Models for Achieving Cooperation Among the Agents in Complex Systems

1. Models for Achieving Cooperation Among the Agents in Complex Systems Dr. Chartchai Leenawong Department of Mathematics and Computer Science Faculty of Science

2. 0 1 0 1 0 1 0 1 1 Team Replacement Problem 0  p(x = … )  1 … N 1 2 Positions • Only two candidates for each job position (0 or 1). • p(x) is the performance of Team x. • Objective: To choose, for each position, a candidate (0 or 1) so that the resulting team x has the best performance. Question: How is the team performance determined?

3. … ... ) / N p(x) = ( + + + Determining the Team Performance • The NK Model x = ... N 1 2 p1(x) p2(x) pN(x) • A positive integer Nrepresents the team size. • The contribution of the person in position i, pi(x), depends on that person and on K (0  K N  1)other people on the team, say, the K people to the right of position i, wrapping around when necessary. • pi(x)~ U[0,1].

4. 2K+1 Combinations = (0.1 + 0.6 + 0.5)/3 = 0.4. Determining the Team Performance (Cont’d) Example N = 3, K = 1, and

5. The NK Problem NK Problem:Given integers N > 0 and 0 KN – 1, and given polynomial computable functions p1, …, pN, find a team (binary n-vector) x so as to Theorem:The NK problem is NP-hard. • For large N and K, it’s unlikely to find the global maximum. • A heuristic is needed to find a local maximum.

6. The One-Replacement Process 2N teams x 1 110(0.9) 100 (0.4) 111 (0.5) 101 (0.1) x 2 (0.2) 000 010 (0.3) 001 (0.5) 011 (0.6) x 3

7. 0.80 0.70 N =100 Expected Performance 0.60 0.50 0 20 40 60 80 100 K The Interaction Catastrophe • How does the amount of interaction, K, affect • the team performance?

8. Z x = … x1 x2 xN Team Replacement Problem With a Leader pi(x,z) = Contribution to Team Performance of Worker xi with Leader z p(x, z) = Performance of Team x with Leader z E(p(x*,z)) = Expected Performance of a Local Maximum Team x* Question: How does a leader affect pi(x,z)?

9. Roles of Team Leaders • 1. Traditional Management Activities • - Planning • - Decision Making • - Controlling • 2. Human Resource Management Activities • - Motivating • - Disciplining/Punishing • - Seeking Cooperation • - Staffing • - Training/Developing • 3. Communication Activities • - Exchanging Information • - Handling Paperwork • 4. Networking Activities • - Interacting with Outsiders • - Socializing/Politicking

10. Cooperational Leadership Models Model 1: A Model Based on the Leader’s Role in Eliciting Higher Performance Contributions (The NK (,) Model) •   0 represents the cooperational skill level of the leader •  is a number that represents the variation of the skill level pi(xiK,z) = F(y) where y ~ N(,) or pi(xiK,z) ~ SN(,) y ~ N(2,1) pi(xiK,z)=F(y) y

11. Results of Model 1 • The skill level of the leader improves the team performance • The skill of the leader can be more important than • the amount of interaction between the workers. • A more skillful leader can manage more interaction • without adversely affecting team performance, thus, • attenuating the interaction catastrophe.

12. Results of Model 1 (Cont’d) • For a fixed skill level of the Leader, team performance approaches 0.8 as variation approaches infinity.

13. Z bN b1 b2 rN r1 r2 x = … x1 x2 xN ... a1 a2 aN Model 2: A Model Based on the Leader’s Role of Achieving a Good Relationship Among the Workers (The NKLC (,) Model) [ai(xi), bi(xi)] = Contribution Range of Worker xi = [min(u1, u2), max(u1, u2)], u1 and u2 ~ U[0,1] 0  ri(xiK,z)  1 = Relationship Achieved by Leader z Between Worker xi and K Others That Interact with xi. ~ SN (,) Note: The closer ri is to 1, the more cooperation Leader z achieves. pi(xiK,z) = (1- ri(xiK,z)) ai(xi) + ri(xiK,z) bi(xi)

14. Model 1 V.S. Model 2 • The pattern of the curves are similar, thus leading to similar conclusions. • As the skill increases, team performance in Model 2 approaches • 0.8, rather than 1.0 as in Model 1, because the largest possible • contribution of Worker xi in Model 2 is bi(xi) 1. • [ai(xi), bi(xi)] makes Model 2 more realistic by allowing different • Workers to have different ranges of contributions

15. Model 1 V.S. Model 2 (Cont’d) • The variation of the skill leader affects the team performance • in both models similarly. • The curves are shifted down in Model 2 because of the • contribution range effects.

16. Model 3: A Model Based on the Leader’s Ability to Influence Worst-Case and Best-Case Contributions • What happens to the contribution of a worker when K is increased by 1? In the NK Model: i,pi(xiK) ~ U[0,1] and pi(xiK +1) ~ U[0,1] Ex. Team x = For position 1, when K =1, p1(x11) = 0.1 For position 1, when K =2, p1(x12) = 0.9 Far from 0.1 • A more realistic scenario: pi(xiK +1) should be within • some lower bound li and upper bound uiaround • the value of pi(xiK ), where the bounds depend on the • skill of the leader.

17. and uiK +1 and uiK +1 1 Then, Model 3: The NKLC (a) Model • liK +1 and uiK +1 depend on pi(xiK) and a • where 0  a 1 is the cooperational skill level of the leader Define When a  0, liK +1 0 When a 1, liK +1pi(xi K )

18. Results of Model 3 • When a is small, the performance decreases monotonically. • As a increases, the performance curves move up. • A more skillful leader can manage more interaction • without adversely affecting team performance, thus, • attenuating the interaction catastrophe.

19. Future Research • Emergence of a Hierarchical Structure • Develop a hierarchical team model that reflects the structure of organization. • Study how the leaders at different levels affect each other and the team performance. • Emergence of a Leader and Team Splitting • Study how a leader is recognized. • Study how large a team should be when split.

20. Future Research (Cont’d) • Other Topics • Develop a mathematical model that includes • simultaneously two or more roles played by • the leader. • Study the role of the leader in the • worker-replacement process, for example, • in deciding who to replace.