Analytic Prediction of Emergent Dynamics for ANTs Systems

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Analytic Prediction of Emergent Dynamicsfor ANTs
Systems

• Utah State University

James Powell powell_at_math.usu.edu Todd Moon moon_at_ece.usu.edu Dan Watson watson_at_cs.usu.edu
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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

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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

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

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Summary (cont.)

• First step Describe with rate equations, verify
  with comparison with simulation

Divisible Nonspatial
Nondivisible Nonspatial
Divisible Spatial
Nondivisible Spatial
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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

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The Screaming Generals simulation was Performed
using different numbers of generals with random
values for their job sizes (Rj) and deadlines
(Dj). The job sizes had a uniform distribution
from 1 to 50, and the deadlines had a
uniform distribution from 1 to 20. There were 10
Resources (M10) available in every run. The
test was run 3 times each test consisted of
1000 simulations. The results are grouped by the
number of failed tasks in each test. The first
test had 3 generals competing for the 10
resources, the second had 4 generals, and the
third had 5 generals.
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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.
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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.
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Summary (cont.)

• Second step Describe nondivisible/nonspatial.
  Rate equations difficult to apply

Divisible Nonspatial
Nondivisible Nonspatial
Divisible Spatial
Nondivisible Spatial
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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.

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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

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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) gt
  bpR(u) is satisficing.
• Where b is the boldness lowering the
  boldness results in retaining more decisions in u

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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
• Aces 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
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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
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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

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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

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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

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Application in context of existing ANT projects

• N pilots X1,XN to collectively fly M lt 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 pilots satisfaction, 0 ? gi(s)
  ? 1
• Skill increase
• where g(s(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.

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Application in context of existing ANT projects

• Let Ui 1, 0, indicating fly or dont fly.
  Group decision U 0, 1N
• 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

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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.

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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

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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

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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