WEBVTT

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Suppose an online auction offers a lamp

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that is worth exactly one
hundred dollars to you.

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That number is personal:

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owning the lamp rather than keeping one
hundred dollars leaves you indifferent.

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Should you bid more?

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Here is the rule.

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Everyone submits one sealed number.

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The highest number wins,
but the winner does not pay their own bid.

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They pay the highest
bid among everybody else.

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Your payoff is simple.

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If you win at price p, you receive
something worth one hundred and give up p,

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so your gain is one hundred minus p.

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If you lose, your gain is zero.

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Notice the division of labor.

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Your bid helps decide whether you win.

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Somebody else's bid
determines what you pay.

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Your bid is a gate,
not the number printed on the bill.

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Now hold your value fixed

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at one hundred and inflate
your bid to one hundred thirty.

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The green point is what
the lamp is worth to you.

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The yellow point is what you wrote.

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Every possible best rival bid
falls into one of three ranges.

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We will walk all three, so there is
nowhere for a hidden advantage to escape.

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First, suppose the best rival
bid is above one hundred thirty.

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Let it be one hundred forty-five.

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Your inflated bid loses, and your truthful
bid of one hundred would also lose.

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Nothing changes.

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Second, suppose the best
rival bid is below your value.

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Move it to sixty.

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With a bid of one hundred
you win and pay sixty.

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With a bid of one hundred
thirty you also win and pay sixty.

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Your payoff is forty dollars either way.

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Raising your bid did not raise the price,
but it did not improve anything either.

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This entire range is another tie.

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Third, put the rival between
your value and your inflated bid.

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Say one hundred fifteen. With the
truthful bid of one hundred, you lose.

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That loss is good news.

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The rival is willing to pay one hundred
fifteen for a lamp worth only one hundred

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to you, so you should let them have it.

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But the inflated bid
of one hundred thirty wins.

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The second price
is the rival's one hundred fifteen,

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so you pay one hundred fifteen
for something worth one hundred.

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Your payoff is minus fifteen.

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Only this interval changes the winner,
and every change inside it is bad for you.

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The higher bid can turn a sensible
loss into an unwanted purchase,

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but it can never create
a profitable purchase.

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So bidding above your value is never
better than bidding your value.

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In two ranges it changes nothing.

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In the remaining range it
makes you strictly worse off.

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Now try the opposite deviation. The lamp
is still worth one hundred dollars to you,

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but you shade your bid down to eighty.

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Again, every rival bid lies
in one of three ranges.

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First, suppose the best
rival bid is above your value.

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Put it at one hundred twenty.

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A bid of eighty loses, and a truthful
bid of one hundred also loses.

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Nothing changes.

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Second, suppose the best
rival bid is below eighty.

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Move it to fifty-five.

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Your low bid wins and pays fifty-five.

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Your truthful bid would also win
and pay exactly the same fifty-five.

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The payoff is forty-five
dollars under either bid.

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Once more, changing your bid changes
neither the allocation nor the price.

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The only interesting range lies
between your low bid and your value.

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Put the best rival bid at ninety.

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With a truthful bid of one
hundred, you win and pay ninety.

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The lamp is worth one hundred to you,

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so that purchase produces
ten dollars of profit.

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With the shaded bid of eighty, you lose.

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You do not hand over money,

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but you throw away a purchase that would
have made you ten dollars better off.

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So bidding below your value has the
same logical shape as bidding above.

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It ties the truthful bid in two ranges.

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In the only range where it changes the
result, it changes the result against you.

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Combine the two halves.

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A bid above value risks buying at a loss.

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A bid below value risks missing a gain.

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The truthful bid avoids both errors.

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This is why economists call
truthful bidding weakly dominant

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in a second-price auction.

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Whatever the rival bids turn out to be,

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truth is at least
as good as the deviation.

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Weakly matters.

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Often a different bid produces exactly
the same winner, price, and payoff.

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Dominance does not mean truth
wins strictly in every case.

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It means no alternative
can do better in any case.

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The case proof treats your one
hundred dollars as a private value.

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It is your own payoff from the lamp,
and it does not rise or fall merely

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because somebody else
values the lamp differently.

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It also takes the auction rule seriously:

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the highest bid wins
and the winner pays the highest rival bid,

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apart from a fixed tie rule.

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The proof does not require
you to predict the rivals.

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Risk neutrality, symmetry,

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and independent value draws are not needed
for this elementary dominance comparison.

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They will matter

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when we predict first-price bids
and compare the seller's expected revenue.

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Now keep the same lamp,

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the same one-hundred-dollar
value, and the same sealed bids.

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Change one rule:

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the highest bidder still wins,
but the winner pays their own bid.

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Try the truthful bid of one hundred.

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If you lose, your payoff is zero.

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If you win, you pay one hundred
for something worth one hundred,

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so your payoff is still zero.

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Truth now destroys every dollar
of profit conditional on winning.

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To make a positive gain,
you must bid below your value.

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But moving down also makes it
easier for a rival to beat you.

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Let us make that tension numerical.

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There are two bidders.

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Your value is one hundred,

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and the rival's value is equally likely
to be anywhere from zero to one hundred.

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Assume the rival follows
the symmetric strategy we are testing.

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With two uniform bidders, that strategy
will turn out to be bidding half of value.

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Equivalently, the rival's bid
is uniform from zero to fifty.

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It is convenient to rescale
the horizontal calculation.

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If b denotes a candidate
cutoff from zero to one hundred,

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the chance of beating a uniform
rival cutoff is b over one hundred.

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Your expected gain is the chance
of winning times the profit if you win.

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The curve shows that product
for every candidate bid.

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Start at one hundred.

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You win for sure, but keep zero,
so the average gain is zero.

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Shade to seventy.

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You win about seven times in ten
and keep thirty dollars when you win.

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Seven tenths of thirty gives
an average gain of twenty-one.

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Shade to fifty.

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Now the chance and the profit
are both one half of their scales.

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You win half the time, keep fifty
when you win, and average twenty-five.

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Shade farther to thirty.

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The profit after a win rises to seventy,
but the win chance falls to three tenths.

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The average drops back to twenty-one.

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So shading is not free money. Too
little shading leaves almost no profit.

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Too much shading
throws away too many wins.

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The best bid balances those two effects.

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For this two-bidder uniform example,
that balance occurs at half your value.

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A bidder worth one hundred bids fifty.

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A bidder worth eighty bids forty.

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The higher value still
submits the higher bid.

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With n symmetric risk-neutral bidders
whose private values are independent

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and uniform, the same calculation
gives this equilibrium rule.

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The bid is n minus one over n times value.

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With two bidders that is one half.

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With more rivals the fraction rises,

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because stronger competition makes
aggressive bidding more valuable.

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This formula is not a universal
law of first-price auctions.

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It uses uniform independent values,
symmetric bidders, and risk neutrality.

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Change those assumptions
and the amount of shading changes.

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The durable conclusion is the incentive,
not this particular fraction.

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Because your own bid becomes your price,

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you trade a greater chance of winning
against a smaller profit when you win.

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We have reached the seller's question.

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Bidders behave differently

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under the two rules, so does one
format reliably collect more money?

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Use the same two bidders,

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with independent values uniformly
distributed from zero to one hundred.

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In second price they bid their values.

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In first price the symmetric
equilibrium bid is half of value.

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Take one draw: values ninety and twenty.

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Second price charges
the lower value, twenty.

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First price receives half
of the higher value, forty-five.

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The blue first-price result
is larger in this particular auction.

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Revenue equivalence does not deny that.

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Take values eighty and forty.

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Second price collects forty.

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The high-value bidder shades
eighty to forty in first price,

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so that format also collects forty.

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Take values fifty and thirty.

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Second price collects thirty.

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First price collects twenty-five.

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This time second price earns more.

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Sale by sale, either format can lead.

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The theorem concerns the average
across all possible value draws,

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with bidders using equilibrium strategies.

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Average over every pair.

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Both markers land
at thirty-three dollars and a third.

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That equality
of expectations is the claim.

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Here is the calculation
behind that landing.

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In second price, seller revenue
is the lower of the two values.

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Its expected value
is one hundred over three.

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In first price, the winner
has the higher value but bids half of it.

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The expected maximum of two uniform
values is two hundred over three.

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Half of two hundred over three
is again one hundred over three.

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Different bidding behavior

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and different payment rules produce
the same expected seller revenue.

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That numerical match is one
instance of a broader theorem,

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but the theorem has assumptions.

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First, values are private
and independently drawn.

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Second, bidders are symmetric: their
values come from the same distribution

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and they face the same opportunities.

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Third, bidders are risk neutral.

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They compare strategies

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by expected money payoff, without an
extra preference for safer outcomes.

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Fourth, the allocation is efficient: the
highest-value bidder receives the object.

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The formats also need matched
participation and the same expected payoff

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for the lowest possible type.

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Under those conditions, efficient
auction formats with the same payoff

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for the lowest type generate the same
expected payment from every bidder type.

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That is the precise content
of revenue equivalence.

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Real auctions can violate the conditions.

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Values may be correlated
or partly common, bidders may differ,

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and people may care about risk.

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Budgets, entry costs, reserve prices,

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platform fees, imperfect learning, and
collusion can also change participation,

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allocation, strategies, and revenue.

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The theorem is a benchmark,
not a promise about every marketplace.

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So what should the seller hear?

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Under the standard
assumptions and equilibrium play,

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first-price and second-price auctions
deliver the same expected revenue.

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They need not collect the same
amount in a particular sale.

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They need not expose
the seller to the same variation,

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and the conclusion can fail
when the assumptions fail.

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For bidders, the contrast remains sharp.

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Second price separates
winning from payment,

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so truthful bidding is weakly dominant.

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First price makes your bid your payment,
so equilibrium bids are shaded.

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For the seller, revenue equivalence
says something narrower and more careful:

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after averaging over value draws,
under the ideal conditions,

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neither standard format
has an expected-revenue advantage.