Testing the Δ‐∑ hypothesis in the suboptimal choice task: Same delta with different probabilities of reinforcement
The Δ-Σ model of sub-optimal choice fails when one reinforcement probability nears zero.
01Research in Context
What this study did
González et al. (2020) tested pigeons in a two-key choice task. The birds picked between a sure thing and a gamble that gave the same size reward.
The gamble key flashed a stimulus that sometimes led to food and sometimes did not. The team varied how likely food was after that flash.
They wanted to know if a math rule called the Δ-Σ model could predict which key the birds would peck.
What they found
When both food chances were clearly above zero, the birds picked the key with the bigger Δ value. That fit the model.
But when one chance neared zero, the birds flipped and picked the other key. The model broke down at that edge.
So the Δ-Σ rule only works when neither probability is close to zero.
How this fits with other research
Allan et al. (1994) saw the same flip. Their pigeons also preferred the risky 50 % key when delays were short, just like here.
Vaughan (1985) showed pigeons care more about delay than probability. That older finding helps explain why tiny probability drops can swing choice so fast.
Miranda-Dukoski et al. (2014) went further and cycled probabilities in real time. Their birds could track shifting odds, showing the pigeon brain is flexible, not stuck on one rule.
Why it matters
If you use probabilistic reinforcement in therapy, keep both chances well above zero or the learner may quit the task. Watch for near-zero edges where rules break. When in doubt, add a brief signal or shorten delays to keep the learner engaged.
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02At a glance
03Original abstract
In a concurrent-chain procedure, pigeons choose between 2 initial-link stimuli; one is followed by terminal link stimuli that signal reliably whether food will be delivered after a delay; the other is followed by terminal link stimuli that do not signal whether food will be delivered after the delay. Pigeons prefer the former alternative even when it yields a lower overall probability of food. Recently, we proposed the Delta-Sigma (∆-∑) hypothesis to explain the effect: Preference depends on the difference (∆) between the reinforcement probabilities associated with the terminal link stimuli, and the overall probability of reinforcement (∑) associated with the alternative. The hypothesis predicts that, for constant ∑, animals should prefer alternatives with greater ∆ values regardless of the specific probabilities of reinforcement that determine ∆. In 2 experiments, we tested this prediction by comparing a ∆ = .5 against a ∆ = 0 alternative, with the former obtained with different pairs of reinforcement probabilities across conditions. The results supported the hypothesis when the 2 probabilities defining ∆ were significantly greater than 0, but not when one of them was close to 0. The results challenge our theoretical accounts of suboptimal choice and the variables considered to determine pigeons' preference.
Journal of the Experimental Analysis of Behavior, 2020 · doi:10.1002/jeab.621