Combinatorial Agency Moshe Babaioff ( UC Berkeley)

1. Combinatorial AgencyMoshe Babaioff (UC Berkeley) Joint with: Michal Feldman, Noam Nisan (Hebrew University)

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

3. Example • Routing Quality of Service (QoS) [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 Projects succeeds with high probability Projects 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) Nash Equilibrium Agent i’s utility exerts effort Does not exert effort • Principal’s best contract to induce eq. a=(a1,…,an): • pi= 0 for agent i withai=0 • pi= c / Di(a-i)for agent i with ai=1 • Optimal contract: The principal chooses a profile a*(v) that maximizes her optimal equilibrium utility ui( 1,a-i ) =pi· t( 1,a-i )– c ui( 0,a-i) =pi· t(0,a-i )

8. 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 agency” • What is the complexity of computing the optimal contract?

9. 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

10. 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

11. 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 and with n transitions.

12. v ɣ Majority, 5 agents

13. 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?

14. The Collection of Optimal Contracts • Given t we like 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 • At most 2n changes to the optimal contract. • Thm: There exists a tech. with optimal contracts • Open question 1: is there a read-once network with exponential number of optimal contracts? Is there a structure on the collection of optimal contracts of t? Can a technology have exponentially many different optimal contracts?

15. 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 agency” • What is the complexity of computing the optimal contract?

16. Non-Strategic Benchmark • Benchmark: non-strategic case (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*NS, the profile with maximum social welfare.

17. Price of Agency – Definition • Definition: The Price Of Agency (POA) of a technology is the worst ratio (over v) between the social welfare in the non-strategic case, and the social welfare in the agency case: • a* - optimal contract for v in the agency case • a*NS - optimal contract for v in the non-strategic case • Example: AND of 2 agents: s t v agency 0 2 non-strategic 0 2

18. Price of Agency - Results • Theorem: The POA of AND technology is • unbounded for any fixed n≥2, when g0 • unbounded for any fixed g<½ when n • The POA for any OR technology with 2 agents is ≤ 2 • Open question 2: Is POA bounded for any n? (new result found after BAGT: Yes, by 5/2)

19. 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 agency” • What is the complexity of computing the optimal contract?

20. 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). • Theorem: For read-once networks, the optimal contract problem is #p-complete (under Turing reduction) • Open problem 3: is it polynomial for serial-parallel networks? • Open problem 4: does it have a good approximation?

21. v x1 s t x2 ɣ Mixed Nash Equilibrium • Can the principal gain by inducing a Mixed Nash Equilibrium? • In the non-strategic case, she can never gain. • In the agency case: • AND – no. • OR – yes! But at most a factor of 2

22. Summary • We study “Combinatorial Agency”: hidden actions in combinatorial settings. • We consider other problems not presented in this talk, for example: • Can the principal gain by removing edges? When? • What are the implications of asymmetry in the agents’ individual success probabilities? • Many open questions remain, we have only scratched the surface!