Analytic Prediction of Emergent Dynamics for ANTs Systems

1. Analytic Prediction of Emergent Dynamicsfor ANTs Systems Utah State University

2. Summary of Proposed Approach • Given: • A set of tasks to perform, each with start times and deadlines • A set of resources that can be scheduled to perform tasks • A negotiating strategy between task and resource agents • Analytically Determine: • Behavior of the overall system • Conditions under which task completion is not feasible

3. Summary (cont.) • It would be uninteresting to: • Develop a new negotiating paradigm -or- • Develop a model for one specific negotiating strategy -or- • Develop a model for one specific problem • Instead, we would rather have: • An analytical model that reflects many negotiation strategies • For a broad class of problems • To determine global resulting behavior of individual actions

4. Summary (cont.) Divisible Nonspatial Nondivisible Nonspatial Divisible Spatial Nondivisible Spatial

5. Summary (cont.) • Allocation problem taxonomy • Divisible: jobs that can be performed in fractional amounts, such as digging a ditch. Rate equations easily apply. • Nondivisible: task performed completely, or not at all. Difficult to assess amount of doneness. • Nonspatial: physical juxtaposition of resources not considered • Spatial: physical juxtaposition of agents and objectives is critical

6. Summary (cont.) First step: Describe with rate equations, verify with comparison with simulation Divisible Nonspatial Nondivisible Nonspatial Divisible Spatial Nondivisible Spatial

7. Screaming Generals • Example of Divisible Nonspatial class • Number of generals each with a task to complete • Require resource to complete task • No time-ordering of operations (e.g., ditch digging) • Each “day”, each general: • Determines own stress (work_remaining/time_left) • Makes request for resources • Is allocated resource for that day based on need/availability

8. Figure 1: Typical Results - front-loaded work schedule. More tasks and deadlines are scheduled for the beginning of the simulation, with correspondingly higher rates of failure earlier. Stresses increase asymptotically and work completion rates are characterized by positive concavity (diminishing returns in time). When tasks are removed (time index 23) stresses flatten out, concavity changes Through linear to negative and the number of failures plateaus.

9. Figure 2: Typical Results - rear-loaded work schedule. More tasks and deadlines Are scheduled for the end of the simulation, with correspondingly higher rates of failure. Stresses increase asymptotically and work completion rates are characterized by positive concavity (diminishing returns in time) in the regions during which many tasks fail. Before tasks are added (time index 23) stresses flatten out, concavity is zero and the system is unstressed.

10. Summary (cont.) Second step: Describe nondivisible/nonspatial. Rate equations difficult to apply Divisible Nonspatial Nondivisible Nonspatial Divisible Spatial Nondivisible Spatial

11. Summary (cont.) • Wish to avoid building a different model for each new negotiating strategy • Focus instead on the ability for all parties to find satisficing solutions given the tasks required and resources available • Need a lingua franca of negotiation • Use praxeic utility theory to describe negotiation strategies and problem sets • Praxeic utility theory useful for determining jointly satisficing solutions • Build analytical tools that model results of choices over time and task spaces.

12. Summary (cont.) • Praxeic Utility Theory - An approach to decision making and control • Satisficing Games and Decisions, Wynn Stirling 2000 • Provides for locality of decisions • Avoids over-proscription • Each alternative weighed on basis of own merits • Retain candidates whose utility for approaching goal outweighs its cost • Because each choice based only on its own merits, breaks the “grip of optimality”

13. Summary (cont.) • Basic framework built on two functions, each of which follows the axioms of probability: Selectability – pS(u) – Utility of a decision w.r.t. moving toward a desired goal. Rejectability – pR(u) – Cost associated with that decision. Out of the set of possible decisions U, retaining all choices u for which pS(u) >= bpR(u) is satisficing. Where b is the boldness – lowering the boldness results in retaining more decisions in u

14. An Example Ace has the option to go to the game (G), stay home (H), or go to the museum (M). However, there is the probability of rain = . The set of outcomes are: Ace’s ordered preferences: u6  u1  u4  u3  u5  u2 Rejectability: Enjoyment is a resource Ace likes to conserve (unfavorable options have a high degree of rejectability). Selectability: Not going to game in sunshine is failure. Going to game in rain is failure. Staying home is failure

15. An Example Marginals: pR(G) = pR(R,G) + pR(S,G) = 0.25. Similarly: Taking b = 1

16. Tie Breaking • By the stated criteria, any element in Sbis acceptable. However, when it is necessary to reduce the choices to a single one, one of several tie breakers may be used: • Most selectable • Least rejectable • Maximally discriminating (max of pS(u) – bpR(u)) • Arbitrary

17. Application in context of existing ANT projects • Example: Pilot Scheduling Problem (CAMERA) • MICANTS-style examples also exist • Each mission incurs risk to flyers, risk function depends on least skilled flyer in group • Individual Goal: each pilot increases skill level (satisfaction) • Group Goal: all participants to increase skill levels • Mission Goal: reduce mission risk

18. Application in context of existing ANT projects • The view from the Praxeic Utilitarian • My estimate of your selectability and rejectability may affect my own selectability and rejectability • Group decisions require formulation of joint selectability and rejectability • Agents must negotiate to obtain a group decision • Boldness becomes a tool for negotiation

19. Application in context of existing ANT projects • N pilots X1,…XN to collectively fly M < N aircraft for mission k • Let I(k) = {i1,…,iM} denote the set of indices of participants • Each Xi has skill level si(k). • Let s(k) = { s1(k),…,sN(k)} • Let s(k) = {si1(k),…,siM(k)}, ij I(k) is the skill level of each pilot chosen for mission k • Let gi(s) denote pilot’s satisfaction, 0  gi(s)  1 • Skill increase: where g(s(k) ) denotes the joint satisfaction of the group

20. Application in context of existing ANT projects • The individual goal of each pilot agent is to increase its skill level (i.e., its satisfaction) • The goal of the group is for all participants to increase their skills uniformly • Each mission incurs some danger, or risk, to its participants. Under the assumption that a group is as vulnerable as its least skilled member

21. Application in context of existing ANT projects • Since all agents agree on who will participate in mission, some form of negotiation must occur. • Agents who have low skill levels will be willing to drive harder bargains than those with higher skill levels.

22. Application in context of existing ANT projects • Let Ui = {1, 0}, indicating fly or don’t fly. Group decision: U = {0, 1}N • The decision vector (of length N) must have exactly M 1s in it; there are N choose M possible choices in this set, designated UN. • For a u UN, we can write s(k) = F(u(k)s(k)), where F(u) maps the vector to a matrix. And the goal function can be written as

23. Application in context of existing ANT projects • Joint Selectability: • Joint Rejectability:

24. Application in context of existing ANT projects • Can describe negotiation problems abstractly, makes possible the analytical study. • For this case, can determine that for arbitrary initial skill levels, all pilots converge to same skill level over time.

25. General Conditions for applicability • Nondivisible / Nonspatial • Praxeic utility theory provides more appropriate representation • Define explicit goals (ps) – follows axioms of probability (normalization) • Define explicit costs (pr) – follows axioms of probability • Define individual versus group satisfiability and rejectability

26. General Conditions for applicability • Divisible / Nonspatial • “work completed” can be represented as a fraction of total work required • No time-ordering or execution-ordering • Rate equations provide reasonable description • Can predict ability to accomplish goals

27. Integration approaches • Need to have better understand of both CAMERA and MICANTS projects. • First step: Obtain license for CAMERA scheduler • Use code hooks in scheduler to “break out” problem descriptions • Model agents in CAMERA world, or other types • Verify validity of approach • Identify over-constrained problems, given agents behaviors and constraints • “Close the loop,” examine predictive utility • Extend predictive model to include spatially and temporally juxtaposed elements (more applicable to MICANTS). • Extend to allow individual satisfiability and rejectability based on partial “localized” information