Understanding the Karni Belief-Revelation Mechanism

Suppose you wanted to find out someone’s subjective belief in the likelihood of a given event. Your first instinct might be to offer them betting odds on the event. If they would accept odds of 2:1 but no lower, you might conclude that they believe the event to have a 33.3% chance of occurring.

This would be correct if the person you’re offering the bet to is risk neutral. But if they’re risk averse, they might actually believe the event has a 50% likelihood of occurring, and they just require more favourable odds to compensate them for accepting risk.

Luckily, there’s a better way to elicit subjective probabilities. It comes from Karni (2009), a short paper published in Econometrica.*

The Karni mechanism works as follows:

  1. You tell me the probability you think the event will occur, p.
  2. I draw a random number, r.
  3. Depending on the value of r, one of two things occur:
    1. If r>p, you get a lottery that pays a high payoff with probability r.
    2. If r<p, you get a high payoff if the event occurs.

This guarantees that your incentive is to bid your true value regardless of your risk preference.**

Get it? It works like a second-price auction, except with probabilities instead of prices. When I first learned this mechanism it was in a class full of graduate students and we all had difficulty wrapping our heads around it. I’m working on a paper that builds on Karni (2009), so I’ve been trying to explain it to lots of people and they all have trouble understanding.

For this reason, I’ve come up with an analogy to help people understand the mechanism.

Suppose I have a box full of coins. These are not fair coins, they are all weighted to different degrees in favour of heads or tails. Printed on the face of each coin is the percentage probabiliy it lands heads. So I might pull out a coin that says “78.94” and you know that it has a 78.94 percent chance of landing heads on a given flip.

I want to know what subjective probability you place on the likelihood that the Prime Minister of Canada will be reelected. You tell me that you believe he has a 67% chance of reelection. Then I draw a coin at random from the box. If it’s higher than 67, you get to bet on the coin. If it’s lower than 67, you get to bet on the reelection. The payoff of winning or losing a bet is the same regardless of what you’re betting on, so the probability of winning is all that matters.

Your choice of 67 tells me that you prefer a bet on the reelection of the Prime Minister to any bet on a coin worse than 67, but you’d prefer to bet on any coin higher than 67 over betting on the reelection of the Prime Minister.

The brilliance of this mechanism is that a 68% chance of winning money is better than a 67% chance of winning the same amount of money regardless of how risk averse you are. If you bid below your true belief, you’re just risking a coin toss with worse odds. If you bid above, you’re sacrificing coin tosses with better odds. Betting your true belief is a weakly dominant strategy.

* Karni, E. (2009). A mechanism for eliciting probabilities. Econometrica, 77(2), 603-606.

** There’s one exception. If you have a personal stake in the outcome of the event you’re betting on, you might choose not to be truthful. For instance, if I ask you what’s the likelihood your house will burn down in the next year, you might actually give a higher number than your true belief, since a bet on your house burning down is valuable as insurance.