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Combinatorial Agency Michal Feldman ( Hebrew University)

Combinatorial Agency Michal Feldman ( Hebrew University). Joint with: Moshe Babaioff (UC Berkeley) Noam Nisan (Hebrew University). Hidden Actions. Algorithmic Mechanism Design: computational mechanisms to handle Private Information . (Classical) Mechanism Design Private Information

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Combinatorial Agency Michal Feldman ( Hebrew University)

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  1. Combinatorial AgencyMichal Feldman(Hebrew University) Joint with: Moshe Babaioff (UC Berkeley) Noam Nisan (Hebrew University)

  2. Hidden Actions • Algorithmic Mechanism Design: computational mechanisms to handle Private Information. • (Classical) Mechanism Design • Private Information • Hidden Actions • We study hidden actions in multi-agents computational settings

  3. Example • Quality of Service (QoS) Routing [FCSS’05]: • We have some value from message delivery. • Each agent controls an edge: • succeeds with low probability by default. • succeeds with high probability if exerts costly effort • Message delivered if there is a successful source-sink path. • Effort is not observable, only the final outcome. source sink

  4. Modeling: Principal-Agent Model exerts effortcost: c >0 Project succeeds with high probability Project succeeds with low probability Does not exert effortcost: 0 Agent Principal Motivating rational agents to exert costly effort toward the welfare of the principal, when she cannot contract on the effort level, only on the final outcome “Success Contingent” contract. The agent gets a high payment if project succeeds, gets a low payment if project fails Our focus is on multi-agents technologies

  5. Our Model The Principal’s “input” parameter. • n agents • Each agent has two actions (binary-action): • effort (ai=1), with cost c>0 (ci(1)=c) • noeffort (ai=0), with cost 0 (ci(0)=0) • There are two possible outcomes (binary outcome): • project succeeds, principal gets value v • project fails, principal gets value 0 • Monotone technology functiont: maps an action profile to a success probability: • t: {0,1}n [0,1] t(a1,…,an)=success probability given (a1,…,an) • i t(1, a-i) > t(0,a-i) (monotonic) • Principal designs a contract for each agent • Project succeeds agent i receives pi(otherwise he gets 0) • Players’ utilities, under action profile a=(a1,…,an) and value v: • Agent i: ui(a) = t(a)·pi – ci(ai) • Principal: u(a,v) = t(a)·(v –Σipi) • Agents are in a game, reach Nash equilibrium. The Principal’s design parameter: Used to induce the desired equilibrium

  6. Example: Read-Once Networks • A graph with a given source and sink • Each agent controls an edge, independently succeeds or fails in his individual task (delivering on his edge) • Succeeds with probability ɣ<½ with no effort • Succeeds with probability 1-ɣ (>½>ɣ) with effort • The project succeeds if the successful edges form a source-sink path. example: t(1, 1, 0) = Pr { x1 (x2 x3) =1 | a=(1,1,0) } = (1- ɣ) (1- ɣ(1-ɣ)) a2=1 a1=1 Pr {x2=1}=1- ɣ sink source a3=0 Pr {x1=1}=1- ɣ Pr {x3=1}=ɣ

  7. Di(a-i) P2=0 Nash Equilibrium Agent i’s utility exerts effort Does not exert effort • Principal’s best contract to induce eq. a=(a1,…,an): • pi= c / Di(a-i)for agent i with ai=1 • pi= 0 for agent i with ai=0 • e.g., (1,0) (1,1) ui( 1,a-i ) =pi· t( 1,a-i )– c ui( 0,a-i) =pi· t(0,a-i )

  8. Optimal Contract • the principal chooses a profile a*(v) that maximizes her optimal equilibrium utility Probability of success Total payments

  9. Research Questions • How does the technology affect the structure of the optimal contracts? • Several examples (AND, OR, Majority …) • General technologies • What is the damage to the society due to the inability to monitor individual actions? • “price of unaccountability” • What is the complexity of computing the optimal contract? • Can the principal gain utility from mixed strategies? • Can the principal gain utility from a-priory removing edges from the graph?

  10. Optimal Contracts: simple AND technology 2 agents, g = ¼, c=1 • t(0,0) = g2 = (¼)2=1/16 • t(1,0) =t(0,1)= g(1-g) = 3/16; D0 =t(1,0)-t(0,0)=3/16 - 1/16 = 1/8 • t(1,1) = (1-g)2 = 9/16 Principal’s Utility • 0 agents exert effort: u((0,0),v) =t(0,0)·v = v/16 • 1 agentexerts effort: u((1,0),v) = t(1,0)·(v-c/D0) = =3/16(v-1/(1/8))=(3/16)v-3/2 • 2 agents exert effort: u((1,1),v) = t(1,1)·(v-2c/D1) = 9v/16-3 s t x1 x2 At value of 6 there is a “jump” from 0 to 2 agents

  11. AND ɣ=1/4 optimal to contract with 0 agents up to 6, then with 2 agent Optimal Contract Transitions in AND and OR OR x1 x1 x2 s t s t x2 v v 2 g g

  12. Optimal Contract Transitions in AND and OR • Theorem: For any AND technology, there is only one transition, from 0 to n agents. • Theorem: For any OR technology, there are always n transition (any number of agents is optimal for some value). • We characterize all technologies with 1 transition and with n transitions.

  13. Proofs Idea-AND’s single transition • Observation (monotonicity): number of contracted agents monotonically non-decreasing in v. • Proof for AND’s single transition: • At the indifference value between 0 and n agents, contracting with 0<k<n agent has lower utility. • By the above observation, a single transition. The 0 and n indifference value

  14. Transitions in AND and OR • Proof (AND):k: number of contracted agents this function has a single minimum point, thus maximized at one of the edges 0 or n

  15. Proofs Idea – OR’s n transitions • Let vk be the indifference point between k and k+1 agents ( u(k,vk) = u(k+1,vk) ) • We show that for OR: vk+1> vk • This ensures that k is optimal from vk-1 to vk v1:The 1 ,2 indifference value. v0:The 0 ,1 indifference value. v1>v0

  16. Transitions in AND and OR • k: number of contracted agentssolve for v: u(k) = u(k+1), and let v(k) be the solutionwe have to show: v(k+1) > v(k)  d • E.g., n=3 v(2) v(1) v(0) d

  17. General Technologies • In general we need to know which agents exert effort in the optimal contract • Examples: • In potential, any subset of agents (out of 2n subsets) that exert effort could be optimal for some v. • Which subsets can we get as an optimal contract?

  18. f = hg R T And-of-Ors (AOO) Technology • Example: 2x2 AOO technology • Theorem: The optimal contract in any AOO network (with identical OR components) has the same number of agents in each OR-component • Proof: by induction based on following lemmas: • Decomposition lemma: if S=TUR is optimal on f=hg on some v, then T is optimal for h on v·tg(R) and R is optimal for g on v·th(T) • Component monotonicity lemma: the function vth(T) is monotone non-decreasing (same for vtg(R) )  {A1,B1} {A1,B1,A2,B2} v

  19. f = hg R T Decomposition Lemma if S=TUR is optimal on f=hg on some v, then T is optimal for h on v·tg(R) and R is optimal for g on v·th(T) • Proof:

  20. Component Monotonicity Lemma The function vth(T) is monotone non-decreasing (same for vtg(R) ) • Proof: • S1 = T1 U R1 optimal on v1 • S2 = T2 U R2 optimal on v2<v1 • By monotonicity lemma: f(S1) ≥ f(S2) • Since f=g·h, f(S1)=h(T1)·g(R1) ≥ h(T2)·g(R2) = f(S2) • Assume in contradiction that h(T1) < h(T2). Since h(T1)·g(R1) ≥ h(T2)·g(R2) , we get g(R1) > g(R2). • By decomposition lemma, T1 is optimal for h on v1·g(R1), and T2 is optimal for h on v2·g(R2) • As v1 > v2, and g(R1) > g(R2),T1 is optimal for h on a larger value than T2. • Thus, by monotonicity lemma, h(T1) ≥ h(T2) f: R1 h g R2 T1 T2

  21. And-of-Ors • Theorem: The optimal contract in any AOO network, composed of nc OR-components (of size nl) contracts with the same number of agents in each OR-component. Thus, |orbit(AOO)| ≤ nl+1 • Proof: by induction on nc • Base: nc=2assume (k1,k2) is optimal on some v, assume by contradiction k1>k2 (wlog), thus h(k1)>h(k2).By decomposition lemma: k1 optimal for h on v·h(k2) k2 optimal for h on v·h(k1)>v·h(k2)but if k2 optimal for a larger value, k2≥k1. in contradiction.

  22. h2 g And-of-Ors k1 k3 k3 k2 k2 k2 = = = h h h h • assume (induction) that claim holds for any number of OR components < nc • Assume 1st component has k1 contracted agents • Let g be the conjunction of the other (nc-1) comp. • By decomposition lemma, contract on g is optimal at v·h(k1), thus by induction hypothesis has same number of agents, k2, on each OR component. • Let h2 be conjunction of first two comp. • By decomp. Lemma, contract on h2 is optimal for some value and by induction hypothesis has same number of agents, k3 • We get k1=k3 (in first comp. k1 agents contracted), and k2=k3 (in second comp. k2 agents contracted), thus k1=k2

  23. The Collection of Optimal Contracts • Given t we wish to understand how the optimal contract changes with v (the “orbit”). • Monotonicity Lemma: The optimal contract success probability t(a*(v)) is monotonic non-decreasing with v • So is the utility of the principal, and the total payment • Thus, there are at most 2n-1 changes to the optimal contracts (|Orbit(t)| ≤ 2n) Is there a structure on the collection of optimal contracts of t?

  24. The Collection of Optimal Contracts • Observation 1: in the observable-actions case, only one set of size k can be optimal (set with highest probability of success) • Observation 2: not all 2nsubsets can be obtained • Only a single set of size 1 can be optimal (set with highest probability of success) • Thm: There exists a tech. with optimal contracts • Open question 1: is there a read-once network with exponential number of optimal contracts? Can a technology have exponentially many different optimal contracts?

  25. S1 S3 S4 S2 Exponential number of optimal contracts (1) • Thm: There exists a tech. with optimal contracts • Proof sketch: • Lemma 1: all k-size sets in any k-admissible collection can be obtained as optimal contracts of some t • Lemma 2: For any k, there exists a k-admissible collection of k-size sets of size • Based on error correcting code • Lemma 3: for k=n/2 we get a k-admissible collection of k-size sets of size , as required. Collection of sets of size k, in which every two sets in it differ by at least two elements

  26. t(S’)= ½ - eS’ Define t to ensure that the marginal contribution of at least one agent is very small S’ S\i S\i S’\i S’\i t(S’\i)= ½ - 2eS’ Proof of Lemma 1 n • marginal contribution of i  S is: t(S) – t(S\i) = eS t(S)= ½ - eS k S k-1 t(S\i)= ½ - 2eS • Claim: at vs=(ck) / 2eS2,the set S is optimal: • S better than any other set in col. (by derivative of u(S,v)) • S better then any other set not in col. (too high payments) 1

  27. Let vs be v s.t.

  28. Exponential number of optimal contracts (2) • Lemma: For any n ≥ k, there exists an admissible collection of k-size sets of size • Proof:take error correcting code that corrects 1 error. • Hamming distance ≥ 3  admissible • Known:  codes with W(2n/n) code words. • Construct a code with sufficient # of k-weight words • XOR every code word with a random word r. weight k w/ prob • Expected number of k-weight code words • There exists r such that the expectation is achieved or exceeded

  29. Research Questions • How does the technology affect the structure of the optimal contracts? • What is the damage to the society / principal due to the inability to monitor individual actions? • “price of unaccountability” • What is the complexity of computing the optimal contract? • Can the principal gain utility from mixed strategies? • Can the principal gain utility from a-priory removing edges from the graph?

  30. Observable-Actions Benchmark (first best) • Actions are observable • Payment: an agent that exerts effort is paid his cost (c) • Principal’s utility: u(a,v) = v·t(a) – Si|ai=1c • Principal’s utility = social welfare sw(a,v). • The principal chooses a*OA, the profile with maximum social welfare.

  31. Social Price of Unaccountability • Definition: The Social Price Of Unaccountability (POUS) of a technology is the worst ratio (over v) between the social welfare in the observable-action case, and the social welfare in the hidden-action case: • a* - optimal contract for v in the hidden-action case • a*OA - optimal contract for v in the observable-action case • Example: AND of 2 agents: s t v Hidden actions 0 2 Observable actions 0 2

  32. Principal’s Price of Unaccountability • Definition: The Principal’s Price Of Unaccountability (POUP) of a technology is the worst ratio (over v) between the principal’s utility in the observable-action case, and the principal’s utility in the hidden-action case: • a* - optimal contract for v in the hidden-action case • a*OA - optimal contract for v in the observable-action case

  33. Price of Unaccountability - Results • Theorem: The POU of AND technology is • unbounded for any fixed n≥2, when g0 • unbounded for any fixed g<½ when n • Theorem: The POU of OR technology is bounded by 2.5 for any n

  34. Research Questions • How does the technology affect the structure of the optimal contracts? • What is the damage to the society due to the inability to monitor individual actions? • “price of unaccountability” • What is the complexity of computing the optimal contract? • Can the principal gain utility from mixed strategies? • Can the principal gain utility from a-priory removing edges from the graph?

  35. setsof sizen t(S)=1 S’ sets of size n/2 0 0 1 0 0 0 t(S)=0 sets of size 1 Complexity of Finding the Optimal Contract • Input: value v, description of t • Output: optimal contract: (a*,p) • Theorem: There exists a polynomial time algorithm to compute (a*,p), if t is given by a table (exponential input). • Theorem: If t is given by a black box, exponentially many queries may be required to find (a*,p). • Proof: • for value v = c(k+ ½), S’ is optimal • Any algorithm must query all sets of size k=n/2 to find S’ in the worst case

  36. Complexity of Finding the Optimal Contract • Input: value v, description of t • Output: optimal contract: (a*,p) • Theorem: For read-once networks, the optimal contract problem is #p-hard • Proof: reduction from network reliability problem • Open problem 3: is it polynomial for series-parallel networks? • Open problem 4: does it have a good approximation?

  37. s G t Best Contract Computationin Read-Once Networks • Proof (sketch): an algorithm for this problem can be used to compute t(E) (probability of success) • Player x will enter the contract only for very large value of v (only after all other agents are contracted), call this value vc • At vc, principal is indifferent between E and EU{x} G’ t gx ½

  38. Research Questions • How does the technology affect the structure of the optimal contracts? • What is the damage to the society due to the inability to monitor individual actions? • “price of unaccountability” • What is the complexity of computing the optimal contract? • Can the principal gain utility from mixed strategies? • Can the principal gain utility from a-priory removing edges from the graph?

  39. Mixed Strategies Can mixed-strategies help the principal ? What is the price of purity ? • In the non-strategic case: NO (convex combination) • What about the agency case? • Extended game: • qi : probability that agent i exerts effort • t( qi,q-i ) = qi·t(1,q-i )+ (1-qi )·t(0,q-i ) • Marginal contribution: Di(q-I ) = t(1,q-i ) - t(0,q-i ) ≥ 0

  40. Nash Equilibrium in Mixed Strategies • Claim: agent i’s best-response is to mix with probability q  (0,1) only if she is indifferent between 0 and 1 • Agent i’s utility: • Principal’s utility: Agent i’s utility High effort Low effort ui( 0,q-i ) =pi· t(0,q-i ) ui( 1,q-i ) =pi· t( 1,q-i ) – ci

  41. Example:OR with two agents • Optimal contract for v=110 • Pure strategies: both agents contracted: u = 88.12... • Mixed strategies: q1=q2=0.96..: u=88.24... • Two observations: • q1=q2 in optimal contract • Principal’s utility is improved, but only slightly • How general are these observations?

  42. (qi-ε,qj+yε,q-ij) Optimal Contract in OR Technology • Lemma: For any anonymous OR (any g,n,c,v), k{0,1,…,n} agents exert effort with equal probabilities q1=…=qk  (0,1], and n-k agents shirk. i.e. optimal profile: (0n-k, qk) • Proof (skecth): suppose by contradiction that (qi,qj,q-ij) s.t. qi,qj (0,1) and qi > qj is optimal (qi,qj,q-ij) qj For a sufficiently small ε , success probability increases, and total payments decrease. In contradiction to optimality qi

  43. Optimal Contract in OR Technology Example: OR with 2 agents:

  44. Price of Purity (POP) • Definition: POP is the ratio between principal’s utility in mixed strategies and in pure strategies Optimal mixed contract Optimal pure contract

  45. Price of Purity • Definition: technology t exhibits • increasing returns to scale (IRS) if for any i and any b ≥ at(bi,b-i)-t(ai,b-i) ≥ t(bi,a-i)-t(ai,a-i) • decreasing returns to scale (DRS) if for any i and any b ≥ at(bi,b-i)-t(ai,b-i) ≥ t(bi,a-i)-t(ai,a-i) • Observations: AND exhibits IRS, OR exhibits DRS • Theorem: for any technology that exhibits IRS, optimal contract is obtained in pure strategies • e.g., AND

  46. Price of Purity • For any anonymous DRS technology, POP ≤ n • For anonymous OR with n agents, POP ≤ 1.154.. • For any anonymous technology with 2 agents, POP ≤ 1.5 • For any technology (not necessarily anonymous, but with identical costs) with 2 agents, POP ≤ 2 • Observation: the payment to each agent in a mixed profile is greater than the min payment in a pure profile and smaller than the max payment in a pure profile

  47. Research Questions • How does the technology affect the structure of the optimal contracts? • What is the damage to the society due to the inability to monitor individual actions? • “price of unaccountability” • What is the complexity of computing the optimal contract? • Can the principal gain utility from mixed strategies? • Can the principal gain utility from a-priory removing edges from the graph?

  48. Free-Labor • So far, technology was exogenously given • Now, suppose the principal has control over the technology in that he can ex-ante remove some agents from the graph • Example: OR with 2 agents • Action set of agent i: ai {1,0,} • 1: exert effort– succeed with probabilityd. cost=c • 0: do not exert effort - succeed with probability g< d. cost=0 • : do not participate – succeed with probability 0. cost=0 • Action  “wastes free-labor” since action “0” increases the success probability with no additional cost as before

  49. Free-Labor Are there scenarios in which the principal gains utility from “wasting free-labor”? • The answer is: YES • Example: OR technology, n=2, g=0.2 • Theorem: for technologies with increasing marginal contribution (e.g., AND), utilizing all free-labor is always optimal v 0 1 2 1 removed

  50. Analysis of OR • Lemma: for any OR with n agents and g which is small enough, there exists a value for which in the optimal contract one agent exerts effort and no other agent participates g=0.01 g=0.25 g=0.49

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