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Characterizing Mechanism Design Over Discrete Domains Ahuva Mu’alem and Michael Schapira

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Mechanisms : elections, auctions (1 st / 2 nd price, double, combinatorial, …), resource allocations … social goal vs. individuals’ strategic behavior. Main Problem: Which social goals can be “achieved”? Motivation

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Social Choice Function (SCF( f : V 1 × … × V n → A A is the finite set of possible alternatives. Each player has a valuation v i : A → R. f chooses an alternative from A for every v 1,…, v n. –1 item Auction: A = {player i wins | i=1..n}, V i = R +, f (v) = argmax(v i ) –[Nisan, Ronen]’s scheduling problem: find a partition of the tasks T 1..T n to the machines that minimizes max i cost i (T i ). 3

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Truthful Implementation of SCFs Dfn: A Mechanism m(f, p) is a pair of a SCF f and a payment function p i for every player i. Dfn: A Mechanism is truthful ( in dominant strategies ) if rational players tell the truth: v i, v -i, w i : v i ( f(v i, v -i )) – p i (v i, v -i ) ≥ v i ( f(w i, v -i )) – p i (w i, v -i ). 4

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Truthful Implementation of SCFs Dfn: A Mechanism m(f, p) is a pair of a SCF f and a payment function p i for every player i. Dfn: A Mechanism is truthful ( in dominant strategies ) if rational players tell the truth: v i, v -i, w i : v i ( f(v i, v -i )) – p i (v i, v -i ) ≥ v i ( f(w i, v -i )) – p i (w i, v -i ). - If the mechanism m(f, p) is truthful we also say that m implements f. - First vs. Second Price Auction. - Not all SCFs can be implemented: e.g., Majority vs. Minority between 2 alternatives. 5

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Truthful Implementation of SCFs Dfn: A Mechanism m(f, p) is a pair of a SCF f and a payment function p i for every player i. Dfn: A Mechanism is truthful ( in dominant strategies ) if rational players tell the truth: v i, v -i, w i : v i ( f(v i, v -i )) – p i (v i, v -i ) ≥ v i ( f(w i, v -i )) – p i (w i, v -i ). Main Problem: Which social choice functions are truthful? 6

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Truthfulness and Monotonicity

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Truthfulness vs. Monotonicity Example: 1 item Auction with 2 bidders [Myerson] v1v1 v2v2 Mon. Truthfulness player 2 wins and pays p 2. p2p2 2 wins 1 wins v1v1 v2v2 v' 2 p2p2 ●● ● Mon. Truthfulness the curve is not monotone - player 2 might untruthfully bid v’ 2 ≤ v 2. 8

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Truthfulness “Monotonicity” ? Monotonicity refers to the social choice function alone (no need to consider the payment function). Problem: Identify this class of social choice functions. 9

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Thm [Roberts]: Every truthfully implementable f :V → A is Weak-Monotone. Thm [Rochet]: f :V → A is truthfully implementable iff f is Cyclic-Monotone. Dfn : V is called WM-domain if any social choice function on V satisfying Weak-Monotonicity is truthful implementable. Truthfulness vs. Monotonicity 10 Cyclic-MonotonicityWeak-Monotonicity“Simple”-Monotonicity

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Thm [Roberts]: Every truthfully implementable f :V → A is Weak-Monotone. Thm [Rochet]: f :V → A is truthfully implementable iff f is Cyclic-Monotone. Dfn: V is called WM-domain if any social choice function on V satisfying Weak-Monotonicity is truthful implementable. Truthfulness vs. Monotonicity 11 Cyclic-MonotonicityWeak-Monotonicity“Simple”-Monotonicity

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WM-Domains Dfn: V is called WM-domain if any social choice function on V satisfying Weak-Monotonicity is truthful implementable. Thm [Bikhchandani, Chatterji, Lavi, M, Nisan, Sen],[Gui, Muller, Vohra 2003]: Combinatorial Auctions, Multi Unit Auctions with decreasing marginal valuations, and several other interesting domains (with linear inequality constraints) are WM-Domains. Thm [Saks, Yu 2005]: If V is convex, then V is a WM-Domain. Thm [Monderer 2007]: If closure(V) is convex and even if f is randomized, then Weak-Monotonicity Truthfulness. 12

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WM-Domains Dfn: V is called WM-domain if any social choice function on V satisfying Weak-Monotonicity is truthful implementable. Thm [Bikhchandani, Chatterji, Lavi, M, Nisan, Sen],[Gui, Muller, Vohra 2003]: Combinatorial Auctions, Multi Unit Auctions with decreasing marginal valuations, and several other interesting domains (with linear inequality constraints) are WM-Domains. Thm [Saks, Yu 2005]: If V is convex, then V is a WM-Domain. Thm [Monderer 2007]: If closure(V) is convex and even if f is randomized, then Weak-Monotonicity Truthfulness. 13

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14 Cyclic- Monotonicity Truthfulness [Rochet] Convex Domains [Saks+Yu] Combinatorial Auctions with single minded bidders [LOS] Essentially Convex Domains [Monderer] WM-Domains 1 item Auctions [Myerson]

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15 Cyclic- Monotonicity Truthfulness [Rochet] Convex Domains [Saks+Yu] Combinatorial Auctions with single minded bidders [LOS] Essentially Convex Domains [Monderer] Discrete Domains?? WM-Domains 1 item Auctions [Myerson]

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16 Monge Domains Integer Grid Domains WM-Domains 0/1 Domains Strong-Monotonicity Truthfulness

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Weak / Strong / Cyclic – Monotonicity Cyclic-MonotonicityWeak-Monotonicity

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Dfn1: f is Weak-Monotone if for any v i, u i and v -i : f (v i, v -i ) = a and f (u i, v -i ) = b implies v i (a) + u i (b) > v i (b) + u i (a). Dfn2: f is 3 -Cyclic-Monotone if for any v i, u i, w i and v -i : f (v i, v -i ) = a, f (u i, v -i ) = b and f (w i, v -i ) = c implies v i (a) + u i (b) + w i (c) > v i (b) + u i (c) + w i (a). Dfn3: f is Strong-Monotone if for any v i, u i and v -i : f (v i, v -i ) = a and f (u i, v -i ) = b implies v i (a) + u i (b) > v i (b) + u i (a). Monotonicity Conditions 18

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Example: A single player, 2 alternatives a, and b, and 2 possible valuations v 1, and v 2. Majority satisfies Weak-Mon. f(v 1 ) = a, f(v 2 ) = b. Minority doesn’t. f(v 1 ) = b, f(v 2 ) = a. v1v1 v2v2 a 1 0 b 0 1 v1v1 v2v2 a 1 0 b 0 1 19

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Dfn1: f is Weak-Monotone if for any v i, u i and v -i : f (v i, v -i ) = a and f (u i, v -i ) = b implies v i (a) + u i (b) > v i (b) + u i (a). Dfn2: f is 3 -Cyclic-Monotone if for any v i, u i, w i and v -i : f (v i, v -i ) = a, f (u i, v -i ) = b and f (w i, v -i ) = c implies v i (a) + u i (b) + w i (c) > v i (b) + u i (c) + w i (a). Dfn3: f is Strong-Monotone if for any v i, u i and v -i : f (v i, v -i ) = a and f (u i, v -i ) = b implies v i (a) + u i (b) > v i (b) + u i (a). Monotonicity Conditions 20

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Example: single player A = {a, b, c}. V 1 = {v 1, v 2, v 3 }. f(v 1 )=a, f(v 2 )=b, f(v 3 )=c. v1v1 v2v2 v3v3 a 0 1-2 b 0 1 c 1 0 21

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Example: single player A = {a, b, c}. V 1 = {v 1, v 2, v 3 }. f(v 1 )=a, f(v 2 )=b, f(v 3 )=c. f satisfies Weak-Monotonicity, but not Cyclic-Monotonicity: v1v1 v2v2 v3v3 a 0 1-2 b 0 1 c 1 0 22 v1v1 v2v2 a 0 1 b-2 0

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Discrete Domains: Integer Grids and Monge

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Integer Grid Domains are SM-Domains but not WM-Domains Prop [Yu 2005]: Integer Grid Domains are not WM-Domains. Thm: Any social choice function on Integer Grid Domain satisfying Strong-Monotonicity is truthful implementable. Similarly: Prop : 0/1-Domains are SM-Domains, but not WM-Domains.

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Dfn: B=[b r,c ] is a Monge Matrix if for every r < r’ and c < c’: b r, c + b r’, c’ > b r’, c + b r, c’. Example: 4X5 Monge Matrix 1 2 2 00 0 1 5 44 -2 0 8 88 1 9 910 25 Monge Matrices

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Dfn : V= V 1 ×...× V n is a Monge Domain if for every i ∈ [n]: there is an order over the alternatives in A: a 1, a 2,... and an order over the valuations in V i : v i 1, v i 2,..., such that the matrix B i =[b r,c ] in which b r,c = v i c ( a r ) is a Monge matrix. Examples: Single Peaked Preferences Public Project(s) v i 1 v i 2 v i 3 v i 4 v i 5 a1a1 1 2 2 0 0 a2a2 0 1 5 4 4 a3a3 -2 0 8 8 8 a4a4 1 9 910 Monge Domains

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Dfn: f is Weak-Monotone if for any v i, u i and v -i : f (v i, v -i ) = a and f (u i, v -i ) = b implies v i (a) + u i (b) > v i (b) + u i (a). There are two cases to consider: … Monotonicity on Monge Domains v i 1 v i 2 v i 3 v i 4 v i 5 a1a1 1 2 2 0 0 a2a2 0 1 5 4 4 a3a3 -2 0 8 8 8 a4a4 1 9 910

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A simplified Congestion Control Example: Consider a single communication link with capacity C > n. Each player i has a private integer value d i that represents the number of packets it wishes to transmit through the link. For every vector of declared values d’= d’ 1, d’ 2,..., d’ n, the capacity of the link is shared between the players in the following recursive manner (known as fair queuing [ Demers, Keshav, and Shenker ]): If d’ i > C / n then allocate a capacity of C / n to each player. Otherwise, perform the following steps: Let d’ k be the lowest declared value. Allocate a capacity of d’ k to player k. Apply fair queuing to share the remaining capacity of C - d’ k between the remaining players. 28

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A simplified Congestion Control Example (cont.): Assume the capacity C=5, then V i : v i 1 v i 2 v i 3 v i 4 v i 5 a1a1 1 1 1 1 1 a2a2 1 2 2 2 2 a3a3 1 2 3 3 3 a4a4 1 2 3 4 4 a5a5 1 2 3 4 5

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A simplified Congestion Control Example (cont.): Clearly, a player i cannot get a smaller capacity share by reporting a higher v i j. And so, The Fair queuing rule dictates an “alignment”. Claim: Every social choice function that is aligned with a Monge Domain is truthful implementable. Thm: Monge Domains are WM-Domains. Proof: … v i 1 v i 2 v i 3 v i 4 v i 5 a1a1 1 1 1 1 1 a2a2 1 2 2 2 2 a3a3 1 2 3 3 3 a4a4 1 2 3 4 4 a5a5 1 2 3 4 5

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Monge Domains Claim: Every social choice function that is aligned with a Monge Domain is truthful implementable. Thm: Monge Domains are WM-Domains. Proof: … 31

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Related and Future Work [Archer and Tardos]’s setting: scheduling jobs on related parallel machines to minimize makespan is a Monge Domain. [Lavi and Swamy]: unrelated parallel machine, where each job has two possible values: High and Low (it’s a special case of [Nisan and Ronen] setting). It’s a discrete, but not a Monge Domain. They use Cyclic-monotonicity to show truthfulness. Find more applications of Monge Domains (Single vs. Multi- parameter problems). Relaxing the requirements of Monge Domains: a partial order on the alternatives/valuations instead of a complete order.

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