Some problems related to the Progressive Second Price Auction Mechanism
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1 Patrick Maillé Slide 1 Workshop ARC PRIXNeT, March 2003 Some problems related to the Progressive Second Price Auction Mechanism Patrick Maillé
2 Patrick Maillé Slide 2 Outline
3 Patrick Maillé Slide 2 Outline The PSP mechanism: principle and results (Lazar, Semret)
4 Patrick Maillé Slide 2 Outline The PSP mechanism: principle and results (Lazar, Semret) An incitation problem
5 Patrick Maillé Slide 2 Outline The PSP mechanism: principle and results (Lazar, Semret) An incitation problem Some solutions
6 Patrick Maillé Slide 2 Outline The PSP mechanism: principle and results (Lazar, Semret) An incitation problem Some solutions Prohibit some bids
7 Patrick Maillé Slide 2 Outline The PSP mechanism: principle and results (Lazar, Semret) An incitation problem Some solutions Prohibit some bids Play on off-game players bids
8 Patrick Maillé Slide 2 Outline The PSP mechanism: principle and results (Lazar, Semret) An incitation problem Some solutions Prohibit some bids Play on off-game players bids Imperfect information
9 Patrick Maillé Slide 3 The PSP auction mechanism Introduced by Lazar and Semret (1999) Principle : each user pays for the social opportunity cost he imposes on others
10 Patrick Maillé Slide 3 The PSP auction mechanism Introduced by Lazar and Semret (1999) Principle : each user pays for the social opportunity cost he imposes on others each player i submits a bid s i =(q i,p i ) 8 < : q i = asked quantity of resource p i = proposed unit price
11 Patrick Maillé Slide 3 The PSP auction mechanism Introduced by Lazar and Semret (1999) Principle : each user pays for the social opportunity cost he imposes on others each player i submits a bid s i =(q i,p i ) 8 < : q i = asked quantity of resource p i = proposed unit price Allocation a i and price to pay c i : computed based on the bid profile s =(s i ) i I
12 Patrick Maillé Slide 4 p p 6 q 6 p 5 p i q 5 q 1 p 3 p 2 p 1 q 2 q 3 0 q i a i = Q,q 6, q 5 c i = Q q a i (s) =q i h Q P i + p k p i,k i q k c i (s) = P j i p j [a j (s i ) a j (s)]
13 Patrick Maillé Slide 5 User behaviour Set I of users (players)
14 Patrick Maillé Slide 5 User behaviour Set I of users (players) Users preferences: determined by their utility function
15 Patrick Maillé Slide 5 User behaviour Set I of users (players) Users preferences: determined by their utility function u i (s) =θ i (a i (s)) c i (s)
16 Patrick Maillé Slide 5 User behaviour Set I of users (players) Users preferences: determined by their utility function u i (s) =θ i (a i (s)) c i (s) θ i =player i s valuation function, assumed non-decreasing and concave
17 Patrick Maillé Slide 5 User behaviour Set I of users (players) Users preferences: determined by their utility function u i (s) =θ i (a i (s)) c i (s) θ i =player i s valuation function, assumed non-decreasing and concave Each user i chooses his bid s i, knowing the others bids, in order to maximize his utility θ i (a i ) c i
18 Patrick Maillé Slide 6 ɛ-best reply p u i ((v i ;w i );s,i )= θ 0 i(q) w i = θ 0 i(v i ) q v i = q,ε=θ 0 i(0) Q q u i ((v i,w i ),s i ) max si u i (s i,s i ) ɛ
19 Patrick Maillé Slide 7 Properties of PSP mechanism (Lazar, Semret 1999) Incentive compatibility: there is no immediate gain to lie about one s valuations p i = θ i (q i) If all players make their ɛ-best reply if and only if their utility gain is greater than ɛ :
20 Patrick Maillé Slide 7 Properties of PSP mechanism (Lazar, Semret 1999) Incentive compatibility: there is no immediate gain to lie about one s valuations p i = θ i (q i) If all players make their ɛ-best reply if and only if their utility gain is greater than ɛ : The bid profile converges after a finite number of bids, to a (ɛ + ɛ )-Nash equilibrium s
21 Patrick Maillé Slide 7 Properties of PSP mechanism (Lazar, Semret 1999) Incentive compatibility: there is no immediate gain to lie about one s valuations p i = θ i (q i) If all players make their ɛ-best reply if and only if their utility gain is greater than ɛ : The bid profile converges after a finite number of bids, to a (ɛ + ɛ )-Nash equilibrium s Efficiency of the corresponding allocation: P i θ i(a i ) max ai : P i a i Q P i θ i(a i ) K ɛ + ɛ
22 Patrick Maillé Slide 8 An incitation problem Semret s strategy p θ 0 1 (q) p 0 Q q
23 Patrick Maillé Slide 9 An incitation problem Semret s strategy p θ 0 1 (q) p 0 Q q
24 Patrick Maillé Slide 10 An incitation problem Semret s strategy p θ 0 2 (q) p 0 Q q
25 Patrick Maillé Slide 11 An incitation problem Semret s strategy p θ 0 2 (q) p 0 Q q
26 Patrick Maillé Slide 12 An incitation problem Semret s strategy p p 0 Q q
27 Patrick Maillé Slide 13 An incitation problem Semret s strategy p p 0 Q q
28 Patrick Maillé Slide 14 An incitation problem Semret s strategy p p 0 Q q u 1 =0, u 2 = θ 2 (Q) Qθ 1 (Q)
29 Patrick Maillé Slide 15 An incitation problem A dominant strategy for player 1
30 Patrick Maillé Slide 15 An incitation problem A dominant strategy for player 1 p θ 0 1 (q) p 0 Q q
31 Patrick Maillé Slide 16 An incitation problem A dominant strategy for player 1 p p max θ 0 1 (q) p 0 Q q
32 Patrick Maillé Slide 17 An incitation problem A dominant strategy for player 1 p p max θ 0 2 (q) p 0 Q q
33 Patrick Maillé Slide 18 An incitation problem A dominant strategy for player 1 p p max p 0 Q q
34 Patrick Maillé Slide 19 An incitation problem A dominant strategy for player 1 p p max p 0 Q q u 1 = θ 1 (Q) p 0 Q u 2 =0
35 Patrick Maillé Slide 20 A solution against the incitation problem: forbid some bids p Q q
36 Patrick Maillé Slide 21 A solution against the incitation problem: forbid some bids p Q q
37 Patrick Maillé Slide 22 A solution against the incitation problem: forbid some bids p θ 0 1 (q) p 0 Q q
38 Patrick Maillé Slide 23 Consequences of the restriction of possible bids The incitation problem is solved The equilibrium reached is the same as that wanted efficient allocation Problem of the convergence time
39 Patrick Maillé Slide 24 Another solution: play on excluded players bids p θ 0 i(q i ) θ 0 i(q) p 0 q i = argmaxq(θ 0 i(q), p 0 ) Q q
40 Patrick Maillé Slide 25 Another solution: play on excluded players bids p θ 0 1 (q) θ 0 i (q) p 0 Q q
41 Patrick Maillé Slide 26 Another solution: play on excluded players bids p θ 0 1(q) p 0 u 1 = Q q
42 Patrick Maillé Slide 27 Another solution: play on excluded players bids p θ 0 1 (q) θ 0 i (q) p 0 Q q
43 Patrick Maillé Slide 28 Another solution: play on excluded players bids p θ 0 1(q) p 0 u 1 = Q q
44 Patrick Maillé Slide 29 Consequences The first player can not win all the resource anymore, as being honest is better for him : the dishonest strategy is not a dominant strategy anymore Problem: some more complicated strategies can be played, depending on the player s knowledge about the other players
45 Patrick Maillé Slide 30 Some other possible solutions One-bid mechanism, submitted at the arrival
46 Patrick Maillé Slide 30 Some other possible solutions One-bid mechanism, submitted at the arrival Less signaling information
47 Patrick Maillé Slide 30 Some other possible solutions One-bid mechanism, submitted at the arrival Less signaling information No convergence phase
48 Patrick Maillé Slide 30 Some other possible solutions One-bid mechanism, submitted at the arrival Less signaling information No convergence phase Imperfect information
49 Patrick Maillé Slide 30 Some other possible solutions One-bid mechanism, submitted at the arrival Less signaling information No convergence phase Imperfect information No more incitation problem
50 Patrick Maillé Slide 30 Some other possible solutions One-bid mechanism, submitted at the arrival Less signaling information No convergence phase Imperfect information No more incitation problem Less signaling information
51 Patrick Maillé Slide 30 Some other possible solutions One-bid mechanism, submitted at the arrival Less signaling information No convergence phase Imperfect information No more incitation problem Less signaling information Efficiency of the allocation?
52 Patrick Maillé Slide 31 Thank you for your attention Click here to go back Click here to close, or here to exit
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