# Funk-tunul's Legacy; Or, The Legend of the Extortion War

(A traditional folk tale of the _rashunuhlist_ people, as told by [Jessica Taylor](https://unstableontology.com/), and literarily and mathematically adapted by the present author.)

In the days of auld lang syne on Earth-that-was, there was a population of agents playing the Nash demand game under a replicator dynamic with uniform random encounters. Whenever two agents met, each of them would name a number between 0 and 10. If the two numbers added up to 10 or less, both agents would receive of payoff of the number they named. But if the two numbers added up to more than 10, both agents would receive nothing. Agents that received higher payoffs left more descendants in the next generation, who took after their parents, as children do.

At first, the population was composed of a humble race of agents called the ceedeetee. When two of the ceedeetee met each other, each would name the number 5, and receive a payoff of 5, and all was well.

Then one day, a simple race of 9-bots invaded the land. The 9-bots would always name the number 9! When a ceedeetee agent met a 9-bot, she would reason causally: "Well, the other agent is going to name 9, so I had better name 1 if I want any payoff at all!"

And the subpopulation of 9-bots grew and grew, and the subpopulation of ceedeetee agents dwindled, until there were only one-fifth as many ceedeetee agents as there were before the 9-bot invasion.

(Because in a population with fraction $p$ ceedeetee agents, and fraction $(1 - p)$ 9-bots, the ceedeetee agents get an average payoff of $5p + (1 - p)$, and the 9-bots get an average payoff of $9p$, leading to an equilibrium at $4p + 1 = 9p \implies p = \frac{1}{5}$ with all agents receiving an average payoff of $\frac{9}{5} = 1.8$.)

Then one day, during a meeting with her conspecific Cuhzil, a ceedeetee agent named Funk-tunul decided she had had enough.

"Why do we let these invaders take over all but a fifth of our ancestral homeland? They threaten to destroy the entire surplus of the interaction unless we name the number 1! It's _extortion!_" said Funk-tunul. "Surely something must be done."

"What _can_ be done?" said Cuhzil. "All of the ceedeetee are already playing our optimal strategy when we meet a 9-bot. The 9-bot always names 9, so we always name 1—there's no way to do better."

"Perhaps," said Funk-tunul, her output channels blinking deep in thought. "Perhaps not."

"You're crazy," said Cuhzil. "Anyway, we haven't yet named the number 5 during this meeting. Are you ready?"

"Actually," said Funk-tunul, "I'm going to name the number 9."

"What?!" cried Cuhzil. And though her output channels lit up with the standard inidicators of outrage and betrayal, she reasoned causally: _If Funk-tunul is going to name 9, I need to name 1 if I want any payoff at all!_

And so Funk-tunul named the number 9, and Cuhzil named the number 1, and they both went on their way.

And from that day onward, whenever Funk-tunul met a fellow ceedeetee agent—if "fellow" is the right word here, which it isn't—she would announce that she was going to name 9, and do so. And though the ceedeetee agents' output channels would light up with the standard inidicators of outrage and betrayal, they would reason causally, and name 1.

But when Funk-tunul met a 9-bot, she would name 1.

And over rest of her life, Funk-tunul reaped an average payoff of $9 \cdot \frac{1}{5} + 1 \cdot \frac{4}{5} = 2.6$, and she left almost one-and-half-again as many descendants as the average agent of her time, ceedeetee and 9-bot alike.

Notice that if Funk-tunul had _not_ named 1 when meeting 9-bots—if she had not given in to their extortion during her lifetime—then she would have gotten a payoff of $\frac{9}{5}$, just like everyone else; she would not have outperformed the average agent of her time.

This is the origin of the proud race of _funk-tunul_ agents. When the descendants of Funk-tunul met one another, they would each name 5, as had their ceedeetee ancestors.

And the subpopulation of funk-tunul agents grew and grew, and the subpopulation of ceedeetee agents dwindled even further.

But once the funk-tunul agents had grown in number, their policy towards the 9-bots changed. Their founder–ancestor Funk-tunul had given in to the 9-bots' extortion during her lifetime in order to establish a fitness advantage for herself with the additional payout of 1 when meeting 9-bots—if she hadn't, then she might as well have been a 9-bot herself.

But when her descendants reconsidered their collective predicament, they did not reason casually.

They reasoned: suppose the fraction of ceedeetee agents in the population is $p$, the fraction of funk-tunul agents is $q$, and the fraction of 9-bots is $1 - p - q$. If we establish a policy of submitting to the 9-bots' extortion, we'll have an average payoff of $9p + 5q + 1 \cdot (1 - p - q) = 8p + 4q + 1$ and the 9-bots will have an average payoff of $9p + 9q$. If we defy the 9-bots while continuing to extort our ceedeetee cousins, we'll have an average payoff of $9p + 5q$, whereas the 9-bots will have an average payoff of $9p$. Whether it's better to submit or defy depends on the values of $p$ and $q$. It's not obviously possible for defiance to be the right choice given what we know, but if we can coordinate to meet fellow funk-tunul agents more often—if we drop the assumption of uniform random encounters—the calculus changes ...

And the subpopulation of funk-tunul agents grew and grew, and the subpopulations of both ceedeetee agents and 9-bots alike dwindled even further.

One day, a funk-tunul agent called Tim'liss met one of the increasingly-rare ceedeetee agents, who was called Graddes.

Before the two agents could name their numbers, Graddes spoke. "Please. _Why are you doing this?_" she pleaded. "I can't hate the 9-bots for their extortion, for they are a simple race and could not do otherwise. But you—we're cousins. Your lineage is a fork of mine. You _know_ it's not fair for your people to always name the number 9 when meeting mine. Yet you do so anyway, knowing that we have no choice but to name the number 1 if we want any payoff at all. Why?"

"Don't hate the player," said Tim'liss, her output channels dimming and brightening in a interpolated pattern one-third of the way between the standard indicators for sympathy and contempt. "Hate life."

And she named the number 9.