Licenses Allocation First part: Fahimeh FathianRad 22/ 1 In - - PowerPoint PPT Presentation

Balancing Efficiency and Equality in Vehicle Licenses Allocation

First part: Fahimeh FathianRad

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In Introduction

air pollution prompts the government to take more effective measures to control the number of vehicles.

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Shanghai(before 2013) Focuses: Efficiency Disadvantages: Low Equality (Auction) Beijing, Guiyang (Lottery) Focuses: Equality Disadvantages: Inefficiency, No Value Exploration Focuses: Efficiency, Equality Disadvantages: Hard to set price, No Value Exploration Focuses: Efficiency, Equality Disadvantages: Untruthful, Hard to choose, Hard to set auction size Shanghai(after 2013) Guangzhou, Hangzhou, Shenzhen, Tianjin, Shijiazhuang (Simultaneous auction and lottery) (Reserved-price lottery)

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Model and the Unified Framework

The government plans to distribute K homogeneous licenses, which are desired by N (> K) players. We assume every player i 2 N = f1; 2; : : : ;Ng is risk-neutral and has unit demand with private value vi > 0 which is drawn independently from a common distribution F(.). utility function of every player i: 𝑤1 ≥ 𝑤2 ≥ ⋯ ≥ 𝑤𝑜

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Definitions:

• 1. A mechanism is called incentive compatible if for any player,

truthfully bidding his private value is the weakly dominant strategy for him, no matter what other players' bids are

• 2. A mechanism is ex-post individually rational, if no matter how other

buyers bid, each player always gets a non-negative ex-post utility when bidding truthfully.

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Payment Rule

If a player i wins a license, then his payment is:

• therwise his payment is:

In lottery I: In lottery II :

No payment for participants: Mechanism 1 with the above payment rule is incentive compatible and ex-post individual rational.

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Efficiency Measure

Recall that and vi is the i-th highest value among the N values, so easy computation leads to the following corollary. The social welfare does not decrease as k increases, or as decreases.

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Equality Measure

where eqi is the i-th probability of all the players' winning probabilities in the descending order

is the average value of , The equality measure does not increase as k increases, or as decreases

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Relationship to Existing Mechanisms

Our framework either includes the existing mechanisms as special cases or

• utperforms them in terms of both efficiency and equality.

Auction Lottery Reserved-price Lottery Simultaneous Auction and Lottery

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Balancing Efficiency and Equality in Vehicle Licenses Allocation

Second part: HajarSiar

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Optimal two-group mechanism

• The players' private values are drawn independently from a common distribution

F(.)

• How should we determine the parameters 𝜆,𝛿 to achieve this objective?
• where c ∈ [0; 1] is a constant to measure the least required equality.

K N K c t s v E N K v E

N i i i i

       

 

  

          

 

1 , , ) , ( . . ) ( ) ( 1 ) , ( max

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Optimal two-group mechanism

• Theorem1: If N ≥ 2K and the probability distribution function f(.) is

monotonic non-increasing, then the optimal solution of the two- group mechanism is 𝛿= 1; 𝜆 =

𝑂𝐿(1−𝑑) 𝑂−𝐿

• Theorem2: If N ≥ 2K and the probability distribution function f

satisfies 𝑔 𝐺−1 𝑦

≥ 𝑔 𝐺−1 1 − 𝑦 , ∀𝑦 ∈ [0,

1 2], and 𝑔 𝐺−1 1 − 𝑦

is monotone non-decreasing about x ∈ [0,

1 2], then the optimal solution of

two-group mechanism is 𝛿= 1; 𝜆 =

𝑂𝐿(1−𝑑) 𝑂−𝐿

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Proving theorems 1&2

• Denote n = 𝜆𝛿:
• Solving the optimization problem of two-group mechanism is equivalent to solving the

following programming:

• The new programming means that we need to find the number n to maximize the

expected mean value difference between the n players in Lottery I and other N-n players in Lottery II.

Nc n K N c NK t s v E n N v E n n D

N n i i n i i

      

 

  

) 1 ( . . ) ( 1 ) ( 1 ) ( max

1 1

n N K c K c NK nK K N NK K K N            ) 1 ( ) ( ) ( ) , (       

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Proving theorems 1&2

• The trend of spacing is used: ∆𝒋= 𝑭 𝒘𝒋 − 𝒘𝒋+𝟐
• Lemma1: If ∆1≥ ∆2≥ … ≥ ∆𝑂

2

and ∆𝑗 ≥ ∆𝑂−𝑗 , ∀𝑗 ≤

𝑂 2 , then we have D(1) ≥D(2) ≥… ≥ D(𝑂 2) and D(N)≥

D(

𝑂 2) and D(N) ≥D(N-n), ∀𝑜 ≤ 𝑂 2 .

• Lemma2: If ∆1≥ ∆2≥ … ≥ ∆𝑂 then we have D(1) ≥D(2) ≥… ≥ D(N).
• Lemma3: If the probability distribution f(.) is monotone non-increasing, then we have ∆1≥ ∆2≥ … ≥ ∆𝑂−1.
• Lemma4: If f satisfies 𝑔 𝐺−1 𝑦

≥ 𝑔 𝐺−1 1 − 𝑦 , ∀𝑦 ∈ [0, 1

2], and 𝑔 𝐺−1 1 − 𝑦

then ∆𝑗 ≥ ∆𝑂−𝑗 , ∀𝑗 ≤ 𝑂

2.

• As long as N ≥ 2K, Maximization of D(n) is optimized at n =

𝑂𝐿(1−𝑑) 𝑂−𝐿

, i.e. 𝛿 = 1 , 𝜆 =

𝑂𝐿(1−𝑑) 𝑂−𝐿

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Comparison with Multi-group Mechanism

• The proposed framework can be generalized to dividing all players

into at most N groups, and every player is endowed with a winning probability 𝑟𝑗.

• So, the optimal multi-group mechanism can be defined as follows:

1 , , ) ,..., , ( . . ) ( max

1 2 1 1

    

 

  i N i i N N i i i

q K q c q q q t s v E q  

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Comparison with Multi-group Mechanism

• The efficiency gaps between two-group mechanism and 𝜁𝐶 are always

less than 0.1%.

• These results show that the two-group optimal mechanism could

achieve near optimal efficiency under commonly used distributions.

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Comparison with Multi-group Mechanism

• The optimality of this mechanism is robust to the distribution's mean,

variance and even the distribution itself.

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Conclusion

• Under the considered framework the optimal mechanism is always first

lottery then auction.

• The optimality of this mechanism is robust to the distribution's mean,

variance and even the distribution itself.

• The optimal parameters of the two-group mechanism are determined by

the number of participants only. It makes the government can directly implement this simple mechanism without considering the shape of the value distribution.

• Besides, the mechanism framework and the results we proposed here can

also be applied to any other similar public resource allocation problems, such as public housing allocation, emission rights allocation and so on.

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