A Primer of the Parallel Constraint-Satisfaction Network Model of Decision Making

Participants are asked to choose the more profitable stock in a sequence of \( z = \{1,\ldots,Z\} \) pairwise comparisons between stocks.

Which stock is more profitable?

1. Trial: Stock 1 or stock 2
2. Trial: Stock 3 or stock 4

\( \vdots \)

Z. Trial: Stock \( (2 \times Z)-1 \) or stock \( 2 \times Z \)

In each comparison, there are stock-market experts that either speak for or against a stock.

Experts differ in their ability to predict the better stock. The predictive accuracy of the experts is unknown to the participants.

In the first comparison between stock 1 and 2, expert A and B speak for stock 1 and against stock 2 while expert C speaks against stock 1 and for stock 2.

The Structure of the Parallel-Constraint-Satisfaction Network Model of Decision Making (PCS-DM)

Modelling the Decision Process in PCS-DM

The model allows deriving predictions for several measures:

• Participants choose the option with the higher activation.
• Participants' decision time increases with an increasing number of iterations \( I \) needed for reaching a stable solution.
• Participants' confidence in a decision increases with increasing difference in activations for option-nodes (stocks).

An animation of the decision process for the exemplary cue-pattern can be found here.

In the left display, the current activation of nodes is shown; in the right display, activations are plotted over iterations.

Note that activations of cues are driven by the source node until cues are influenced by backward activation around iteration #36 (i.e., activation for expert 3 slightly decreases).

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Node activations in each iteration are updated in three steps:

1. Nodes receive input from other nodes. The input is calculated as the weighted sum of activations of nodes connected to the node.
2. Input is transformed into an activation to let node-activations vary between \( -1 \) and \( +1 \).
3. The overall coherence of the PCS-network is calculated. If the extent of coherence does not change significantly over iterations, iterative updating is stopped.

The input of a node p at iteration i is calculated as the weighted sum of all nodes linked to the node:

\[ input_{node_p,i} = \sum_{q = 1, q \neq p}^{Q} w_{node_q-node_p} \times a_{node_{q},i}. \]

Example

In case Stock 1 has an activation of \( .3 \) and Stock 2 an activation of \( -.3 \) at iteratrion # 30, the input for Expert A is:

\[ input_{\text{Expert A},i = 30} = 2 \times .003 + 1 \times .05 = .056. \]

The input of a node is transformed into an activation:

\[ a_{node_{i+1}} = a_{node_{i}} \times (1-decay) + input_{node_{p,i}} \times x \]

with

\[ x = \left\{ \begin{array}{lr} a_{node_p,i} + 1 & \text{if } input_{node_p,i} < 0\\ 1 - a_{node_p,i} & \text{if } input_{node_p,i}\geq 0 \end{array} \right. \]

Example

In case Expert A had an activation of .2 at iteration # 30 and \( decay \) is .1, the input results in an activation of:

\[ a_{\text{Expert A},i = 31} = .2 \times .9 + .056 \times (1 -.2) = .22. \]

The coherence (or negative energy) measures how well the activations of the nodes at iteration \( i \) in the network fit to each other:

\[ energy_{i} = - \sum_{p = 1}^{P = N} \sum_{q = 1, q \neq p}^{Q = N} w_{node_p-node_q,i} \times a_{node_p,i}\times a_{node_q,i} \]

Low values of energy indicate a coherent network.

Example

A network is more (less) coherent when two positive activations are connected by a positive (negative) link to each other.

Updating Cue-Validities in PCS-DM Through Learning

Updating of validity weights consists of three steps:

1. Validity weights are changed in proportion to the difference between desired and observed activations for stocks.
2. Changes in weights are transformed to let weights vary between \( -1 \) and \( +1 \).
3. Transformed changes of validity weights are added to the validity weights.

Change in validity weights is determined by the difference between desired and observed activations of stocks and an individual learning rate \( \lambda \):

\[ \Delta w_{cue_q,t}= \lambda \times \sum_{p=1}^{P}[(d_{a_{opt_{p},I,t}}-a_{opt_{p},I,t}) \times w_{cue_q-opt_p,t}] \]

Example

In case Stock 1 had an activation of \( .6 \) and Stock 2 had an activation of \( -.6 \) and the desired activation is \( -1 \) and \( 1 \) for stocks, and \( \lambda = 1 \), the validity weight of Expert 1 is updated by:

\[ \Delta w_{\text{Expert A},t} = -1.6 \times .01 \times 2 = -.032. \]

To let validity weights vary between \( -1 \) and \( +1 \), changes in validity weights are transformed:

\[ f(\Delta w_{cue_p,t}) = \Delta w_{cue_p,t} \times \left\{ \begin{array}{lr} (1 - w_{cue_p,t}) & \text{if } w_{cue_p,t} \geq 0\\ (1 + w_{cue_p,t}) & \text{if } w_{cue_p,t} < 0 \end{array} \right. \]

Example

The transformed change in the validity weight of Expert 1 is:

\[ f(\Delta w_{\text{Expert A},t}) = -.032 \times (1 - .05) = -.03. \]

The validity weights are updated according to:

\[ w_{cue_p,t+1}=w_{cue_p,t}+f(\Delta w_{cue_p,t}). \]

Transformed changes in validity weights are added to the validity weights.

Example

The validity weight of Expert 1 is adjusted to:

\[ w_{\text{Expert A},t+1} = .05 - .03 = .02. \]

More detailed introductions to PCS-DM and learning in PCS-DM can be found here:

• Jekel, M., Gloeckner, A., & Broeder, A. (to be submitted). Learning in dynamic probabilistic environments: A Parallel-constraint satisfaction network-model approach. (link to draft)

• Gloeckner, A., Hilbig, B. E., & Jekel, M. (2014). What is adaptive about adaptive decision making? A parallel constraint satisfaction theory for decision making. Cognition, 133, 641–666. (link to draft, link to article)

• A web-based graphical user interface to derive predictions for PCS-DM can be found here.
• R-functions for deriving predictions for PCS-DM can be found here.