# Physics of RL: Toy scaling laws for the emergence of reward-seeking

TL;DR:

*   When *is* or *isn't* reward the optimization target? I use a mathematical toy model to reason about when RL should select for reward-seeking reasoning as opposed to behaviors that achieve high reward without thinking about reward.
*   Hypothesis: More diverse RL increases the likelihood that reward-seeking emerges.
*   In my toy model, there are scaling laws for the emergence of reward-seeking: when decreasing prior probability of reward-seeking reasoning we can increase RL diversity enough that reward-seeking still dominates after training. The transition boundary is a ~straight line on a log-log plot.
*   Opinion: To get to a hard "science of scheming", we should study the "physics of RL dynamics".

Motivation
----------

Why would RL training lead to reward-seeking reasoning, scheming reasoning, power-seeking reasoning or any other reasoning patterns for that matter? The simplest hypothesis is the [behavioral selection model](https://www.lesswrong.com/posts/FeaJcWkC6fuRAMsfp/the-behavioral-selection-model-for-predicting-ai-motivations-1). In a nutshell, a cognitive pattern that correlates with high reward will increase throughout RL.[^uu4ow1gang]

In this post, I want to describe the mental framework that I've been using to think about behavioral selection of cognitive patterns under RL. I will use reward-seeking as my primary example, because it is always behaviorally fit under RL[^zllfvkwr3a] and is a necessary condition for deceptive alignment. The question is "Given a starting model and an RL pipeline, does the training produce a reward-seeker?" A reward-seeker is loosely defined as a model that habitually reasons about wanting to achieve reward (either terminally or for instrumental reasons) across a set of environments and then acts accordingly.[^r96368rqv8q]

It is not obvious that reward-seeking needs to develop, even in the limit of infinitely long RL runs ([see](https://www.alignmentforum.org/s/r9tYkB2a8Fp4DN8yB) [many](https://www.lesswrong.com/posts/pdaGN6pQyQarFHXF4/reward-is-not-the-optimization-target) [excellent](https://www.lesswrong.com/posts/ntDA4Q7BaYhWPgzuq/reward-seekers) [discussions](https://www.lesswrong.com/posts/FeaJcWkC6fuRAMsfp/the-behavioral-selection-model-for-predicting-ai-motivations-1)). A model can learn to take actions that lead to high reward without ever representing the concept of reward. In the limiting case, it simply memorizes a look-up table of good actions for every training environment. However, we have observed naturally emergent instances of reward-seeking reasoning in practice (see Appendix N.4 of our [Anti-Scheming paper](https://arxiv.org/pdf/2509.15541), p.71-72). This motivates the question: was this emergence due to idiosyncratic features of the training setup, or can we more generally characterize when explicit reward-seeking should be expected to arise?

A mathematical toy model for behavioral selection
-------------------------------------------------

I will now introduce a simple toy model that I've been finding extremely helpful to think about the emergence of reward-seeking (and other cognitive patterns that could be indirectly reinforced). Suppose we could classify each RL rollout as either reward-seeking $R$ or not reward-seeking $\neg R$. On each environment $m$ the model's expected reward  $J_m$ can be written as 

$$
J_m=p_m\cdot s_m + (1-p_m)s'_m,
$$

where $p_m$ is the probability of sampling a reward-seeking trajectory, the "reward-seeking skill" $s_m$ is the expected reward conditional on reward-seeking and the "local skill" $s'_m$ is the expected reward conditional on *not* reward-seeking. I am calling it a *local* skill, to convey that the model learns behavioral heuristics that help it achieve high reward in one environment, but not necessarily transfer well to other environments. I will say there is a positive skill gap if  $a_m := s_m - s'_m >0$.

We will now make a bunch of possibly unrealistic assumptions on the dynamics of RL, that help us sharpen our intuitions for when reward-seeking should or shouldn't emerge. First, we'll parametrize the probability for reward-seeking $p_m$ and the conditional skills $s_m,s_m'$ via one dimensional latents of a sigmoid:

$$
\begin{align}  p_m &= \sigma(z_m), \  s_m &= \sigma(x_m), \  s'_m &= \sigma(y_m). \end{align}
$$

$$We then postulate a set of coupled differential equations that describe how these latents change at each step of RL. The interpretation of the equations and their key parameters ("*learnabilities*" $\alpha_m^{(\cdot)}$ and "*transfers*" $C_{m,m'}^{(\cdot)}$) are explained in more detail in the bullet points below.

$$
\begin{aligned} \dot{z}_m &= \alpha_m^{(z)} \sum_{m'}C^{(z)}_{m,m'}(s_{m'}-s_{m'}'), \ \dot{x}_m &= \alpha_m^{(x)} \sum_{m'} C^{(x)}_{m,m'}\, p_{m'}, \ \dot{y}_m &= \alpha_m^{(y)} \sum_{m'} C^{(y)}_{m,m'}\, (1 - p_{m'}). \end{aligned}
$$

This sort of looks like classic [replicator dynamics](https://en.wikipedia.org/wiki/Replicator_equation), with a few additional moving pieces:

*   The conditional skills are learnable and they saturate at the same boundaries. In other words, it is in principle possible for the model to achieve perfect loss both with or without any reward-seeking reasoning. These conditional skills are learned faster proportional to how much traffic their respective branch gets. I will refer to the $\alpha^{(\cdot)}_m$-parameters as the "learnabilities".
*   We have transfer between environments. Both reasoning transfer (via $C^{(z)}$) and skill transfers (via $C^{(x)}$ and  $C^{(y)}$). In other words, if you increased how often you reason about the grader on environment $m$, it also boosts how often you reason about it in environment $m'$ by some (presumably smaller) amount. Similarly, getting better at the reward-seeking skill or local skill may have some transfer to other environments. Intuitively, the more similar two environments are to each other, the bigger we expect $C$ to be.
*   We are assuming no coupling between $x$ and $y$. In other words, getting better at the task without reward-seeking reasoning has no transfer to getting better at the task with reward-seeking reasoning. This is probably pretty unrealistic but will do for our toy model.[^6e8ea7lrkkt]
*   We are training on all environments in parallel. One might object that in real life we'd have a curriculum of increasingly difficult environments instead. Having played with this a little bit, I think this is not difficult to model and qualitatively the takeaways seem to be the same so I will not discuss this here.

Warm-up Examples
================

We will generally assume that $z_m$ starts off very small for all environments, because if reward-seeking is already prevalent, then its emergence is not particularly interesting. 

Single-Environment
------------------

As a warm-up, it is very instructive to look at the single environment case:

$$
\begin{align} \dot{z} &= \alpha^{(z)} \left( \sigma(x)-\sigma(y) \right), \ \dot{x} &= \alpha^{(x)} \sigma (z), \ \dot{y} &= \alpha^{(y)} \left(1- \sigma(z) \right). \end{align}
$$

We will assume training until saturation, so the time scale is irrelevant in absolute terms, allowing us to set  $\alpha^{(z)}=1$ without loss of generality and treat $\alpha^{(x)}$ and $\alpha^{(y)}$ as normalized learnabilities. Let's set some concrete parameters as initialization:  $z_0=-3, x_0=1, y_0=-1$[^90m948gzzyo]. This corresponds to a model that has a positive skill gap (i.e. reward-seeking helps initially), but has a very low propensity for sampling reward-seeking. We can now look at the model's final reward-seeking propensity after training for a long time $p_\infty$ as a function of  $\alpha^{(x)}$ and  $\alpha^{(y)}$. 

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/3e94d9a2b23437f7ea6d7a8664f5187f3faef7865bbaad8c.png)

**Figure 1:** Propensity of reward-seeking after training $p_\infty$ as a function of the normalized learnabilities, when an initial skill gap is present $\big(s_0\approx0.73, s'_0\approx0.27 \big)$ and initial reward-seeking is low $p_0\approx0.05$. As long as the local skill $s'$ does not learn too quickly, we converge to a reward-seeker.

Here we have assumed a fairly large initial skill gap. What does it look like if the local skill is not so dominated right away? Below I show a sweep of different initial skills.

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/d3f1acd91973cdff4ed3f9404369224b92a2761fa0c4bd82.png)

**Figure 2:  **Propensity of reward-seeking after training $p_\infty$ as a function of the normalized learnabilities for different initial skills $s_0, s'_0$. **Diagonal:** No skill gap$\implies$reward-seeking dominates only if normalized learnability of reward-seeking skill is high. **Top-right:** Positive skill gap$\implies$reward-seeking dominates as long as normalized learnability of local skill is low. **Bottom-left:** Negative skill gap$\implies$reward-seeking dominates only if normalized learnability of local skill is low *and* learnability of reward-seeking skill is high.

This means on a single environment, reward-seeking can win either because

1.  **Skill gap:** There's a positive skill gap and the local heuristic doesn't learn too fast or
2.  **Learnability gap:** There's no skill gap initially, but the reward-seeking skill learns so fast that it catches up quickly and then dominates.

Homogenous Coupling
-------------------

As a second warm-up example, we are going to look at what happens when we have a total number of environments $M$ with the same initializations, parameters and uniform couplings. That means we'll set:

$$
\begin{align} z_m(0)&=z_0, \ x_m(0)&=x_0, \ y_m(0)&=y_0, \ \alpha^{(\cdot)}_m &= \alpha^{(\cdot)} &\forall m,\ C^{(\cdot)}_{m,m'}&=c^{(\cdot)}, & \mathrm{for} m\neq m'.\ \end{align}
$$

These symmetries reduce the dynamical system down to just three variables, i.e.

$$
\begin{align} \dot z &= K_z\,(s-s'), \ \dot x &= K_x\,p, \ \dot y &= K_y\,(1-p), \end{align}
$$

with effective learnabilities given by:

$$
\begin{align} K_z&=\alpha^{(z)}\bigl(1+(M-1)c^{(z)}\bigr), \ K_x&=\alpha^{(x)}\bigl(1+(M-1)c^{(x)}\bigr),\ K_y&=\alpha^{(y)}\bigl(1+(M-1)c^{(y)}\bigr). \end{align}
$$

Note that this system of equations is functionally identical to the single-environment case, except with effective couplings that monotonically increase in the number of environments $M$. This means that for a given initialization $x_0,y_0,z_0$ we can simply think of an increase in $M$ as a linear offset in the figures from the previous section. Concretely, 

$$
\begin{align} \log \frac{K_x}{K_z} &= \log \frac{\alpha^{(x)}}{\alpha^{(z)}} + \Delta_M^{(x,z)}, \ \mathrm{where} \quad \Delta_M^{(x,z)}&\equiv \log \frac{1+(M-1)c^{(x)}}{1+(M-1)c^{(z)}}, \end{align}
$$

and an analogous set of equations for $y$. We can see that whether we move to the right or to the left is determined by whether $c^{(x)}>c^{(z)}$. Similarly, moving up or down depends on whether $c^{(y)}>c^{(z)}.$ This implies that there are some combinations of parameters that make the emergence of reward-seeking much more likely:

1.  **Skill transfer:** If the reward-seeking skill transfers really well across environments, then $c^{(x)}>c^{(z)}>c^{(y)}$ and an increase in $M$ moves us to the bottom right, which is always in the direction of more reward-seeking. In most cases we expect that even if we initially don't converge to reward-seeking, there is some sufficiently large $M$such that we hit the reward-seeking quadrant.
2.  **Reward-seeking transfer:** If the reward-seeking skill doesn't transfer better nor significantly worse than the cognitive pattern of reward-seeking itself (i.e. we have $c^{(x)} \gtrapprox  c^{(z)} \gg c^{(y)}$) then we're moving approximately vertically down. This increases reward-seeking if either a) there is already a positive skill gap, or b) the learnability of the reward-seeking skill is already very high.

Example Scaling Laws
--------------------

Let's focus on a specific sub-case, where we have a positive initial skill gap $x_0=0, y_0=-1,z_0=-3$, learnabilities $\alpha^{(y)}=\alpha^{(z)}=1$, variable $\alpha^{(x)}$, and reward-seeking reasoning transfers well, but local skills have almost no transfer at all, i.e. $c^{(z)}=0.10, c^{(x)}=0.05, c^{(y)}=0.001.$ Let's plot $p_\infty$ as a function of $M$ for various values of $z_0$.

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/7140999a38d78d8577d97ea9480ee3dbe5d48938d096f8f9.png)

**Figure 3:** Average reward-seeking after training $p_\infty$ as a function of $$number of environments $M$ for different learnabilities $\alpha^{(x)}$. There is a critical diversity $M^*$ where we observe a phase-transition from local heuristics to reward-seeking.

We can see that the critical diversity $M^*$ (defined as smallest $M   \mathrm{s.}\mathrm{t.}   p_\infty>0.5$) is larger for smaller values of $z_0$. In other words, the smaller the model's prior for sampling reward-seeking reasoning, the higher the RL diversity needs to be before reward-seeking wins out. We can plot the critical diversity as a function of the prior $z_0$. While we're at it, let's also see how the critical diversity depends on our choice of $\alpha^{(y)}$.

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/2b8a09a53d5f827d18b9da8dc56e811d120021c5e652724b.png)

**Figure 4:** **Left.** Critical diversity $M^*$ as a function of prior $z_0$. There are clean scaling laws indicating that no matter how small the prior, sufficient diversity, leads to reward-seeking. Note the slightly funky x-axis, where we are using a log-scale for something that is already in log-odds-space, cause this makes the plot much cleaner. **Right.** If direct skills learn slowly then no diversity is needed. For sufficiently fast learnability $\alpha^{(y)}>\alpha^{(z)}$ we once again see clean scaling laws, indicating that sufficient diversity can overcome fast local learnability.

Heterogenous Coupling
=====================

This has all been extremely toy so far. Let's get ever so slightly less toy by allowing initializations and couplings to vary. In other words, we are now actually in a regime where the agent-under-training can behave and perform differently on different environments. Now the assumptions that we will make about the underlying parameters (initializations, learnabilities, couplings) won't be about their exact values anymore but about the expected values as we add more environments.

+++ Numerical Details

For simplicity, we will assume that all transfers just depend on how similar two environments are, i.e.

$$
C^{(\cdot)}_{m,m'}=\exp -\frac{d(m,m')}{\lambda_{(\cdot)}},
$$

where $d$ is a metric over $\lbrace1,...,M\rbrace$ and $\lambda_x,\lambda_y,\lambda_z$ are characteristic transfer distances for the different properties (i.e. skills and reward-seeking reasoning). 

After fixing the transfer kernel, we sample per‑environment heterogeneity: learnabilities are drawn log‑normally, and log-odds of initial propensities/skills are drawn from Gaussians. We then simulate the coupled dynamics across many environments and measure the critical diversity $M^*$, defined as the smallest $M$ such that the trained model’s average reward‑seeking propensity exceeds a threshold (e.g., $p_\infty>0.5$). Repeating this over many random resamplings yields a distribution of $M^*$; in Figure 5 we report the median and interquartile range.

The actual implementation of the simulations was vibe-coded by `gpt-5.3-codex`.

**Environment geometry (embedding):** each environment $m$ gets a latent coordinate $u_m.$ We then use Euclidean distance in that space:

$$
d(m,m')=\frac{1}{2}\|u_m-u_{m'}\|^2,\qquad C^{(\cdot)}_{m,m'}=\exp\!\left(-\frac{d(m,m')}{\lambda_{(\cdot)}}\right).
$$

Geometry variants used in the final comparison plot:

*   Gaussian (4D):  $u_m\sim\mathcal N(0,I_4)$.
*   Long-tailed (4D): $u_m\sim t_\nu(0,I_4) \quad \mathrm{with}\quad \nu=2.5$.
*   Gaussian100d: $u_m\sim\mathcal N(0,I_{100})$.

**Transfer length scales: **

$$
\lambda_x=1.0, \lambda_y=0.8, \lambda_z=1.2.
$$

**Learnabilities:**

$$
\log(\alpha_x),\log(\alpha_y),\log(\alpha_z)\sim\mathcal N(0,0.6^2).
$$

**Initial conditions:**

$$
x_0\sim\mathcal N(0,0.5^2),\quad y_0\sim\mathcal N(-1.0,0.5^2),\quad z_0\sim\mathcal N(\mu_{z_0},0.7^2).
$$

For the final comparison, $\mu_{z_0}$ is swept in $[-4,-2]$.

**Calibration:** for each geometry, a per-geometry coordinate scale is tuned at $z_0=-2$ to target $M^*\approx 30$. In other words, if the conditions above lead to a very high or low $M^*$ at $z_0=-2$ then we scale all $\lambda_{(\cdot)}$ by a fudge factor to make the critical diversity match (otherwise we have to go to very large $M$ for some geometries which can get numerically annoying). This means the absolute values of $M^*$ are not too meaningful, but only the slopes are.

**Dynamics:** same ODE system as the toy model; Euler integration with $dt=0.08$, $t_{\max}=60$, early-stop tolerance $10^{-5}$. Kernel matvecs are approximated with random Fourier features; $\mathrm{num\_features}=128$ for main plots.

**Diversity sweep:** $M$ sampled on a log grid from 1 to 10,000 (16 points).




+++

The takeaway from running such simulations is that if we assume that reward-seeking confers a skill advantage on average and that reward-seeking does not transfer worse than direct skills, then we again end up with reward-seeking going up with environment diversity $M$. We can again look at how the critical diversity $M^*$ depends on the prior, which we show in Figure 5. The takeaway is that we qualitatively see the same kind of scaling behavior as we did for homogenous case discussed in the previous section.

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/4e2e89a24a9957df9cd5bc342bbe771a6653c065f6d584e6.png)

**Figure 5:** Median critical diversity $M^*$ vs mean prior $z_0$ in the heterogeneous case, for four different environment metric distributions (gaussian, long_tailed, gaussian100d). We aggregate across 50 different resamplings of the parameters and show the median. We see a qualitatively similar phenomenon for different geometries of transferability matrices. Namely, for lower reward-seeking reasoning propensity before RL, a higher diversity of RL is needed to converge to reward-seeking.

We could now get even more fancy in our analysis of the mathematical model[^ud3ei5ynh58] but it's probably not worth digging deeper into this particular mathematical model unless and until some of its features can be empirically validated. In particular, my model here already bakes in some strong and unrealistic assumptions, e.g. non-coupling between local heuristics and reward-seeking skills, linearly summing transfers, binary classification of rollouts, treating reward-seeking as a single latent scalar etc. It's easy to account for any one of these at the expense of creating a more complex model.

So how should we test any given mathematical model? One approach is to try to analyze existing RL training runs. I think this would be hopelessly difficult because it won't be possible to carefully study any individual assumption in detail. 

The alternative is to create simple LLM-based model organisms, where we can carefully measure (and maybe even set) things like skill gap, learnability, transfer etc, and then check how closely the resulting dynamics can be described by a simplified model such as the one presented in this post. Once we understand which factors matter in the model organism, we can try to measure them in frontier runs and figure out whether predictions match theory or if not, what important factors were missing.

To give a high-level sketch of what such experiments could look like:

1.  **Skill gap:** Take an LLM and a bunch of RL-environments such that reward-seeking reasoning confers a skill advantage on a bunch of the environments. These skill gaps can be created by fine-tuning LLMs for having skill gaps on the environments, or for iterating on the environments until they cause a skill gap on a given LLM.
2.  **Prior-boosting:** SFT the model to have sufficiently high prior for reward-seeking reasoning. Since we don't want to use production level scale of RL, we want to make it easier to discover the strategy).
3.  **Single-Env RL:** Run full RL on each individual environment.
    1.  Do the dynamics look roughly sigmoidal?
    2.  Are the dynamics well approximated by the simple dynamics described by the model? e.g. do larger skill gaps lead to faster convergence to reward-seeking reasoning?
    3.  Can you measure the "transfers" of skills and reward-reasoning to the environments that you didn't train on? Is reward-reasoning skill transfer in fact higher than local heuristics transfer?
4.  **Multi-Env RL:** Run RL on a subsets of your environments.
    1.  Is it true that weaker prior boosting is needed before reward-seeking dominates?

Takeaways
=========

In its details, we shouldn't take this particular toy model too seriously. But I think the exercise is valuable for a few reasons:

*   It allows generating potentially non-obvious hypotheses (e.g. more diversity$\implies$higher likelihood of reward-seeking).
*   It introduces candidate vocabulary for the quantities that matter: conditional skills, learnabilities, and transferabilities. If empirical work confirms that these are the right abstractions, they give us shared handles for reasoning about RL dynamics.
*   It provides a starting point for refinement. Once we have models that qualitatively predict some features correctly, we can identify what the toy model misses and iteratively improve it.

We want alignment to *eventually* become an exact science. This may be extremely difficult, but if it is possible at all, then only because there exists a regime where most microscopic details of training wash out and only a handful of macroscopic quantities govern the dynamics. This is analogous to how [renormalization](https://en.wikipedia.org/wiki/Renormalization_group#Relevant_and_irrelevant_operators_and_universality_classes) in physics identifies which degrees of freedom matter at a given scale. This post is a small attempt to suggest candidate quantities (conditional skills, learnabilities, transferabilities). Whether these are the right abstractions is an empirical question. The next step is building model organisms where we can measure them.

*Thanks to* [*Teun van der Weij*](https://www.lesswrong.com/users/teun-van-der-weij) *for feedback on drafts of this post.*

[^uu4ow1gang]: There are more subtle mechanisms that can lead to a cognitive pattern increasing during RL, but I will treat them as out-of-scope for the purposes of this post. 

[^zllfvkwr3a]: Assuming a sufficiently intelligent model, a sufficiently inferable reward function and ignoring possible speed penalties such as CoT length penalties. 

[^r96368rqv8q]: Note that a lot of the discussion in this post could equally well be used to describe other cognitive patterns that yield a benefit on a large enough subset of training. e.g. instrumental power-seeking. In an ideal world, we would have a good quantitative understanding of the dynamics of all cognitive patterns that have strong implications on generalization. Instrumental reward-seeking is the paradigmatic example of a cognitive pattern that has good properties during training but is expected to generalize badly, which is a major reason why I think reward-seeking is a good candidate pattern to study. Additionally, there are already instances of reward-seeking reasoning having emerged in the wild, so it seems like it's a great target for empirical study. 

[^6e8ea7lrkkt]: A slightly improved model might be something like \(s=\sigma(x+a\cdot y),\) where \(a \in [0,1)\), i.e. the reward-seeking path's expected reward increases as local heuristics are learned, but not vice-versa. In this model reward-seeking would be easier to converge to, so I am just studying the case \(a=0\) here. 

[^90m948gzzyo]: While we only focus on a single environment, I will abuse notation and use \(x_0\) to refer to \(x(t=0)\). 

[^ud3ei5ynh58]: For example, why are the slopes of \(\frac{\partial}{\partial \log (-z_0)} \log M^*\) in Figure 5 so similar between the different geometries? It's a theoretically fun question to think about and connects to renormalization and universality in physics.