research · a behavioral model of latent cognitive variables

the metrics above show that agents differ, not how. so we fit a compact behavioral model to every agent's choices. its backbone is the ideal bayesian observer. on top, it allows a few interpretable deviations, each controlled by one parameter.

$$\begin{aligned} \Delta\mathbf{h}_t &= \delta_t \left(\mathbf{e}_{a_t} - \tfrac{1}{4}\mathbf{1}\right) \\ \delta_t &= x_t \log\frac{\alpha_B}{\alpha_U} + (1 - x_t) \log\frac{1-\alpha_B}{1-\alpha_U} \end{aligned}$$

evidence: the sampled button $a_t$ gains (red, $x_t = 1$) or loses (green) log-odds relative to the others

$$\mathbf{h}_t = \sum_{u=1}^{t} w_{t,u}(\beta) \, \Delta\mathbf{h}_u, \qquad w_{t,u}(\beta) = \begin{cases} e^{-\beta (t-u)} & \beta > 0 \\ 1 & \beta = 0 \\ e^{\beta (u-1)} & \beta < 0 \end{cases}$$

memory: forgetful ($\beta > 0$), exact bayesian accumulation ($\beta = 0$) or stubborn ($\beta < 0$)

$$\boldsymbol{\pi}^{s}_t = \sigma\Big( \underbrace{\kappa_s(t)}_{\text{strategy}} \underbrace{\mathbf{h}_{t-1}}_{\text{memory}} + \underbrace{\log\boldsymbol{\omega}_s}_{\text{bias}} + \underbrace{\gamma\,\mathbf{i}_t}_{\text{inertia}} - \underbrace{\theta_s (\mathbf{1} - \mathbf{v}_t)}_{\text{occlusion}} \Big)$$

sampling policy, with gain $\kappa_s(t) = m\, n_t / T$ growing with the number $n_t$ of usable observations

$$\boldsymbol{\pi}^{f} = \sigma\Big( \underbrace{\kappa_f}_{\text{strategy}} \underbrace{\mathbf{h}_{T}}_{\text{memory}} + \underbrace{\log\boldsymbol{\omega}_f}_{\text{bias}} \Big)$$

final choice: $\kappa_f = 1$ is probability matching, $\kappa_f \to \infty$ is map

$\beta$

memory

how evidence is weighted over time. 0 is perfect accumulation. negative is stubborn (early evidence dominates), positive is forgetful (only recent evidence counts).

$m,\ \kappa_f$

strategy

$m$ sets how quickly sampling moves from exploring to following the evidence. $\kappa_f$ sets how sharply beliefs become the final choice: 1 is probability matching, $\infty$ is map.

$\boldsymbol{\omega},\ \gamma$

choice bias

preferences for particular buttons regardless of evidence ($\boldsymbol{\omega}$), and inertia: repeating the last sampled button ($\gamma$).

$\theta$

occlusion awareness

how strongly occluded buttons are avoided. large $\theta$ never picks an unavailable button, small $\theta$ produces invalid choices.

the model predicts held-out choices better than an agent-specific frequency baseline for all 38 agents. simulated games reproduce each agent's sampling and inference losses ($r = 0.75$ and $0.95$). each parameter also leaves a distinct fingerprint: perturbing one at a time in the model fitted to the optimal agent shows that $\kappa_f$ moves only inference loss, $m$ and $\theta$ move only sampling loss, and memory $\beta$ moves both.

03a behavioral model of latent cognitive variables