Graph Theoretic Characterization of Revenue Equivalence

Transcription

1 Graph Theoretic Characterization of University of Twente joint work with Birgit Heydenreich Rudolf Müller Rakesh Vohra

2 Optimization and Capitalism Kantorovich [... ] problems of which I shall speak, relating to the organization and planning of production, are connected specifically with the Soviet system of economy and in the majority of cases do not arise in [... ] a capitalist society. [... ] (1939) There [capitalism], the choice of output is determined not by the plan but by the interests and profits of individual capitalists.

3 Optimization and Capitalism Myerson In situations where individuals private information and actions are difficult to monitor, the need to give people an incentive to share information and exert efforts may impose constraints on economic systems [... ] The theory of mechanism design is the [... ] mathematical methodology for analyzing these constraints. (1988)

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5 Scope of this Talk No overview talk (Mechanism design Microeconomic Theory by Mas-Colell et al., 1995, and Algorithmic Game Theory, N. Nisan et al., 2007 some basic concepts of mechanism design focus on one classic result, revenue equivalence message: discrete math approach really helps LP approach to mechansim design, manuscript, R. Vohra 2009

6 Mechanism = Algorithm + Incentive Constraints Incentive Constraints? Data of optimization problem X is distributed among agents Therefore, need to incentivize agents to behave (truthfully) New Challenges game theoretic dimension (solution = equilibrium) not all algorithms are feasible ( implementable ) analogue to competitive analysis ( PoA,... ) new complexity issues ( PPAD hardness,... )

7 Introducing : Single Item Auction Bidders have valuation & utility n bidders bidder i has valuation v i = willingness to pay private! looses utility is 0 wins utility=valuation-price Auction who will win? what will be the price? allocation rule payment scheme Optimization problem: Maximize auction revenue

8 2nd Price Auction (Vickrey 61) Allocation & payment rule Bidders submit bids b i (by ) allocate item to highest bid payment π i = 2nd highest bid Bidders strategy? truthtelling b i = v i, even if all other b j known (i.e., truthtelling is a dominant strategy) Result Allocation rule is efficient (allocates to v max ), however auctioneer s revenue is only v n 1... can we get more revenue?

9 1st Price Auction Allocation & payment rule Bidders submit bids b i by allocate item to highest bid payment π i = b i Bidders strategy? trivial: bid below v i (bid-shading), but by how much? (now depends on given information on other bidders!) Result Allocation rule is efficient (allocates to v max ), to compare (expected) revenues, look at simple example...

10 Two Auctions: Revenues assume 2 bidders only both valuations v j are i.i.d., uniform on [0, 1] 2nd price auction (Vickrey) bid b j = v j (dominant strategy equilibrium) revenue collected E[min{v 1, v 2 }] = 1 3 1st price auction bid b j = n 1 n v j = 1 2 v j (Bayes-Nash equilibrium) revenue collected 1 2 E[max{v 1, v 2 }] = 1 ( 2 ) 2 3 = 1 3 Auctions are quite different, expected revenues are equivalent

11 (RE) auctioning a single item bidders uncertain about other bidders valuations Textbook Theorem Suppose bidders valuations are i.i.d. and bidders are risk neutral (maximizing expected utility). Then any [... ] standard auction a yields the same (expected) revenue to the seller. Example: 1st price auction 2nd price auction a Efficient: bidder with v max wins Individual rational: losers pay 0 see: Vickrey 61/ 62, Riley & Samuelson 81, Myerson 81

12 & Optimization As auction designer natural approach for revenue optimization: optimize over the payments but, whenever revenue equivalence holds... to increase revenue need to modify the allocation rule Example Using reserve prices in auctions increases expected revenue (at the expense of possibly not allocating the item)

13 Mechanism Design: Setting agents i = 1,..., n types t i T i, private information outcomes a A valuations v i : A T i R, (or: v i : T i R A ) Direct revelation mechanism given reports t 1,..., t n of all agents mechanism: (f, π) allocation rule payment scheme f (t 1,..., t n ) = a π i (t 1,..., t n ) R payment from i utility = valuation - payment, u i = v i (f (t), t i ) π i (t)

14 Non-Auction Example: Scheduling p j S j C j = S j + p j agents = jobs, types w i = individual cost for waiting outcomes = all n! sequences σ of jobs valuations v i = w i S i (σ) maximize total happyness = minimize σ j w jc j (σ)

15 Implementability Definition (truthful mechanism) (f, π) truthful iff for all agents i, reports t i = (..., t i 1, t i+1,... ), utility from truth-telling t i utility from lying t i Definition (implementable) Allocation rule f is called (truthfully) implementable if there exists a payment function π such that (f, π) is truthful Why care about truthful f s? By Myerson s revelation principle, this restriction is w.l.o.g.

16 Definition (RE) Let f truthfully implementable. f satisfies RE iff for all truthful implementations of f, (f, π) and (f, π ), for all agents i, π i (t i, t i ) π i(t i, t i ) = const. t i Payment function is (essentially) determined by f alone

17 Notation Utilitarian Maximizer f = argmax a A { i v i (a) } Auction example: bidder with highest valuation v max wins Scheduling example: Schedule that maximizes i w ic i

18 : Literature Sufficient conditions on agents preferences (T, v) I (Green+Laffont 77, Holmström 79): f = utilitarian maximizer (Myerson 81, Krishna+Maenner 01, Milgrom+Segal 02): all implementable f Characterization of agents preferences (T, v) II III (Suijs 96): on finite outcome spaces, f = utilitarian maximizer satisfies RE (Chung+Olszewski 07): on finite outcome spaces, all implementable f satisfy RE Our result characterize agents preferences (T, v) and f s.t. RE holds, arbitrary outcome space

19 Link to Graph Theory: Allocation Graph G f fix one agent i and reports t i of others (notation: drop index i) Allocation graph G f for agent i complete directed graph node set: possible outcomes a, b A (may be infinite) arc lengths l ab = inf [v(b, t) v(a, t)] t f 1 (b) if true type is any t with f (t) = b, l ab = (least) gain in valuation for truthtelling instead of lying to get outcome a l ab a b

20 Node Potentials Remark: Payments for outcomes (f, π) truthful and f (s) = f (t) = a for two reports s and t, then π(s) = π(t) w.l.o.g. define payments π(a) for outcomes a A only Definition (node potential) π : G f R such that (shortest path) -inequality holds for all arcs (a, b): π(b) π(a) + l ab

21 Truthful Mechanism Node Potential Observation (Rochet, 1987) (f, π) truthful π( ) node potential in G f (f, π) truthful iff for any outcomes a, b: utility truth-telling t f 1 (b) utility lying false s f 1 (a) v(b, t) π(b) v(a, t) π(a) t f 1 (b) π(a) + [v(b, t) v(a, t)] π(b) t f 1 (b) π(a) + inf t f 1 (b)[v(b, t) v(a, t)] π(b) π(a) + l ab π(b)

22 Node Potentials Observation (f, π) truthful π node potential in G f Consequence f is implementable Rochet 87 well known G f has node potential G f has no negative cycle Revenue equivalence? f satisfies RE node potential in G f unique (up to constant)

23 Unique Node Potential - Characterization Proposition 1 Any two node potentials differ only by a constant dist(v, w) + dist(w, v) = 0 Proof: dist(v, ) and dist(w, ) are node potentials, so dist(v, w) = dist(w, w) + c and dist(v, v) = dist(w, v) + c }{{}}{{} =0 =0 π(w) π(v) dist(v, w) and π(v) π(w) dist(w, v) so π(w) = dist(v, w) + π(v), for all w so π( ) and dist(v, ) differ by constant π(v)

24 Main Result: Characterization of RE Theorem (Characterization of RE) Truthfully implementable f satisfies revenue equivalence For all outcomes a, b, dist Gf (a, b) + dist Gf (b, a) = 0 Proof. payment scheme π node potential in G f dist Gf (a, b) + dist Gf (b, a) = 0 necessary and sufficient condition for unique node potential in G f (± constant)

25 Analytical Theorems Demand Rationing Application I: Sufficient Conditions for RE Theorem 1 (A finite) agents types T (topologically) connected for all a A, valuations v(a, ) continuous on T Then any truthfully implementable f satisfies revenue equivalence Theorem 2 (A infinite, countable) agents types T R k, (topologically) connected valuations v(a, ) equicontinuous on T Then any truthfully implementable f satisfies revenue equivalence Theorems 1 and 2 aren t new yet had heavier proofs before

26 Analytical Theorems Demand Rationing Proof Idea (A finite) Pick any partition of A: A 1 A 2 T connected: t f 1 (A 1 ) f 1 (A 2 ) A finite v continuous f truthful partition of T a 1 A 1, a 2 A 2 : dist(a 1, a 2 ) + dist(a 2, a 1 ) = 0 Exercise: sufficient for dist(a, b) = dist(b, a) in G f.

27 Analytical Theorems Demand Rationing Application II: Demand Rationing Problems Setting distribute 1 unit of divisible good among n agents agent i has demand t i (0, 1], f i = amount allocated to i, { 0, if fi t v i (f i, t i ) = i ; f i t i, if f i < t i. Dictatorial allocation rule Let f 1 = t 1, split rest equally among agents 2,..., n this f is implementable but RE doesn t hold: π 1 (t) = 0 and π 1 (t) = t 1 1 are both truthful for agent 1 All known results (..., all implementable f satisfy RE ) silent!

28 Analytical Theorems Demand Rationing Proportional Rule Can show: The proportional rule n f i (t) = t i /( t j ) is implementable & satisfies RE j=1 fixing t i, the report-outcome function f i (t i ) is one of the cases j i t j 1 j i t j < 1 f i f i t i 1 j i t j t i

29 Analytical Theorems Demand Rationing Demand Rationing: RE Theorem If report-outcome functions f i (t i ) are continuous, and any of cases (i), (ii) or (iii) holds for every agent i (and t i ), then RE holds. f i f i f i (i) (ii) (iii) t i t i t i Proof. Explicitly compute dist functions in G f and case distinction - tedious but not too hard x

30 Bottom Line previous characterizations Suijs 96 is a special case of ours Chung & Olszewski (C&O 07) can be derived quite easily previous sufficient conditions Green+Laffont 77, Holmström 79, Krishna+Maenner 01, Milgrom+Segal 02 can be derived, too (as also done by C&O 07) new RE results are conceivable

31 Summary simple(!) characterization of RE, graph theory is key first condition on preferences and allocation rule together applies also in settings, where all previous results are silent works same way for other equilibrium concepts Bayes-Nash, Ex-post with externalities

32 Myerson, R. (1981). Optimal auction design. Mathematics of Operations Research 6, Heydenreich, Müller, Uetz, Vohra (2009). Characterization of revenue equivalence. Econometrica 77, both are online