Balancing Efficiency and Equality in Vehicle Licenses Allocation First part: Fahimeh FathianRad 22/ 1
In Introduction air pollution prompts the government to take more effective measures to control the number of vehicles. 22/ 2
Shanghai(before 2013) (Auction) Focuses: Efficiency Disadvantages: Low Equality Beijing, Guiyang (Lottery) Focuses: Equality Disadvantages: Inefficiency, No Value Exploration Shanghai(after 2013) (Reserved-price lottery) Focuses: Efficiency , Equality Disadvantages: Hard to set price, No Value Exploration Guangzhou, Hangzhou, Shenzhen, Tianjin, Shijiazhuang (Simultaneous auction and lottery) Focuses: Efficiency, Equality Disadvantages: Untruthful, Hard to choose, Hard to set auction size 22/ 3
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: 𝑤 1 ≥ 𝑤 2 ≥ ⋯ ≥ 𝑤 𝑜 22/ 4
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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. 22/ 6
Payment Rule In lottery I: If a player i wins a license, then his payment is: otherwise his payment is: In lottery II : No payment for participants: Mechanism 1 with the above payment rule is incentive compatible and ex-post individual rational . 22/ 7
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. 22/ 8
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 22/ 9
Relationship to Existing Mechanisms Our framework either includes the existing mechanisms as special cases or outperforms them in terms of both efficiency and equality. Auction Lottery Reserved-price Lottery Simultaneous Auction and Lottery 22/ 10
Balancing Efficiency and Equality in Vehicle Licenses Allocation Second part : HajarSiar 22/ 11
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? N 1 K max ( , ) E ( v ) E ( v ) i i N i 1 i 1 s . t . ( , ) c , K , N 1 K • where c ∈ [0; 1] is a constant to measure the least required equality. 22/ 12
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- 𝑂𝐿(1−𝑑) group mechanism is 𝛿 = 1; 𝜆 = 𝑂−𝐿 • Theorem2: If N ≥ 2K and the probability distribution function f satisfies 𝑔 𝐺 −1 𝑦 ≥ 𝑔 𝐺 −1 1 − 𝑦 2 ] , and 𝑔 𝐺 −1 1 − 𝑦 1 is , ∀𝑦 ∈ [0, 1 monotone non-decreasing about x ∈ [0, 2 ], then the optimal solution of 𝑂𝐿(1−𝑑) two-group mechanism is 𝛿 = 1; 𝜆 = 𝑂−𝐿 22/ 13
Proving theorems 1&2 • Denote n = 𝜆𝛿 : N ( K ) K N ( K ) nK K ( , ) ( 1 ) c K c n NK NK N • Solving the optimization problem of two-group mechanism is equivalent to solving the following programming: n N 1 1 max ( ) ( ) ( ) D n E v E v i i n N n i 1 i n 1 NK ( 1 c ) s . t . n Nc N K • 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. 22/ 14
Proving theorems 1&2 • The trend of spacing is used: ∆ 𝒋 = 𝑭 𝒘 𝒋 − 𝒘 𝒋+𝟐 𝑂 2 , then we have D(1) ≥ D(2) ≥ … ≥ D( 𝑂 • Lemma1: If ∆ 1 ≥ ∆ 2 ≥ … ≥ ∆ 𝑂 and ∆ 𝑗 ≥ ∆ 𝑂−𝑗 , ∀𝑗 ≤ 2 ) and D(N) ≥ 2 𝑂 𝑂 D ( 2 ) and D(N) ≥ D(N-n), ∀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 . 𝑂𝐿(1−𝑑) 𝑂𝐿(1−𝑑) • As long as N ≥ 2K, Maximization of D(n) is optimized at n = , i.e. 𝛿 = 1 , 𝜆 = 𝑂−𝐿 𝑂−𝐿 22/ 15
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 𝑟 𝑗 . • So, the optimal multi-group mechanism can be defined as follows: N max ( ) q E v i i i 1 s . t . ( q , q ,..., q ) c , 1 2 N N q K , i 1 i 0 1 q i 22/ 16
Comparison with Multi-group Mechanism • The efficiency gaps between two-group mechanism and 𝜁 𝐶 are always less than 0.1%. • These results show that the two-group optimal mechanism could achieve near optimal efficiency under commonly used distributions. 22/ 17
Comparison with Multi-group Mechanism • The optimality of this mechanism is robust to the distribution's mean, variance and even the distribution itself. 22/ 18
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. 22/ 19
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