Analytic Prediction of Emergent Dynamicsfor ANTs Systems
Utah State University
James [email protected]
Todd [email protected]
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
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
Summary (cont.)
DivisibleNonspatial
NondivisibleNonspatial
DivisibleSpatial
NondivisibleSpatial
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
Summary (cont.)
First step: Describe with rate equations, verify with comparison with simulation
DivisibleNonspatial
NondivisibleNonspatial
DivisibleSpatial
NondivisibleSpatial
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
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.
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 increaseasymptotically 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.
Summary (cont.)
Second step: Describe nondivisible/nonspatial. Rate equations difficult to apply
DivisibleNonspatial
NondivisibleNonspatial
DivisibleSpatial
NondivisibleSpatial
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.
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”
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
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
u1 : (R,G) game & rains u2 : (S,G) game & shines u3 : (R,M) museum & rains
u4 :(S,M) museum & shines u5 : (R,H) home & rains u6 : (S,H) home & shines
pR(R,G) = 0.25 pR(S,G) = 0.0 pR(R,M) = 0.15
pR(S,M) = 0.20 pR(R,H) = 0.10 pR(S,H) = 0.30
pS(R,G) = 0 pS(S,G) = 1 - pR(R,M) =
pR(S,M) = 0 pR(R,H) = 0 pR(S,H) = 0`
An Example
Marginals: pR(G) = pR(R,G) + pR(S,G) = 0.25. Similarly:
Taking b = 1
pR(G) = 0.25 pR(M) = 0.35 pR(H) = 0.40
pS(G) = 1 - pS(M) = pS(H) = 0
75.035.0 },{
0.35 }{75.0 }{
GM
G
M
sb
Tie Breaking
By the stated criteria, any element in Sb is 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
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
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
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 (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((k) ) denotes the joint satisfaction of the group
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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
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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.
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 (k) = u(k)s(k)), where umaps the vector to a matrix
And the goal function can be written as
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Application in context of existing ANT projects
Joint Selectability:
Joint Rejectability:
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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.
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
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
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