De Finetti’s Game: How to Quantify Belief
August 15, 2011 9 Comments
What do people really mean when they say they’re “sure” of something? Everyday language is terrible at describing actual levels of confidence – it lumps together different degrees of belief into vague groups which don’t always match from person to person. When one friend tells you she’s “pretty sure” we should turn left and another says he’s “fairly certain” we should turn right, it would be useful to know how confident they each are.
Sometimes it’s enough to hear your landlord say she’s pretty sure you’ll get towed from that parking space – you’d move your car. But when you’re basing an important decision on another person’s advice, it would be better describe confidence on an objective, numeric scale. It’s not necessarily easy to quantify a feeling, but there’s a method that can help.
Bruno de Finetti, a 20th-century Italian mathematician, came up with a creative idea called de Finetti’s Game to help connect the feeling of confidence to a percent (hat tip Keith Devlin in The Unfinished Game). It works like this:
Suppose you’re half a mile into a road trip when your friend tells you that he’s “pretty sure” he locked the door. Do you go back? When you ask him for a specific number, he replies breezily that he’s 95% sure. Use that number as a starting point and begin the thought experiment.
In the experiment, you show your friend a bag with 95 red and 5 blue marbles. You then offer him a choice: he can either pick a marble at random and, if it’s red, win $1 million. Or he can go back and verify that the door is locked and, if it is, get $1 million.
If your friend would choose to draw a marble from the bag, he preferred the 95% chance to win. His real confidence of locking the door must be somewhere below that. So you play another round – this time with 80 red and 20 blue marbles. If he would rather check the door this time, his confidence is higher than 80% and perhaps you try a 87/13 split next round.
And so on. You keep offering different deals in order to hone in on the level where he feels equally comfortable selecting a random marble and checking the door. That’s his real level of confidence.
The thought experiment should guide people through the tricky process of connecting their feeling of confidence to a corresponding percent. The answer will still be somewhat fuzzy – after all, we’re still relying on a feeling that one option is better than another.
It’s important to remember that the game doesn’t tell us how likely we are to BE right. It only tells us about our confidence – which can be misplaced. From cognitive dissonance to confirmation bias there are countless psychological influences messing up the calibration between our confidence level and our chance of being right. But the more we pay attention to the impact of those biases, the more we can do to compensate. It’s a good practice (though pretty rare) to stop and think, “Have I really been as accurate as I would expect, given how confident I feel?”
I love the idea of measuring people’s confidence (and not just because I can rephrase it as measuring their doubt). I just love being able to quantify things! We can quantify exactly how much a new piece of evidence is likely to affect jurors, how much a person’s suit affects their persuasive impact, or how much confidence affects our openness to new ideas.
We could even use de Finetti’s Game to watch the inner workings of our minds doing Bayesian updating. Maybe I’ll try it out on myself to see how confident I feel that the Ravens will win the Superbowl this year before and after the Week 1 game against the rival Pittsburgh Steelers. I expect that my feeling of confidence won’t shift quite in accordance with what the Bayesian analysis tells me a fully rational person would believe. It’ll be fun to see just how irrational I am!