The Paradox of Similarity in Decision‑Making

 

Regular readers may have heard of Fredkin’s Paradox – which says that the more similar two choices are, the less the decision should matter - yet the harder it is to choose between them. I mentioned this at a recent social event, and received an interesting response back:

“But surely that isn't true, and it doesn't bear out in my experience. If I have to choose between a Chinese restaurant and an Indian restaurant, it's a harder choice than a Chinese and a Thai, or Indian and Bangladeshi?”

On the off chance that readers might find it interesting, I’ll share my response here – which is to say that the above retort actually strengthens Fredkin’s Paradox more than it undermines it. This is because Fredkin wasn’t talking about similarity of category - Chinese vs Indian, Thai vs Bangladeshi, etc - he was talking about similarity of value. That is, two choices are “similar” in the paradox only when they satisfy your goal equally (or nearly equally) well.

Think about your own experience of walking around a capital city trying to help your wife decide where she wants to eat. If the goal is simply “eat something tasty,” and both Chinese and Indian hit that spot near-equally on that day, the decision becomes hard. Not because the cuisines are similar, but because your desire for them is equally strong.

This is why the restaurant example doesn’t contradict the paradox. It actually illustrates it. If in Chinese vs. Indian they are both equally appealing, then the decision is hard. If in Chinese vs. Thai they are not close to being equally appealing, because, say, you had Chinese last weekend and you really fancy a Pad Thai today, then the decision is easy. Similarly, Indian vs Bangladeshi might be similar cuisines but not equally appealing for a number of reasons, making the decision easy, and not a close call.

Of course, a key thing to add about Fredkin’s Paradox is that most human decisions aren’t of this nature. Humans rarely experience true indifference, especially with socio-cultural factors and identity driving so many decisions. Fredkin’s Paradox applies only when you do care, but the options seem equally good in terms of value.

At this point in writing this blog post, I then just started to ponder Fredkin’s Paradox in terms of the dating pool and people’s search for their optimum beloved. If Fredkin is right about the difficulty applying in situations with agonising similarity of value, and I think broadly he is, then if you’re agonising between two prospective beloveds, it’s likely that neither is your optimum beloved. Because as anyone who has fallen in love knows, despite the impression one might form from being a single person immersed in the dating pool search space, the optimum beloved is not the result of a difficult optimisation problem - the optimum beloved is the person who ends the optimisation problem.

Another reason for this is found in Pascal’s great wisdom that “The heart has its reasons, of which reason knows nothing.” While the beloved is chosen through many rational and empirical-based decision processes, the heart is not really captured by weighing marginal gains or running a mental cost–benefit analysis; it is captured by a kind of intuitive clarity that smothers mechanical deliberation. Yes, reason tries to optimise, compare, and evaluate, but it’s the heart that does the falling.

That’s why, when the heart knows, the decision process comes to a natural cessation, as the optimum beloved becomes the one who makes the very act of further choosing feel unnecessary.

An Initial Absurdity That Becomes Interesting

 

Sometimes economist logic takes us to funny places. Here’s an example. Consider this question: Why do we think it’s fine for a couple to choose not to have children, but not fine for them to have children and periodically treat them badly? It’s quite easy to elicit an intuitive response, something along the lines of: “The two situations are very different. If you choose not to have children, that’s your right. But once you have a child, your responsibilities change, and you are compelled to look after that child and treat them well.”

Everyone can relate to that answer, and I do think it’s right. But it doesn’t entirely settle the matter – because, given the astronomically low probability of any individual being born, I assume most children would prefer to be born and be periodically treated badly than not be born at all. Suppose there is a 14 year old boy called Tom, who is quite unhappy at being mistreated by his parents, Jack and Jill, but still glad he has been born, compared with the alternative of never having been born. Given Tom’s preference, it would be worse for him to have never been born, and Jack and Jill remained childless, than it would to be born and treated badly by Jack and Jill. Yet a childless Jack and Jill would receive no criticism, even though not having Tom is worse for potential Tom than having him and mistreating him.

It’s quite easy to elicit a further intuitive response, something along the lines of: “Yes, but a childless Jack and Jill face no moral charge, because they don’t know about the hypothetical Tom that never got born – so no situation where it’s a worse option for Tom actually exists”.

This is also true – Tom’s non-existence isn’t a worse option for him if he’s not alive to recognise it thus. But it’s not a wholly satisfactory closure either, and does show a strange paradoxical nature (although not quite a paradox in this case), whereby a negative moral judgement is conferred upon a less bad outcome for an agent, and a neutral judgement is conferred upon a worse outcome for an agent, even though we can all follow both logical paths very easily.

This isn’t really one of those linguistic peculiarities, such that I covered in this blog How To Make Sense Of Language Paradoxes; I think it’s more in keeping with the kind of economic considerations that are fun to explore in blog posts like The Absurdity Of Quota-Based Discrimination. 

How To Make Sense Of Language Paradoxes

There are many statements that make no sense in their entirety but contain enough conjuncts to appear intelligible to a reader. For example, consider this statement:

“Tony Blair’s first movie appearance was in Full Metal Jacket.”

The statement is false in a number of ways – namely that Tony Blair never appeared in movies, so wasn’t in Full Metal Jacket. But it is a coherent statement in that one could assess its veracity based on the information contained.

Now consider another statement:

“The dog on Westminster Bridge is fed up with all these terrorist attacks.”  

This statement is less coherent because we don’t really know what it means for dogs to be fed up, let alone fed up with human things like terrorism. As Bertrand Russell said, a statement can be true only if none of its propositions are false. It could be true that there is a dog on Westminster Bridge; and it could be true that people are fed up with terrorist attacks; and it is true that there is someone called Tony Blair; and it is true that there is a movie called Full Metal Jacket - but when expressed as a conjunction of claims, neither statement is true.

My perception of language paradoxes is that they belong in the same family as the conjunction problems above: they are rather like linguistic versions of Escher drawings. Paradoxes are about a limitation in defining a perception or definition of a statement. Either the language employed is not precise enough to encapsulate that which is being described, or we are attempting to define something and getting our first and second order terms mixed up.

The liar paradox is a famous statement that seems to present a problem. The statement 'This sentence is false' has the paradox: if it's true then it's false, and if it's false then it's true. Mathematician Alfred Tarski sought to resolve the dilemma by talking about levels of language and how they predicate truth or falsehood. When one sentence refers to the truth or falsehood of another sentence, then according to Tarski it is 'semantically higher'. If I said "It rained on Westminster Bridge at mid-day on March 23rd 2014" and called that statement Statement 1, then there is a higher level proposition attached to it "Statement 1 is False". Here the truth or falsity of the proposition clearly is predicated on whether there was rainfall on Westminster Bridge at mid-day on March 23rd 2014.

But when it comes to statements like 'This sentence is false', while the language employed makes sense on a word-by-word basis, the level at which it is employed doesn't, because it is stated as a higher level statement, when in fact it isn't about anything related to a lower level proposition. Because of this we can construct sentences that accord with our ordinary semantic rules, but they cannot consistently be assigned a truth value because they are in isolation from a concomitant statement.

Statement A: “Every even number is the sum of two prime numbers"

Statement B: “The statement that every even number is the sum of two prime numbers cannot be proven.”

Either Statements A and B are both true or they are both false. If they are both true then there is a statement in arithmetic that cannot be proven. And if they are both false then we have proof that we can prove a false statement. If upon reading the statement 'This sentence is false" you decide to say that it is neither true nor false, you come smack up against the Godelian problem that there is no complete system of rules of inference in mechanised logic, and that any formally mechanised system in which a categorical set of axioms exists cannot be captured in one grand slam rationale without leaving a brute residue of incompleteness. But if on the other hand upon reading the statement "Every even number is the sum of two prime numbers” you decide to say that it's neither true nor false you find this cannot be allowed because it must be either true or false.

Another one: here’s one of Zeno’s famous paradoxes; If I fire an arrow directly at you, the arrow will never reach you. Suppose the distance the arrow travels is 10 metres – Zeno shows how it will never reach its target, because it first has to travel half that distance (1/2), then half again (1/4), and then half again (1/8), an so on, ad infinitum. Zeno’s ‘logic’ told him that the arrow would carry on travelling indefinitely, but his senses told him that it would reach its target.  Then it was later shown (principally by Leibniz) that this sequence of common ratios (1/2,1/4,1/8,1/16, etc) converges into 1 as a geometric series. Despite Zeno’s logic of infinite travelling, the mathematics supports what Zeno’s senses showed, even if physical reality does not, as King Harold would attest. 

 

Logical paradoxes can give the impression of an illogical world – but as Wittgenstein said in his Tractatus, we could not say what an illogical world would look like. It is because language is a human construction that we get into these semantic situations. The statement ‘I am lying’ which as we've said, is false if it’s true and true if it’s false - but why this paradox occurs should be easily seen when we treat language as a mere invention with first-order, second-order (and so on) statements. 

Clearly to avoid self-contradiction, ‘I am lying’ has to be a statement that refers to a statement other than the one being made. If John is lying about where he was last night, then the statement John makes which says “I was round Terry’s last night” needs to be related to the second order statement about the first-order perspective, which is “It is true that I, John, was round Terry’s last night”, which is a statement about a statement. The second order statement is a statement about the first order statement, and here John can be lying by relating his whereabouts to the truth or falsity of his whereabouts – but with ‘I am lying’ he would be mistakenly conflating first and second order statements without the other level with which to correlate the ‘lie’ in question. 

Locke, in his Essay Concerning Human Understanding, talked about the vitalness of language in that it maps words to ideas, concepts or representations in each person's mind. This is one of the principal reasons why humans are the most advanced of all the animals; the ability for one man to share the concepts in his head with the concepts in someone else's head by some form of mutual consent is a key foundation in our being able to construct moral systems, as well as build skyscrapers and jumbo jets. To conjoin private concepts to a word, sentence or paragraph in our common language is to take us one huge step forward in realising the potential of our minds in a shared human reality. To that end, language paradoxes represent us at our most brilliant and at our most frivolous simultaneously.