Combinatorial Exchanges. David C. Parkes Harvard University

Transcription

1 Combinatorial Exchanges David C. Parkes Harvard University

2 What is a combinatorial exchange? Two-sided Complex valuations (swaps, contingent swaps, all-or-nothing sells, etc.)

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4 Fragmented Spectrum (E.Kwerel) Channel TV Incumbents Within 100 Miles of New York City C1 D1 PS C2 D2 PS WBNE WRNNTV WHSPTV WHSETV WBPHTV WMBCTV WHSITV WPPX WTICTV WEDY WFMZTV WACI Spectrum encumbered by: One station: Two stations: Three or more:

5 Challenges Expressive bidding language Preference Elicitation / Price discovery Scalable Winner Determination Payments: Incentive compatibility Stability Fairness

6 Computational MD Economic constraints e.g., incentive compatibility, core, etc. Computational constraints e.g., scalable winner determination, minimal preference elicitation, etc. Mechanism = Algorithm

7 Exchange Example -10 sell A -5 sell B +5 swap A for B exchange +35 buy AB +15 buy A

8 Exchange Example -10 sell A -5 sell B +5 swap A for B +5 exchange +35 buy AB +15 buy A

9 Exchange Example -10 sell A -5 sell B +5 swap A for B +15 exchange +35 buy AB +15 buy A

10 Exchange Example -10 sell A -5 sell B +5 swap A for B +20 exchange +35 buy AB +15 buy A

11 Exchange Example -10 sell A -5 sell B +5 swap A for B +20 exchange Payments? +35 buy AB +15 buy A

12 Tree Based Bidding Language Defines change in value for a trade (Cavallo et al. 05) entirely symmetric for buyers and sellers sell AB, value -$100 ; buy A, value +$20 Generalizes XOR, OR, XOR/OR (Sandholm 99, Nisan 00)

13 Example 1: and [3,3] $ am +10am +11am

14 Example 2: xor [1,1] $200 $180 $150 +9am +10am +11am

15 Example 3: xor of and [1,1] [3,3] $200 [3,3] $150 +9am +10am +11am +9pm +10pm +11pm

16 Example 4: choose IC[x,y]: accept an allocation in which at least x and at most y children are satisfied [2,3] $220 $200 $180 $150 $120 +8am +9am +10am +11am +12pm

17 Example 5: swap [2,2] -$50 +9am -11am

18 Example 6: contingent sale [2,2] [1,1] -$200 [1,3] -9am -10am -11am $300 $200 $150 +9pm +10pm +11pm

19 Winner Determination Goods: {1,,m}. Agents: {1,,n} Trades: l ij Z Initial allocation: x 0 ij Z Winner determination: max i v i (l i ) s.t. l ij +x 0 ij 0, i j i l ij 0, j l ij Z l feas(x 0 )

20 Value given l i? max sati { } T v i ( )sat i ( ) s.t. Leaf(i) q ij ( )sat i ( ) l ij, j (3) IC x,i ( )sat i ( ) child( ) sat i ( ) IC y,i ( )sat i ( ), Leaf(i) (4) sat i ( )2 {0,1}

21 Concise WD Formulation max l,sat i Ti v i ( )sat i ( ) s.t. (feas), (TBBL semantics) Linear in size of TBBL trees

22 ICE: Proxied Exchange (Parkes et al. 2007)

23 + another activity rule in later stages Activity Rule Show one trade is weakly better then all others

24 Scalability (I)

25 Scalability (II)

26 Price Feedback

27 Payments? -10 sell A -5 sell B +5 swap A for B +20 exchange? +35 buy AB +15 buy A

28 Payments: VCG? sell A +20 buy AB -5 exchange +15 sell B buy A swap A for B

29 Myerson-Satterthwaite impossibility EFF, No-deficit, IR and BNIC

30 utility i report

31 Approx SP: GSP utility i report

32 How should we set payments in a CE that clears straightforwardly based on bids? ( how should we make the mechanism maximally incentive compatible?)

33 Relaxing away from SP We like SP for reasons of equity (Roth 03, Pathak and Sonmez 08) simplify reasoning can predict properties of the mechanism

34 Relaxing away from SP We like SP for reasons of equity (Roth 03, Pathak and Sonmez 08) simplify reasoning can predict properties of the mechanism But it is generally hard to obtain And, can be provably bad along other dimensions e.g., CAs with complements (Ausubel & Milgrom 06, Rastegeri, Condon, & Leyton-Brown 10)

35 Example: Course Allocation (Budish and Cantillon 08) Random Serial Dictatorship basically unique amongst SP mechanisms (Papai 01) HBS mechanism: snake back and forth, pick one at a time callousness of RSD (allocating 10 courses)

36 Old Favorite: Min Max Regret Regret = best utility actual utility Maximally SP: minimizes max regret across agents on every instance ²-SP: max regret ²

37 payoff vcg agents

38 payoff vcg agents

39 regret payoff vcg agents

40 Two mechanism rules (Parkes, Kalagnanam and Eso 01) vcg vcg agents Threshold rule (min max regret) Small rule max #(regret=0) agents

41 Back to Example vcg = (5,15,5) Surplus 20 Threshold: 1 = 3.33, 2 =13.33, 3 =3.33 payments (-13.33, , ) regret = 1.33 for all agents Small: 1 = 5, 2 =10, 3 =5 payments (-15, -15, +30) regret = 0, 5, 0 Theorem. Threshold rule minimizes ex post opportunity for gain across simple CEs truthful most often (assume cost C d ) [Milgrom]

42 Compute approx BNE Single-minded CEs (Related: Vorobeychik et al.) Need a way to compute approximate, restricted BNE Approach: assume piecewise linear, symmetric strategy profiles Buy: bid (1+ ) v Sell: bid (1+ ) v Use ( 1, 2, 3 )

43 Approximate BNE Analysis (Lubin & Parkes 09) strategy efficiency (For BNE, see Vorobeychik & Wellman 08, Rabinovich, Gerding, Polukarov & Jennings 09) Parkes AAAI 10 64

44 Distributional View: Payoffs

45 Regret Quantiles (Lubin PhD 10)

46 Regret Quantiles (Lubin PhD 10) look at F(regret ²) and max regret

47 Hypothesis I Maximizing the number of agents with zero regret provides less ex ante incentive for strategic behavior than minimizing the maximum regret. ( proof would require reasoning about distributional properties)

48 Hypothesis II Consider a strategyproof reference mechanism M* with the same allocation rule but a different payment rule Reducing the divergence between the distribution on payoffs in M and the distribution on payoffs in M* reduces the ex ante incentive for strategic behavior.

49 Distributional View: Payoffs Parkes AAAI

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51 Discriminative power of metrics

52 Other Approx SP Concepts SPITL: SP in a large market (Budish 09) get best outcome in choice set choice set becomes agent-independent in limit of continuum market Counting manipulations (Pathak & Sonmez 09) (roughly) B manipulable by less agents in less instances than A Marginal gain (Erdil & Klemperer 09) minimize ¼ i (v i, b -i ) / v i

53 Conclusion Incentive-efficient CEs are only know for simple settings (e.g., known single-minded bidders; one buyer, one seller, single unit) ICE = bidding language, winner determination, price feedback, proxy agents Payment design small >> threshold maximize # agents with zero regret or, minimize divergence to VCG payoffs