Efficient Handicap Auction

(Old version: The 40%-Handicap Auction)

Yosuke YASUDA

Osaka University, Dept. of Economics

yasuda@econ.osaka-u.ac.jp

December, 2015

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The Paper is About ...

Efficient design of license auctions�� ��Ex Spectrum auctions

Auctions design: Competition in the auction⇐ Game Theory

Entry regulation: Competition after the auction⇐ Industrial Organization

Connecting two competitions in a single unified model.�� ��Q What is efficient way to provide licenses?�� ��A Use the (40%) handicap auction!

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Introduction

Government can influence markets by allocating licenses.

(Direct Control ⇒) “Beauty Contest” ⇒ License Auctions

Advantages: efficiency, revenue, transparency, speed . . .

Real life examples of license auctions:

Bus routes

Radio spectrum rights

Airport slots

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How many licenses to provide/sell?

The government can choose the # of licenses to provide.

Auctions likely achieve efficiency given the # of licenses.

Usual Efficiency = maximizing winners’ valuations

Efficiency in this paper = maximizing total welfare

Impossible to decide the optimal # prior to the auction.

market competition(many licenses)

vs. production efficiency(few licenses)

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Free entry is NOT efficient

Fact 1 (Excess Entry Theorem)

The equilibrium # of firms in an oligopoly market under free entryis greater than the efficient # of firms.

Symmetric firms with fixed costs:

Mankiw and Whinston (1986, Rand)

Suzumura and Kiyono (1987, REStud)

Asymmetric firms (with no fixed cost):

Lahiri and Ono (1988, EJ)

⇒ A weak rationale for entry regulation.

⇒ How to implement the optimal regulation in practice?

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Motivating Example

European UMTS (3G) Auctions

Spectrum auctions in European countries in 2000-01.

Each country sets a number of licenses

={

# of incumbent firms in 2G services# of — + 1

Effectively, “accepting a new entrant” or “not.”

⇒ Can we better choose the number?

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Simplest Setting: Monopoly or Duopoly

Consider a monopoly market.

The monopolist already has a license.

Government provides a second license or not:{No additional licenseProviding a license

⇒ Monopoly⇒ Duopoly

Either a monopoly or a duopoly can be efficient.

The costs are private information of the firms.

⇒ How to implement an optimal policy?

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The Benchmark Model

Homogenous good

Linear demand: p = a− bq

2 firms

{IncumbentNewcomer

: firm 1: firm 2

Cournot competition

Constant marginal costs: ci, i = 1, 2

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Welfare-Reducing Entry

Fact 2 (Lahiri and Ono, 1988)

Duopoly is more efficient than monopoly iff

c2 < c∗

where c∗ =5a + 17c1

22.{

c2 < c∗ (low-cost)c2 > c∗ (high-cost)

⇒ Social Welfare ↑⇒ Social Welfare ↓

See the next figures.

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High-Cost or Low-Cost

Figure: c2 is high (Left) c2 is low (Right)

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The Beauty Contest

If the government knew the parameters a, c1, c2

⇒ Optimal policy can be implemented.

Otherwise, what can we do?

How about auctions?

English or second-price auction:{Firm 1 wins:Firm 2 wins:

⇒ Monopoly⇒ Duopoly

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Benchmark assumptions

The government maximizes social welfare (total surplus).

⇒ Generalized social welfare

The firms’ costs are common knowledge among firms.

⇒ Asymmetric information

The government can only control entry decision.

= Other (direct) regulations are excluded.

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Valuations for the license

Each firm’s valuation for the license:

v1 = πm − π1

v2 = π2

Truthful bidding is optimal (a dominant strategy).

Entry occurs iff

v1 < v2 ⇐⇒ πm − π1 < π2

⇒ Does this mechanism achieve efficiency?

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Symmetric Firms (c1= c2)

Duopoly is more efficient than monopoly.

However, the incumbent wins (Gilbert and Newbery, 1982).

⇐ The monopoly profit is larger than the duopoly (joint-)profit.

πm > π1 + π2 ⇐⇒ πm − π1 > π2 ⇐⇒ v1 > v2

Some kind of handicap favoring a newcomer is needed.

⇒ Think about “handicap” auctions!

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Handicap (English) Auction with H

The auction stops when one firm drops out.

The remaining bidder is the winner who obtains the license.

⇒ Similar to an English auction.

Only the payment of the newcomer is different.{Incumbent:Newcomer:

Pay the winning price if it wins.Pay only H of the winning price if it wins

⇒ The newcomer’s optimal strategy becomes “biddingv2

H.”

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Benchmark Result

Theorem 3

The 40% handicap auction (H = 0.4) described as followsimplements entry iff duopoly is more efficient than monopoly.{

Incumbent:Newcomer:

Pay the winning price if it wins.Pay only 40% of the winning price if it wins

⇒ Why “40%”?

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The sketch of the proof

∆SW = ∆CS + ∆PS = ∆CS + π1 + π2 − πm

= ∆CS + v2 − v1 (SW)

∆CS can be expressed as follows (Lemma 3).

∆CS =v1

2+

v2

4

Substituting it into (SW), we obtain the result.

∆SW > 0 ⇐⇒ v1

2+

v2

4+ v2 − v1 > 0

⇐⇒ v1 <v2

0.4

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Lemma 3: ∆CS =v1

2+

v2

4

Figure: Valuations (Left) ∆CS in two parts (Right)

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Remarks

Our auction is independent of the parameters, a, b, c1, c2.

Demand functions need NOT be globally linear.

Robustness: welfare loss caused by introducing non-linearity issecond order effect. (Akerlof and Yellen, 1985)

Efficiency is achieved by dominant strategies.

⇒ Independent of the cost distributions.

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Generalization

Asymmetric Information among firms

Non-linear costs & demand

Wealfare Loss: Numerical Results

Multiple incumbents (← if time remains)

No incumbent firm, i.e., new market (← Skipped)

General social welfare functions (← Skipped)

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Asymmetric Information

Two cases of asymmetric information.

Case 1: The newcomer’s cost is only privately known.

⇒ Reasonable situation.

Case 2: The both firms’ costs are private information.

⇒ Impossibility result. (← Skipped)

A government cannot observe firms’ costs.

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One-Sided Private Information

Theorem 4

The 40% handicap auction continues to achieves efficiency even ifthe newcomer’s cost becomes private information.

Solved by iterative dominance.

The newcomer has a dominant strategy, “biddingv2

0.4.”

⇒ What is the incumbent’s best response?

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Efficiency Result

For the incumbent, it is optimal to take the following biddingstrategy b1 (Lemma 4).

b1 = πm − π1(c1, c∗)

⇒ b1 = v1 iff b1 = b2

(=

v2

0.4

).

The same outcome and payoff as in the benchmark case.

⇒ No efficiency loss.

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Fixed Costs

Suppose a newcomer has a fixed set-up cost F .

C2(q2) ={

F + c2q2

0if q2 > 0if q2 = 0

The 40% handicap auction fails to achieve the first best whenF > 0 (Lemma 5).

⇒ Is there any efficient mechanism?

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Fixed Costs: Result

Theorem 5

Suppose the government can observe F . Then, the 40% handicapauction with the following conditional subsidy achieves efficiency.{

If the newcomer winsIf the incumbent wins

⇒ The subsidy of 0.2F⇒ No subsidy

The government need not know F prior to the auction.Instead, it is sufficient to observe it ex-post.

Implemented by “investment tax credit.”

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Quadratic Costs

Suppose firms have the following quadratic cost functions.

Ci(qi) =αq2

i

2+ ciqi i = 1, 2

The slope of the marginal costs α is common across the firms.{α < 0: concaveα > 0: convex

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Quadratic Costs: Result

Theorem 6

Suppose firms have the above quadratic costs, and the governmentknows α. Then the handicap auction with Hq achieves efficiency.

Hq =2

5 + 2α + α2+α

The optimal handicap Hq depends on α.α < 0: concave

α = 0: linear

α > 0: convex

⇒ Hq ↑

⇒ Hq = 0.4

⇒ Hq ↓

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When α is not known

The first best cannot be achieved.

What if the 40% auction is employed?{α < 0: concaveα > 0: convex

⇒ Never deter welfare increasing entry⇒ Never accept welfare decreasing entry

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Efficiency Loss

The optimal handicap H∗ is different from 40% in non-linearcases.

If we employ the 40% handicap auction in non-linear cases,then{

How likely is an inefficiency?How big is the expected efficiency loss?

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When Does Inefficiency Happen?

Under the optimal rule{ v2H∗ < v1 ⇔ v2

v1< H∗

v2H∗ > v1 ⇔ v2

v1> H∗

⇒ Monopoly⇒ Duopoly

Under the 40% handicap auction{ v2v1

< 0.4v2v1

> 0.4⇒ Monopoly⇒ Duopoly

Inefficiency happens iff{H∗ < v2

v1< 0.4 if H∗ < 0.4

0.4 < v2v1

< H∗ if H∗ > 0.4⇒ Deter desirable entry⇒ Accept undesirable entry

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Numerical Results

Fixed costs with F =12π2.

Then, the optimal handicap H∗ is13.

Assume the following distribution.

v1 = 1, v2 ∼ U [0, 1]

⇒ No entry under the English auction.

An inefficient outcome occurs with 7% .

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Expected Social Welfare

The previous analysis did not consider the impact ofinefficient outcomes.

⇒ How big is the expected efficiency loss?

Expected social welfare under each policy.

V ∗ =∫ H∗

0SWmdv2 +

∫ 1

H∗SW ddv2

V 40 =∫ 0.4

0SWmdv2 +

∫ 1

0.4SW ddv2

V m =∫ 1

0SWmdv2

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Expected Welfare Loss

Relative performance of the 40% auction.

RP ≡ V 40 − V m

V ∗ − V m≤ V 40

V ∗ ≤ 1

In our example, RP is 0.99.

⇒ The expected loss is just 1%!

The expected loss of social welfare is much smaller than 7%.

⇒ |∆SW | is relatively small when inefficient outcomeshappen.

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Multiple Incumbents

Suppose there are n incumbents.

European UMTS Auctions{The number of incumbents in 2G services ( = n)One more than it ( = n + 1)

⇒ Can we apply our handicap auction?

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Multiple Incumbents: Result 1

Theorem 7

Suppose the incumbents are allowed to bid as a group and toprovide a single bid. If they can fully cooperate for bidding, thenthe handicap auction with Hm achieves efficiency.

Hm =n + 12n + 3

We do NOT assume incumbents are symmetric.

Even so, Hm depends only on n, not on c1, ..., cn.

Cooperative bidding resolves “free-rider problem.”

⇒ Fixed costs?

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Multiple Incumbents: Result 2

Theorem 8

Suppose all the conditions stated in Theorem 7 are satisfied and Fis observed by the government. Then, the combination of thehandicap auction with Hm and the following conditional subsidyachieves efficiency.{

If the newcomer wins

If the incumbent wins

⇒ Subsidized by 12n+3F

⇒ No subsidy

The subsidy converges to 0 as n→∞.

⇒ Is cooperation possible?

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Pre-Auction Mechanism (among the incumbents)

Each incumbent chooses bi ∈ [0, vi].

The incumbents bidn∑

i=1bi in the auction and each incumbent

paysvi

n∑j=1

vj

of the winning price it they win.

⇒ This mechanism implements cooperative bidding.

Risk of collusion after the auction.

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Conclusion

An extremely simple efficient license auction is proposed.

Can be generalized in many situations.

Efficiency losses in non-linear cases are quite small.

⇒ Contribution to practical market design.

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Future Works

Other type of competition

⇒ Bertrand with differentiated goods

Uncertainty

⇒ Future demand, or costs after the auction

Incentive issues

⇒ Cost reducing investment

Robustness check in the above cases

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