Addendum to Killeen's (2019) Bidding for Delayed Rewards

Killeen (2020) is a short follow-up to his 2019 paper on how animals and people "bid" for bigger-later rewards. He fixed a math error in the old expenditure function. No new data were collected.

The update keeps the same economic flavor but now predicts preference flips that the old formula missed.

The revised equation lines up better with classic delay-discounting curves. It shows when a subject will switch from a small-immediate reward to a large-delayed one.

The model gives exact numbers you can test in your next discounting probe.

Killeen (2015) offered an older arithmetic discounting model that adds, not multiplies, delay values. Killeen’s 2020 update keeps multiplication but fixes the exponent, so it now supersedes that additive idea.

Animal data line up behind the fix. Grosch et al. (1981) and Cullinan et al. (2001) showed pigeons like signaled delays and switch less when signals differ. The new function predicts those same switch points without extra fudge factors.

Kuroda et al. (2014) found full signals keep stimulus control sharp. Killeen’s model folds signal salience into one delay parameter, so the papers talk about the same variable in different vocabularies.

If you run delay-discounting assessments, plug the new expenditure function into your Excel sheet. It gives a clear cutoff for when a client might choose the bigger-later reinforcer. Use that cutoff to set token-exchange times, schedule progressions, or explain self-control to parents in plain numbers.