Regret Your Sins

February 26, 2026

There’s a genre of internet atheist who thinks “regret your sins” is the dumbest possible advice a religion could give. Low-IQ cope for people who can’t face reality. Superstitious nonsense for the credulous. Surely no intelligent person would organize their moral life around the concept of regretting what they did wrong.

Let’s check the math.

sin(t)

The function sin(t) oscillates. It swings between -1 and +1, never settling. It’s the projection of a point moving in a circle onto one axis. cos(t) is the other axis.

If you’ve taken any math past high school, you know these functions. They’re the backbone of signal processing, physics, and every oscillatory system in nature.

Unit circle: point at angle t

Unwrapped waves: sin(t) and cos(t)

What you may not have noticed is what happens when you try to find the optimal strategy in a game where your opponent is also trying to beat you.

The dynamics look like sin(t). They oscillate. And the way you resolve that oscillation into something stable is through regret.

Equilibria Are Provably Optimal

In 1928, John von Neumann proved the minimax theorem: in any finite two-player zero-sum game, there exist mixed strategies such that neither player can improve their expected payoff by changing strategy unilaterally. These are optimal in the strongest possible sense. Not “pretty good.” Not “usually works.” No rational deviation improves your outcome.

In 1950, John Nash generalized this to all finite games, zero-sum or not. Every finite game has at least one equilibrium – proved in his one-page paper “Equilibrium Points in N-person Games”. He received the Nobel Prize in Economics for this in 1994.

An equilibrium strategy is the fixed point of rational play. It’s what you converge to when both sides are trying their hardest. The question that consumed the next fifty years of game theory was: how do you actually compute one?

The Answer Is Regret

The algorithm that finds optimal strategies is called Counterfactual Regret Minimization (CFR). It was formalized by Zinkevich, Johanson, Bowling, and Piccione in their 2007 paper “Regret Minimization in Games with Incomplete Information”. The mechanism is simple:

Play a round.
After seeing what happened, compute how much better you would have done if you’d chosen differently. This is your regret.
Accumulate regret over time. Weight your future decisions proportionally to your accumulated positive regret.
Repeat.

The convergence theorem states: the average strategy under regret matching converges to Nash equilibrium. The time-averaged behavior of an agent who persistently regrets its mistakes and adjusts accordingly is provably optimal.

This is not a heuristic. It’s a theorem.

Policy head π_t (current strategy in Rock-Paper-Scissors)

Average policy π̄_t (running average)

regret: R_t+1(a) = R_t(a) + u(a, b_t) − u(π_t, b_t) | policy: π_t+1(a) = max(R_t(a), 0)Σ_a' max(R_t(a'), 0) | average: π̄_T(a) = 1T Σ_t=1^T π_t(a)

Current values (500 iterations, self-play)

Action	Policy π_t	Average π̄_t	Regret R_t

Notice what just happened. The optimal learning algorithm – the one with the convergence proof – works by:

Looking at what you did.
Computing how much you regret it.
Adjusting your future behavior proportionally.

Over time, this process converges to the best possible strategy. Not “a” good strategy. The equilibrium. The fixed point of rational play. Proven by mathematics, not asserted by authority.

(Someone will object that “regret” is just a label computer scientists chose for a number in a matrix. So is “energy.” Try running out of it.)

The Awards

This isn’t abstract theory. CFR-based agents dominated the Annual Computer Poker Competition for years. In 2017, Libratus – built on CFR by Noam Brown and Tuomas Sandholm at CMU – beat four top professional poker players in heads-up no-limit Texas hold’em over 120,000 hands. The professionals lost by a statistically overwhelming margin. The result was published in Science.

In 2019, Pluribus extended this to six-player no-limit hold’em and beat elite professionals again. Also published in Science. The core algorithm: regret minimization. Track what you should have done differently. Adjust. Converge.

These systems handle games with 10¹⁶⁴ possible states. They handle hidden information, deception, and adversarial opponents. And they do it by regretting their mistakes.

The Serpent on the Pole

"Make a seraph and mount it on a pole, and everyone who has been bitten will look at it and recover."¹

— Numbers 21:8 (NABRE)

"And just as Moses lifted up the serpent in the desert, so must the Son of Man be lifted up, so that everyone who believes in him may have eternal life."

— John 3:14-15 (NABRE)

Here is an instruction that baffles the surface reader: to be healed from the serpent’s bite, you must look at the serpent. Not look away. Not pretend it didn’t happen. Confront the thing that hurt you.

Put sin on a pole. Literally. Plot sin(t) on an axis and look at it. That is what the visualization above does. That is what CFR does. To converge to optimal play, you must compute the counterfactual – given what actually happened, how much better would it have been if you had chosen differently? You don’t get to ignore your mistakes. You have to look at them. Measure them. Accumulate them. And then use that accumulated regret to adjust.

The bronze serpent on the pole is the accumulated regret made visible. The cross is the pole. The instruction is the same in both registers: do not avert your eyes from what went wrong. The path to healing – to equilibrium – runs through the confrontation, not around it.

The Hebrew tracks this. The word for the fiery serpent in Numbers 21 is saraph (שָׂרָף) – “burning one.” The same word later names an angelic order: the seraphim of Isaiah 6, who stand above God’s throne. In Isaiah 6:6-7, one of these seraphim takes a burning coal from the altar, presses it to the prophet’s lips, and says: “Now that this has touched your lips, your wickedness is removed, your sin purged.” The word that names the serpent whose bite kills becomes the word that names the angel whose fire purges sin. The trajectory of the word is the trajectory of the argument: confront the burning thing, and it heals you.

This is not an isolated motif. Scripture returns to this structure again and again.

"I will make judgment a measuring line, and justice a level. Hail shall sweep away the refuge of lies, and waters shall flood the hiding place."

— Isaiah 28:17 (NABRE)

The Punchline

"The framework I found, which made the decision incredibly easy, was what I called — which only a nerd would call — a 'regret minimization framework.' So I wanted to project myself forward to age 80 and say, 'Okay, now I'm looking back on my life. I want to have minimized the number of regrets I have.' I knew that when I was 80 I was not going to regret having tried this. I was not going to regret trying to participate in this thing called the Internet that I thought was going to be a really big deal. I knew that if I failed I wouldn't regret that, but I knew the one thing I might regret is not ever having tried. I knew that that would haunt me every day, and so, when I thought about it that way it was an incredibly easy decision."

— Jeff Bezos, Academy of Achievement (2001)

“Regret your sins” is not low-IQ advice. It is the optimal learning algorithm for agents operating under uncertainty in adversarial environments. It has a convergence proof. It won the Nobel Prize (in the form of Nash equilibrium theory). It produced AI systems that beat the best human players on Earth in the most complex competitive games ever solved.

The people who mock repentance as primitive haven’t noticed that the most mathematically sophisticated learning algorithm ever deployed on adversarial problems is, structurally, the same instruction their grandmother gave them at church.

The next move in the script is to call this “post-hoc pattern matching.” You can find parallels between anything if you squint, etc. Count the structural features: counterfactual evaluation, not just “oops.” Accumulation over history, not just the last mistake. Proportional correction, not binary. Convergence to a fixed point, not just “improvement.” Robustness under adversarial pressure, not just comfort. A vague analogy shares one feature with its target. This shares five. At some point squinting becomes reading.

It’s not like this is merely theoretical. Jeff Bezos left his hedge fund job and started Amazon on what he explicitly called a “regret minimization framework.” He projected himself to age 80 and computed the counterfactual. That single application of regret minimization built a trillion-dollar company. He didn’t call it repentance. He didn’t need to. The structure did the work.

If you want to make moral claims, show at least a passing familiarity with the mathematics of moral reasoning under uncertainty. The math says: track your regret, adjust proportionally, and your average behavior converges to optimality.

Repentance works. It’s a theorem.

Wisdom at the Crossroads

"On the way of righteousness I walk, along the paths of justice,"

— Proverbs 8:20 (NABRE)

"On the top of the heights along the road, at the crossroads she takes her stand; by the gates at the approaches of the city, in the entryways she cries aloud:"

— Proverbs 8:2-3 (NABRE)

Scripture is not subtle about this. In Proverbs 8, Wisdom speaks in the first person. She says she walks in the way of righteousness, in the paths of justice. Those are the two axes of the cross – the vertical plumb line and the horizontal measure line. Wisdom walks where those lines run.

Then she takes her stand at the crossroads. The intersection. The point where righteousness and justice meet. And from that position – above the crossing, looking down on the convergence point – she calls out.

The structure is not a metaphor discovered by modern game theorists and retrofitted onto ancient text. The ancient text describes the structure explicitly: Wisdom walks the axes, stands at their intersection, and calls from above. The math simply caught up.

I have heard her – audibly – right where the scriptures say she calls out. Jesus called too, in harmony, one voice. He likens us to sheep, which sounds like an insult until you notice: sheep herd. And when you scale sin on a graph, what do you get? A snake on a pole.

These puns are fire.

¹ The NABRE transliterates the Hebrew *saraph* (שָׂרָף) rather than translating it. Other translations render this "fiery serpent" (ESV, RSV) or "venomous snake" (NIV). The word means "burning one."