In one of my books, I have
a chapter called To Err Is Human, To Agree Is Divine, which talks a lot
about Aumann’s Agreement Theorem (which I’ve also blogged about before - see here
and here).
While Robert Aumann proved the original Agreement Theorem using ideal Bayesian
rationality, the brilliant Scott Aaronson explored what happens when the agents
are computationally bounded, and analysed the efficiency of communication
required to reach (near) agreement (see here
for a discussion I had with Scott on my Philosophical Muser YouTube channel).
In this blog, I want to consider an aspect of Scott Aaronson’s paper The Complexity of Agreement (which can be found here) - the "knife-edge" problem - in relation to domestic situations in marriage (my emphasis), where we have the agents “smooth their messages by adding random noise to them”. This use of noise not only helps stabilise communication in the face of minute belief differences (which is the knife-edge problem), but it also ties into Aaronson’s broader result; where he introduces a protocol that allows agents to agree within ε after exchanging O(1/ε²) messages, where the O notation describes how something grows, ε is how close two agents need to come to agreement, and therefore, the number of messages (or amount of communication) needed grows like the square of 1 divided by ε*.
Let me give you a few life examples from chez Knight, where my sweetheart and I humorously navigate the adventurous terrain of domestic harmony. What this will show is a way in which smudging the strength of our preferences through "noisy messaging" can help avoid deadlocks over desires that are only trivially different.
1) Shall we have a
Domino’s pizza or buy a Tesco Finest?
Me: It’s only worth having high quality pizzas, so Domino’s.
Zosia: Tesco Finest is very good quality too.
Noisy message: The outer edge of the Domino’s pizza has no toppings at all, to the circumferential measurement of about 450mm from the outer edge of the dough, and all the toppings are tightly packed within the centre area of the dough, with an area measuring no more than a third of the entire pizza.
Agreement: Let’s go for Tesco Finest.
2) Frequency of husband hoovering
Me: Now we have Robbie the Robot Vacuum Cleaner, does that mean I can hoover less often?
Zosia: No, Robbie doesn’t get in the corners or move furniture around.
Me: Neither do I (just kidding!)
Noisy message: Hoovering is a good opportunity to say prayers and be thankful for things in our home.
Agreement: James doesn’t outsource his hoovering to a robot.
3) Shall we have dessert straight after dinner or wait a bit?
Zosia: I’m full, I don’t think I can eat dessert just yet.
Me: I could eat it now, because eating it later means getting up out of our comfy living room chairs to make it.
Noisy message: If we model the post-dinner energy curve as an exponential decay function, the likelihood of voluntary dessert assembly drops sharply after every easily-accessible After Eight consumed in the living room.
Agreement: Let’s have dessert now, while morale is still high.
4) What shall we watch on TV tonight?
Zosia: How about The Crown?
Me: That’s more of a Sunday afternoon programme, not a Saturday night one.
Noisy message: It’s a bit early in the weekend to be reminded of institutional decline and quiet heartbreak through repressed sighs, ancestral tension and softly-lit drawing rooms.
Agreement: Let’s watch something where no one wears gloves indoors.
5) I’ll bet in a blind test you can’t tell the difference between Aldi peach yoghurts and Lidl peach yoghurts
Me: Yes I can!
Zosia: Right, I’ll grab two spoons and a blindfold!
Noisy message: While initial assumptions rested on a shared flavour profile across discount supermarket brands, empirical testing revealed statistically significant variance in mouthfeel viscosity and synthetic peach note latency.
Agreement: “James, I was wrong, you’re the winner at identifying different peach yoghurts that I mistakenly thought were the same.”
6) It’s important to hit peak boil when making a cuppa
Me: We should hit the pour-ometer within seconds, hun!
Zosia: It doesn’t make that much difference.
Noisy message: We simply can’t be those people who pause mid-pour for idle chat, creating a thermodynamic class system in the mug hierarchy - where the last pour is barely warmer than a laptop monitor, and where some unlucky recipient gets, at best, a tepid travesty of a brew, engendering the quite unjust result of tea temperature inequality.
Agreement: Peak boil, baby!
Those were largely real life domestic situations that occurred in chez Knight, but with amusing embellishments, to show the theory in action. In domestic cases like the above, agreement can be reached by mutual consent through noisy messaging, as long as it applied properly. In a household where laughs are aplenty – as all households should ideally be – couples can blur the strength of preferences, embed a touch of absurdity, or lean on gentle tangents to tip the scales toward shared resolution.
*For example, if ε = 0.1 (we want 90% agreement), then 1/ε² = 1 / (0.1)² = 100, so the process might need about 100 messages. On the other hand, if ε = 0.01 (we want 99% agreement), then 1/ε² = 10,000, so the number of messages might shoot up dramatically. The closer you want the agents to agree, the more work it takes - and it gets expensive fast.
