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Efficient Handicap Auction
(Old version: The 40%-Handicap Auction)
Yosuke YASUDA
Osaka University, Dept. of Economics
yasuda...
The Paper is About ...
Efficient design of license auctions
§
¦
¤
¥Ex Spectrum auctions
Auctions design: Competition in the ...
Introduction
Government can influence markets by allocating licenses.
(Direct Control ⇒) “Beauty Contest” ⇒ License Auction...
How many licenses to provide/sell?
The government can choose the # of licenses to provide.
Auctions likely achieve efficienc...
Free entry is NOT efficient
Fact 1 (Excess Entry Theorem)
The equilibrium # of firms in an oligopoly market under free entry
...
Motivating Example
European UMTS (3G) Auctions
Spectrum auctions in European countries in 2000-01.
Each country sets a num...
Simplest Setting: Monopoly or Duopoly
Consider a monopoly market.
The monopolist already has a license.
Government provide...
The Benchmark Model
Homogenous good
Linear demand: p = a − bq
2 firms
Incumbent
Newcomer
: firm 1
: firm 2
Cournot competitio...
Welfare-Reducing Entry
Fact 2 (Lahiri and Ono, 1988)
Duopoly is more efficient than monopoly iff
c2 < c∗
where c∗ =
5a + 17c1...
High-Cost or Low-Cost
Figure: c2 is high (Left) c2 is low (Right)
10 / 39
The Beauty Contest
If the government knew the parameters a, c1, c2
⇒ Optimal policy can be implemented.
Otherwise, what ca...
Benchmark assumptions
The government maximizes social welfare (total surplus).
⇒ Generalized social welfare
The firms’ cost...
Valuations for the license
Each firm’s valuation for the license:
v1 = πm − π1
v2 = π2
Truthful bidding is optimal (a domin...
Symmetric Firms (c1= c2)
Duopoly is more efficient than monopoly.
However, the incumbent wins (Gilbert and Newbery, 1982).
⇐...
Handicap (English) Auction with H
The auction stops when one firm drops out.
The remaining bidder is the winner who obtains...
Benchmark Result
Theorem 3
The 40% handicap auction (H = 0.4) described as follows
implements entry iff duopoly is more effic...
The sketch of the proof
∆SW = ∆CS + ∆PS = ∆CS + π1 + π2 − πm
= ∆CS + v2 − v1 (SW)
∆CS can be expressed as follows (Lemma 3...
Lemma 3: ∆CS =
v1
2
+
v2
4
Figure: Valuations (Left) ∆CS in two parts (Right)
18 / 39
Remarks
Our auction is independent of the parameters, a, b, c1, c2.
Demand functions need NOT be globally linear.
Robustne...
Generalization
Asymmetric Information among firms
Non-linear costs & demand
Wealfare Loss: Numerical Results
Multiple incum...
Asymmetric Information
Two cases of asymmetric information.
Case 1: The newcomer’s cost is only privately known.
⇒ Reasona...
One-Sided Private Information
Theorem 4
The 40% handicap auction continues to achieves efficiency even if
the newcomer’s cos...
Efficiency Result
For the incumbent, it is optimal to take the following bidding
strategy b1 (Lemma 4).
b1 = πm − π1(c1, c∗)...
Fixed Costs
Suppose a newcomer has a fixed set-up cost F.
C2(q2) =
F + c2q2
0
if q2 > 0
if q2 = 0
The 40% handicap auction ...
Fixed Costs: Result
Theorem 5
Suppose the government can observe F. Then, the 40% handicap
auction with the following cond...
Quadratic Costs
Suppose firms have the following quadratic cost functions.
Ci(qi) =
αq2
i
2
+ ciqi i = 1, 2
The slope of th...
Quadratic Costs: Result
Theorem 6
Suppose firms have the above quadratic costs, and the government
knows α. Then the handic...
When α is not known
The first best cannot be achieved.
What if the 40% auction is employed?
α < 0: concave
α > 0: convex
⇒ ...
Efficiency Loss
The optimal handicap H∗ is different from 40% in non-linear
cases.
If we employ the 40% handicap auction in n...
When Does Inefficiency Happen?
Under the optimal rule
v2
H∗ < v1 ⇔ v2
v1
< H∗
v2
H∗ > v1 ⇔ v2
v1
> H∗
⇒ Monopoly
⇒ Duopoly
U...
Numerical Results
Fixed costs with F =
1
2
π2.
Then, the optimal handicap H∗ is
1
3
.
Assume the following distribution.
v...
Expected Social Welfare
The previous analysis did not consider the impact of
inefficient outcomes.
⇒ How big is the expected...
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...
Multiple Incumbents
Suppose there are n incumbents.
European UMTS Auctions
The number of incumbents in 2G services ( = n)
...
Multiple Incumbents: Result 1
Theorem 7
Suppose the incumbents are allowed to bid as a group and to
provide a single bid. ...
Multiple Incumbents: Result 2
Theorem 8
Suppose all the conditions stated in Theorem 7 are satisfied and F
is observed by t...
Pre-Auction Mechanism (among the incumbents)
Each incumbent chooses bi ∈ [0, vi].
The incumbents bid
n
i=1
bi in the aucti...
Conclusion
An extremely simple efficient license auction is proposed.
Can be generalized in many situations.
Efficiency losses...
Future Works
Other type of competition
⇒ Bertrand with differentiated goods
Uncertainty
⇒ Future demand, or costs after the...
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Efficient Handicap Auction

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Presentation slides for a workshop at Chulalongkorn University (Dec. 1st, 2015).

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Efficient Handicap Auction

  1. 1. Efficient Handicap Auction (Old version: The 40%-Handicap Auction) Yosuke YASUDA Osaka University, Dept. of Economics yasuda@econ.osaka-u.ac.jp December, 2015 1 / 39
  2. 2. 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! 2 / 39
  3. 3. 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 3 / 39
  4. 4. 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) 4 / 39
  5. 5. Free entry is NOT efficient Fact 1 (Excess Entry Theorem) The equilibrium # of firms in an oligopoly market under free entry is 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? 5 / 39
  6. 6. 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? 6 / 39
  7. 7. Simplest Setting: Monopoly or Duopoly Consider a monopoly market. The monopolist already has a license. Government provides a second license or not: No additional license Providing 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? 7 / 39
  8. 8. The Benchmark Model Homogenous good Linear demand: p = a − bq 2 firms Incumbent Newcomer : firm 1 : firm 2 Cournot competition Constant marginal costs: ci, i = 1, 2 8 / 39
  9. 9. 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. 9 / 39
  10. 10. High-Cost or Low-Cost Figure: c2 is high (Left) c2 is low (Right) 10 / 39
  11. 11. 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 11 / 39
  12. 12. 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. 12 / 39
  13. 13. 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? 13 / 39
  14. 14. 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! 14 / 39
  15. 15. 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 “bidding v2 H .” 15 / 39
  16. 16. Benchmark Result Theorem 3 The 40% handicap auction (H = 0.4) described as follows implements 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%”? 16 / 39
  17. 17. 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 17 / 39
  18. 18. Lemma 3: ∆CS = v1 2 + v2 4 Figure: Valuations (Left) ∆CS in two parts (Right) 18 / 39
  19. 19. 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 is second order effect. (Akerlof and Yellen, 1985) Efficiency is achieved by dominant strategies. ⇒ Independent of the cost distributions. 19 / 39
  20. 20. 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) 20 / 39
  21. 21. 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. 21 / 39
  22. 22. One-Sided Private Information Theorem 4 The 40% handicap auction continues to achieves efficiency even if the newcomer’s cost becomes private information. Solved by iterative dominance. The newcomer has a dominant strategy, “bidding v2 0.4 .” ⇒ What is the incumbent’s best response? 22 / 39
  23. 23. Efficiency Result For the incumbent, it is optimal to take the following bidding strategy 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. 23 / 39
  24. 24. Fixed Costs Suppose a newcomer has a fixed set-up cost F. C2(q2) = F + c2q2 0 if q2 > 0 if q2 = 0 The 40% handicap auction fails to achieve the first best when F > 0 (Lemma 5). ⇒ Is there any efficient mechanism? 24 / 39
  25. 25. Fixed Costs: Result Theorem 5 Suppose the government can observe F. Then, the 40% handicap auction with the following conditional subsidy achieves efficiency. If the newcomer wins If 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.” 25 / 39
  26. 26. 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: concave α > 0: convex 26 / 39
  27. 27. Quadratic Costs: Result Theorem 6 Suppose firms have the above quadratic costs, and the government knows α. Then the handicap auction with Hq achieves efficiency. Hq = 2 5 + 2α + α 2+α The optimal handicap Hq depends on α.    α < 0: concave α = 0: linear α > 0: convex ⇒ Hq ↑ ⇒ Hq = 0.4 ⇒ Hq ↓ 27 / 39
  28. 28. When α is not known The first best cannot be achieved. What if the 40% auction is employed? α < 0: concave α > 0: convex ⇒ Never deter welfare increasing entry ⇒ Never accept welfare decreasing entry 28 / 39
  29. 29. Efficiency Loss The optimal handicap H∗ is different from 40% in non-linear cases. If we employ the 40% handicap auction in non-linear cases, then How likely is an inefficiency? How big is the expected efficiency loss? 29 / 39
  30. 30. When Does Inefficiency Happen? Under the optimal rule v2 H∗ < v1 ⇔ v2 v1 < H∗ v2 H∗ > v1 ⇔ v2 v1 > H∗ ⇒ Monopoly ⇒ Duopoly Under the 40% handicap auction v2 v1 < 0.4 v2 v1 > 0.4 ⇒ Monopoly ⇒ Duopoly Inefficiency happens iff H∗ < v2 v1 < 0.4 if H∗ < 0.4 0.4 < v2 v1 < H∗ if H∗ > 0.4 ⇒ Deter desirable entry ⇒ Accept undesirable entry 30 / 39
  31. 31. Numerical Results Fixed costs with F = 1 2 π2. Then, the optimal handicap H∗ is 1 3 . Assume the following distribution. v1 = 1, v2 ∼ U[0, 1] ⇒ No entry under the English auction. An inefficient outcome occurs with 7% . 31 / 39
  32. 32. Expected Social Welfare The previous analysis did not consider the impact of inefficient outcomes. ⇒ How big is the expected efficiency loss? Expected social welfare under each policy. V ∗ = H∗ 0 SWm dv2 + 1 H∗ SWd dv2 V 40 = 0.4 0 SWm dv2 + 1 0.4 SWd dv2 V m = 1 0 SWm dv2 32 / 39
  33. 33. 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 outcomes happen. 33 / 39
  34. 34. 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? 34 / 39
  35. 35. Multiple Incumbents: Result 1 Theorem 7 Suppose the incumbents are allowed to bid as a group and to provide a single bid. If they can fully cooperate for bidding, then the handicap auction with Hm achieves efficiency. Hm = n + 1 2n + 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? 35 / 39
  36. 36. Multiple Incumbents: Result 2 Theorem 8 Suppose all the conditions stated in Theorem 7 are satisfied and F is observed by the government. Then, the combination of the handicap auction with Hm and the following conditional subsidy achieves efficiency. If the newcomer wins If the incumbent wins ⇒ Subsidized by 1 2n+3 F ⇒ No subsidy The subsidy converges to 0 as n → ∞. ⇒ Is cooperation possible? 36 / 39
  37. 37. Pre-Auction Mechanism (among the incumbents) Each incumbent chooses bi ∈ [0, vi]. The incumbents bid n i=1 bi in the auction and each incumbent pays vi n j=1 vj of the winning price it they win. ⇒ This mechanism implements cooperative bidding. Risk of collusion after the auction. 37 / 39
  38. 38. 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. 38 / 39
  39. 39. 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 39 / 39

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