Characterizing Mechanism Design Over Discrete Domains

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

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

Social Choice Function(SCF( f : V1 × … × Vn→ A • A is the finite set of possible alternatives. • Each player has a valuationvi: A →R. • fchooses an alternative from A for every v1 ,…, vn. • 1 item Auction:A = {playeri wins | i=1..n}, Vi =R+, f (v) = argmax(vi) • [Nisan, Ronen]’s scheduling problem:find a partition of the tasks T1..Tn to the machines that minimizes maxicosti(Ti ).

Truthful Implementation of SCFs Dfn: A Mechanismm(f, p)is a pair of a SCF fand a payment function pi for every player i. Dfn: A Mechanism is truthful (in dominant strategies) if rational players tell the truth:  vi , v-i , wi : vi ( f(vi , v-i))– pi(vi , v-i) ≥ vi ( f(wi , v-i))– pi(wi , v-i).

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

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

Truthfulness and Monotonicity

Truthfulness vs. Monotonicity Example: 1 item Auction with 2 bidders [Myerson] 1 wins v1 ● ● v1 2 wins ● p2 v2 p2 v'2 v2 • Mon. Truthfulness • player 2 wins and pays p2. Mon.Truthfulness the curve is not monotone - player 2 might untruthfully bid v’2 ≤ v2.

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.

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

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

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.

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 convexand even if f is randomized, then Weak-Monotonicity Truthfulness.

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

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

Monge Domains  Integer Grid Domains  0/1 Domains WM-Domains Strong-Monotonicity Truthfulness

Weak / Strong / Cyclic – Monotonicity  Weak-Monotonicity Cyclic-Monotonicity

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

Example: A single player, • 2 alternatives a, and b, and • 2 possible valuations v1, and v2. • Majoritysatisfies Weak-Mon. • f(v1) = a, f(v2) = b. • Minoritydoesn’t. • f(v1) = b, f(v2) = a.

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

Example: • single player • A = {a, b, c}. • V1 = {v1, v2, v3}. • f(v1)=a, f(v2)=b, f(v3)=c.

Example: • single player • A = {a, b, c}. • V1 = {v1, v2, v3}. • f(v1)=a, f(v2)=b, f(v3)=c. • f satisfies Weak-Monotonicity , but not Cyclic-Monotonicity:

Discrete Domains: Integer Grids and Monge

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

Monge Matrices Dfn: B=[br,c] is a Monge Matrix iffor every r < r’ and c < c’: br, c + br’, c’ >br’, c+ br, c’. Example: 4X5 Monge Matrix

Monge Domains • Dfn: V= V1× . . .×Vnis a Monge Domain if for every i∈[n]: • there is an order over the alternatives in A:a1,a2,. . . • and an order over the valuations in Vi: vi 1,vi 2,. . . , • such that the matrix Bi=[br,c] • in which br,c=vi c( ar) • is a Monge matrix. • Examples: • Single Peaked Preferences • Public Project(s)

Monotonicity on Monge Domains Dfn: f is Weak-Monotone if for any vi ,uiand v-i: f (vi , v-i) = a and f (ui, v-i) = b implies vi (a) + ui(b) > vi (b) + ui(a). There are two cases to consider: …

A simplified Congestion Control Example: • Consider a single communication link with capacity C > n. • Each player i has a private integer value dithat 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’kto player k. Apply fair queuing to share the remaining capacity of C - d’k between the remaining players.

A simplified Congestion Control Example (cont.): Assume the capacity C=5, then Vi:

A simplified Congestion Control Example (cont.): Clearly, a player icannot get a smaller capacity share by reporting a higher vi 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: …

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

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.

Thank you

Characterizing Mechanism Design Over Discrete Domains