Everyone says honesty is the best policy. But when you put aside virtues and focus on material gains, it quickly becomes evident that a bit of deception might be the best tactic.
Let’s assume you love ice cream (a safe assumption), and that there was a neat little parlor in your neighborhood that served the most amazing chocolate ice cream that could ever be conceived. You frequented the place and enjoyed the ice cream and life was bliss until they closed up shop, leaving you devastated.
So imagine your delight when the owner announces he’ll be putting on his apron one last time for a special treat. A treat that will go to the highest bidder. You’re excited about the opportunity, but know that your wallet isn’t the biggest one around, and you need to pay rent so you can actually have a home to eat ice cream in.
You walk into the auction, and they announce that everyone needs to seal their bid in an envelope, and whoever makes the highest bid gets the ice cream. So how much do you bid? If you spend more than $100, you won’t have enough to pay off the loan sharks and this might turn out to be your last ice cream ever. Not a bad way to go, but you’re rational.
And since you are rational, you believe no ice cream is worth $100, and, delicious as it may be, the ice cream is only worth $70 to you. You’re about to put in the $70 bid when something hits you.
What if no one else believes the ice cream is worth $70 — the ignoramuses. What if the next highest bid is only $50? Then you’ve spent a lot more than you needed to and end up feeling shortchanged, even though you were happy paying this amount initially. And besides, everyone else is afraid of overpaying too. So everyone ends up bidding less than they were originally willing to, and the maker of the world’s most delicious ice cream leaves with chump change and a possibly disgruntled buyer.
This is an example of the first-price sealed-bid auction.
However, before the auction even starts, the owner adds in a little extra rule — the highest bidder still gets the ice cream, but only needs to pay as much as the next highest bidder. Now you need to reconsider your strategy.
First off, do you pay more than $70? Of course not, the value of the ice cream to you doesn’t change simply because the rules changed, and you definitely don’t want to risk paying more than it’s worth.
But what about underbidding? This seemed attractive, even optimal, earlier. If you bid less than 70$, you risked ending up with no ice cream. But you definitely didn’t want to feel like you could’ve gotten away with paying a smaller amount.
And this auction, known as the Vickrey auction, is designed to prevent you from overpaying. In the earlier first-price sealed-bid auction, if you stayed true to yourself, and bid $70, and the next highest bidder bid $50, you’d end up overpaying by almost $20. The Vickrey auction means you only have to spend $50 and still get the ice cream as a reward for being truthful and a true connoisseur of ice cream.
The Vickrey bid ensures everyone remains truthful if they want to maximize their potential gains. And this is the Nash Equilibrium of the auction — no one gains by lying and placing a deceptive bid. In fact, it’s the dominant strategy — no matter what anyone else does, this is still the best strategy for you.
In fact, the Revelation Principle proves that any system where people gain through deception can be converted to one where the optimal strategy for everybody is to play honestly. Here’s how it works:
- Consider a system (called a mechanism in Game Theory). Everyone has their optimal strategies based on the information they have. And no one wants to change their strategies, resulting in equilibrium.
- Consider another mechanism. This mechanism takes a different set of strategies and applies strategies for the old mechanism to the new ones. You can assume the strategies to be similar to functions, and applying strategies to other strategies as function composition.
- The composed strategies are then passed into the old mechanism, and the outcome is the same as the outcome of the old mechanism with these new strategies.
- So what happens in case someone decides to lie in the new mechanism? The mechanism takes the lie and applies the strategy for the original mechanism to this false information. This results in a different strategy getting passed into the original mechanism, which isn’t what the player would want since he had a different optimal strategy for the original mechanism.
Assume a player has some private information x and an optimal strategy s(x). In the new mechanism, he’d have a different optimal strategy s'(x). The new mechanism applies the original strategy s to s’(x) resulting in s(s'(x)) and feeds this into the old mechanism.
If s'(x) isn’t representative of the truth x, then the strategy fed into the original system might not be optimal. In other words, the new mechanism encourages the truth by lying for the player.
Emboldened by this realization, you proceed to place your bid of $70. But as you walk up to hand over the envelope, a tragic sight leaves you distraught. While you and your fellow bidders were pondering over the cleverest strategy to maximize your gains, the owner had comfortably sat down and proceeded to gobble up the entire bowl. His excuse? The ice cream was melting.