Vickrey-Clark-Groves auctions First price vs second price Recap - PowerPoint PPT Presentation

Algorithmic game theory Ruben Hoeksma December 3, 2018 Vickrey-Clark-Groves auctions First price vs second price

Recap Last week: ◮ Introduction mechanism design ◮ Single item auctions ◮ First price auction ◮ Vickrey (second price) auction ◮ Truthfulness ◮ Individual rationality ◮ Revelation principle Today: ◮ Bayes-Nash equilibrium ◮ Revenue equivalence ◮ Vickrey-Clarke-Groves auctions

Equilibria in mechanism design Mechanism design ◮ Players have private types. ◮ Player i ’s strategy depends on their type. ◮ NE is not well suitable Instead, ◮ let F i be the distribution of player i ’s type t i ◮ utility functions u i ( t i , s ( t )) Definition (Bayes-Nash equilibrium (BNE)) The strategy vector s ( t ) is a Bayes-Nash equilibrium (BNE) if E s − i ∼ F − i [ u i ( t i , s i ( t i ) , s − i ( t − i )] ≥ E s − i ∼ F − i [ u i ( t i , x i , s − i ( t − i )] .

First price vs second price auctions Single item auction ◮ Alice, valuation a ∈ [0 , 1] ◮ Bob, valuation b ∈ [0 , 1] ◮ a , b ∼ U (0 , 1): P [ a ≤ x ] = x for all x ∈ [0 , 1] Claim In a first price auction where both players have valuation distribution U (0 , 1), it is a Bayes-Nash equilibrium when both players bid half their valuation. Proof. Let x denote Alice’s bid and y Bob’s bid. TP: if y = b 2 then x = a 2 is a best response (visa versa by symmetry).

First price vs second price auctions Proof cont. � if x ≤ b 0 u A ( x , b 2 2 ) = a − x otherwise � if 0 ≤ x ≤ 1 2 x P [ x > b 2 2 ] = P [ b ≤ 2 x ] = if 1 1 2 < x ≤ 1 E [ u A ( a , x , b )] = 2 x ( a − x ) = 2 xa − 2 x 2 Minimum at dx (2 xa − 2 x 2 ) = 2 a − 4 x d a 0 = ⇒ x = 2 .

Revenue equivalence Remember: Myerson’s revelation principle Let M be a mechanism such that there is an equilibrium strategy vector for the players. Then, there exists a mechanism M ′ in which the strategies of the players are just to report a type, and M ′ has an equilibrium in which all players report their type truthfully. Revenue equivalence Two auctions that have the same allocation in BNE, for any player, if they have a type for which the expected payment is equal in both auctions, then the expected payment is equal for each type of that player. If this is true for each player, the expected revenue of the auctions is equal.

Revenue equivalence Example: Two player first price and second price auction with U (0 , 1). ◮ Both allocate the item to the player with highest val. in BNE. 2 1 Analysis: E [max { a , b } ] = 3 and E [min { a , b } ] = 3 for a , b ∼ U (0 , 1). When a = 0 (or b = 0), the expected payment is 0. 1 E [revenue(SPA( A , B ))] = E [min { a , b } ] = 3 E [revenue(FPA( A , B ))] = E [max { a 2 , b 1 1 2 } ] = 2 E [max { a , b } ] = 3

Vickrey-Clark-Groves auctions Sponsored search auctions

Sponsored search Model: ◮ A search has k sponsored slots ◮ Each slot j has a click trough rate (CTR) α j ◮ n bidders have value v i for a click ◮ Valuation of bidder i for slot j is v i α j ◮ Each bidder is assigned at most one slot ◮ Price are set per click Question: Can we achieve an auction similar to the Vickrey auction? ◮ truthtelling ◮ maximizes the social welfare in equilibrium ◮ individually rational

Sponsored search - welfare maximization Welfare maximization: max � i ∈ N v i α s ( i ) Assume truthfulness. What is the optimal allocation? Claim: Assigning greedily highest v i to highest α j is optimal. Proof Suppose not. Let s be an optimal allocation of bidders to slots. Then, there are two bidders i , h ∈ N such that v h > v i and α s ( h ) < α s ( i ) . We compare the objective of s to the objective when the allocations of i and h are switched. The difference in objective value is v i α s ( i ) + v h α s ( h ) − ( v i α s ( h ) + v h α s ( i ) ) = v i ( α s ( i ) − α s ( h ) ) + v h ( α s ( h ) − α s ( i ) ) = ( v i − v h )( α s ( i ) − α s ( h ) ) < 0 So, such two bidders cannot exist in an optimal allocation.

Sponsored search - A payment rule Payment rule idea: The ℓ -th highest bidder pays the ( ℓ + 1)-st highest bid. (Generalization of second price auction) Observation: Individual rationality holds Truthtelling?: No! Proof b i > v i : Not advantageous b i < v i : Example: 2 slots, 3 bidders; α 1 = 1 10 , α 2 = 1 20 , v 1 = 10, v 2 = 9, v 3 = 6. Suppose b 2 = v 2 , b 3 = v 3 . Best response for bidder 1: Utility of bidder 1; if b 1 > v 2 : u 1 ( b 1 , v 2 , v 3 ) = α 1 ( v 1 − v 2 ) = 1 10 (10 − 9) = 1 10 if v 2 > b 1 > v 3 : α 2 ( v 1 − v 3 ) = 1 20 (10 − 6) = 4 20 = 2 10

Vickrey auction - another interpretation Social welfare (= total value of the allocation) = v 1 Social welfare of bidders 2 , . . . , n = 0 Social welfare of bidders 2 , . . . , n if Bidder 1 did not participate = v 2 Bidder 1 imposes a “cost” of v 2 on the other players by their presence. Definition (Externalities) The externalities of a player are all costs imposed and/or benefits gained by others from that player’s actions. In the SPA, the payment of the highest bidder is equal to (an approximation of) their externalities..

Vickrey-Clarke-Groves (VCG) auction Idea: payments are equal to externalities. Definition (Vickrey-Clarke-Groves (VCG) auction) For a set of possible allocations A , the VCG auction is the following 1. Bidders “bid”, b i ( a ) , ∀ a ∈ A , value for each possible allocation. 2. Allocation, a ∗ , maximizes the total reported value � a ∗ = argmax b i ( a ) . a ∈ A i ∈ N 3. Bidder i ’s payment:     � � b ℓ ( a ∗ ) . max b ℓ ( a )  − a ∈ A  ℓ � = i ℓ � = i

Sponsored search - VCG auction Let v 1 ≥ v 2 ≥ . . . ≥ v n and α 1 ≥ α 2 . . . ≥ α k . Welfare: k � v j α j j =1 Without Bidder i : i − 1 k +1 � � v j α j + v j α j − 1 j =1 j = i +1 Player i ’s externalities:   i − 1 k +1 i − 1 k +1 k +1 � � � � �  = v j α j + v j α j − 1 − v j α j + v j α j v j ( α j − 1 − α j )  j =1 j = i +1 j =1 j = i +1 j = i +1

Sponsored search - VCG auction VCG auction for Sponsored search 1. Bidders submit bids, order them b 1 ≥ b 2 ≥ . . . ≥ b n 2. Assign slots 1 , . . . , k to bidders 1 , . . . , k , respectively 3. Bidder i ∈ { 1 , . . . , k } pays k +1 1 � b j ( α j − 1 − α j ) α i j = i +1 per click, where α k +1 ≡ 0, and b j ≡ 0 for all j > n . Note: k +1 1 α j − 1 − α j = α i − α k +1 = α i − 0 � = 1 α i α i α i j = i +1

Sponsored search - VCG auction Theorem The VCG auction for sponsored search is truthtelling. Proof. Consider bidder i . Let s i ( b ) be the slot of bidder i for bid vector b . Bidder i maximizes   k +1 1 � � �  v i − α s i (0 , b − i ) − α s i ( b i , b − i ) α s i ( b i , b − i ) b j  α s i ( b i , b − i ) j = i +1 k +1 k +1 � � = α s i ( b i , b − i ) v i + b j α s i ( b i , b − i ) − b j α s i (0 , b − i ) j = i +1 j = i +1 � � = α s i ( b i , b − i ) v i + b j α s i ( b i , b − i ) − b j α s i (0 , b − i ) . j � = i j � = i � �� � Constant w.r.t. b i Player i maximizes the social welfare minus a constant.