Chapter 28: Optimal Auctions, Multi-Unit Sales, and Package Bidding. The Encyclopedia of Economic Principals, interactive companion

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

When VCG prices invite a blocking deal

Why can VCG revenue be so low that the seller would rather sell to the losing package bidder?

VCG charges each winner the harm it does to the others. With two complementary locals, each one's harm is small, so their combined payments can fall short of the losing package bid. A core-selecting auction raises payments until no coalition with the seller can block.

Equation, written in LaTeX: 60+60=120>100.

$Equation, written in LaTeX: p_A^{VCG}=100-60=40.$

$Equation, written in LaTeX: p_A+p_B\geq100.$

Equation, written in LaTeX: (p_A,p_B)=(50,50).

Scroll sideways for the whole equation

Alice bids 60 for license A, Ben bids 60 for license B, and Nora bids N for the package AB. p_A and p_B are the payments of Alice and Ben. The core asks that no coalition of the seller and bidders could do better on its own.

Predict first. Does raising Nora's bid from 100 to 110 raise or lower the locals' VCG payments?

Your prediction It raises them It lowers them

Nora's package bid

Figure: When VCG prices invite a blocking deal. Bars: Nora's bid 100, VCG revenue 80, core revenue 100. A dashed line at 100 shows the blocking gap of 20 above VCG revenue. — Nora's package bid: 100
Constructed example: the chapter's hypothetical license auction (Alice 60, Ben 60, Nora 100, counterfactual 130); Nora bids of 90 and 110 are added for comparison.

Winners: Alice and Ben
VCG payment each: 40
VCG revenue: 80
Blocking gap: 20
Core payments (Alice, Ben): (50, 50)
Nora's payment: none (she loses)

The locals create 60 + 60 = 120, more than Nora's 100, so they win. Removing Alice leaves Nora's 100 as the best alternative, so Alice pays 100 - 60 = 40, and Ben pays the same. VCG revenue is 80, which is 20 below Nora's bid, so the seller and Nora could block. The core requires p_A + p_B >= 100; the symmetric minimum-revenue point is (50, 50), within each local's bid of 60.

Worked steps

60 + 60 = 120 > 100
p_A = 100 - 60 = 40, p_B = 100 - 60 = 40
VCG revenue = 40 + 40 = 80 < 100
Core: p_A + p_B = 100, symmetric (50, 50)

Use the idea

When package bidders compete with several small bidders, compare VCG revenue with the best losing package bid before relying on VCG prices.

Where the conclusion applies

Single-minded bids, truthful reports and a symmetric rule for splitting the core adjustment. Core pricing gives up the exact truth-telling incentive of VCG.

Check your understanding: With Nora's bid at 110, what are the VCG and symmetric core payments for each local?

VCG: 110 - 60 = 50 each, revenue 100 < 110, so the outcome is blocked. The core needs p_A + p_B >= 110, so the symmetric point is (55, 55), within each bid of 60.

Chapter 28 source: section "Core-selecting package auctions".

Demonstration 2 of 4

The optimal reserve sometimes refuses to sell

Why does the revenue-maximizing auction sometimes keep an object a bidder values?

The virtual value subtracts the information rent the seller must leave to higher types. Selling only when it is positive trades a few lost sales for higher prices on the rest, which raises expected revenue from 33.33 to 41.67.

$Equation, written in LaTeX: \phi(v)=v-\frac{1-v/100}{1/100}=2v-100.$

$Equation, written in LaTeX: \frac12(50)+\frac14(\frac{200}{3})=\frac{125}{3}\approx41.67.$

Scroll sideways for the whole equation

Two bidders' values v are independent and uniform on [0, 100]. phi(v) is the virtual value; the optimal mechanism sells to the highest positive virtual value, which is a second-price auction with reserve 50.

Predict first. For the profile (45, 40), how much realized revenue does the reserve of 50 cost?

Your prediction 40 5 Nothing

Higher value

Lower value

Reserve

phi(higher value): 60
phi(lower value): -20
Outcome: sale
Payment: 50
Winner utility: 30
Realized revenue change from the reserve: +10
Expected revenue of this format: 41.67

With values uniform on [0, 100], phi(v) = 2v - 100, so phi(80) = 2 x 80 - 100 = 60 and phi(40) = 2 x 40 - 100 = -20. With reserve 50, the bidder with value 80 wins and pays max(40, 50) = 50. Without a reserve the payment would be 40, so the reserve changes realized revenue by 10 in this profile. Expected revenue is 41.67 with this format, against 41.67 with reserve 50 and 33.33 with none.

Worked steps

phi(80) = 2 x 80 - 100 = 60
phi(40) = 2 x 40 - 100 = -20
Payment = max(40, 50) = 50, utility = 80 - 50 = 30
Change from no reserve = 50 - 40 = 10

Use the idea

Set a reserve where the virtual value of the value distribution crosses the seller's own value, not at the seller's value itself.

Where the conclusion applies

Independent private values from a known uniform distribution, risk-neutral bidders and a seller who can commit not to resell after a no-sale.

Check your understanding: With values (60, 30), what does each format collect?

No reserve: 30. Reserve 50: 60 >= 50, so the sale is at max(30, 50) = 50, a gain of 20. phi(60) = 20 > 0.

Chapter 28 source: section "Myerson optimal-auction theorem".

Demonstration 3 of 4

Correlated types let the seller take everything

How can a side bet on a rival's report remove every bidder's information rent?

Different types hold different beliefs about the rival. A bet the low type expects to break even on costs the high type exactly its rent, so the seller collects the full surplus. The weaker the correlation, the larger the bets needed.

$Equation, written in LaTeX: \Pr(B=H\mid A=H)=\frac{0.4}{0.5}=0.8, \Pr(B=H\mid A=L)=\frac{0.1}{0.5}=0.2.$

$Equation, written in LaTeX: 0.8(\frac{80}{3})-0.2(\frac{20}{3})=20,$

$Equation, written in LaTeX: 0.2(\frac{80}{3})-0.8(\frac{20}{3})=0.$

Scroll sideways for the whole equation

Each bidder values the license at 100 (type H) or 0 (type L). Pr(H,H) = Pr(L,L) = a and each off-diagonal pair has probability 0.5 - a. The charge applies when the rival reports H and the rebate when it reports L.

Predict first. As correlation weakens, do the required side bets grow or shrink?

Your prediction They grow They shrink

Pr(H,H) = Pr(L,L)

Figure: Correlated types let the seller take everything. Bars: efficient surplus 60.00, second-price revenue 40.00, revenue with the lottery 60.00. — Pr(H,H) = Pr(L,L): 0.4
Constructed example: the chapter's hypothetical license auction (0.4 and 0.1, independent counterfactual 0.25); a = 0.3 is added.

Pr(B=H | A=H): 0.80
Pr(B=H | A=L): 0.20
Interim rent of H: 20.00
Charge if rival reports H: 26.67
Rebate if rival reports L: 6.67
Efficient surplus: 60.00
Second-price revenue: 40.00
Total revenue: 60.00

With Pr(H,H) = 0.40, Pr(B=H | A=H) = 0.40/0.5 = 0.80 and Pr(B=H | A=L) = 0.10/0.5 = 0.20. The high type's second-price rent is 0.20 x 100 = 20.00. A charge of 26.667 when the rival reports H and a rebate of 6.667 when it reports L cost the high type 0.80 x 26.667 - 0.20 x 6.667 = 20.000 and the low type 0.20 x 26.667 - 0.80 x 6.667 = 0. Revenue rises from 40.00 to 60.00, the whole surplus.

Worked steps

Pr(B=H | A=H) = 0.40 / 0.5 = 0.80
Rent of H = 0.20 x 100 = 20.00
0.80 x 26.667 - 0.20 x 6.667 = 20.000
0.20 x 26.667 - 0.80 x 6.667 = 0.000
Total = 40.00 + 2 x 0.5 x 20.00 = 60.00 = surplus 60.00

Use the idea

Treat full extraction as a warning about modeling assumptions: it needs known correlation, risk neutrality and unlimited liability.

Where the conclusion applies

Common knowledge of the joint distribution, risk-neutral bidders and no limit on how large a charge or rebate can be.

Check your understanding: With a = 0.3, what charge and rebate extract the high type's rent?

Pr(H|H) = 0.6, Pr(H|L) = 0.4; the H rent is 0.4 x 100 = 40. Solve 0.6x - 0.4y = 40 and 0.4x - 0.6y = 0: x = 120, y = 80. Surplus 70 = 30 + 2 x 0.5 x 40.

Chapter 28 source: section "Cremer-McLean full-surplus extraction".

Demonstration 4 of 4

Bidding for less to pay less

Why would a firm bid zero for a unit it values more than its rival does?

In a uniform-price auction a large bidder's own marginal bid can set the price it pays on every unit. Withdrawing a unit lowers that price, which can outweigh the unit's surplus. Vickrey pricing charges each bidder its externality instead, which removes the motive.

Equation, written in LaTeX: (100-70)+(90-70)=50.

Equation, written in LaTeX: 100-0=100.

Equation, written in LaTeX: 100+90+80=270.

Equation, written in LaTeX: 100+80+70=250.

Equation, written in LaTeX: 100+90-70=120.

Scroll sideways for the whole equation

Three identical permits; the top three marginal bids win and every winner pays the highest rejected bid. Firm A values two units at 100 and a2; firm B values two units at 80 and b2.

Predict first. If B's second unit were worth only 50, would A still gain by shading?

Your prediction Yes No

A's second-unit value

B's second-unit value

Figure: Bidding for less to pay less. Bars of A's utility: 50 truthful under the uniform price, 100 when shading, 120 truthful under Vickrey. — A's second-unit value: 90, B's second-unit value: 70
Constructed example: the chapter's hypothetical permit auction (A 100 and 90, B 80 and 70); second-unit values a2 = 60 and 75 and b2 = 50 are added.

Truthful clearing price: 70
A's truthful utility: 50
A's utility when shading: 100
Efficiency loss from shading: 20
A's truthful utility under Vickrey: 120
Vickrey vs shading: truthful under Vickrey

Truthful bids 100, 90, 80 and 70: the top three win and the price is the highest rejected bid, 70. A earns (100 - 70) + (90 - 70) = 50. Bidding 0 for its second unit makes the price 0, so A earns 100 - 0 = 100 and gains by shading. Value falls from 270 to 250, a loss of 20. Under Vickrey, A's truthful utility is 120, more than the 100 from shading.

Worked steps

Truthful price = highest rejected bid = 70
A truthful: (100 - 70) + (90 - 70) = 50
A shades: 100 - 0 = 100
Value: 270 - 250 = 20 lost
Vickrey: A pays 150 - 80 = 70, utility 120

Use the idea

In a multi-unit sale, check whether any bidder's second bid is likely to be the price-setting rejected bid; if so, expect demand reduction.

Where the conclusion applies

Known values, zero reserve, and B bidding truthfully. Under Vickrey pricing the comparison uses A requesting one unit at a price of 0.

Check your understanding: With a2 = 75 and b2 = 50, what are A's truthful and reduced payoffs?

Truthful winners 100, 80 and 75; price 50; A earns (100 - 50) + (75 - 50) = 75. Shading: 100 - 0 = 100 > 75, with an efficiency loss of 75 - 50 = 25.

Chapter 28 source: section "Demand reduction".

When VCG prices invite a blocking deal

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

The optimal reserve sometimes refuses to sell

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Correlated types let the seller take everything

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Bidding for less to pay less

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies