analytic prediction of emergent dynamics for ants systems utah state university james powell...

Analytic Prediction of Emergent Dynamicsfor ANTs Systems

Utah State University

James [email protected]

Todd [email protected]

Dan [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


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


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


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

)( if )]([)( if )]()),(([

)1(2

1

kIiksf

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i

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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

))((min )( ksr ikIi


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.


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

00100001

)1100(

))](s)(u([)]([ kkgkg


Joint Selectability:

Joint Rejectability:

N

N

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Uu

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ksug

ksup N

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))(;(

N

N

UvR

Uu

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ksup N

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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

analytic prediction of emergent dynamics for ants systems utah state university james powell...

Documents

critical slide

needavailability slide

feasible slide

screaming generals simulation

task completion

work completion rates

set of tasks

complete task