The Physics of Agency, Part 3: The Kybit — A New Unit of Control

How to Quantify Intentional Influence on Future Outcomes

The Kybit: A Fundamental Measure of Control

In the previous post, we explored the thermodynamic struggle between agency and drift.

Now we introduce a precise way to quantify intentional control: the kybit.

Just as the bit measures information, the kybit measures an agent's intentional influence on future outcomes.

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Control: Kybits as KL Divergence

The kybit measure can quantify the total control exerted when shifting probability from an initial distribution to a final distribution using the Kullback–Leibler (KL) divergence:

\(D_{KL}(P_2 \parallel P_1) = \sum_{i} P_2(i)\log_2\frac{P_2(i)}{P_1(i)}\)

This measure:

• Is always non-negative, being zero only if the initial and final distributions are identical.

• Represents the total thermodynamic work required to rearrange the probability distribution.

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Illustrative Examples

Example: KL Divergence Calculation for a Forced Coin Flip

Initial Probabilities:

• Heads: 0.5

• Tails: 0.5

Final Probabilities (forced to heads):

• Heads: 1.0

• Tails: 0.0

KL Divergence Calculation

Substituting the values for our coin flip into the KL Divergence equation:

\(D_{KL} = (1.0) \log_2 \frac{1.0}{0.5} + (0.0) \log_2 \frac{0.0}{0.5}\)

Evaluating each term:

• First term:

  \((1.0) \log_2 \frac{1.0}{0.5} = 1.0 \times \log_2(2) = 1.0\)

• Second term (zero probability outcome):

  \((0.0) \log_2 \frac{0.0}{0.5} = 0 \quad \text{(by the standard limit convention)}\)

Thus, the total KL divergence is:

\(D_{KL}(P_{final} || P_{initial}) = 1.0 \text{ kybit}\)

Interpretation

Forcing a previously uncertain (50/50) outcome to complete certainty (heads) incurs exactly 1 kybit of control. The zero-probability outcome (tails) contributes zero by convention, making KL divergence robust and reliable.

Example: KL Divergence for a Biased Die Roll

Initial Probabilities (fair die):

• Faces 1 through 6: each 1/6 ≈ 0.1667

Final Probabilities (biased die):

• Face 1: 0.5

• Face 2: 0.25

• Faces 3–6: each 0.0625 (1/16)

KL Divergence Calculation

The KL divergence is defined as:

\(D_{KL}(P_{final} || P_{initial}) = \sum_{O} P_{final}(O) \log_2 \frac{P_{final}(O)}{P_{initial}(O)}\)

Substituting the probabilities:

\(\begin{aligned} D_{KL} &= (0.5) \log_2 \frac{0.5}{1/6} + (0.25) \log_2 \frac{0.25}{1/6} + 4 \times (0.0625) \log_2 \frac{0.0625}{1/6} \\ &= 0.5 \times \log_2(3) + 0.25 \times \log_2(1.5) + 0.25 \times \log_2(0.375) \\ &\approx 0.7925 + 0.1463 - 0.3538 \\ &\approx 0.585 \text{ kybits} \end{aligned}\)

Interpretation

Biasing a fair die roll to favor certain outcomes requires about 0.585 kybits of control. KL divergence robustly quantifies the total measure of control exerted to reshape the probability distribution from an initial fair state to a biased final state.

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Physical Reality of Kybits

Kybits represent real thermodynamic costs. Each kybit corresponds to a minimum energy expenditure required to intentionally bias outcomes, analogous to Landauer's principle in thermodynamics:

\(Wmin= k_B T \ln 2 \quad \text{per kybit}\)

where k is Boltzmann's constant and T is the temperature.

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Where We're Headed

In the next post, we will formalize the Law of Control Work, explicitly connecting kybits of control to measurable thermodynamic work.

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Reflective Question

How might explicitly quantifying total intentional control using KL divergence reshape the understanding of agency in decision theory and related fields?

(Please share your reflections in the comments.)

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Next post: "The Law of Control Work: Agency Costs Energy"