Cognitive Move Diagrams (CMDs)

• Cognitive Move Diagrams (CMDs) are graphical tools that compress high-dimensional cognitive data into clusters representing mental states and their transitions.
• They utilize systematic clustering and reliability encoding to map individual shifts in decision-making processes to group-level cognitive moves.
• CMDs are applied in fields like military operations and project planning, though challenges arise in high-dimensional contexts requiring careful methodological choices.

A Cognitive Move Diagram (CMD) is a graphical tool designed to visualize the evolution of mental states for one or more decision-makers (agents) as they respond to changing information or stimuli. CMDs utilize nodes to represent Group Cognitive States (GCS) at successive time steps and edges to denote Group Cognitive Moves (GCM)—the transitions between these states. By compressing high-dimensional cognitive data into a state-transition graph, CMDs facilitate detection and analysis of consensus, dissent, and perceptual bias, particularly as key experimental conditions such as information reliability are manipulated (Iorio et al., 2014).

1. Formal Structure and Notation

The formalism underpinning CMDs distinguishes between individual and aggregate cognitive constructs:

• Agent Cognitive State (ACS): At time $t$, outcome measurements for $M$ agents are represented as $C^t \in \mathbb{R}^{N \times M}$, where $N$ is the number of dependent variables. The ACS of agent $j$ is $s^t_j = [c^1_j^t, ..., c^N_j^t] \in \mathbb{R}^N$.
• Agent Cognitive Move (ACM): When the stimulus or condition changes from $t$ to $t+1$, the shift in ACS is $\Delta s^t_j = s^{t+1}_j - s^t_j$.
• Group Cognitive State (GCS): The set $\{s^t_j\}_{j=1...M}$ is clustered or aggregated. Formally, a mapping $$f: \mathbb{R}^{N \times M} \rightarrow \mathbb{R}^{N \times L}$$ (with $L \leq M$) produces representative cluster centroids or states.
• Group Cognitive Move (GCM): The vector transition is $\Delta S^t = S^{t+1} - S^t$.
• Reliability Encoding: Agents' perceived reliability $r^t_{k,j} \in [0,1]$ for stimulus $k$ at time $t$ can augment ACS, e.g., $s^t_j = g(c^t_j, r^t_j)$ or modify clustering weights.

CMDs are constructed as directed graphs where nodes are cluster centroids $\mu_\ell^t$ for $t=1...T$ and edges link $\mu_\ell^t$ to $\mu_{\ell'}^{t+1}$ according to agent transitions.

2. Methodology: Constructing a CMD

A seven-step protocol organizes the CMD construction:

1. Data Collection: For each agent $j=1...M$ and scenario $t=1...T$, measure dependent variables $c^i_j^t$ and log all manipulated independent variables $v^t_{k,j}$, such as different reliability framings.
2. Form ACSs: Specify $s^t_j \leftarrow [c^1_j^t, ..., c^N_j^t]$.
3. Group-State Mapping: Employ clustering (e.g., k-means, hierarchical, PCA+clustering) to partition $\{s^t_j\}$.
4. Compute GCSs: For each $t$, partition into clusters $C_1,...,C_L$ with centroids $$\mu_\ell^t = (1/|C_\ell|) \sum_{j \in C_\ell} s^t_j$$.
5. State-Transition Graph: Each $\mu_\ell^t$ is a node labeled by $(t,\ell)$ and cluster size $|C_\ell|$. Draw directed edges from $\mu_\ell^t$ to $\mu_{\ell'}^{t+1}$ for agents transitioning between clusters.
6. Reliability/Risk Attitude Encoding: Average reliability $\bar{r}_\ell^t$ and shift magnitude $|\Delta \mu_\ell^t|$ are encoded via node color/border and edge thickness.
7. Encoding Scheme Selection: Three main categories:
   • Scheme 1: Unique letter labels for clusters.
   • Scheme 2: Numeric, encoding cluster size only.
   • Scheme 3: Color and border reflect continuous encoding of size and time.

3. Theoretical Context: Information Reliability and Risk Attitude

CMDs are fundamentally sensitive to uncertainty in information reliability and individual risk attitudes. In decision settings such as project planning and military operations, agents weigh incoming data by perceived reliability, and risk attitudes modulate how much trust is placed in unreliable information. Manipulating the framing of reliability—whether qualitative or quantitative—can produce divergent or convergent cognitive trajectories, observable as branching or merging patterns in the CMD.

This conceptualization allows CMDs to visually qualify biases rooted in either information processing or risk-sensitive interpretation. A plausible implication is that CMDs can systematically expose risk-weighting heterogeneity in the population.

4. Illustrative Example: Memory and Weapon-Noise Experiment

Consider three agents ($M=3$) evaluated under three noise levels (0, 1, 2 hours), with two outcome variables (recall %, association %):

Time (t)	Agent 1	Agent 2	Agent 3
0	[0.20, 0.50]	[0.25, 0.50]	[0.45, 0.70]
1	[0.70, 0.70]	[0.75, 0.70]	[0.70, 0.75]
2	[0.25, 1.00]	[0.50, 0.60]	[0.75, 0.55]

Clustering output:

• $t=0$: $\mu_a^0 = [0.23, 0.50]$ ({1,2}); $\mu_b^0 = [0.45, 0.70]$ ({3})
• $t=1$: $\mu_c^1 = [0.72, 0.72]$ (all agents)
• $t=2$: $\mu_d^2 = [0.25, 1.00]$, $\mu_e^2 = [0.50, 0.60]$, $\mu_f^2 = [0.75, 0.55]$ (singletons)

CMD construction yields nodes $${\mu_a^0, \mu_b^0, \mu_c^1, \mu_d^2, \mu_e^2, \mu_f^2}$$ with edges: $\mu_a^0 \rightarrow \mu_c^1$, $\mu_b^0 \rightarrow \mu_c^1$, then $\mu_c^1 \rightarrow \mu_d^2$, $\mu_c^1 \rightarrow \mu_e^2$, $\mu_c^1 \rightarrow \mu_f^2$. This configuration reveals consensus at $t=1$ (all agents in one cluster) and subsequent divergence at $t=2$ (distinct singleton clusters).

5. Interpreting CMDs: Patterns and Cognitive Implications

CMD visualization patterns hold direct analytical implications:

• Convergence: Multiple edges into a single node indicate consensus under specific conditions.
• Divergence: Single node branching to many indicates dissent or heterogeneous response.
• Stability: Cycles or repeated node visits reflect cognitive stability or inertia.
• Magnitude: Edge thickness and color gradients convey extent of cognitive shift or information dimension reliance.
• Trajectory Shape: Straight paths suggest monotonic bias; zig-zag paths encode oscillatory or unstable biasing.

A plausible implication is that CMD trajectory topology can be used to quantify the group’s susceptibility to framing effects or cognitive volatility.

6. Empirical Evaluation and Applications

CMDs have been empirically validated in a military resource allocation context involving three subject-matter experts tasked to allocate trucks under successively introduced reliability framings: none, qualitative, and quantitative. Outcomes included normalized allocation and deviation from expected value. The resultant CMD (Scheme 1) exhibited movement in every agent’s allocation when reliability was referenced and revealed no shared interpretation—no cluster unified all agents at any stage. CMDs thereby efficiently exposed individual differences in risk weighting and unreliable information perception (Iorio et al., 2014).

CMD applicability extends to domains where consensus, dissent, and bias detection are critical. Their utility is marked in military command and control, project planning, and any experimental framework necessitating analysis of high-dimensional cognitive trajectories under variable information reliability.

7. Limitations and Practical Considerations

Interpretability of CMDs decreases in high-dimensional outcome spaces ($N \gg 3$); PCA or factor analysis can pre-process data for tractability. Clustering function choice materially affects CMD structure, necessitating sensitivity analysis. Scalability issues arise for large $M$ (number of agents) and $T$ (number of time steps), potentially demanding cluster aggregation or selective trajectory coding. The effectiveness of CMDs also relies on meaningfulness of proxy outcome measures relative to latent cognitive states.

A plausible implication is that the methodological choices in preprocessing and clustering govern CMD informativeness and susceptibility to misrepresentation.

Table: Core CMD Elements

Element	Mathematical Notation	Purpose
ACS	$s^t_j \in \mathbb{R}^N$	Agent's cognitive state at time $t$
GCS	$S^t = \{\mu_\ell^t\}_{\ell=1}^L$	Aggregate (clustered) group state at time $t$
ACM/GCM	$\Delta s^t_j$, $\Delta S^t$	State change vector for agents/groups
Reliability	$r^t_{k,j} \in [0,1]$	Perceived reliability weights for information
CMD Node/Edge	$\mu_\ell^t \rightarrow \mu_{\ell'}^{t+1}$	Transition between clusters over time

For comprehensive elaboration and source methodology, see Iorio, Abbass, Gaidow & Bender (Iorio et al., 2014).