Mechanism DesignMechanism Design

How to get agents to reveal private information

King Solomon’s “Wisdom”And the king said, Bring me a sword. Any they brought a sword

before the king.And the king said, Divide the living child in two, and give half to

the one, and half to the other.Then spake the woman whose the living child was, unto the king,

for her bowels yearned upon her son, and she said, O my lord, give her the living child, and in no wise slay it. But the other said, Let it be neither mine nor thine, but divide it.

Then the king answered and said, give her [the first woman] the child, and in no wise slay it; she is the mother thereof.

And all Israel heard of the judgment which the king had judged; and they feared the king: for they saw that the wisdom of God was in him, to do judgment.

Mechanism Design

Task: to design rules of the game such that a desired set of outcomes is an equilibrium of the game.

Design constraints– Participation: agents can opt not to play so

need to be given an incentive to participate– Incentive compatibility: given mechanism,

desired outcome must be self-enforcing

Some Examples Commons Problem

– Government auctions rights to use of common resource every year

– Government auctions once-and-for-all rights to the common resource

Selling the da Vinci Diaries– Make a take-it-or-leave-it offer– Offer two prices with the higher more certain sale than the lower

Auctions– Highest bidder gets the diaries– Highest bidder gets the diaries but only has to pay the amount of

the next highest bid

Buyer with Unknown WTP

Suppose Gates may have a WTP for the diaries of either $40 million or $10 million.

Seller places equal weight on either possibility.

If could determine Gates’ WTP, then could tailor a take-it-or-leave-it sales price.– Expected revenue = $25 million

Option 1: Ask the buyer

Suppose Sotherbys asked Gates what is WTP was and gave him a contingent price.– If WTP = 40, price = P– If WTP = 10, price = p

What happens?

Option 2: Set a flat price

What is the highest flat price Sotherbys can charge?

Could it gamble? Is this renegotiation proof?

Option 3: Guaranteed Purchase at Higher Price

Sotherbys sets:– P = 20 with guaranteed sale– p = 10 with 50% probability of sale

What will Gates do?– If his WTP = 40?– If his WTP = 10?

Option 4: Guarantee sale at a higher price and offer a

probabilistic sale at lower price Sotherbys sets:

– P = 20 with guaranteed sale– p = 10 with probability of sale of

What will maximise sales revenue?

Option 5: Tell me who you are

Gates is asked whether he has a WTP of 40 or 10.– If says 40, then guaranteed P– If says 10, then gets it for p with probability .

This is an example of a direct revelation mechanism

Direct Revelation

Imagine that if given information of an agent’s type, you could program a computer to play that agent’s optimal strategy

Example: if give WTP, the computer will undertake the optimal bidding strategy

Would this type of direct revelation mechanism result in the same outcome for the designer?

Revelation Principle

For any mechanism and an incentive-compatible, individually rational assignment, there is a direct-revelation mechanism in which truth telling is incentive compatible, individually rational, and which produces an identical assignment.

Designer can restrict attention to direct-revelation mechanisms and truth-telling assignments within those mechanisms.

Designer can, therefore, safely save agents the trouble of working out optimal strategies.

Revelation Principle: Many Agents

For any mechanism and any (Bayes-Nash) equilibrium of that mechanism, there is a direct-revelation mechanism with truth telling as a Bayes-Nash equilibrium that has an identical assignment.

Designer can restrict attention to direct-revelation mechanisms and truth-selling equilibria within those mechanisms.

Focus on implementable outcomes.

Issues

Mechanism Design is high commitment– Agents must believe the mechanism will hold

as true– May not be immune to ex post renegotiation

How much knowledge does the designer require?– Types– Distribution of types

Three Examples

Solomon Two-sided information asymmetry Public goods

Solomon Revisited Suppose the two women are Anna and Bess and

consider the following mechanism:1. Solomon decides on a fine or punishment of F; this need not

be monetary or drastic2. Anna is chosen to go first. Either she gives up her claim to

the child (in which case it goes to Bess and the game ends) or she asserts her claim, in which case the game continues to …

3. Bess either accepts Anna’s claim or challenges her claim. In the latter case, Bess must put in a bid, B, of her own choosing for the child, and Anna must pay the fine, F, to Solomon.

4. Anna now either matches Bess’s bid (she gets the child and pays B while Bess pays F) or chooses not to match.

Divided Ownership

What happens if there is two-sided information asymmetry?

Usually, it is argued that clear ownership is more likely to give rise to efficient trade. However, with private information, we know that there is a chance of strategic misrepresentation leading to bargaining breakdown.

Suppose Smith and Jones are potential developers of Blackacre as a mall. It is currently farm land. Who is the more efficient developer is private information.

Now, suppose that Smith owns the land but that Jones has a share in the land if it is ever used for anything but farming.

Identity Crises

• Consider Smith’s thought in considering whether to purchase Jones’ share. Smith as a buyer wishes to offer a low price, but the possibility that Smith could become a seller complicates Smith’s decision.

• If Smith offers too low a price, Jones is liable to turn the tables by suggesting that Smith should sell her own claim. Jones would saying, “Where did you get that price? If that’s all you think Blackacre is worth, I’ll buy your claim.”

• Thus, divided ownership might lead to more efficient trade as it provides an incentive to reveal information.

• Suggests that liability laws may be better than simple property rights rules.

Public Goods

Mayor of a town is considering installing a streetlight that costs $100

Wants to tax 5 closest households, $20 each but will only install it if the sum of their valuations exceeds $100

That is, each pays $20 if sum of reports exceeds $100 and nothing otherwise

If asks them, each has an incentive to lie

Groves Mechanism

Let mi be the valuation report of household i.

Then i pays:

Or else nothing. Makes it a dominant strategy to be truthful.

5

1

100 if 100j jj i j

m m

Example

Suppose Smith’s valuation is 40 and that the sum of all valuations is 100. If others report their truthful sum of 70, then if Smith tells the truth he receives 40 less a tax of 30.

Reporting more would not change this outcome Reporting less than 30 would reduce Smith’s surplus to 0. But what if others’ lie?

– If they reported 50 rather than 70, Smith could over-report (say 60) but his payoff would be –10. Better to report 40 and kill the project.

– If they reported 80, Smith should report at least 20 as this gives him the most value. This includes 40 as a truthful revelation.