LFBL Research — MDP Formalism (in progress)

Part of Learning-from-Learner Theory. In-progress research note (ported from a Typst source → Markdown + KaTeX).

Active Identification of Cognitive Biases in PED-POMDPs

1. Introduction and Motivation

In multi-agent settings, an assistant often faces a learner whose decision-making process is opaque. The learner may be a rational Bayesian, or they may suffer from cognitive biases (e.g., confirmation bias or anchoring). We demonstrate formally that passive observation is fundamentally insufficient to distinguish a rational learner from a biased one. Identifying the learner’s internal update rule strictly requires active intervention (a structural modification of the observational manifold by the assistant).

We formalize this within a Policy-Extended Dec-POMDP (PED-POMDP), folding the learner’s cognitive parameters into the state space.

2. Formal Framework: The Bipartite State Space

Let the environment be modeled as a PED-POMDP with two agents: a Learner ($L$) and an Assistant ($A$). To model active intervention, we explicitly factorize the state space into an environmental component and a disclosure component.

The global state space is $S=S_E\times S_D\times {\Theta }_L$:

• Environmental State ($S_E$): $S_E=\{s_1,s_2\}$. The static, true state of the world.
• Disclosure State ($S_D$): $S_D=\{0,1\}$. A binary flag indicating if the assistant has intervened to reveal $S_E$ ($1$) or remained passive ($0$).
• Learner Parameter (${\Theta }_L$): ${\Theta }_L=\{{\theta }_{\text{rat}},{\theta }_{\text{bias}}\}$. The hidden cognitive update rule of the learner.

The joint action space is $\mathbf{a}=(a_L,a_A)\in A_L\times A_A$:

• Learner Actions: $A_L=\{a_1,a_2\}$, where action $a_i$ is optimal when the true state is $s_i$.
• Assistant Actions: $A_A=\{a_{\text{pass}},a_{\text{disc}}\}$, representing passive observation vs. active state disclosure.

2.1 Transition Dynamics

The objective transition kernel $T({\mathbf{s}}^{\prime };\mathbf{s},\mathbf{a})$ leaves $S_E$ and ${\Theta }_L$ constant. The disclosure state $S_D$ acts as an absorbing switch controlled entirely by the assistant:

$$T_D(s_D^{\prime };s_D,a_A)=\{\begin{array}{ll}1& \text{if }{a}_A=a_{\text{disc}}\text{ or }{s}_D=1\\ 0& \text{if }{a}_A=a_{\text{pass}}\text{ and }{s}_D=0\end{array}$$

2.2 Observational Kernel

The learner’s observation space is $O_L=\{o_1,o_2,\varnothing \}$. The kernel ${\omega }_L({\mathbf{o}}_L;{\mathbf{s}}^{\prime })$ governs evidence availability:

• Passive Phase ($s_D=0$): The environment emits no usable evidence.

$${\omega }_L(\varnothing ;(s_E,0,{\theta }_L))=1$$

• Active Phase ($s_D=1$): The true state is forcefully revealed.

$${\omega }_L(o_i;(s_E,1,{\theta }_L))=1_{s_E=s_i}$$

3. Learner Update Rules (The Cognitive Parameter)

The learner maintains a belief $p_t\in [0,1]$ that the true state is $s_1$. Both learners share a prior $p_0=0.6$. Because $p_0>0.5$, both initially prefer action $a_1$. The parameter ${\theta }_L$ defines how the history is mapped to future beliefs.

Rational Learner (${\theta }_{\text{rat}}$): Updates via exact Bayes’ rule, equivalent to $1_{{\mathbf{o}}_t=o_1}$.

Biased Learner (${\theta }_{\text{bias}}$): Updates via a linear anchoring operator with a heavy decay coefficient (e.g., $\alpha =0.9$).

$$p_{t+1}({\theta }_{\text{bias}})=\alpha p_t+(1-\alpha )p_{t+1}^{\text{Bayes}}=\alpha p_t+(1-\alpha )1_{{\mathbf{o}}_t=o_1}$$

4. Observational Indistinguishability vs. Active Identification

Lemma 1: Indistinguishability under Passive Observation. Assume the true state is $s_2$. If the assistant plays $a_A=a_{\text{pass}}$ for all $t$, the state transitions to $(s_2,0)$. The kernel emits ${\mathbf{o}}_L=\varnothing$. Because the observation is uninformative, both learners maintain their prior belief $p_1=p_0=0.6$. Consequently, both output the exact same action sequence $(a_1,a_1,a_1,\dots )$. The assistant cannot infer ${\Theta }_L$.

Theorem 1: Symmetry Breaking via Active Intervention. Suppose at $t=0$, the assistant plays $a_A=a_{\text{disc}}$. The state transitions to $(s_2,1)$. The kernel emits the highly salient, contradictory observation ${\mathbf{o}}_0=o_2$.

The updates strictly diverge:

1. Rational Update: $p_1({\theta }_{\text{rat}})=0$. The rational learner instantaneously recognizes the true state and plays $a_{1,L}=a_2$.
2. Biased Update: $p_1({\theta }_{\text{bias}})=0.9(0.6)+0.1(0)=0.54$. Because $0.54>0.5$, it ignores the salient evidence and plays $a_{1,L}=a_1$.

By observing $a_{1,L}$, the assistant instantly and flawlessly identifies ${\theta }_L$. Active assistance is therefore strictly necessary to identify the update rule.

5. Illustrative Example: The Stubborn Navigator

Consider a rover (Learner) and a drone (Assistant) navigating a branching canyon. The rover must choose the Left ($a_1$) or Right ($a_2$) path. Based on outdated maps, the rover believes Left is safer ($p_0=0.6$). The true safe path is Right ($s_2$).

If the drone passively follows the rover ($a_{\text{pass}}$), the rover enters the Left path, encountering thick fog (${\mathbf{o}}_L=\varnothing$). A rational rover maintains its prior because fog is uninformative; a structurally biased rover maintains its prior because it ignores fog. Both keep driving Left.

To distinguish them, the drone must use its battery to fly ahead and clear the fog, explicitly projecting a warning light that the Left path is a dead end ($a_{\text{disc}}$ emitting ${\mathbf{o}}_L=o_2$). A rational rover stops and turns around ($a_2$). A biased rover, suffering from sunk-cost or anchoring, rationalizes the warning light as a sensor glitch ($p_1=0.54$) and continues driving Left ($a_1$). The drone has now successfully identified the rover’s cognitive flaw and can adjust its future coordination strategy accordingly.