Two modern developments in matching theory.

Marr (1989) wrote a math paper, not an experiment. He took the old matching law and gave it curves. The new equations use exponents, so small ratio changes have big effects at the extremes.

The goal was to predict human choice when one option pays a lot and the other pays little. Linear matching always misses those tails; power functions catch them.

The curved equations fit old data sets better than straight lines. They show why people look "indifferent" when they really just face lopsided pay-offs.

One equation handles both under-matching and extreme bias without extra rules.

Pliskoff et al. (1981) already showed that under- and over-matching flip with change-over costs. Marr (1989) keeps their data but swaps the math, so the same numbers now trace a curve instead of a bent line.

Campbell (2003) later found that pigeons in matching-to-sample ignore the sample when reinforcer ratios hit 9:1. The power-function idea predicts this collapse before it happens.

Locurto et al. (1980) proved reinforcing errors worsens accuracy in signal detection. J’s model adds a single exponent that drops accuracy faster once error payoff rises—same story, tighter formula.

If your client stalls between two tasks, check the payoff gap. When one task earns 5 tokens and the other earns 1, the old matching law says 5:1 responding. The power version says you may see 8:1 or total abandonment of the lean side. Build in quick alternate rewards or reduce the rich-task rate until the curve flattens. You’ll spend less time guessing why "indifference" looks like avoidance.