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Designing a market feels open-ended: sealed bids, open outcry, reserve prices, endless rules. The revelation principle tames it. Whatever any clever mechanism achieves in equilibrium, you can also achieve by simply asking everyone to tell the truth.

Mechanism design and the revelation principle

Mechanism design runs game theory backward: fix the outcome you want, then build the rules that deliver it. The revelation principle is the key simplification. Any equilibrium outcome of any mechanism can be reproduced by a direct mechanism in which telling the truth is a best response, so you never have to search beyond honest, direct rules.

From earlier lessons

Two quick ones before we start.

In a first-price auction with two bidders (uniform values), the equilibrium bid is...

A mechanism is incentive compatible if truth-telling is...

Predict, then play

Two bidders, values uniform on $[0,1]$ . In the first-price auction you shade your bid to half your value. Toggle to a direct mechanism where you simply report your value, and compare the outcome.

Why did that happen?

Take a guess, then reveal the reason

In the first-price auction you shade because your bid sets your price. The designer can absorb that shading into the rules: instead of asking for a bid and making you shave it, ask for your value and charge you the payment your shaded bid would have produced. Facing that direct rule, you have no reason to lie, because any misreport just moves you to the outcome of a different type, which you did not prefer. The strategic shading is folded into the mechanism, leaving honesty optimal.

The math

With two bidders and values uniform on $[0,1]$ , the first-price equilibrium bid is $b(v) = v/2$ . A bidder of value $v$ wins when the rival’s value is below $v$ , which happens with probability $v$ , and pays its bid $v/2$ when it wins. Its interim expected payment is

m(v) = v \cdot \tfrac{v}{2} = \tfrac{v^2}{2}.

The direct mechanism: each bidder reports a value, the highest report wins, and a bidder reporting $v$ pays $m(v) = v^2/2$ in expectation. This reproduces the same allocation and the same payments, and truth-telling is a best response.

Definition, the revelation principle

For any mechanism and any Bayes-Nash equilibrium of it, there is a direct mechanism (players report their types) in which truthful reporting is an equilibrium and which yields the same outcome. So without loss of generality the designer can restrict attention to incentive-compatible, individually rational direct mechanisms. It characterises what is achievable; it says nothing about robustness or simplicity in practice.

Work it out

From shaded bids to truthful reports

In the two-bidder first-price auction, a type $v$ bids $v/2$ . Probability of winning is $\Pr(\text{rival} < v) = v$ , and the payment on winning is the bid $v/2$ , so the interim expected payment is $v \cdot v/2 = v^2/2$ , and interim surplus is $v \cdot v - v^2/2 = v^2/2$ .

Now build the direct mechanism: ask each bidder for a number, award the item to the highest, and charge a reporter of $v$ the amount $v^2/2$ in expectation. A bidder considering reporting $r \ne v$ would receive the win probability and payment of type $r$ , but since $v$ ‘s own bundle was already optimal in the original auction, deviating cannot help. Truth-telling is optimal, and every number, the allocation, the payment, the revenue, matches the first-price auction.

Key idea

The revelation principle lets a designer study only direct, truthful mechanisms without losing any achievable outcome. It is a lens for what is possible, not a claim that direct mechanisms are the best to run in practice.

Check yourself

Check yourselfapply

The direct mechanism charges a reporter of value v the payment v squared over two. Where does that number come from?

Check yourselfanalyse

What does the revelation principle let a designer ignore?

Check yourselfbrings back: bayes-nash

In what sense is truthful reporting in the direct mechanism a Bayes-Nash equilibrium?

Sources

Myerson, R. B. (1979). Incentive compatibility and the bargaining problem. Econometrica, 47(1), 61-73.
Gibbard, A. (1973). Manipulation of voting schemes: a general result. Econometrica, 41(4), 587-601.