What Logic And Game Theory Says About The Crocodile Paradox

“Your kid is in my mouth,” says the crocodile, smiling around little Billie’s wriggling body. “And I’m going to eat them.”

It pauses for a second. “Or am I?” it says. “The choice is up to you – all you have to do to save them… is to correctly guess whether I eat them or let them go.”

What do you do?

This is the problem known as the crocodile paradox – and here’s how it works.

What’s up with the crocodile paradox?

The crocodile paradox, played right, is a true paradox, in that it’s not possible to resolve it in a way that’s logically coherent. It’s a puzzle in the same vein as the liar’s paradox – the old “this sentence is a lie” statement. 

And we really mean it when we say “old”, by the way: this paradox goes back at least to Eubulides of Miletus, a fourth-century Greek philosopher and contemporary of Aristotle whose entire legacy is the paradoxes he invented. Since then, it was picked up in various forms by Christian thinkers such as Saint Jerome, Hindu philosophers like Bhartrhari, and Muslim scholars like Naṣīr al-Dīn al-Ṭūsī; in more modern times, it even turned up in a more general form at the hands of set theorists like Russell and Zermelo.

The key to all these paradoxes is that they’re what’s called “self-referential” – that is, a statement that asserts something about itself. It’s most obvious in the setup of the liar paradox, but you can see it in the crocodile version too, albeit slightly malformed: “[T]here is a crucial difference between the Liar and the Crocodile Paradox,” points out Stamatios Gerogiorgakis, a philosopher at the University of Erfurt, in a 2016 paper; “[U]nlike what happens in the Liar Paradox, the loop in the Crocodile Paradox is triggered only once a certain prediction is chosen: the prediction that the reward will not be granted.” 

In other words, it’s only some of the time that the crocodile paradox is – well, paradoxical. If you tell the croc he’s going to let the kid go, then there’s no problem: he releases the child, then you were correct, and his prescribed action is to release the child; he eats them, and you got it wrong, so he had to eat them in any case. No paradox here – if anything, it’s a crocodile tautology.

If, on the other hand, you decide to play the pessimist, and tell the crocodile he’s going to eat your kid. Well, then the whole thing glitches out: he can’t eat the child, because then you’ll have guessed correctly, so he has to give the kid back – but if he gives the kid back, then you guessed incorrectly, and he gets to eat them.

“Unlike the Liar who gets automatically involved in the loop as soon as he utters the paradoxical sentence, in the Crocodile Paradox, it is a choice among others to trigger the cyclic reference of the choice to the choosing itself,” explains Gerogiorgakis.

Is it possible to beat the crocodile paradox?

So, your kid’s being chomped by a giant and highly logical reptilian. What do you do?

Well, it’s true that you can’t really logic your way out of this particular pickle – but sometimes, game theory can do what pure reason can’t. Could this be one of those instances?

“In a unique way among the paradoxes of ancient logic, the Crocodile Paradox can be reformulated as a game with four possible strategy pairs, not all of which are cyclic,” writes Gerogiorgakis. “One of them is the Nash equilibrium of the game.” 

Now, that’s a technical term from game theory: the Nash equilibrium of a game is the point where no player has any incentive to change their strategy, even knowing their opponent’s moves too. 

Let’s take the classic prisoner’s dilemma as an example. Two prisoners, each held in solitary confinement with no way of communicating between them, are asked to betray each other. The bribe is a shorter sentence: if prisoner A betrays B, and B remains silent, then A goes free and B serves 10 years; the same is true in reverse if B betrays A and A stays silent; if both narc on each other then they both serve five years, and if neither blab then both serve one year. 

In that case, the Nash equilibrium is given when both prisoners rat the other out. Sure, neither talking would be better for both – but if prisoner A knows that prisoner B is keeping mum, then it makes sense for A to switch to talking. It’s only if both know that the other is betraying them that neither has any incentive to change their strategy.

You can see, then, that the Nash equilibrium is… well, not the optimal outcome, that’s for sure – but perhaps we could call it the least bad overall. So, what’s the Nash equilibrium in the crocodile paradox?

Well, there’s a brutal answer – and an annoying one. The former, Gerogiorgakis suggests, is to just tell the croc it’s going to let your baby go, and take the chance on whether it agrees. That is, of course, unlikely – “It is much less probable that [the crocodile] is a vegetarian teaser,” Gerogiorgakis points out – but perhaps, he says, “a horrible end is better than an endless horror.”

Of course, there’s a problem with that strategy. “The parents’ utility if the crocodile devours the baby is, obviously, extremely negative,” Gerogiorgakis says, “because the parents love their baby and [this] strategy […] makes it a crocodile’s meal.” 

“It is extremely implausible,” he reasons, “that [any] mother or father would endorse” such a move. But here’s the thing: switching strategy doesn’t just fail to solve the paradox. It creates a whole new one. 

Aw, snap.

“Independently of the choice which the crocodile makes, the parents will not regret their choice to predict ‘You will devour our baby’ as their strategy,” Gerogiorgakis calculates. “Alas, the presented choice is cyclic!” 

In other words, all you’ve done is move the paradox from the outcome of the decision to the decision itself. The Nash equilibrium is given by the parents choosing to think the worst of the croc, and the croc choosing to go ahead and prove them right – but as we’ve seen, doing that would mean he would have to let the child go free. 

It is, in philosophical terms, a “rationality gap” – a situation in which there’s no way to know which option is the rational one. “The rational choice turns out to be irrational, then rational, then irrational and so on,” Gerogiorgakis explains. “The parents’ choice involves a rationality gap since there are two strategy pairs each of which makes the other the best choice but eo ipso itself a suboptimal choice in every iteration. There is no way to determine once and for all which choice is rational.”

So, after all that, is it possible to beat the crocodile paradox? Well, yes – but only if you’d rather see your child devoured in front of you than get stuck in a logical loop. 

Otherwise, the best option might just be to avoid the reptile house – and any nearby philosophy departments.