome with the same probability which is even less preferred. The probability of getting an undesired final outcome must be at least as high in the dominated prospect as in the dominating prospect. In any compelling violation of Independence for Constant Outcomes, no individual preference between two prospects violates the Weak Principle of Equiprobable Unidimensional Dominance. For instance, the Weak Principle of Equiprobable Unidimensional Dominance does not assume the point at issue against Allais and Ellsberg Preferences. None of the pairwise preferences in Allais and Ellsberg Preferences violates the Weak Principle of Equiprobable Unidimensional Dominance.

From (39) or more simply from (40), we have, by the Weak Principle of Equiprobable Unidimensional Dominance,

1. (41) $G_1\succ G_1^{-}\succ G_2$.

Now, consider the decision problem in Figure 13, the Constant-Outcomes Money Pump.Footnote ¹²⁷

Figure 13

The Constant-Outcomes Money Pump

At the initial choice node, you have a choice whether to accept a trade from G₁ to $G_1^{-}$. If you turn down this trade, chance node 5 determines (depending on E₁) whether you will face node 6 or 9. If E₁ occurs, you are offered a trade at node 6 from ApD to  $B{}^{--}p{D}^{--}$. And, if E₁ does not occur, you are offered a trade at node 9 from BpC to  $A{}^{--}p{C}^{--}$.

At node 6, you would accept the trade from ApD to $B{}^{--}p{D}^{--}$, since you prefer $B{}^{--}p{D}^{--}$ to ApD. And, at node 9, you would also accept the trade from BpC to $A{}^{--}p{C}^{--}$, since you prefer $A{}^{--}p{C}^{--}$ to BpC.

Taking this into account at node 1, the choice at that node is effectively between $G_1^{-}$ (accepting the trade) and G₂ (turning it down). Since you prefer $G_1^{-}$ to G₂, you accept the initial trade. So you end up with $G_1^{-}$ when you could have kept G₁ for free. Moreover, note that $G_1^{-}$ is less preferred than G₁ in every state of nature.

Hence we have a money-pump argument that preferences of the kind in (36) are irrational. Moreover, since the Constant-Outcomes Money Pump is BI-terminating, this argument need not assume that you retain your rationality at nodes that can’t be reached without making irrational choices.

Accordingly, we have a money-pump argument that the weak strict-preference version of Independence for Constant Outcomes is a requirement of rationality, and this argument relies on the following requirements of rationality:

• Backward induction at nodes that can be reached without making irrational choices

• The Principle of Unexploitability

• Unidimensional Continuity of Preference

• The Weak Principle of Equiprobable Unidimensional Dominance

And, moreover, the argument relies on the following principles:

• Decision-Tree Separability

• The possibility of the Constant-Outcomes Money Pump

• The Principle of Preferential Invulnerability

Since we can show that the weak strict-preference version of Independence for Constant Outcomes is a requirement of rationality and thereby that Allais and Ellsberg Preferences are irrational, we can rebut the most prominent objections to Independence.

5.2 The weak strict-preference version of Independence

Having rebutted the most prominent objections to Independence (that is, the alleged rationality of Allais and Ellsberg Preferences), let us explore whether there are any compelling positive arguments that Independence is a requirement of rationality. We begin with the weakest version of Independence, namely,

This version of Independence can be shown to be a requirement of rationality with the help of a money-pump argument with even weaker assumptions than those we relied on for the argument that Independence for Constant Outcomes is a requirement of rationality.

Violations of the weak strict-preference version can only be of the following kind, where p is a probability such that $0<p<1$:

1. (42) $A\succ B$, and $BpC\succ ApC$.

So suppose that you violate the weak strict-preference version of Independence by having the preferences in (42). From (42), we have, by Unidimensional Continuity of Preference,

1. (43) $BpC\succ B^{-}p{C}^{-}\succ ApC$,

where $B^{-}p{C}^{-}$ is a souring of BpC.

Now, consider the decision problem in Figure 14, the Independence Money Pump.Footnote ¹²⁸

Figure 14

The Independence Money Pump

Here, the two chance nodes depend on the same event E, which occurs with probability p. You start off with BpC. At node 1, you are offered a trade from BpC to $B^{-}p{C}^{-}$. If you accept this trade, then, if E occurs, you end up with $B^{-}$ and, if E does not occur, you end up with  $C^{-}$. If you turn the trade down and E occurs, you will be offered a trade from B to A at node 4. And, if you turn down the trade at node 1 and E does not occur, you end up with C.

Since you prefer A to B, you would accept the trade at node 4. Using backward induction at node 1, the prospect of going down is then effectively ApC and the prospect of going up is $B^{-}p{C}^{-}$. So you go up at node 1, since you prefer $B^{-}p{C}^{-}$ to ApC. But then you end up with $B^{-}p{C}^{-}$ when you could have kept BpC for free if you had followed the plan to go down at each choice node. And, since the chance nodes depend on the same event, we find that, in every state of nature, the prospect of going up at node 1 is a souring of the prospect of following the plan to go down at each choice node.

Accordingly, we have a money-pump argument that the weak strict-preference version of Independence is a requirement of rationality, and this argument is based on the following requirements of rationality:

• Backward induction at nodes that can be reached without making irrational choices

• The Principle of Unexploitability

• Unidimensional Continuity of Preference

And, in addition, the argument relies on the following principles:

• Decision-Tree Separability

• The possibility of the Independence Money Pump

• The Principle of Preferential Invulnerability

Still, axiomatizations of Expected Utility Theory typically need a stronger version of Independence, like the strong strict-preference version.

As mentioned earlier, Expected Utility Theory can be axiomatized by Completeness, Transitivity, Continuity, and the strong strict-preference version of Independence. Can we strengthen this standard axiomatization so that it relies on the weak strict-preference version of Independence rather than the strong one? We cannot. Likewise, we cannot replace the strong strict-preference version of Independence with the weak strict-preference version of Independence for Constant Outcomes in the axiomatization.Footnote ¹²⁹

5.3 The strong strict-preference version of Independence

So let us turn to

The good news is that there is a money-pump argument that this version of Independence is a requirement of rationality; the bad news is that this argument requires notably stronger assumptions than the argument for the weak strict-preference version. In order to show that the strong strict-preference version is a requirement of rationality, it’s not enough to show that preferences of the following kind are irrational:

1. (42) $A\succ B$, and $BpC\succ ApC$.

We also need to show the irrationality of violations of the following kinds, where (as before) p is a probability such that $0<p<1$:

1. (44) $A\succ B$, and $ApC\sim BpC$.

2. (45) $A\succ B$, and $ApC\parallel BpC$.

The money-pump argument in Section 5.2 doesn’t work against the preferences in (44) and (45). Preferences of the kind in (45) are ruled out by Completeness. The preferences in (44) are more challenging. These preferences violate the strong strict-preference version of Independence, but they do not violate the other standard axioms of Expected Utility Theory.Footnote ¹³⁰ And, since the biconditional weak-preference version of Independence is logically stronger than the strong strict-preference version, the preferences in (44) violate that version too. Hence, to have a cogent argument that these versions of Independence are requirements of rationality, we must show that the preferences of the kind in (44) are irrational.

To establish the irrationality of preferences of the kind in (44), we assume that the following dominance principle is a requirement of rationality:

Just like the Weak Principle of Equiprobable Unidimensional Dominance, this requirement should be acceptable even if one is risk-averse. The probability of getting an undesired final outcome must be at least as high in the dominated prospect as in the dominating prospect.Footnote ¹³¹ In any compelling violation of Independence, the individual pairwise preferences do not violate the Strong Principle of Unidimensional Stochastic Dominance.

We will show that preferences of the kind in (42) can be derived from (44) – given that the Strong Principle of Unidimensional Stochastic Dominance, Transitivity, and Unidimensional Continuity of Preference are requirements of rationality.

From (44), we have, by Unidimensional Continuity of Preference,

1. (46) $A\succ A^{-}\succ B$,

where $A^{-}$ is a souring of A. From (46), we have, by the Strong Principle of Unidimensional Stochastic Dominance,

1. (47) $ApC\succ A^{-}pC$.

From (44) and (47), we have, by Transitivity,

1. (48) $BpC\succ A^{-}pC$.

Finally, from (46) and (48), we have

1. (49) $A^{-}\succ B$, and $BpC\succ A^{-}pC$.

Hence, from (44), we have derived preferences of the kind in (42). Since preferences of that kind can be shown to be irrational by the money-pump argument for the weak strict-preferences version (Section 5.2), we can show that preferences of the kind in (44) are irrational.

Accordingly, we have a money-pump argument that the strong strict-preference version of Independence is a requirement of rationality, and this argument relies on the following requirements of rationality:

• Backward induction at nodes that can be reached without making irrational choices

• Completeness (defended in Section 3)

• The Principle of Unexploitability

• The Strong Principle of Unidimensional Stochastic Dominance

• Transitivity (defended in Section 4.1)

• Unidimensional Continuity of Preference

And, in addition, the argument relies on the following principles:

• Decision-Tree Separability

• The possibility of the Independence Money Pump

• The Principle of Preferential Invulnerability

Note that this argument requires stronger assumptions than those needed in the argument for the weak strict-preference version of Independence, since that argument did not need Completeness, the Strong Principle of Unidimensional Stochastic Dominance, and Transitivity.

5.4 The biconditional weak-preference version of Independence

Finally, let us turn to

With only slightly stronger assumptions than those for the argument that the strong strict-preference version is a requirement of rationality, we can show that the biconditional weak-preference version is a requirement of rationality.

In addition to preferences of the kind in (42), (44), and (45) which we have already shown are irrational (with the arguments in Sections 5.2 and 5.3), violations of the biconditional weak-preference version of Independence can also be of the following kinds, where again p is a probability such that $0<p<1$:

1. (50) $A\sim B$, and $ApC\succ BpC$.

2. (51) $A\sim B$, and $ApC\parallel BpC$.

3. (52) $A\parallel B$, and $ApC\succsim BpC$.

Two of these violations – namely, (51) and (52) – are ruled out by Completeness. So, to finish the argument that the biconditional weak-preference version is a requirement of rationality, we need only show that preferences of kind in (50) are irrational.

From (50), we have, by Unidimensional Continuity of Preference,

1. (53) $ApC\succ A^{-}p{C}^{-}\succ BpC$,

where $A^{-}p{C}^{-}$ is a souring of ApC. Since $A^{-}p{C}^{-}$ is a souring of ApC, it follows that $A^{-}$ is a souring of A and that $C^{-}$ is a souring of C. So we have, by the Souring Principle,

1. (54) $A\succ A^{-}$, and $C\succ C^{-}$.

From (50) and (54), we have, by Transitivity,

1. (55) $B\succ A^{-}$.

From (54), we have, by the Strong Principle of Unidimensional Stochastic Dominance,

1. (56) $A^{-}pC\succ A^{-}p{C}^{-}$.

From (53) and (56), we have, by Transitivity,

1. (57) $A^{-}pC\succ BpC$.

Finally, from (55) and (57), we have

1. (58) $B\succ A^{-}$, and $A^{-}pC\succ BpC$.

We have, once more, derived preferences of the same kind as those in (42). And, since such preferences can be shown to be irrational by the money-pump argument in Section 5.2, we can show that preferences of the kind in (50) are irrational.

Hence we have a money-pump argument that the biconditional weak-preference version of Independence is a requirement of rationality, and this argument relies on the following requirements of rationality:Footnote ¹³²

• Backward induction at nodes that can be reached without making irrational choices

• Completeness (defended in Section 3)

• The Principle of Unexploitability

• The Souring Principle

• The Strong Principle of Unidimensional Stochastic Dominance

• Transitivity (defended in Section 4.1)

• Unidimensional Continuity of Preference

And the argument also relies on the following principles:

• Decision-Tree Separability

• The possibility of the Independence Money Pump

• The Principle of Preferential Invulnerability

Note that, apart from the addition of the Souring Principle, this argument for the biconditional weak-preference version does not require stronger assumptions than the argument for the strong strict-preference version in Section 5.3.