why we rescale the value map

step 1: from dollars to value

first, we use your choices to estimate two curves. one turns each chance into a ‘subjective probability’: how much a chance seems when you choose. the other turns each amount into ‘utility’: how much that amount is worth to you. switch the chart to after preferences to watch every offer move.

subjective probabilities have fixed ends: a 0% chance always counts as 0, and a 100% chance always counts as 1. so each one means something on its own. with the example preferences (make a few choices below to see your own), an 80% chance seems like 64.5%.

utility is different: it has no natural unit. we could count $1 as 1 unit, or $10 as 10 units, and every choice the model predicts would stay exactly the same. so it isn’t quite right to say that $3 feels like $2.41. what is meaningful is the shape of the curve. $3 and $5 are $2 apart, and so are $8 and $10. counted in utility with the example preferences, the first gap is 1.21 units (3.62 − 2.41), but the second is only 1.03 units (6.31 − 5.28): the same $2 is worth less higher up. economists call this ‘diminishing marginal utility’.

step 2: rescaling

since utility has no natural unit, we rescale both axes to the range of offers in the task: the least likely chance and the smallest prize sit at 0, and the surest chance and the biggest prize sit at 1. so a 10% offer at 0 doesn’t mean a 0% chance. it’s just the least likely offer in the set.

this matters for the map. the grid-like signal we looked for depends on the direction of each comparison across the map, and a direction only means something when both axes are on the same scale: stretch one axis, and every angle changes.

it also fits how the brain seems to handle value. dopamine neurons (Tobler and colleagues, 2005) and neurons in orbitofrontal cortex (Padoa-Schioppa, 2009) adapt their value signals to the range of options on offer, rather than coding amounts on a fixed scale, called 'range adaptation'. and in our study, the brain’s grid-like signal fit the rescaled map. when we try our same analyses on a map that does not rescale each axis, we do not see effects as strong as when we do rescale.

watch your map change

the more choices you make, the better we can estimate your map. keep choosing below, and watch your map shift with each answer.