Why Truthful Bidding Works—and Why First-Price Bidders Shade

Suppose an online auction offers a lamp that is worth exactly one hundred dollars to you. That number is personal: owning the lamp rather than keeping one hundred dollars leaves you indifferent. Should you bid more? Here is the rule. Everyone submits one sealed number. The highest number wins, but the winner does not pay their own bid. They pay the highest bid among everybody else. Your payoff is simple. If you win at price p, you receive something worth one hundred and give up p, so your gain is one hundred minus p. If you lose, your gain is zero. Notice the division of labor. Your bid helps decide whether you win. Somebody else's bid determines what you pay. Your bid is a gate, not the number printed on the bill. Now hold your value fixed at one hundred and inflate your bid to one hundred thirty. The green point is what the lamp is worth to you. The yellow point is what you wrote. Every possible best rival bid falls into one of three ranges. We will walk all three, so there is nowhere for a hidden advantage to escape. First, suppose the best rival bid is above one hundred thirty. Let it be one hundred forty-five. Your inflated bid loses, and your truthful bid of one hundred would also lose. Nothing changes. Second, suppose the best rival bid is below your value. Move it to sixty. With a bid of one hundred you win and pay sixty. With a bid of one hundred thirty you also win and pay sixty. Your payoff is forty dollars either way. Raising your bid did not raise the price, but it did not improve anything either. This entire range is another tie. Third, put the rival between your value and your inflated bid. Say one hundred fifteen. With the truthful bid of one hundred, you lose. That loss is good news. The rival is willing to pay one hundred fifteen for a lamp worth only one hundred to you, so you should let them have it. But the inflated bid of one hundred thirty wins. The second price is the rival's one hundred fifteen, so you pay one hundred fifteen for something worth one hundred. Your payoff is minus fifteen. Only this interval changes the winner, and every change inside it is bad for you. The higher bid can turn a sensible loss into an unwanted purchase, but it can never create a profitable purchase. So bidding above your value is never better than bidding your value. In two ranges it changes nothing. In the remaining range it makes you strictly worse off.

Now try the opposite deviation. The lamp is still worth one hundred dollars to you, but you shade your bid down to eighty. Again, every rival bid lies in one of three ranges. First, suppose the best rival bid is above your value. Put it at one hundred twenty. A bid of eighty loses, and a truthful bid of one hundred also loses. Nothing changes. Second, suppose the best rival bid is below eighty. Move it to fifty-five. Your low bid wins and pays fifty-five. Your truthful bid would also win and pay exactly the same fifty-five. The payoff is forty-five dollars under either bid. Once more, changing your bid changes neither the allocation nor the price. The only interesting range lies between your low bid and your value. Put the best rival bid at ninety. With a truthful bid of one hundred, you win and pay ninety. The lamp is worth one hundred to you, so that purchase produces ten dollars of profit. With the shaded bid of eighty, you lose. You do not hand over money, but you throw away a purchase that would have made you ten dollars better off. So bidding below your value has the same logical shape as bidding above. It ties the truthful bid in two ranges. In the only range where it changes the result, it changes the result against you. Combine the two halves. A bid above value risks buying at a loss. A bid below value risks missing a gain. The truthful bid avoids both errors. This is why economists call truthful bidding weakly dominant in a second-price auction. Whatever the rival bids turn out to be, truth is at least as good as the deviation. Weakly matters. Often a different bid produces exactly the same winner, price, and payoff. Dominance does not mean truth wins strictly in every case. It means no alternative can do better in any case. The case proof treats your one hundred dollars as a private value. It is your own payoff from the lamp, and it does not rise or fall merely because somebody else values the lamp differently. It also takes the auction rule seriously: the highest bid wins and the winner pays the highest rival bid, apart from a fixed tie rule. The proof does not require you to predict the rivals. Risk neutrality, symmetry, and independent value draws are not needed for this elementary dominance comparison. They will matter when we predict first-price bids and compare the seller's expected revenue.

Now keep the same lamp, the same one-hundred-dollar value, and the same sealed bids. Change one rule: the highest bidder still wins, but the winner pays their own bid. Try the truthful bid of one hundred. If you lose, your payoff is zero. If you win, you pay one hundred for something worth one hundred, so your payoff is still zero. Truth now destroys every dollar of profit conditional on winning. To make a positive gain, you must bid below your value. But moving down also makes it easier for a rival to beat you. Let us make that tension numerical. There are two bidders. Your value is one hundred, and the rival's value is equally likely to be anywhere from zero to one hundred. Assume the rival follows the symmetric strategy we are testing. With two uniform bidders, that strategy will turn out to be bidding half of value. Equivalently, the rival's bid is uniform from zero to fifty. It is convenient to rescale the horizontal calculation. If b denotes a candidate cutoff from zero to one hundred, the chance of beating a uniform rival cutoff is b over one hundred. Your expected gain is the chance of winning times the profit if you win. The curve shows that product for every candidate bid. Start at one hundred. You win for sure, but keep zero, so the average gain is zero. Shade to seventy. You win about seven times in ten and keep thirty dollars when you win. Seven tenths of thirty gives an average gain of twenty-one. Shade to fifty. Now the chance and the profit are both one half of their scales. You win half the time, keep fifty when you win, and average twenty-five. Shade farther to thirty. The profit after a win rises to seventy, but the win chance falls to three tenths. The average drops back to twenty-one. So shading is not free money. Too little shading leaves almost no profit. Too much shading throws away too many wins. The best bid balances those two effects. For this two-bidder uniform example, that balance occurs at half your value. A bidder worth one hundred bids fifty. A bidder worth eighty bids forty. The higher value still submits the higher bid. With n symmetric risk-neutral bidders whose private values are independent and uniform, the same calculation gives this equilibrium rule. The bid is n minus one over n times value. With two bidders that is one half. With more rivals the fraction rises, because stronger competition makes aggressive bidding more valuable. This formula is not a universal law of first-price auctions. It uses uniform independent values, symmetric bidders, and risk neutrality. Change those assumptions and the amount of shading changes. The durable conclusion is the incentive, not this particular fraction. Because your own bid becomes your price, you trade a greater chance of winning against a smaller profit when you win.

We have reached the seller's question. Bidders behave differently under the two rules, so does one format reliably collect more money? Use the same two bidders, with independent values uniformly distributed from zero to one hundred. In second price they bid their values. In first price the symmetric equilibrium bid is half of value. Take one draw: values ninety and twenty. Second price charges the lower value, twenty. First price receives half of the higher value, forty-five. The blue first-price result is larger in this particular auction. Revenue equivalence does not deny that. Take values eighty and forty. Second price collects forty. The high-value bidder shades eighty to forty in first price, so that format also collects forty. Take values fifty and thirty. Second price collects thirty. First price collects twenty-five. This time second price earns more. Sale by sale, either format can lead. The theorem concerns the average across all possible value draws, with bidders using equilibrium strategies. Average over every pair. Both markers land at thirty-three dollars and a third. That equality of expectations is the claim. Here is the calculation behind that landing. In second price, seller revenue is the lower of the two values. Its expected value is one hundred over three. In first price, the winner has the higher value but bids half of it. The expected maximum of two uniform values is two hundred over three. Half of two hundred over three is again one hundred over three. Different bidding behavior and different payment rules produce the same expected seller revenue. That numerical match is one instance of a broader theorem, but the theorem has assumptions. First, values are private and independently drawn. Second, bidders are symmetric: their values come from the same distribution and they face the same opportunities. Third, bidders are risk neutral. They compare strategies by expected money payoff, without an extra preference for safer outcomes. Fourth, the allocation is efficient: the highest-value bidder receives the object. The formats also need matched participation and the same expected payoff for the lowest possible type. Under those conditions, efficient auction formats with the same payoff for the lowest type generate the same expected payment from every bidder type. That is the precise content of revenue equivalence. Real auctions can violate the conditions. Values may be correlated or partly common, bidders may differ, and people may care about risk. Budgets, entry costs, reserve prices, platform fees, imperfect learning, and collusion can also change participation, allocation, strategies, and revenue. The theorem is a benchmark, not a promise about every marketplace. So what should the seller hear? Under the standard assumptions and equilibrium play, first-price and second-price auctions deliver the same expected revenue. They need not collect the same amount in a particular sale. They need not expose the seller to the same variation, and the conclusion can fail when the assumptions fail. For bidders, the contrast remains sharp. Second price separates winning from payment, so truthful bidding is weakly dominant. First price makes your bid your payment, so equilibrium bids are shaded. For the seller, revenue equivalence says something narrower and more careful: after averaging over value draws, under the ideal conditions, neither standard format has an expected-revenue advantage.