Mathematical Psychology
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Rescorla-Wagner Model

The Rescorla-Wagner model formalizes associative learning as error-driven updating, where the change in associative strength is proportional to the discrepancy between expected and actual outcomes.

ΔV = α·β·(λ − ΣV)

The Rescorla-Wagner model, published in 1972, is the most influential formal model of classical conditioning. It captures the key insight that learning is driven by prediction error — organisms learn when events are surprising and stop learning when outcomes are fully predicted.

The Learning Rule

Rescorla-Wagner Learning Rule ΔVₐ = αₐ · β · (λ − ΣV)

Vₐ = associative strength of CSₐ
αₐ = salience of CSₐ (0 to 1)
β = learning rate for the US (0 to 1)
λ = maximum conditioning supported by the US
ΣV = total associative strength of all CSs present

The term (λ − ΣV) is the prediction error: the discrepancy between what the US supports and what is currently predicted. When multiple conditioned stimuli are present, their combined associative strengths compete for a fixed amount of associative strength, naturally explaining phenomena like blocking and overshadowing.

Predictions and Successes

The model elegantly explains acquisition (V approaches λ asymptotically), extinction (V approaches 0 when λ = 0), blocking (no learning to a redundant predictor), and the fact that learning decelerates as V approaches λ. Its prediction error formulation directly inspired the temporal difference (TD) learning algorithm in reinforcement learning and the discovery that dopamine neurons encode reward prediction errors.

Limitations

The model cannot account for latent inhibition (pre-exposure retards conditioning), spontaneous recovery, or configural learning (where a compound AB has different properties than A or B alone). These limitations motivated successor models including Pearce-Hall (variable attention) and Mackintosh (attention to the best predictor).

Interactive Calculator

Each row represents a conditioning trial: cs_present (1 = CS present, 0 = absent) and us_present (1 = US present, 0 = absent). The calculator simulates the Rescorla-Wagner learning rule: ΔV = α·β·(λ − ΣV). Parameters: α=0.3 (CS salience), β=0.4 (US salience), λ=1.0 (max conditioning).

Click Calculate to see results, or Animate to watch the statistics update one record at a time.

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References

  1. Rescorla, R. A., & Wagner, A. R. (1972). A theory of Pavlovian conditioning: Variations in the effectiveness of reinforcement and nonreinforcement. In A. H. Black & W. F. Prokasy (Eds.), Classical conditioning II: Current research and theory (pp. 64–99). Appleton-Century-Crofts. https://doi.org/10.1037/a0030892
  2. Miller, R. R., Barnet, R. C., & Grahame, N. J. (1995). Assessment of the Rescorla-Wagner model. Psychological Bulletin, 117(3), 363–386. https://doi.org/10.1037/0033-2909.117.3.363
  3. Schultz, W., Dayan, P., & Montague, P. R. (1997). A neural substrate of prediction and reward. Science, 275(5306), 1593–1599. https://doi.org/10.1126/science.275.5306.1593

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