Explain why a payment rule changes incentives even when the object and bidders stay fixed.
By the end: justify truthful bidding in a private-value second-price auction and recognize its limits.
One object goes to the highest bid. In a first-price sealed-bid auction the winner pays their own bid. In a second-price sealed-bid auction they pay the highest other bid. Losing bidders pay zero. Here utility is private value minus payment if you win and zero otherwise. The example assumes no budgets, fees, resale motive, collusion, or externalities.
Ties go to the earlier listed bidder, independent of the value of the bid. Your private value is 8; rival bids initially are 5 and 7. Raising your bid changes whether you win, and changes your payment in the first-price rule. In the second-price rule it does not change the payment conditional on beating the same rival bids.
An auction has an allocation rule that decides who receives the object and a payment rule that decides what each participant pays. Here both formats allocate to the highest bid, with earlier listed bidders winning ties. They differ only in the winner's payment. Your value of 8 is the stipulated benefit of receiving the object, expressed on a scale where paying one unit reduces utility by one. It is not your bid or a guaranteed resale price. Losing gives zero because this model omits entry fees and any consequences of who else wins.
Against rival bids 5 and 7, begin with your bid at 6. The rival who bid 7 wins. That winner pays 7 under first price and 6 under second price, since your bid is now the highest losing bid. Your own payment and utility remain zero. This distinction matters when reading the panels. Winner pays reports the winner's charge, not necessarily yours. A change in your losing bid can change someone else's second-price payment even while your own utility stays unchanged. The stipulated preferences give you no benefit from that effect.
At your bid of 7, you tie the highest rival and win because you are listed first. First-price payment is 7, leaving utility 1. Second-price payment is also 7 because the tied rival bid remains the highest other bid. Raising your bid to 8 preserves the allocation. First-price utility falls to zero as your payment rises by one, but second-price utility stays 1. Raising it to 11 gives first-price utility negative 3 while second-price utility remains 1 against these rivals. Winning more emphatically creates no extra benefit in this one-object model.
Now replace the rivals with 9 and 10 while retaining your bid of 11. You still win, but second-price payment becomes 10 and utility becomes negative 2. The fact that the payment is below your own bid does not make the purchase beneficial. Compare payment with value, not with the bid ceiling you chose. Under first price you still pay 11 and lose 3 utility units. Both negative values are meaningful outputs. Replacing them with zero would hide the consequence of voluntarily winning at a price above your stipulated value.
If your bid is below both rivals, the winner's second-price payment is the lower rival bid rather than your bid. For example, with bids 3,5,7 the ranking is 7 first, 5 second, and 3 third. The winner pays 5 under second price. The rule is not always pay the learner's bid and not always pay the lower of two rivals. It is pay the highest bid among everyone other than the winner. Sorting all three bids before calculating payment prevents those shortcuts from failing when the identity of the winner changes.
| Your bid; rivals | Winner | First payment; your utility | Second payment; your utility |
|---|---|---|---|
| 7; 5,7 | You by priority | 7; 1 | 7; 1 |
| 8; 5,7 | You | 8; 0 | 7; 1 |
| 11; 5,7 | You | 11; −3 | 7; 1 |
| 11; 9,10 | You | 11; −3 | 10; −2 |
| 3; 5,7 | Rival 2 | 7; 0 | 5; 0 |
A sealed-bid rule means participants submit without seeing the other current bids before choosing. The explorer deliberately reveals and fixes rivals to isolate arithmetic. Knowing their bids allows an exact best response for that profile, but a participant in the corresponding sealed auction ordinarily lacks that knowledge. Keep that informational difference explicit. The model can answer what this bid earns against 5 and 7 without answering what to submit before learning their bids. The later probability example supplies a separate assumed distribution to explain what that additional decision would require.
Call the highest rival bid m and your value v. If m is below v, winning at price m is beneficial, and bidding v wins. If m is above v, winning is harmful, and bidding v loses. If m equals v, winning and losing both give utility zero. These cases prove weak dominance of truthful bidding under this rule.
Weak dominance does not mean truthful bidding uniquely wins every comparison. Many alternative bids produce the same allocation and utility against a particular m. In a first-price auction, bidding your full value gives zero if you win; a lower winning bid can give positive utility. Optimal shading requires beliefs about rival bids, not just the rule name.
To prove a dominance claim, compare truthful bidding with any alternative while holding the other bids fixed. Let the highest other bid be the threshold that determines whether you win. Under second price, crossing that threshold changes allocation but does not let you choose a lower winning payment. With value 8 and threshold 7, bidding 8 wins and pays 7. Bidding 6 loses, giving zero instead of 1. Bidding 10 also wins and pays 7, matching truthful utility. Truth is better than the low bid here, but only tied with the high bid.
Use a higher threshold to test overbidding. At threshold 10, bidding 8 loses and gives zero. Bidding 11 wins, pays 10, and gives negative 2. An alternative bid that won extra cases has added a harmful case. At a lower threshold such as 4, both bids 8 and 11 win and pay 4, giving utility 4. Raising the bid therefore cannot improve a beneficial winning payment in this format, yet it can add purchases whose price exceeds value. This comparison holds rivals fixed rather than assuming they react to the false bid.
| Your bid | Rival maximum 4 | Rival maximum 7 | Rival maximum 10 |
|---|---|---|---|
| 6, underbid | 4, win | 0, lose | 0, lose |
| 8, truthful | 4, win | 1, win | 0, lose |
| 11, overbid | 4, win | 1, win | −2, win |
Underbidding has the opposite risk. A bid of 6 agrees with truth whenever the highest rival bid is below 6, because both bids win at the same payment. It can lose beneficial opportunities when the rival threshold lies above 6 but below 8. At threshold 7 the lost benefit is 1. Above 8 both bids lose and agree again. These cases exhaust the places where lowering your bid can change your own outcome. The payment independence conditional on winning is doing the work; a general instruction to bid aggressively would not prove the result.
Equality needs its own check. If the highest rival bid equals your value 8, a winning truthful bid pays 8 and earns zero. A losing truthful bid would also earn zero under a different fixed tie priority. Thus the proof survives either deterministic priority when priority does not depend on how you manipulate the bid. At a rival threshold 7 and your underbid 7, however, the tie rule affects whether you collect positive utility 1. The explorer gives you that tie, so 7 is sufficient to win in that particular displayed profile.
This is weak dominance, not strict dominance. Truth gives at least as much utility against every rival profile, and strictly more against some profiles than a specified false bid. It need not produce a strict improvement against the rivals currently selected. With threshold 4, bids 5, 6, 7, and 8 all win and give utility 4 under second price. The earlier dominance lesson used the same distinction between a best response in one case and comparison across all cases. Several best responses to one fixed rival profile do not refute a weak-dominance statement.
| Your bid; rival maximum | Earlier priority | Later priority |
|---|---|---|
| 8; 8 | Win, pay 8, utility 0 | Lose, utility 0 |
| 7; 7 | Win, pay 7, utility 1 | Lose, utility 0 |
| 8; 7 | Win, pay 7, utility 1 | Win, pay 7, utility 1 |
The claim also requires a feasible truthful bid and the stated utility model. Here value 8 is among the thirteen available bids. A budget below a beneficial payment, a fee for winning, or a benefit from making another bidder pay more would alter the comparisons. Those are paper changes to assumptions, not switches in the lab. Under the supplied private-value, no-fee rules, truth is a best response regardless of beliefs about rival bids. That is why this second-price argument does not need the probability distribution required for first-price bid shading.
Winner: You.
Winner pays: 8.
Your utility: 0.
Winner: You.
Winner pays: 7.
Your utility: 1.
Your bid 8, value 8. Utilities: first price 0, second price 1.
| Your bid | First-price utility | Second-price utility |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 0 | 0 |
| 2 | 0 | 0 |
| 3 | 0 | 0 |
| 4 | 0 | 0 |
| 5 | 0 | 0 |
| 6 | 0 | 0 |
| 7 | 1 | 1 |
| 8 | 0 | 1 |
| 9 | -1 | 1 |
| 10 | -2 | 1 |
| 11 | -3 | 1 |
| 12 | -4 | 1 |
| Your bid; rivals 5,7 | First-price utility at value 8 | Second-price utility |
|---|---|---|
| 6 | 0, lose | 0, lose |
| 8 | 0, win | 1, win |
| 10 | −2, win | 1, win |
Select rival bids 5 and 7 and open All thirteen possible bids against these rivals. Read the first-price column as a schedule, not thirteen unrelated examples. Bids 0 through 6 lose and earn zero. Bid 7 wins by priority and earns 1. Bid 8 wins and earns zero; each bid above 8 earns a further negative unit. The best first-price response to these known rivals is therefore 7. The useful boundary is the smallest winning bid, because every extra unit paid above that boundary reduces utility without changing receipt of the object.
Read the second-price column beside it. Bids 0 through 6 again lose, but every bid 7 through 12 wins, pays 7, and earns 1. There is a plateau, meaning a consecutive range of bids with the same utility. Truth at 8 lies on this plateau without being its only member. Use the slider to move between 7 and 12 while watching both panels simultaneously. The first-price utility changes on every move while the second-price utility does not. The complete table keeps this comparison available without requiring a sequence of remembered slider states.
Retain bid 8 and select rivals 2 and 4. Your first-price utility remains zero because you pay your value when winning. Your second-price utility rises to 4 because the highest other bid is now 4. In the full table, first-price utility is largest at bid 4, where your priority wins the tie and leaves utility 4. Second-price utility is 4 for every bid from 4 upward. This changed best response is not evidence that value changed. It isolates how a lower competitive threshold changes both the cheapest winning bid and the second-price charge.
Now retain bid 8 and select rivals 9 and 10. You lose under both formats and receive zero. Raising the bid to 10 wins the tie but gives utility negative 2 in both formats. Raising it again to 11 makes first-price utility negative 3 while second-price utility stays negative 2. All losing bids are better than those winning bids for your value 8. An objective such as maximize probability of winning would choose differently from maximize utility. The original utility specification rewards receiving the object only after subtracting its payment.
The display's two formats use exactly the same bid profile at each setting. That is a controlled comparison of payment rules, not two predicted bidding equilibria. Suppose your bid is 8 against rivals 5 and 7. First-price seller receipts are 8 and second-price receipts are 7. But bidders might submit different bids if the announced rule changed before bidding. Holding their numbers fixed deliberately excludes that response. Consequently the panels cannot establish that one auction design produces greater expected revenue once everyone adapts, even if one panel repeatedly shows a larger payment during exploration.
| Rivals | Smallest winning bid | Best first-price bids | Second-price utility at bid 8 |
|---|---|---|---|
| 2, 4 | 4 | 4, utility 4 | 4 |
| 5, 7 | 7 | 7, utility 1 | 1 |
| 9, 10 | 10 | 0 through 9, utility 0 | 0 |
Treat preset changes as changes to the highest rival threshold, not to a probability estimate. The controls supply three particular rival pairs and retain your own bid. They do not assign frequencies to the pairs, generate repeated auctions, or reveal a distribution of private values. To transfer the calculation, first specify value and costs, determine which bid wins under the actual tie rule, then calculate payment and utility. If the bid must be chosen before rivals are known, add an explicit belief model. Without it, the best-response table remains conditional on known rival bids.
Private value means your value does not change when you learn the others' information about the object. With a common uncertain resale value, winning can reveal that your estimate was unusually optimistic. The winner's curse concerns conditioning on that selection, not a universal fixed percentage to subtract.
The explorer holds rival bids fixed to isolate payment arithmetic. It does not compare equilibrium seller revenue across auction formats, model an online platform's full rules, or recommend an actual bid. A real auction needs its own value model, tie policy, participation costs, and information assumptions.
Bid shading means submitting less than your value in order to retain a positive benefit when you win a first-price auction. Choosing the amount requires balancing winning probability against retained utility. Consider a separate paper example, not a random mode in the explorer. Your value remains 8, and the highest rival bid is equally likely to be 4, 6, or 9. You have the same tie priority. Bid 4 wins in one of three cases and earns 4 there. Its expected utility is therefore one third of 4, or four thirds.
Bid 6 wins against thresholds 4 and 6, earning 2 in either case. Two thirds of 2 is again four thirds. Bid 5 wins only against 4 and earns 3, for expected utility 1. Bid 7 wins twice but earns only 1 each time, for expected utility two thirds. Bid 8 gives zero whether it wins or loses. Among all integer bids 0 through 12, bids 4 and 6 tie for the highest expected first-price utility in this paper distribution. More winning opportunities do not by themselves establish a better bid.
Under second price in that same paper example, truth at 8 earns 4 against threshold 4, earns 2 against 6, and loses against 9. Averaging gives 2. Bid 4 instead wins only against threshold 4 and averages four thirds. Bid 10 wins all cases but adds utility negative 1 against threshold 9, reducing its average to five thirds. These averages illustrate the earlier case-by-case proof rather than replacing it. Truth wins the comparison for any probabilities on those thresholds because no individual threshold makes a false bid better under the stated second-price rules.
A common-value auction changes the meaning of information. Suppose on paper that an object will eventually pay the same uncertain amount to whoever owns it. Two observers each estimate that amount, with independent errors of either negative 2 or positive 2, equally likely. Across the four equally likely error pairs, the largest error is negative 2 only when both errors are negative. In the other three pairs it is positive 2. If bids equal estimates, the winning estimate therefore has average error positive 1, even though each observer's own error averages zero before selection.
That calculation shows selection among estimates, not a complete bidding solution. For illustration, if the object's true amount happens to be 8, the winning estimate is 6 in one error pair and 10 in three. A first-price winner bidding that estimate has utility 2 in the first case and negative 2 in the others, averaging negative 1. Bidders do not know the true 8 in this common-value story. Computing a sensible bid before submission would require beliefs about possible true amounts and about how observations relate to them, not simply subtracting the illustrated average error.
| Observer errors | Highest error | Winning estimate if true amount 8 | First-price winner utility |
|---|---|---|---|
| −2, −2 | −2 | 6 | 2 |
| −2, +2 | +2 | 10 | −2 |
| +2, −2 | +2 | 10 | −2 |
| +2, +2 | +2 | 10 | −2 |
| Equal-weight average | 1 | 9 | −1 |
The private-value explorer never updates your value when rivals change. That is appropriate for its stipulated consumption benefit, but inappropriate if winning reveals that your resale estimate exceeded everyone else's. The winner's curse is this conditioning problem, distinct from the first-price tradeoff between probability and payment. Second-price payment does not automatically remove common-value inference, just as it does not protect an overbidder from paying above value. Use the source's distinction between private and common values to identify which mechanism needs analysis. Neither the fixed-rival lab nor these paper calculations predict an online platform's complete auction behavior.
Your value is 8 and the highest rival bid is 10. Compare bidding 8 with bidding 11 in a second-price auction.
The winner pays the rival bid, not their own.
| Your bid | Outcome | Payment | Utility |
|---|---|---|---|
| 8 | Lose | 0 | 0 |
| 11 | Win | 10 | −2 |
Bidding 8 loses and gives zero. Bidding 11 wins, pays 10, and gives utility −2. Paying the second price does not make unlimited bidding safe. Truthful bidding avoids winning at a price above your private value.
With value 8 and rivals 5,7, a bid of 7 wins by the specified tie priority. First price then gives utility 1. Does this prove 7 is your optimal bid before seeing rivals?
The example supplies rivals' bids, but a sealed-bid participant normally does not observe them.
| Question | Information needed |
|---|---|
| Utility against bids 5,7 | Known bids and tie rule |
| Optimal bid before seeing bids | Beliefs about rivals and utility |
| Revenue comparison between formats | How all bids respond to each rule |
No. It is a best response to this particular fixed rival profile, given the tie rule. Before bidding, uncertainty about rival bids matters, and a lower bid trades a smaller payment against a lower chance of winning. The fixed-profile calculation does not supply that distribution or an equilibrium shading rule.
Selected reference passages support the definitions and algorithms below. The games, explanations, and implementations are original. Terminology and assumptions are stated within each lesson; source review is not an independent audit of this course. Optional links are not runtime dependencies.
Selected private/common-value distinctions, winner's curse discussion, and first-/second-price incentives. Fixed-bid comparisons here do not establish revenue rankings.
Selected definitions of strict and weak dominance, and discussion of beliefs about other players. Weak dominance has different equilibrium-preservation properties from strict dominance.