Mechanism Design Tutorial David C. Parkes, Harvard University Indo-US Lectures Week in Machine Learning, Game Theory and Optimization

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1 1 Mechanism Design Tutorial David C. Parkes, Harvard University Indo-US Lectures Week in Machine Learning, Game Theory and Optimization

2 2 Outline Classical mechanism design Preliminaries (DRMs, revelation principle) Positive results Groves, Single-parameter (Myerson) min makespan task assignment Negative results Gibbard-Satterthwaite Algorithmic mechanism design Knapsack auction Price-of-anarchy analysis

3 3 Mechanism design alternatives; agents, value :, Utility, Design a game Γ,,, attain desiderata in equilibrium 1Agent 1 Agent n,,

4 4 Mechanism design alternatives; agents, value :, Utility, Design a game Γ,,, attain desiderata in equilibrium 1Agent 1 Agent n,,

5 5 Examples Auction; e.g., servers, bandwidth, ad space Coordination; e.g., meetings, tasks Public choice; e.g., build a new school Matching; e.g., residents to hospitals Desiderata: efficiency, maxminfairness, envy-free, participation, revenue, budget-balance

6 6 Game theory for MD Incomplete information game;valuation Behavior ; Strategy Dominant strategy equilibrium,,,all, all, all Bayes-Nash equilibrium! "#$ %, '! "#$ %, all, all ',

7 7 Game theory for MD Incomplete information game;valuation Behavior ; Strategy Dominant strategy equilibrium,,,all, all, all Bayes-Nash equilibrium! "#$ %, '! "#$ %, all, all ',

8 8 Game theory for MD Incomplete information game;valuation Behavior ; Strategy Dominant strategy equilibrium,,,all, all, all Bayes-Nash equilibrium! "#$ %, '! "#$ %, all, all ',

9 9 Implementation Mechanism Γ implementsa social choice function (: if Γ ()*for all,,, in equilibrium. 1 ) * ) *,,

10 10 Direct Revelation Mechanisms Choice rule,; Payment rule. 1 ) * ) *, -. - DRM Γ is (Dom/Bayes) incentive compatible if truthful reporting is a (DSE/BNE). ( Strategyproof, Truthful. )

11 11 Direct Revelation Mechanisms Choice rule,; Payment rule. 1 ) * ) *, -. - DRM Γ is (Dom/Bayes) incentive compatible if truthful reporting is a (DSE/BNE). ( Strategyproof, Truthful. )

12 12 Revelation Principle 1 ) * ) *,, Theorem: Any scf ( implemented by Γcan be implemented by an incentive compatible DRM. *Positive results **Negative results

13 13 Outline Classical mechanism design Preliminaries (DRMs, revelation principle) Positive results Groves, Single-parameter (Myerson) Min makespan Negative results Gibbard-Satterthwaite Algorithmic mechanism design Knapsack auction Price-of-anarchy analysis

14 14 Groves mechanism ( arg max 7 8 9)* 5 ; 7 ; 0 Affine maximization (Simple case, 7 1, 9 0)), = - arg max 5 - )*. =, - - >?@ -? )*, for,) - * (arbitrary fcn) 0 Utility:, - 8?@ -?, - > ) * truthful! (and efficient!) -

15 15 Groves mechanism ( arg max 7 8 9)* 5 ; 7 ; 0 Affine maximization (Simple case, 7 1, 9 0)), = - arg max 5 - )*. =, - - >?@ -? )*, for,) - * (arbitrary fcn) Utility:, - 8?@ -?, - > ) * truthful! (and efficient!) -

16 16 Groves mechanism ( arg max 7 8 9)* 5 ; 7 ; 0 Affine maximization (Simple case, 7 1, 9 0)), = - arg max - )* 5. =, - - >?@ -? )*, for,) - * (arbitrary fcn) 0 Utility:, - 8?@ -?, - > ) * truthful! (and efficient!) -

17 17 VCG mechanism Special case of Groves. Payment rule: Negative externality. "BC, -? ) *?@ -? )*, for, = ) - - *,, = ) Truthful, efficient, participation, no-deficit* (Negative result (Roberts): if 3, F E mechanisms are Groves mechanisms.) *., then only truthful

18 18 VCG mechanism Special case of Groves. Payment rule: Negative externality. "BC, -? ) *?@ -? )*, for, = ) - - *,, = ) Truthful, efficient, participation, no-deficit* (Negative result (Roberts): if 3, F E mechanisms are Groves mechanisms.) *., then only truthful

19 19 VCG Example 1 Single-item Auction Values $10, $4, $2,)*: assign to A1.

20 20 VCG Example 1 Single-item Auction Values $10, $4, $2,)*: assign to A ; zero to others a second-price auction

21 21 VCG Example 2 Combinatorial Auction Items {A,B,C},)*: ),, * t J agent A B AB

22 22 VCG Example 2 Combinatorial Auction Items {A,B,C},)*: ),, * t J M v Agent 1 pays zero. agent A B AB (revenue low, and NP-hard winner determination.)

23 23 VCG Example 3 Agents= Edges; Value = -Cost Externality: )-total cost without) (-total cost with); e.g., for edge 17 this is -90-(-40)=-50.

24 24 VCG Example 4 Double Auction A1: buyer, value $10 A2: seller, value $8,)*: trade

25 25 VCG Example 4 Double Auction A1: buyer, value $10 A2: seller, value $8,)*: trade Payments A1: (pays $8) A2: (paid $10!)

26 26 Single-parameter domains Private info S ;induces S, E.g., Min makespan task assignment Agents A1, A2. Tasks T1,T2,T3 (sizes 1, 2 and 4) Private :: Unit processing time (S J, S J 1* Min make-span = max(2.5,2)=2.5 What would VCG do?

27 27 Single-parameter domains Private info S ;induces S, E.g., Min makespan task assignment Agents A1, A2. Tasks T1,T2,T3 (sizes 1, 2 and 4) Private :: Unit processing time (S J, S J 1* Min make-span = max(2.5,2)=2.5 What would VCG do?

28 28 Single-parameter domains S, S T private S %U, * Allocation rule,: U, Fix S -, monotonic, known T : E summarization fcn (1) Auction: S is value, T is (prob) agent allocated? (2) Task assignment: S is ( processing time), T total load T ), S * S S

29 29 Single-parameter domains S, S T private S %U, * Allocation rule,: U, Fix S -, monotonic, known T : E summarization fcn (1) Auction: S is value, T is (prob) agent allocated? (2) Task assignment: S is ( processing time), T total load T ), S * S S

30 30 Myerson mechanism (s.p. domain) Given monotonic,, then mechanism truthful if:. S - S - T, S - W T, X, S 1 Z $ [ \]^ - - * YX > )S =critical valuein 0-1 domains

31 31 Myerson mechanism (s.p. domain) Given monotonic,, then mechanism truthful if:. S - S - T, S - W T, X, S 1 Z $ [ \]^ value - _-value - * YX > )S =critical valuein 0-1 domains payment _-payment (basically necessary)

32 32 Min makespanscheduling Unit processing time (S J, S J 1* c-approx: (Archer and Tardos 01) `a Z `bcd )Z* p c Thm. VCG is an n-approx, and truthful. Proof. UB: `a e? f g,? / )1/i*? f g,? i `bcd LB: imachines, ntasks (size 1). Machine 1 unit cost 1. Machine 2..n unit cost 1 8 j, j ; 0 Min makespan 1 8 j. VCG make-span n. lim m o i/)1 8 j* i

33 33 Min makespanscheduling Unit processing time (S J, S J 1* c-approx: (Archer and Tardos 01) `a Z `bcd )Z* p c Thm. VCG is an n-approx, and truthful. Proof. UB: `a e? f g,? / )1/i*? f g,? i `bcd LB: imachines, ntasks (size 1). Machine 1 unit cost 1. Machine 2..n unit cost 1 8 j, j ; 0 Min makespan 1 8 j. VCG make-span n. lim m o i/)1 8 j* i

34 34 Min makespanscheduling Unit processing time (S J, S J 1* c-approx: (Archer and Tardos 01) `a Z `bcd )Z* p c Thm. VCG is an n-approx, and truthful. Proof. UB: `a e? f g,? / )1/i*? f g,? i `bcd LB: imachines, ntasks (size 1). Machine 1 unit cost 1. Machine 2..n unit cost 1 8 j, j ; 0 Min makespan 1 8 j. VCG makespann. lim m o i/)1 8 j* i

35 35 What else can we do?

36 36 LexOptmechanism Adopt,)* to min makespan, particular tie-breaking rule. Thm. LexOptis monotonic (S ; S, load ) Suppose r. (Case 1) i Z$ [ i Z$. i Z$ [ e i Z$ e i Z$ ) * i Z$ [) *. Contradiction. (Case 2)i Z$ [ p i Z$. (Archer and Tardos 01) S T e i Z$ e i Z$ ) * S T ) *, since iis bottleneck in at S. Monotone.

37 37 LexOptmechanism Adopt,)* to min makespan, particular tie-breaking rule. Thm. LexOptis monotonic (S ; S, load ) Suppose r. (Case 1) i Z$ [ i Z$. i Z$ [ e i Z$ e i Z$ ) * i Z$ [) *. Contradiction. (Case 2)i Z$ [ p i Z$. (Archer and Tardos 01) S T e i Z$ e i Z$ ) * S T ) *, since iis bottleneck in at S. Monotone.

38 38 LexOptmechanism Adopt,)* to min makespan, particular tie-breaking rule. Thm. LexOptis monotonic (S ; S, load ) Suppose r. (Case 1) i Z$ [ i Z$. i Z$ [ e i Z$ e i Z$ ) * i Z$ [) *. Contradiction. (Case 2)i Z$ [ p i Z$. (Archer and Tardos 01) S T e i Z$ e i Z$ ) * S T ) *, since iis bottleneck in at S. Monotone.

39 39 Aside: Computation min- makespan is NP-hard (Dhangwatnotai et al. 11, Christodoulou and Kovacs 10) Standard PTAS optimizes over a restricted range of candidate assignments, set construction violates monotonicity. Exists a monotone PTAS, both randomized and deterministic.

40 40 Outline Classical mechanism design Preliminaries (DRMs, revelation principle) Positive results Groves, Single-parameter (Myerson) Min makespan Negative results Gibbard-Satterthwaite Algorithmic mechanism design Knapsack auction Price-of-anarchy analysis

41 41 Gibbard-Satterthwaite No money. all strict preferences (e.g., f) 3,,)*onto. Dictatorial: Same agent always gets top choice

42 42 Gibbard-Satterthwaite No money. all strict preferences (e.g., f) 3,,)*onto. Dictatorial: Same agent always gets top choice Theorem. The only truthful mechanisms are dictatorial with all strict prefs, A >=3 onto.

43 43 Simple proof (T u) for 2-agent case Monotonicity (M) abcdef badcfd - If v w x, w x y, then v w - - x, w x y. Proof. If report..cd and get d, then report..dc and get g, if g > {c,d} then cd type deviates; else, dc type deviates. Consistency (C) If every agent z { then don t pick b. Proof. Suppose pick b. Still pick b if all a>b > (M) Onto, so exists v with x(v)=a. Still pick a if all a>b> (M). Contradiction. Impossibility 1 is a dictator on a : if 1 reports a top, a picked P1: a>b>c; b>a>c. Can t pick c (C). Consider a. P2: a > b > c; b > c > a Can t pick c (C). Can t pick b (T). Select a. Consider any P3, top(1)=a Pick a (M, consider P2 -> P3) Argue 1 also dictator on b, c.

44 44 Simple proof (T u) for 2-agent case Monotonicity (M) abcdef badcfd - If v w x, w x y, then v w - - x, w x y. Proof. If report..cd and get d, then report..dc and get g, if g > {c,d} then cd type deviates; else, dc type deviates. Consistency (C) If every agent.. z {.. then don t pick b. Proof. Suppose pick b. Still pick b if all a>b > (M) Onto, so exists v with x(v)=a. Still pick a if all a>b> (M). Contradiction. Impossibility 1 is a dictator on a : if 1 reports a top, a picked P1: a>b>c; b>a>c. Can t pick c (C). Consider a. P2: a > b > c; b > c > a Can t pick c (C). Can t pick b (T). Select a. Consider any P3, top(1)=a Pick a (M, consider P2 -> P3) Argue 1 also dictator on b, c.

45 45 Simple proof (T u) for 2-agent case Monotonicity (M) abcdef badcfd - If v w x, w x y, then v w - - x, w x y. Proof. If report..cd and get d, then report..dc and get g, if g > {c,d} then cd type deviates; else, dc type deviates. Consistency (C) If every agent.. z {.. then don t pick b. Proof. Suppose pick b. Still pick b if all a>b > (M) Onto, so exists v with x(v)=a. Still pick a if all a>b> (M). Contradiction. Impossibility 1 is a dictator on a : if 1 reports a top, a picked P1: a>b>c; b>a>c. Can t pick c (C). Consider a. P2: a > b > c; b > c > a Can t pick c (C). Can t pick b (T). Select a. Consider any P3, top(1)=a Pick a (M, consider P2 -> P3) Argue 1 also dictator on b, c.

46 46 Simple proof (T u) for 2-agent case Monotonicity (M) abcdef badcfd - If v w x, w x y, then v w - - x, w x y. Proof. If report..cd and get d, then report..dc and get g, if g > {c,d} then cd type deviates; else, dc type deviates. Consistency (C) If every agent.. z {.. then don t pick b. Proof. Suppose pick b. Still pick b if all a>b > (M) Onto, so exists v with x(v)=a. Still pick a if all a>b> (M). Contradiction. (Svensson 99) Impossibility 1 is a dictator on a : if 1 reports a top, a picked P1: a>b>c; b>a>c. Can t pick c (C). Consider a. P2: a > b > c; b > c > a Can t pick c (C). Can t pick b (T). Select a. Consider any P3, top(1)=a Pick a (M, consider P2 -> P3) Argue 1 also dictator on b, c.

47 47 Outline Classical mechanism design Preliminaries (DRMs, revelation principle) Positive results Groves, Single-parameter (Myerson) Min makespan Negative results Gibbard-Satterthwaite Algorithmic mechanism design Knapsack auction Price-of-anarchy analysis

48 48 Algorithmic Mechanism Design New concern is to obtain computational tractability as well as incentive compatibility Emphasis also placed on bidding languages, preference elicitation.

49 49 Knapsack auction items, agent value S for } units (known) Goal: maximize total value. 0-1 knaspack problem. NP-hard Can t use VCG. 2-approx: `bcd Z e 2, all S `a Z x: order by decreasing S /}. If ~ S max S sell 1 else sell to > Charge critical value (Myerson) (Mu alem and Nisan 08) Example: $5@2, $6@1, $6@3, $12@5; supply 5 units Compare (6+5,12) -> allocated to A4. Pay $11. Suppose A2 reports 8? Now {1,2} allocated. A2 pays $7.

50 50 Knapsack auction items, agent value S for } units (known) Goal: maximize total value. 0-1 knaspack problem. NP-hard Can t use VCG. 2-approx: `bcd Z e 2, all S `a Z x: order by decreasing S /}. If ~ S max S sell 1 else sell to > Charge critical value (Myerson) (Mu alem and Nisan 08) Example: $5@2, $6@1, $6@3, $12@5; supply 5 units Compare (6+5,12) -> allocated to A4. Pay $11. Suppose A2 reports 8? Now {1,2} allocated. A2 pays $7.

51 51 Knapsack auction items, agent value S for } units (known) Goal: maximize total value. 0-1 knaspack problem. NP-hard Can t use VCG. 2-approx: `bcd Z e 2, all S `a Z x: order by decreasing S /}. If ~ S max S sell 1 else sell to > Charge critical value (Myerson) (Mu alem and Nisan 08) Example: $5@2, $6@1, $6@3, $12@5; supply 5 units Compare (6+5,12) -> allocated to A4. Pay $11. Suppose A2 reports 8? Now {1,2} allocated. A2 pays $7.

52 52 Knapsack auction -Analysis Theorem. Truthful and 2-approx. (Mu alem and Nisan 08) Monotone: (Case 1) Allocated and in {1..k}. Still in. (Case 2) Allocated and highest. May cause {1..k} to win but still in. 2-approx: suppose p i. ƒ e ƒ ~?] S? 8 S ~E e ~E?] S? e ~ S? max S?..~ 8 ˆ e 2max )..~, ˆ* 2 g??] 8

53 53 Knapsack auction -Analysis Theorem. Truthful and 2-approx. (Mu alem and Nisan 08) Monotone: (Case 1) Allocated and in {1..k}. Still in. (Case 2) Allocated and highest. May cause {1..k} to win but still in. 2-approx: suppose p i. ƒ e ƒ ~?] S? 8 S ~E e ~E?] S? e ~ S? max S?..~ 8 ˆ e 2max )..~, ˆ* 2 g??] 8

54 54 Knapsack auction -Analysis Theorem. Truthful and 2-approx. (Mu alem and Nisan 08) Monotone: (Case 1) Allocated and in {1..k}. Still in. (Case 2) Allocated and highest. May cause {1..k} to win but still in. 2-approx: suppose p i. ƒ e ƒ ~?] S? 8 S ~E e ~E?] S? e ~ S? max S?..~ 8 ˆ e 2max )..~, ˆ* 2 g??] 8

55 55 Outline Classical mechanism design Preliminaries (DRMs, revelation principle) Positive results Groves, Single-parameter (Myerson) Min makespan Negative results Gibbard-Satterthwaite Algorithmic mechanism design Knapsack auction Price-of-anarchy analysis

56 56 Price of anarchy + MD PoA: worst-case ratio of optimal objto objin equilibrium Extension theorems (Roughgarden, 09, 12; Lucier, PaesLeme11; Syrgkanis12, Syrgkanis Tardos 13) For auctions: PoAfor complete-information auction -> PoAin Bayes Nash equilibrium PoAfor complete-information auction -> PoAfor composition of auctions.

57 57 Price of anarchy + MD PoA: worst-case ratio of optimal objto objin equilibrium Extension theorems (Roughgarden, 09, 12; Lucier, PaesLeme11; Syrgkanis12, Syrgkanis Tardos 13) For auctions: PoAfor complete-information auction under property P -> PoAin Bayes Nash equilibrium PoAfor complete-information auction under property P -> PoAfor composition of auctions. Comment: now worry about all equilibrium

58 58 Example: Extension from NE to BNE For any b, exists - s.t., 8 Š Œ.)S*(smoothness) Do this under P: only depends on S If b is a NE then Ž U,, Ž U 8 Š Œ.)* 8 1 Š Œ.)* 8 1 Œ.)* e Œ.)*, and Œ e / Extends to BNE immediately

59 59 Example: Extension from NE to BNE For any b, exists - s.t., 8 Š Œ.)S*(smoothness) Do this under P: only depends on S If b is a NE then Ž U,, Ž U 8 Š Œ.)S* 8 1 Š Œ.)S* 8 1 Œ.)S* Œ.)S*, and Œ e / Extends to BNE immediately

60 60 Example: Extension from NE to BNE For any b, exists - s.t., 8 Š Œ.)S*(smoothness) Do this under P: only depends on S If b is a NE then Ž U,, Ž U 8 Š Œ.)S* 8 1 Š Œ.)S* 8 1 Œ.)S* Œ.)S*, and Œ e / Extends to BNE immediately

61 61 Apply to FPSB auction PoA for FPSB is 1! But, want to bound under P. Want:, 8 Š Œ.)S* For any bids, Z $ J, 8 S / 2 Either gain Z $ from deviation, or ; Z $ J J Z $, J 8, S Z $, J )S*, 8 Š J Œ.)* ; thus PoA e 2

62 62 Apply to FPSB auction PoA for FPSB is 1! But, want to bound under P. Want:, 8 Š Œ.)S* For any bids, Z $ J, 8 S / 2 Either gain Z $ from deviation, or ; Z $ J J Z $, J 8, S Z $, J )S*, 8 Š J Œ.)* ; thus PoA e 2

63 63 Apply to FPSB auction PoA for FPSB is 1! But, want to bound under P. Want:, 8 Š Œ.)S* For any bids, Z $ J, 8 S / 2 Either gain Z $ from deviation, or ; Z $ J J Z $, J 8, S Z $, J )S*, 8 Š J Œ.)* ; thus PoA e 2

64 64 Apply to FPSB auction PoA for FPSB is 1! But, want to bound under P. Want:, 8 Š Œ.)S* For any bids, Z $ J, 8 S / 2 Either gain Z $ from deviation, or ; Z $ J J Z $, J 8, S Z $, J )S*, 8 Š J Œ.)* ; thus PoA e 2

65 65 Direction for AMD? Lucier and Borodin (2010) First price Single-minded Combinatorial auction Optimal allocation rule, the PoAis m(mitems) But, if the allocation rule is approximate (sqrt-mgreedy), then (½,sqrt(m)) smooth, and O(sqrt(m)) PoA. Design mechanisms that are smooth, and provide good worst-case properties.

66 66 References See Economics and Computation, Parkesand SeukenCUP (forthcoming, 2014) Chapters 8 and 10