When multiple signals are presented simultaneously — such as a visual and an auditory stimulus both indicating the same response — reaction times are typically faster and responses more accurate than for either signal alone. This redundancy gain, first systematically studied by Todd (1912) and formalized by Miller (1982), poses a fundamental question: does the benefit arise because the faster of two independent channels wins a statistical race, or because information from multiple sources is combined (coactivated) before a decision is made?
The Race Model Inequality
P(RT ≤ t | S₁S₂) = P(RT ≤ t | S₁) + P(RT ≤ t | S₂) − P(RT ≤ t | S₁) · P(RT ≤ t | S₂)
Upper bound (assuming maximally negatively dependent):
P(RT ≤ t | S₁S₂) ≤ P(RT ≤ t | S₁) + P(RT ≤ t | S₂)
Violation of this bound → evidence for coactivation
Miller (1982) derived an inequality that places an upper bound on the cumulative distribution function (CDF) of redundant-signal RT under any race model, even those allowing dependencies between channels. If the observed redundant-signal CDF exceeds this bound at any time point t, the pure race model is rejected in favor of coactivation — a process in which signals are combined before the decision threshold is reached. This test has become the standard diagnostic tool for distinguishing race from coactivation in redundant-signal experiments.
Coactivation Models
When the race model inequality is violated, coactivation models provide an account of how information from multiple channels is integrated. In the diffusion superposition model (Schwarz, 1989), evidence from each channel is modeled as a diffusion process, and the redundant condition sums the drift rates, producing a faster-rising combined evidence signal. In the counter model (Raab, 1962), discrete information counts from each channel accumulate toward a common threshold. Both frameworks naturally predict violations of the race model inequality because the combined signal reaches threshold faster than the minimum of two independent accumulators.
From an information-theoretic perspective, redundancy gain reflects the benefit of receiving correlated information from multiple channels. If two channels carry the same information, the redundancy does not increase total information content but does increase the reliability of transmission — analogous to repetition coding in communication theory. The magnitude of redundancy gain depends on the correlation between channels and the signal-to-noise ratio in each, connecting the phenomenon to channel capacity and coding efficiency.
Multisensory Integration
Redundancy gain is a core phenomenon in multisensory integration research. Audiovisual speech perception shows robust redundancy gains: listeners are faster and more accurate when they can see and hear the speaker compared to either modality alone. Neurophysiological studies show that multisensory neurons in the superior colliculus exhibit superadditive responses to redundant stimuli, providing a neural basis for coactivation. The inverse effectiveness principle states that redundancy gains are largest when individual signals are weak — precisely when the additional information is most valuable.
The study of redundancy gain exemplifies the interplay between information theory and experimental psychology. Miller's race model inequality provides a rigorous, distribution-free test derived from probability theory, while coactivation models offer process-level accounts grounded in accumulator frameworks. Together, they reveal how the cognitive system exploits redundant information to improve speed and accuracy.