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Engineering Serendipity Successes and New Investigations
Engineering Serendipity: Successes in SolitaryForaging and New Investigations in Cooperative
Task-Processing Networks
Theodore (Ted) P. Pavlic – The Ohio State University
Department of Electrical & Computer Engineering
Thursday, May 6, 2010
Overview
Introduction
Reflections
Faulty connections
Missed connections
Motivations
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
IntroductionReflections
Faulty connections
Missed connections
Motivations
Solitary foraging: from ecology to engineering and backSpeed choice (→)
Impulsiveness and operant conditioning (←)
Long patch residence times (←)
Cooperative task processingBackground
Task-processing network
Cooperation game
Asynchronous convergence to cooperation
Results
Closing remarks
Manufacturing Serendipity
Introduction
Reflections
Faulty connections
Missed connections
Motivations
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
1
Craig Tovey: “manufacturing serendipity”
1Compliments to XKCD: http://xkcd.com/720/.
Engineering Manufacturing Serendipity
Introduction
Reflections
Faulty connections
Missed connections
Motivations
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
1
Craig Tovey: success with bees and Internet server allocation
1Compliments to XKCD: http://xkcd.com/720/.
Faulty connections
Introduction
Reflections
Faulty connections
Missed connections
Motivations
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
The Biomimicry Institute
David Brazier
Faulty connections
Introduction
Reflections
Faulty connections
Missed connections
Motivations
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
The Biomimicry Institute
David Brazier
Faulty connections
Introduction
Reflections
Faulty connections
Missed connections
Motivations
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
The Biomimicry Institute
David Brazier(R. Wehner)
Missed Abstract Connections
Introduction
Reflections
Faulty connections
Missed connections
Motivations
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
(Important?) Missed Abstract Connections
Introduction
Reflections
Faulty connections
Missed connections
Motivations
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
IFD (Fretwell 1972; Fretwell and Lucas 1969) ⇐⇒ Optimal power
dispatch (Bergen and Vittal 2000)
(Important?) Missed Abstract Connections
Introduction
Reflections
Faulty connections
Missed connections
Motivations
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
IFD (Fretwell 1972; Fretwell and Lucas 1969) ⇐⇒ Optimal power
dispatch (Bergen and Vittal 2000)
Economic dispatch problem:
minimize
n∑
i=1
Ci(Pi) subject to
n∑
i=1
Pi = P
(Important?) Missed Abstract Connections
Introduction
Reflections
Faulty connections
Missed connections
Motivations
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
IFD (Fretwell 1972; Fretwell and Lucas 1969) ⇐⇒ Optimal power
dispatch (Bergen and Vittal 2000)
Economic dispatch problem:
minimize
n∑
i=1
Ci(Pi) subject to
n∑
i=1
Pi = P
Pareto minimization of costs subject to conservation simplex.
(Important?) Missed Abstract Connections
Introduction
Reflections
Faulty connections
Missed connections
Motivations
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
IFD (Fretwell 1972; Fretwell and Lucas 1969) ⇐⇒ Optimal power
dispatch (Bergen and Vittal 2000)
Economic dispatch problem:
minimize
n∑
i=1
Ci(Pi) subject to
n∑
i=1
Pi = P
Pareto minimization of costs subject to conservation simplex.
Solution (from KKT) is an “upside-down” IFD:
dCi(Pi)
dPi= λ ∀i ∈ 1, 2, . . . , n (and truncate appropriately)
Equalization of marginal cost matches IFD equalization of suitability.
(Important?) Missed Abstract Connections
Introduction
Reflections
Faulty connections
Missed connections
Motivations
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
IFD (Fretwell 1972; Fretwell and Lucas 1969) ⇐⇒ Optimal power
dispatch (Bergen and Vittal 2000)
Economic dispatch problem:
minimize
n∑
i=1
Ci(Pi) subject to
n∑
i=1
Pi = P
Pareto minimization of costs subject to conservation simplex.
Solution (from KKT) is an “upside-down” IFD:
dCi(Pi)
dPi= λ ∀i ∈ 1, 2, . . . , n (and truncate appropriately)
Equalization of marginal cost matches IFD equalization of suitability.
[ Conical cost combination with simplex constraint set has simple
solution in dual space (i.e., solve for λ). ]
(Important?) Missed Abstract Connections
Introduction
Reflections
Faulty connections
Missed connections
Motivations
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
IFD (Fretwell 1972; Fretwell and Lucas 1969) ⇐⇒ Optimal power
dispatch (Bergen and Vittal 2000)
IFD as optimization problem (thought experiment):
maximize
n∑
i=1
∫ xi
0si(y) dy subject to
n∑
i=1
xi = N
Pareto maximization of (???) subject to conservation simplex.
Right-side-up IFD:
si(xi) = λ ∀i ∈ 1, 2, . . . , n (and truncate appropriately)
Distribute xi to equalize suitability.
[ Conical cost combination with simplex constraint set has simple
solution in dual space (i.e., solve for λ). ]
(Important?) Missed Abstract Connections
Introduction
Reflections
Faulty connections
Missed connections
Motivations
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
IFD (Fretwell 1972; Fretwell and Lucas 1969) ⇐⇒ Optimal power
dispatch (Bergen and Vittal 2000)
IFD as optimization problem (thought experiment):
maximize
n∑
i=1
Gi(xi) subject to
n∑
i=1
xi = N
Pareto maximization of gain(?) subject to conservation simplex.
Right-side-up IFD:
dGi(xi)
dxi= λ ∀i ∈ 1, 2, . . . , n (and truncate appropriately)
Distribute xi to equalize marginal gain.
[ Conical cost combination with simplex constraint set has simple
solution in dual space (i.e., solve for λ). ]
(Important?) Missed Abstract Connections
Introduction
Reflections
Faulty connections
Missed connections
Motivations
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
IFD (Fretwell 1972; Fretwell and Lucas 1969) ⇐⇒ Optimal power
dispatch (Bergen and Vittal 2000)
IFD as optimization problem (thought experiment):
maximize
n∑
i=1
Gi(ti) subject to
n∑
i=1
ti = T
Maximization of distributed gain subject to limited time inside patch.
Right-side-up IFD:
dGi(ti)
dti= λ ∀i ∈ 1, 2, . . . , n (and truncate appropriately)
Distribute ti to equalize marginal gain.
[ Conical cost combination with simplex constraint set has simple
solution in dual space (i.e., solve for λ). ]
(Important?) Missed Abstract Connections
Introduction
Reflections
Faulty connections
Missed connections
Motivations
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
IFD (Fretwell 1972; Fretwell and Lucas 1969) ⇐⇒ Optimal power
dispatch (Bergen and Vittal 2000)
Risk-sensitive foraging (Stephens and Charnov 1982; Stephens and
Krebs 1986) ⇐⇒ Sharpe ratio/MPT (Sharpe 1966, 1994)
(Important?) Missed Abstract Connections
Introduction
Reflections
Faulty connections
Missed connections
Motivations
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
IFD (Fretwell 1972; Fretwell and Lucas 1969) ⇐⇒ Optimal power
dispatch (Bergen and Vittal 2000)
Risk-sensitive foraging (Stephens and Charnov 1982; Stephens and
Krebs 1986) ⇐⇒ Sharpe ratio/MPT (Sharpe 1966, 1994)
Sharpe (Nobel prize, Economics, 1990) ratio:
E(R)−Rf
σ
(Important?) Missed Abstract Connections
Introduction
Reflections
Faulty connections
Missed connections
Motivations
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
IFD (Fretwell 1972; Fretwell and Lucas 1969) ⇐⇒ Optimal power
dispatch (Bergen and Vittal 2000)
Risk-sensitive foraging (Stephens and Charnov 1982; Stephens and
Krebs 1986) ⇐⇒ Sharpe ratio/MPT (Sharpe 1966, 1994)
Sharpe (Nobel prize, Economics, 1990) ratio:
E(R)−Rf
σ
Exactly the Z-score ranking method of risk-sensitive foraging theory.
(Important?) Missed Abstract Connections
Introduction
Reflections
Faulty connections
Missed connections
Motivations
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
IFD (Fretwell 1972; Fretwell and Lucas 1969) ⇐⇒ Optimal power
dispatch (Bergen and Vittal 2000)
Risk-sensitive foraging (Stephens and Charnov 1982; Stephens and
Krebs 1986) ⇐⇒ Sharpe ratio/MPT (Sharpe 1966, 1994)
Sharpe (Nobel prize, Economics, 1990) ratio:
E(R)−Rf
σ
Exactly the Z-score ranking method of risk-sensitive foraging theory.
MPT (then)→ PMPT (now) (stochastic dominance, Bawa 1982)
(Important?) Missed Abstract Connections
Introduction
Reflections
Faulty connections
Missed connections
Motivations
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
IFD (Fretwell 1972; Fretwell and Lucas 1969) ⇐⇒ Optimal power
dispatch (Bergen and Vittal 2000)
Risk-sensitive foraging (Stephens and Charnov 1982; Stephens and
Krebs 1986) ⇐⇒ Sharpe ratio/MPT (Sharpe 1966, 1994)
Iain Couzin’s (2000+) ⇐⇒ Bertsekas and Tsitsiklis (1997−) (e.g.,
mysterious torus shapes; symmetry; symmetry breaking)
The Tenth Muse
Introduction
Reflections
Faulty connections
Missed connections
Motivations
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Solitary foraging: from ecology toengineering and back
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Foraging theory for speed choice
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Nice homomorphism between solitary foragers and
autonomous vehicles (Andrews et al. 2004; Charnov
1973; Quijano et al. 2006; Stephens and Krebs 1986)
g − c
τ− 1
λ
J∗
−cscs
λ
Foraging theory for speed choice
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Nice homomorphism between solitary foragers and
autonomous vehicles (Andrews et al. 2004; Charnov
1973; Quijano et al. 2006; Stephens and Krebs 1986)
Fitness surrogate (e.g., calories, target value)
Diverse collection of targets
Opportunity cost: some should be ignored
Rate maximization for long runs
Target/task choice ⇐⇒ prey model
g − c
τ− 1
λ
J∗
−cscs
λ
Foraging theory for speed choice
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Nice homomorphism between solitary foragers and
autonomous vehicles (Andrews et al. 2004; Charnov
1973; Quijano et al. 2006; Stephens and Krebs 1986)
Vehicle speed choice is very similar to cryptic prey
problem described by Gendron and Staddon (1983)
Ceteris paribus, encounter rate increases with
search speed
Search cost increases with search speed
Detection mistakes may vary with speed
Non-trivial speed–prey choice coupling
Prey =⇒ speed =⇒ rate =⇒ prey
Bobwhite quail(Gendron andStaddon 1983)
Les Howard
Foraging theory for speed choice
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Nice homomorphism between solitary foragers and
autonomous vehicles (Andrews et al. 2004; Charnov
1973; Quijano et al. 2006; Stephens and Krebs 1986)
Vehicle speed choice is very similar to cryptic prey
problem described by Gendron and Staddon (1983)
Bobwhite quail(Gendron andStaddon 1983)
Les Howard
To match bobwhite quail observations, Gendron and Staddon
choose detection function P di (u) , (1− (u/umax)
Ki)1/Ki that
maps search speed u ∈ [0, umax] to detection probability P di for
tasks of type i with conspicuousness Ki ∈ [0,∞).
No analytical tractability
Chose n = 2 for simulation (1983)
P di is strange at bounds (1 and 0) u/umax
P di (u)
Ki
b
b
b
b
0.4
0.8
1.5
4
On-line prey–speed choice for n ∈ N
(Pavlic 2007; Pavlic and Passino 2009)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Autonomous vehicle faces n-way merged Poisson process
λi: encounter rate for task of type i
(gi, ti): average (value, time) for processing task of type i
pi: probability that task of type i is processed (decision)
cs: cost per-unit-time of searching
On-line prey–speed choice for n ∈ N
(Pavlic 2007; Pavlic and Passino 2009)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Autonomous vehicle faces n-way merged Poisson process
λi: encounter rate for task of type i
(gi, ti): average (value, time) for processing task of type i
pi: probability that task of type i is processed (decision)
cs: cost per-unit-time of searching
Vehicle goes through cycles of searching and processing
G: average per-encounter gain
T : average per-encounter search and processing time
G(t): Markov renewal–reward process for accumulated gain
On-line prey–speed choice for n ∈ N
(Pavlic 2007; Pavlic and Passino 2009)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Autonomous vehicle faces n-way merged Poisson process
λi: encounter rate for task of type i
(gi, ti): average (value, time) for processing task of type i
pi: probability that task of type i is processed (decision)
cs: cost per-unit-time of searching
Long runtime =⇒ maximize rate of return
aslimt→∞
G(t)
t=
G
T=
−cs +n∑
i=1λipigi
1 +n∑
i=1λipiti
, R(p)
As expected, type-II functional response (Holling’s disk equation
without any sandpaper disks).
On-line prey–speed choice for n ∈ N
(Pavlic 2007; Pavlic and Passino 2009)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Autonomous vehicle faces n-way merged Poisson process
λi: encounter rate for task of type i
(gi, ti): average (value, time) for processing task of type i
pi: probability that task of type i is processed (decision)
cs: cost per-unit-time of searching
In general, pi ∈ [0, 1], but
∂R(p)
∂pi=
λigi
(
1 +n∑
j=1λjpjtj
)
− λiti
(
−cs +n∑
j=1λjpjgj
)
(
1 +n∑
i=1λipiti
)2
On-line prey–speed choice for n ∈ N
(Pavlic 2007; Pavlic and Passino 2009)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Autonomous vehicle faces n-way merged Poisson process
λi: encounter rate for task of type i
(gi, ti): average (value, time) for processing task of type i
pi: probability that task of type i is processed (decision)
cs: cost per-unit-time of searching
So KKT reveals optimization is purely O(2n) combinatorial
∂R(p)
∂pi=
λigi
(
1 +n∑
j=1j 6=i
λjpjtj
)
− λiti
(
−cs +n∑
j=1j 6=i
λjpjgj
)
(
1 +n∑
i=1λipiti
)2
So-called zero–one rule because p∗i ∈ 0, 1
On-line prey–speed choice for n ∈ N
(Pavlic 2007; Pavlic and Passino 2009)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Autonomous vehicle faces n-way merged Poisson process
λi: encounter rate for task of type i
(gi, ti): average (value, time) for processing task of type i
pi: probability that task of type i is processed (decision)
cs: cost per-unit-time of searching
So KKT reveals optimization is purely O(2n) combinatorial
∂R(p)
∂pi=
λigi
(
1 +n∑
j=1j 6=i
λjpjtj
)
− λiti
(
−cs +n∑
j=1j 6=i
λjpjgj
)
(
1 +n∑
i=1λipiti
)2
So-called zero–one rule because p∗i ∈ 0, 1
[ Sufficient condition; not necessary ]
On-line prey–speed choice for n ∈ N
(Pavlic 2007; Pavlic and Passino 2009)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Autonomous vehicle faces n-way merged Poisson process
λi: encounter rate for task of type i
(gi, ti): average (value, time) for processing task of type i
pi: probability that task of type i is processed (decision)
cs: cost per-unit-time of searching
Classical prey ranking refines search from O(2n) to O(n+ 1)
Processed types (p∗i = 1)︷ ︸︸ ︷g1t1
>g2t2
> . . . >gk∗
tk∗>
Optimal rate R(p∗)︷ ︸︸ ︷
−cs +k∗∑
i=1λigi
1 +k∗∑
i=1λiti
>
Ignored types (p∗i = 0)︷ ︸︸ ︷gk∗+1
tk∗+1> . . . >
gntn
where optimal p∗i = [i ≤ k∗] with k∗ ∈ 0, 1, . . . , n
On-line prey–speed choice for n ∈ N
(Pavlic 2007; Pavlic and Passino 2009)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Autonomous vehicle faces n-way merged Poisson process
λi: encounter rate for task of type i
(gi, ti): average (value, time) for processing task of type i
pi: probability that task of type i is processed (decision)
cs: cost per-unit-time of searching
Classical prey ranking does not depend on λ (i.e., speed)
Processed types (p∗i = 1)︷ ︸︸ ︷g1t1
>g2t2
> . . . >gk∗
tk∗>
Optimal rate R(p∗)︷ ︸︸ ︷
−cs +k∗∑
i=1λigi
1 +k∗∑
i=1λiti
>
Ignored types (p∗i = 0)︷ ︸︸ ︷gk∗+1
tk∗+1> . . . >
gntn
where optimal p∗i = [i ≤ k∗] with k∗ ∈ 0, 1, . . . , n
On-line prey–speed choice for n ∈ N
Effects of speed
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Speed u ∈ [umin, umax] ⊂ [0,∞) influences each encounter rate
λi(u) = uDiPdi (u)
where Di is the linear density in the population
b
bb
b
On-line prey–speed choice for n ∈ N
Effects of speed
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Speed u ∈ [umin, umax] ⊂ [0,∞) influences each encounter rate
λi(u) = uDiPdi (u)
where Di is the linear density in the population
Detection function is linear interpolation of probability bounds
0
1
u
P di (u)
b
bb
b
umin umax
high
low
P di (u) = P ℓ
i u+ P ai
On-line prey–speed choice for n ∈ N
Effects of speed
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Speed u ∈ [umin, umax] ⊂ [0,∞) influences each encounter rate
λi(u) = uDiPdi (u)
where Di is the linear density in the population
Detection function is linear interpolation of probability bounds
0
1
u
P di (u)
b
bb
b
umin umax
high
low
P di (u) = P ℓ
i u+ P ai
Search cost is also assumed to be affine function
cs(u) = csℓu+ csa
On-line prey–speed choice for n ∈ N
Effects of speed
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Speed u ∈ [umin, umax] ⊂ [0,∞) influences each encounter rate
λi(u) = uDiPdi (u)
where Di is the linear density in the population
Detection function is linear interpolation of probability bounds
0
1
u
P di (u)
b
bb
b
umin umax
high
low
P di (u) = P ℓ
i u+ P ai
Search cost is also assumed to be affine function
ci(u) = cℓiu+ cai
[ Processing costs can be modeled in a similar way ]
On-line prey–speed choice for n ∈ N
Effects of speed
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Speed u ∈ [umin, umax] ⊂ [0,∞) influences each encounter rate
λi(u) = uDiPdi (u)
where Di is the linear density in the population
Detection function is linear interpolation of probability bounds
0
1
u
P di (u)
b
bb
b
umin umax
high
low
P di (u) = P ℓ
i u+ P ai
Search cost is also assumed to be affine function
cs(u) = csℓu+ csa
[ Processing costs can be modeled in a similar way ]
[ . . . but not here. ]
On-line prey–speed choice for n ∈ N
Value of speed
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
After regrouping, new objective function
R(p, u) =G2(p)u
2 +G1(p)u+G0(q)
T2(p)u2 + T1(p)u+ 1
where coefficients
G2(p) ,n∑
i=1
DipigiPℓi
G1(p) ,
n∑
i=1
DipiPai gi − csℓ
G0(p) , −csa
T2(p) ,
n∑
i=1
pitiDiPℓi
T1(p) ,n∑
i=1
pitiDiPai
are constant with respect to u (i.e., biquadratic ratio)
On-line prey–speed choice for n ∈ N
Value of speed
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
After regrouping, new objective function
R(p, u) =G2(p)u
2 +G1(p)u+G0(q)
T2(p)u2 + T1(p)u+ 1
where coefficients
G2(p) ,n∑
i=1
DipigiPℓi
G1(p) ,
n∑
i=1
DipiPai gi − csℓ
G0(p) , −csa
T2(p) ,
n∑
i=1
pitiDiPℓi
T1(p) ,n∑
i=1
pitiDiPai
are constant with respect to u (i.e., biquadratic ratio)
Find optimal u∗ for each p∗ candidate (n+ 1 total)
On-line prey–speed choice for n ∈ N
Finding optimal speed
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Because biquadratic objective, for each p∗ candidate,
∂R(u)
∂u=
(G2T1 −G1T2)u2 + 2(G2 −G0T2)u+ (G1 −G0T1)(
T2u2 + T1u+ 1
)2
By KKT, if quadratic numerator root u∗ ∈ [umin, umax], then u∗ is
optimal speed; otherwise, optimal speed u∗ ∈ umin, umax based
on sign of numerator
On-line prey–speed choice for n ∈ N
Finding optimal speed
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Because biquadratic objective, for each p∗ candidate,
∂R(u)
∂u=
(G2T1 −G1T2)u2 + 2(G2 −G0T2)u+ (G1 −G0T1)(
T2u2 + T1u+ 1
)2
By KKT, if quadratic numerator root u∗ ∈ [umin, umax], then u∗ is
optimal speed; otherwise, optimal speed u∗ ∈ umin, umax based
on sign of numerator
Implement O(n+ 1) algorithm on-line if Di density estimates
available (Dubin’s car AAV simulations with speed filtering, Pavlic
and Passino 2009)
On-line prey–speed choice for n ∈ N
Finding optimal speed
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Because biquadratic objective, for each p∗ candidate,
∂R(u)
∂u=
(G2T1 −G1T2)u2 + 2(G2 −G0T2)u+ (G1 −G0T1)(
T2u2 + T1u+ 1
)2
By KKT, if quadratic numerator root u∗ ∈ [umin, umax], then u∗ is
optimal speed; otherwise, optimal speed u∗ ∈ umin, umax based
on sign of numerator
Implement O(n+ 1) algorithm on-line if Di density estimates
available (Dubin’s car AAV simulations with speed filtering, Pavlic
and Passino 2009)
Non-trivial to guarantee convergence of density estimates on-line
Estimation process =⇒ type-II type-III functional response
Estimation, impulsiveness, and the operant laboratory(Pavlic and Passino 2010c)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Laboratory impulsiveness (Ainslie 1974; Bateson and Kacelnik
1996; Bradshaw and Szabadi 1992; Green et al. 1981; McDiarmid
and Rilling 1965; Rachlin and Green 1972; Siegel and Rachlin 1995;
Snyderman 1983; Stephens and Anderson 2001)
Estimation, impulsiveness, and the operant laboratory(Pavlic and Passino 2010c)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Laboratory impulsiveness
Using starvation, animals are trained to use a Skinner box
Repeat mutually exclusive binary-choice trials (at low weight)
Estimation, impulsiveness, and the operant laboratory(Pavlic and Passino 2010c)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Laboratory impulsiveness
Using starvation, animals are trained to use a Skinner box
Repeat mutually exclusive binary-choice trials (at low weight)
What can be inferred about Skinner box results?
Usually assume simultaneous encounters occur with
probability zero (Poisson assumption)
Mutually exclusive choices when prey is immobile?
Patch impulsiveness vanishes (Stephens et al. 2004)
Attention (Monterosso and Ainslie 1999; Siegel and Rachlin 1995)
Estimation, impulsiveness, and the operant laboratory(Pavlic and Passino 2010c)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Laboratory impulsiveness
Using starvation, animals are trained to use a Skinner box
Repeat mutually exclusive binary-choice trials (at low weight)
What can be inferred about Skinner box results?
Usually assume simultaneous encounters occur with
probability zero (Poisson assumption)
Mutually exclusive choices when prey is immobile?
Patch impulsiveness vanishes (Stephens et al. 2004)
Attention (Monterosso and Ainslie 1999; Siegel and Rachlin 1995)
Worst-case scenario for a robot
Predisposes robots to underestimate
Estimation, impulsiveness, and the operant laboratory(Pavlic and Passino 2010c)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Laboratory impulsiveness
Using starvation, animals are trained to use a Skinner box
Repeat mutually exclusive binary-choice trials (at low weight)
What can be inferred about Skinner box results?
Usually assume simultaneous encounters occur with
probability zero (Poisson assumption)
Mutually exclusive choices when prey is immobile?
Patch impulsiveness vanishes (Stephens et al. 2004)
Attention (Monterosso and Ainslie 1999; Siegel and Rachlin 1995)
Worst-case scenario for an animal?
Predisposes animals to underestimate?
Estimation, impulsiveness, and the operant laboratory(Pavlic and Passino 2010c)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Estimation of per-type densities only necessary for speed regulation
Estimation, impulsiveness, and the operant laboratory(Pavlic and Passino 2010c)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Estimation of per-type densities only necessary for speed regulation
Graphical description of optimal prey choice:
Processing time
Processinggain
For type i: or @ (processing time ti, gain g(ti))
(t1, g1)
(t2, g2)
(t3, g3)
(t4, g4)
(t5, g5)
Process
Ignore
R∗
Estimation, impulsiveness, and the operant laboratory(Pavlic and Passino 2010c)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Estimation of per-type densities only necessary for speed regulation
Graphical description of optimal prey choice:
Processing time
Processinggain
For type i: or @ (processing time ti, gain g(ti))
(t1, g1)
(t2, g2)
(t3, g3)
(t4, g4)
(t5, g5)
Process
Ignore
R∗
=⇒
t: Total search and processing time
G(t):
Accumulatednet
gain
Type-# encounter: # (Process) or # (Ignore)
g1
t1
g2
t2
g3
t3
g4
t4
g5
t5
R∗
1
2
2
1
3 3
4 5
Process encounter k when gi(k)/ti(k) > G(t(k))/t(k)
Estimation, impulsiveness, and the operant laboratory(Pavlic and Passino 2010c)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Estimation of per-type densities only necessary for speed regulation
Graphical description of optimal prey choice:
Processing time
Processinggain
For type i: or @ (processing time ti, gain g(ti))
(t1, g1)
(t2, g2)
(t3, g3)
(t4, g4)
(t5, g5)
Process
Ignore
R∗
=⇒
Time
Accumulatednet
gain
g1
t1
g2
t2
R∗
Process encounter k when gi(k)/ti(k) > G(t(k))/t(k)
Rule (even with mistakes) is optimal facing Poisson encounters (i.e.,
simultaneous w.p.0)
Estimation, impulsiveness, and the operant laboratory(Pavlic and Passino 2010c)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Estimation of per-type densities only necessary for speed regulation
Graphical description of optimal prey choice:
Processing time
Processinggain
For type i: or @ (processing time ti, gain g(ti))
(t1, g1)
(t2, g2)
(t3, g3)
(t4, g4)
(t5, g5)
Process
Ignore
R∗
=⇒
Time
Accumulatednet
gain
g1
t1
g2
t2
R∗
Process encounter k when gi(k)/ti(k) > G(t(k))/t(k)
Rule (even with mistakes) is optimal facing Poisson encounters (i.e.,
simultaneous w.p.0)
Digestive rate constraints (bi: prey bulk) (Hirakawa 1995):
n∑
i=1λipibi
1 +n∑
i=1λipiti
≤ BKKT=⇒
p∗1 = 1
...
p∗k∗−1 = 1
p∗k∗ ∈ [0, 1]
Partial Preferences(rank by gi/bi)
Digression
Estimation, impulsiveness, and the operant laboratory(Pavlic and Passino 2010c)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Estimation of per-type densities only necessary for speed regulation
Graphical description of optimal prey choice:
Processing time
Processinggain
For type i: or @ (processing time ti, gain g(ti))
(t1, g1)
(t2, g2)
(t3, g3)
(t4, g4)
(t5, g5)
Process
Ignore
R∗
=⇒
Time
Accumulatednet
gain
g1
t1
g2
t2
R∗
Process encounter k when gi(k)/ti(k) > G(t(k))/t(k)
Rule (even with mistakes) is optimal facing Poisson encounters (i.e.,
simultaneous w.p.0)
Ecological–physiological hybrid method (Whelan and Brown 2005):
Asymptotic gut constraint ⇐⇒ Rank bygi
ti + tbi
Digression
Estimation, impulsiveness, and the operant laboratory(Pavlic and Passino 2010c)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Estimation of per-type densities only necessary for speed regulation
Graphical description of optimal prey choice:
Processing time
Processinggain
For type i: or @ (processing time ti, gain g(ti))
(t1, g1)
(t2, g2)
(t3, g3)
(t4, g4)
(t5, g5)
Process
Ignore
R∗
=⇒
Time
Accumulatednet
gain
g1
t1
g2
t2
R∗
Process encounter k when gi(k)/ti(k) > G(t(k))/t(k)
Rule (even with mistakes) is optimal facing Poisson encounters (i.e.,
simultaneous w.p.0)
Ecological–physiological hybrid method (Whelan and Brown 2005):
Asymptotic gut constraint ⇐⇒ Rank bygi
ti + tbi
Process encounter k when gi(k)/(ti(k) + tbi(k)) > G(t(k))/t(k)
t: Search and non-ballast handling time
G(t):
Accumulatednet
gain
Type-# encounter: # (Process) or # (Ignore)
g1
t1 + tb1
g2
t2 + tb2
2
2 1
2 2
1
2 2
2
∼
Handling and search time (no ballast time)Accumulatednet
gain
i : Digestive profitability line for type i
g1
t1 + tb1
1
2
3
4
g5
t5 + tb5
5
Digression
Estimation, impulsiveness, and the operant laboratory(Pavlic and Passino 2010c)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Estimation of per-type densities only necessary for speed regulation
Graphical description of optimal prey choice:
Processing time
Processinggain
For type i: or @ (processing time ti, gain g(ti))
(t1, g1)
(t2, g2)
(t3, g3)
(t4, g4)
(t5, g5)
Process
Ignore
R∗
=⇒
Time
Accumulatednet
gain
g1
t1
g2
t2
R∗
Process encounter k when gi(k)/ti(k) > G(t(k))/t(k)
Rule (even with mistakes) is optimal facing Poisson encounters (i.e.,
simultaneous w.p.0)
Ecological–physiological hybrid method (Whelan and Brown 2005):
Asymptotic gut constraint ⇐⇒ Rank bygi
ti + tbi
Process encounter k when gi(k)/(ti(k) + tbi(k)) > G(t(k))/t(k)
t: Search and non-ballast handling time
G(t):
Accumulatednet
gain
Type-# encounter: # (Process) or # (Ignore)
g1
t1 + tb1
g2
t2 + tb2
2
2 1
2 2
1
2 2
2
∼
0
0.25
0.50
0.75
1.00
0 1 2 3 4 5
Types (ordered by digestive profitability)Proportion
ofencounters
accepted
PartialPreferences
Digression
Estimation, impulsiveness, and the operant laboratory(Pavlic and Passino 2010c)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Estimation of per-type densities only necessary for speed regulation
Graphical description of optimal prey choice:
Processing time
Processinggain
For type i: or @ (processing time ti, gain g(ti))
(t1, g1)
(t2, g2)
(t3, g3)
(t4, g4)
(t5, g5)
Process
Ignore
R∗
=⇒
Time
Accumulatednet
gain
g1
t1
g2
t2
R∗
Process encounter k when gi(k)/ti(k) > G(t(k))/t(k)
Rule (even with mistakes) is optimal facing Poisson encounters (i.e.,
simultaneous w.p.0)
Ecological–physiological hybrid method (Whelan and Brown 2005):
Asymptotic gut constraint ⇐⇒ Rank bygi
ti + tbi
Process encounter k when gi(k)/(ti(k) + tbi(k)) > G(t(k))/t(k)
t: Search and non-ballast handling time
G(t):
Accumulatednet
gain
Type-# encounter: # (Process) or # (Ignore)
g1
t1 + tb1
g2
t2 + tb2
2
2 1
2 2
1
2 2
2
∼
Handling and search time (no ballast time)Accumulatednet
gain
i : Digestive profitability line for type i
g1
t1 + tb1
1
2
3
4
g5
t5 + tb5
5
SlidingMode
Digression
Estimation, impulsiveness, and the operant laboratory(Pavlic and Passino 2010c)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Estimation of per-type densities only necessary for speed regulation
Graphical description of optimal prey choice:
t: Total search and processing time
G(t):
Accumulatednet
gain
Type-# encounter: # (Process) or # (Ignore)
g1
t1
g2
t2
g3
t3
g4
t4
g5
t5
R∗
1
2
2
1
3 3
4 5
∼
Time
Accumulatednet
gain
g1
t1
g2
t2
R∗
Process encounter k when gi(k)/ti(k) > G(t(k))/t(k)
Attention: simultaneous encounter (w.p.0) =⇒ low time first
Estimation, impulsiveness, and the operant laboratory(Pavlic and Passino 2010c)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Estimation of per-type densities only necessary for speed regulation
Graphical description of optimal prey choice:
t: Total search and processing time
G(t):
Accumulatednet
gain
Type-# encounter: # (Process) or # (Ignore)
g1
t1
g2
t2
g3
t3
g4
t4
g5
t5
R∗
1
2
2
1
3 3
4 5
∼
Time
Accumulatednet
gain
g1
t1
g2
t2
R∗
R∗type-2-only
Process encounter k when gi(k)/ti(k) > G(t(k))/t(k)
Attention: simultaneous encounter (w.p.1) =⇒ low time first
Estimation, impulsiveness, and the operant laboratory(Pavlic and Passino 2010c)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Estimation of per-type densities only necessary for speed regulation
Graphical description of optimal prey choice:
t: Total search and processing time
G(t):
Accumulatednet
gain
Type-# encounter: # (Process) or # (Ignore)
g1
t1
g2
t2
g3
t3
g4
t4
g5
t5
R∗
1
2
2
1
3 3
4 5
∼
Time
Accumulatednet
gain
g1
t1
g2
t2
R∗
R∗depressed
Process encounter k when gi(k)/ti(k) > G(t(k))/t(k)
Attention: simultaneous encounter (w.p.1) =⇒ either first
Bifurcation; lucky runs accumulate high initial estimate
Estimation, impulsiveness, and the operant laboratory(Pavlic and Passino 2010c)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Estimation of per-type densities only necessary for speed regulation
Graphical description of optimal prey choice:
t: Total search and processing time
G(t):
Accumulatednet
gain
Type-# encounter: # (Process) or # (Ignore)
g1
t1
g2
t2
g3
t3
g4
t4
g5
t5
R∗
1
2
2
1
3 3
4 5
∼
Time
Accumulatednet
gain
gp
tp
g1
t1
g2
t2
R∗
R∗type-2-only
Process encounter k when gi(k)/ti(k) > G(t(k))/t(k)
Attention: simultaneous encounter (w.p.1) =⇒ low time first
Rescue optimality with early ad libitum feeding
Sunk costs and long patch residence times(Pavlic and Passino 2010b)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Nolet et al. (2001) are unable to explain spatial differences in tundra
swan foraging
Sunk costs and long patch residence times(Pavlic and Passino 2010b)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Nolet et al. (2001) are unable to explain spatial differences in tundra
swan foraging
In shallow water, swans feeding on tubers can “head dip”
In deep water, they must “up end,” which requires more energy
Nolet et al. find it strange that swans spend longer at the more
energetic task
Gemma Longman
Derek Tsang
Sunk costs and long patch residence times(Pavlic and Passino 2010b)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Nolet et al. (2001) are unable to explain spatial differences in tundra
swan foraging
In shallow water, swans feeding on tubers can “head dip”
In deep water, they must “up end,” which requires more energy
Nolet et al. find it strange that swans spend longer at the more
energetic task
Other sunk cost/Concorde effects (Arkes and Blumer 1985;
Arkes and Ayton 1999; Dawkins and Carlisle 1976; Kanodia
et al. 1989; Staw 1981)
Sunk costs and long patch residence times(Pavlic and Passino 2010b)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Nolet et al. (2001) are unable to explain spatial differences in tundra
swan foraging
In shallow water, swans feeding on tubers can “head dip”
In deep water, they must “up end,” which requires more energy
Nolet et al. find it strange that swans spend longer at the more
energetic task
Other sunk cost/Concorde effects (Arkes and Blumer 1985;
Arkes and Ayton 1999; Dawkins and Carlisle 1976; Kanodia
et al. 1989; Staw 1981)
Observations consistent with rate maximization when patch entry
costs are modeled
Sunk costs and long patch residence times(Pavlic and Passino 2010b)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Nolet et al. (2001) are unable to explain spatial differences in tundra
swan foraging
Observations consistent with rate maximization when patch entry
costs are modeled. For n = 1,
R(t1) =g1(t1)1λ1
+ t1where a < b < c , g1(0) < 0
g1(t1)
t1−a− 1
λ1
R∗a
t∗1a
Due to entry costs, searching is a less desirable task
Sunk costs and long patch residence times(Pavlic and Passino 2010b)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Nolet et al. (2001) are unable to explain spatial differences in tundra
swan foraging
Observations consistent with rate maximization when patch entry
costs are modeled. For n = 1,
R(t1) =g1(t1)1λ1
+ t1where a < b < c , g1(0) < 0
g1(t1)
t1−a
−b
− 1λ1
R∗b
t∗1bt∗1a
Due to entry costs, searching is a less desirable task
Sunk costs and long patch residence times(Pavlic and Passino 2010b)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Nolet et al. (2001) are unable to explain spatial differences in tundra
swan foraging
Observations consistent with rate maximization when patch entry
costs are modeled. For n = 1,
R(t1) =g1(t1)1λ1
+ t1where a < b < c , g1(0) < 0
g1(t1)
t1
−b
−c
− 1λ1
R∗c
t∗1ct∗1b
Due to entry costs, searching is a less desirable task
Sunk costs and long patch residence times(Pavlic and Passino 2010b)
Introduction
Solitary foraging: fromecology to engineeringand back
Speed choice (→)
Impulsiveness andoperant conditioning(←)
Long patch residencetimes (←)
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Nolet et al. (2001) are unable to explain spatial differences in tundra
swan foraging
Observations consistent with rate maximization when patch entry
costs are modeled. For n = 1,
R(t1) =g1(t1)1λ1
+ t1where a < b < c , g1(0) < 0
g1(t1)
t1
−b
−c
− 1λ1
R∗c
t∗1ct∗1b
Due to entry costs, searching is a less desirable task
May explain overstaying as well (Nonacs 2001)
Cooperative task processing
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
Cooperation for distributed decentralized networks
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
Cooperative control usually involves coordination of agents on
(possibly ad hoc) networks
e.g., Global utility functions to maximize
e.g., Projections onto non-separable spaces (i.e., not
Cartesian products)
Challenges to fast and cheap implementation
Cooperation for distributed decentralized networks
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
Cooperative control usually involves coordination of agents on
(possibly ad hoc) networks
Nash (i.e., competitive) equilibria are solutions to separable
variational inequality problems
Amenable to parallel solvers
Used by communication theorists on networks for congestion
control (Altman et al. 2005a,b; Buttyan and Hubaux 2003;
Shakkottai et al. 2006)
Strong connection to biological (and sociological) models of
emergent cooperation in nature
Cooperation for distributed decentralized networks
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
Cooperative control usually involves coordination of agents on
(possibly ad hoc) networks
Nash (i.e., competitive) equilibria are solutions to separable
variational inequality problems
Used in control to model unknown/unknowable
Typically used in control to model noise or enemy movements
(e.g., worst-case scenarios) or actions of humans in the system
Task conservation is a challenge to communication-like
application of Nash methods to task flow control
Cooperation for distributed decentralized networks
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
Cooperative control usually involves coordination of agents on
(possibly ad hoc) networks
Nash (i.e., competitive) equilibria are solutions to separable
variational inequality problems
Used in control to model unknown/unknowable
Existing task-processing networks (TPN) (Cruz 1991; Perkins and
Kumar 1989) focus on robustness, not optimality:
Flexible manufacturing system, network components =⇒bounded queues/burstiness
Behaviors are static (i.e., no feedback)
Cooperation for distributed decentralized networks
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
Cooperative control usually involves coordination of agents on
(possibly ad hoc) networks
Nash (i.e., competitive) equilibria are solutions to separable
variational inequality problems
Used in control to model unknown/unknowable
Existing task-processing networks (TPN) (Cruz 1991; Perkins and
Kumar 1989) focus on robustness, not optimality:
So here, elements merged from communication, TPN, and possible
analogous systems in nature (e.g., Cooperative breeding, Hamilton
and Taborsky 2005)
Try to design system so that Nash equilibrium has
characteristics that are globally favorable
Definition(Pavlic and Passino 2010a)
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
A task-processing network is a directed graph:
A ⊂ N: Set of task-processing agents
P ⊆ (i, j) ∈ A2 : i 6= j: Directed arcs connecting distinct agents
Vi , j ∈ A : (j, i) ∈ P: Set of conveyors for each i ∈ A
Ci , j ∈ A : (i, j) ∈ P: Set of cooperators for each i ∈ A
V , j ∈ A : Cj 6= ∅: Set of all conveyors
C , i ∈ A : Vi 6= ∅: Set of all cooperators
Task flows at each agent:
Yi ⊂ N: Possibly empty set of task types that arrive at conveyor i ∈ A
λkj ∈ R>0 : Encounter rate of type-k tasks at agent j ∈ A (e.g., Poisson encounters)
πkj ∈ [0, 1]: Probability that conveyor j ∈ A advertises an incoming k-type task to its connected cooperators Cj
γi ∈ [0, 1]: Probability that cooperator i ∈ A volunteers for advertised task from one of its connected conveyors Vi(collected in γ)
TPN examples(Pavlic and Passino 2010a)
Engineering Serendipity Successes and New Investigations
Input streams (k ∈ Yj ⊆ 1, 2, 3):
Conveyors (j ∈ V = 1, 2): V3 = 1 1 V4 = 1, 2 2 V5 = 2
Cooperators (i ∈ C = 3, 4, 5): 3 C1 = 3, 4 4 C2 = 4, 5 5
Y1 = 1, 2 Y2 = 1, 2, 31 2 1 2 3
π11
π21
π12
π22
π32γ3 γ4 γ5
Rate λkj λ1
1 λ21 λ1
2 λ22 λ3
2
Send request @ πkj
Accept request @ γi
Taskarrivals
Nodes andprocessingrequests
Flexible manufacturing system (FMS)
TPN examples(Pavlic and Passino 2010a)
Engineering Serendipity Successes and New Investigations
Input streams (k ∈ Yj ⊆ 1, 2, 3):
Conveyors (j ∈ V = 1, 2): V3 = 1 1 V4 = 1, 2 2 V5 = 2
Cooperators (i ∈ C = 3, 4, 5): 3 C1 = 3, 4 4 C2 = 4, 5 5
Y1 = 1, 2 Y2 = 1, 2, 31 2 1 2 3
π11
π21
π12
π22
π32γ3 γ4 γ5
Rate λkj λ1
1 λ21 λ1
2 λ22 λ3
2
Send request @ πkj
Accept request @ γi
Taskarrivals
Nodes andprocessingrequests
Flexible manufacturing system (FMS)
Cooperative breeders?
TPN examples(Pavlic and Passino 2010a)
Engineering Serendipity Successes and New Investigations
b
b
×
×
×
×
×
×
×
×
×
×
× ×
×
×
×
×
×
×
×
×
×
×
×
×
×
××
×××
×
×
×
×
×
×
×
××
×
×
×
×
××
×
××
×
×
λ11
λ22 λ3
3
1
2 3
AAV patrol scenario
⇐⇒
b
b
1
2 3
πjj
γi
C = V = 1, 2, 3
Ci = Vi = 1, 2, 3 − i
Yj = j
1
2 3
λ11
λ22 λ3
3
AAV TPN
Cooperation gameMetrics of volunteering
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
Need to develop an agent-based metric of performance that
catalyzes cooperation
Cooperation gameMetrics of volunteering
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
Need to develop an agent-based metric of performance that
catalyzes cooperation
Following foraging example, define utility function Ui(γ) based on
rate of gain
Cooperation gameMetrics of volunteering
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
Need to develop an agent-based metric of performance that
catalyzes cooperation
Following foraging example, define utility function Ui(γ) based on
rate of gain
To simplify presentation of combinatorial volunteering analysis,
introduce SOBP and SOMS.
Cooperation gameMetrics of volunteering
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
To simplify presentation of combinatorial volunteering analysis,
introduce SOBP and SOMS.
I : finite index set
Ω , γii∈I : indexed family with γi ∈ [0, 1] for each i ∈ I
For g, h ∈ N and Γ ⊆ I ,
SOBPg(Γ) ,
|Γ|∑
ℓ=0
1
g + ℓ
∑
C⊆Γ|C|=ℓ
((∏
i∈C
γi
)(∏
k∈Γ−C
(1− γk)
))
SOMSh(Γ) ,
|Γ|∑
ℓ=0
(−1)ℓ1
h+ ℓ
∑
C⊆Γ|C|=ℓ
(∏
i∈C
γi
)
Several useful relationships between SOBP and SOMS.
Cooperation gameMetrics of volunteering
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
To simplify presentation of combinatorial volunteering analysis,
introduce SOBP and SOMS. For Γ ⊆ A,
SOBP1(i, k, ℓ − i)
= (1− γk)(1− γℓ) +1
2γk(1− γℓ) +
1
2γℓ(1− γk) +
1
3γkγℓ
(i.e., sum of binomial products)
For conveyor j ∈ V and cooperator i ∈ Cj = i, k, ℓ,SOBP1(i, k, ℓ − i) is probability that i is chosen to process
an advertised task from j ∈ Vi (given that it volunteered)
Cooperation gameMetrics of volunteering
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
To simplify presentation of combinatorial volunteering analysis,
introduce SOBP and SOMS. For Γ ⊆ A,
SOBP1(i, k, ℓ − i)
= (1− γk)(1− γℓ) +1
2γk(1− γℓ) +
1
2γℓ(1− γk) +
1
3γkγℓ
(i.e., sum of binomial products)
For conveyor j ∈ V and cooperator i ∈ Cj = i, k, ℓ,SOBP1(i, k, ℓ − i) is probability that i is chosen to process
an advertised task from j ∈ Vi (given that it volunteered)
SOMS gives curvature information about SOBP
Properties of SOMS and SOBP provide bounds for convergence
analysis (i.e., Lyapunov/non-deterministic set stability)
Cooperation gameAgent utility function – rate of gain
Engineering Serendipity Successes and New Investigations
For i ∈ C, the rate of gain
Ui(γ) ,
Conveyor part — constant with respect to γi︷ ︸︸ ︷
bi +
(
1 −∏
j∈Ci
(1 − γj)
)
︸ ︷︷ ︸
Pr(Volunteer from Ci|Advertisement from i)
ri + γi
∑
j∈Vi
(−
Pr(i awarded task from j|i volunteers)︷ ︸︸ ︷
SOBP1(Cj − i)cij)
︸ ︷︷ ︸
Cooperator part
Cooperation gameAgent utility function – rate of gain
Engineering Serendipity Successes and New Investigations
For i ∈ C, the rate of gain
Ui(γ) ,
Conveyor part — constant with respect to γi︷ ︸︸ ︷
bi +
(
1 −∏
j∈Ci
(1 − γj)
)
︸ ︷︷ ︸
Pr(Volunteer from Ci|Advertisement from i)
ri + γi
∑
j∈Vi
(−
Pr(i awarded task from j|i volunteers)︷ ︸︸ ︷
SOBP1(Cj − i)cij)
︸ ︷︷ ︸
Cooperator part
where
bi ,∑
k∈Yi
λki
(
bki − c
ki
)
ri ,∑
k∈Yi
λki π
ki
(
rki −
(
bki − c
ki
))
are the costs and benefits of local process-
ing on i ∈ V
Cooperation gameAgent utility function – rate of gain
Engineering Serendipity Successes and New Investigations
For i ∈ C, the rate of gain
Ui(γ) ,
Conveyor part — constant with respect to γi︷ ︸︸ ︷
bi +
(
1 −∏
j∈Ci
(1 − γj)
)
︸ ︷︷ ︸
Pr(Volunteer from Ci|Advertisement from i)
ri + γi
∑
j∈Vi
(−
Pr(i awarded task from j|i volunteers)︷ ︸︸ ︷
SOBP1(Cj − i)cij)
︸ ︷︷ ︸
Cooperator part
and
cij ,∑
k∈Yj
λkj π
kj c
kij
are the costs and benefits to i ∈ C for vol-
unteering for tasks exported from j ∈ Vi
Cooperation gameAgent utility function – rate of gain
Engineering Serendipity Successes and New Investigations
For i ∈ C, the rate of gain
Ui(γ) ,
Conveyor part — constant with respect to γi︷ ︸︸ ︷
bi +
(
1 −∏
j∈Ci
(1 − γj)
)
︸ ︷︷ ︸
Pr(Volunteer from Ci|Advertisement from i)
ri + γi
∑
j∈Vi
(−
Pr(i awarded task from j|i volunteers)︷ ︸︸ ︷
SOBP1(Cj − i)cij)
︸ ︷︷ ︸
Cooperator part
where
bi ,∑
k∈Yi
λki
(
bki − c
ki
)
ri ,∑
k∈Yi
λki π
ki
(
rki −
(
bki − c
ki
))
are the costs and benefits of local process-
ing on i ∈ V
and
cij ,∑
k∈Yj
λkj π
kj c
kij
are the costs and benefits to i ∈ C for vol-
unteering for tasks exported from j ∈ Vi
Cooperation gameAgent utility function – rate of gain
Engineering Serendipity Successes and New Investigations
For i ∈ C, the rate of gain
Ui(γ) ,
Conveyor part — constant with respect to γi︷ ︸︸ ︷
bi +
(
1 −∏
j∈Ci
(1 − γj)
)
︸ ︷︷ ︸
Pr(Volunteer from Ci|Advertisement from i)
ri −Qipi(Qi) + γi
∑
j∈Vi
(pij(Qj) −
Pr(i awarded task from j|i volunteers)︷ ︸︸ ︷
SOBP1(Cj − i)cij)
︸ ︷︷ ︸
Cooperator part — γi and Qj vary with γi
where
bi ,∑
k∈Yi
λki
(
bki − c
ki
)
ri ,∑
k∈Yi
λki π
ki
(
rki −
(
bki − c
ki
))
pi(Qi) ,∑
k∈Yi
λki π
ki p
ki (Qi)
are the costs and benefits of local process-
ing on i ∈ V
and
cij ,∑
k∈Yj
λkj π
kj c
kij
pij(Qj) ,∑
k∈Yj
λkj π
kj q
kijp
kj (Qj)
are the costs and benefits to i ∈ C for vol-
unteering for tasks exported from j ∈ Vi
Fictitious payment functions added as stabilizing controls (Qi ,∑
j∈Ciγj )
Cooperation gameAgent utility function – rate of gain
Engineering Serendipity Successes and New Investigations
For i ∈ C, the rate of gain
Ui(γ) ,
Conveyor part — constant with respect to γi︷ ︸︸ ︷
bi +
(
1 −∏
j∈Ci
(1 − γj)
)
︸ ︷︷ ︸
Pr(Volunteer from Ci|Advertisement from i)
ri −Qipi(Qi) + γi
∑
j∈Vi
(pij(Qj) −
Pr(i awarded task from j|i volunteers)︷ ︸︸ ︷
SOBP1(Cj − i)cij)
︸ ︷︷ ︸
Cooperator part — γi and Qj vary with γi
where
bi ,∑
k∈Yi
λki
(
bki − c
ki
)
ri ,∑
k∈Yi
λki π
ki
(
rki −
(
bki − c
ki
))
pi(Qi) ,∑
k∈Yi
λki π
ki p
ki (Qi)
are the costs and benefits of local process-
ing on i ∈ V
and
cij ,∑
k∈Yj
λkj π
kj c
kij
pij(Qj) ,∑
k∈Yj
λkj π
kj q
kijp
kj (Qj)
are the costs and benefits to i ∈ C for vol-
unteering for tasks exported from j ∈ Vi
Fictitious payment functions added as stabilizing controls (Qi ,∑
j∈Ciγj )
Cournot oligopolies on a graph
Nash equilibriumExistence, uniqueness, and asynchronous convergence
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
Natural choice for distributed variational inequality is local gradient
ascent
Nash equilibriumExistence, uniqueness, and asynchronous convergence
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
Natural choice for distributed variational inequality is local gradient
ascent
Asynchronous system is governed by difference inclusion (not
difference equation)
Nash equilibriumExistence, uniqueness, and asynchronous convergence
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
Natural choice for distributed variational inequality is local gradient
ascent
Asynchronous system is governed by difference inclusion (not
difference equation)
For set stability, sufficient to show synchronous system is a
contraction mapping
Also gives existence and uniqueness of Nash equilibrium
Nash equilibriumExistence, uniqueness, and asynchronous convergence
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
Natural choice for distributed variational inequality is local gradient
ascent
Asynchronous system is governed by difference inclusion (not
difference equation)
For set stability, sufficient to show synchronous system is a
contraction mapping
Also gives existence and uniqueness of Nash equilibrium
Because γ ∈ [0, 1]|C| comes from product topology of intervals,
must use block maximum norm (‖γ‖∞ , maxi∈C|γi|)
Nash equilibriumExistence, uniqueness, and asynchronous convergence
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
Natural choice for distributed variational inequality is local gradient
ascent
Asynchronous system is governed by difference inclusion (not
difference equation)
For set stability, sufficient to show synchronous system is a
contraction mapping
Also gives existence and uniqueness of Nash equilibrium
Because γ ∈ [0, 1]|C| comes from product topology of intervals,
must use block maximum norm (‖γ‖∞ , maxi∈C|γi|)
Procedure leads to constraints on payment functions and topology
Asynchronous convergence to Nash equilibriumPayment and topological constraints
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
Assume that (Payment and topological constraints):
Asynchronous convergence to Nash equilibriumPayment and topological constraints
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
Assume that (Payment and topological constraints):
1. For all i ∈ C and j ∈ Vi, pij is a stabilizing payment function
For k ∈ N, p′(Q) , dp(Q)/dQ < 0 for all Q ∈ [0, k]
For k ∈ N, p′′(Q) , d2p(Q)/dQ2 > 0 for all Q ∈ [0, k]
For k ∈ N, γp′′(Q) ≤ −p′(Q) for all Q ∈ [γ, k − (1− γ)]with γ ∈ [0, 1]
Asynchronous convergence to Nash equilibriumPayment and topological constraints
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
Q
pℓ(Q)
m > 0
b
b
0
b
0 1 2 k
−m
Q
pe(Q)
τ > 1
b
b
0
A
0 1 2 k
−Aτ
Q
ph(Q)
κ > 0
ε > κ+ 1
b
b
0
A
0 1 2 k
−Aκεκ
Q
pp(Q)
p > 1
q0 > k + p− 1
b
b
0
A
0 1 2 k
−Apq0
Sample stabilizing payment (inverse demand) functions
Assume that (Payment and topological constraints):
1. For all i ∈ C and j ∈ Vi, pij is a stabilizing payment function
Asynchronous convergence to Nash equilibriumPayment and topological constraints
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
Q
pℓ(Q)
m > 0
b
b
0
b
0 1 2 k
−m
Q
pe(Q)
τ > 1
b
b
0
A
0 1 2 k
−Aτ
Q
ph(Q)
κ > 0
ε > κ+ 1
b
b
0
A
0 1 2 k
−Aκεκ
Q
pp(Q)
p > 1
q0 > k + p− 1
b
b
0
A
0 1 2 k
−Apq0
Sample stabilizing payment (inverse demand) functions
Assume that (Payment and topological constraints):
1. For all i ∈ C and j ∈ Vi, pij is a stabilizing payment function
2. For all j ∈ V , |Cj| ≤ 3 (i.e., no conveyor can have more than 3
outgoing links to cooperators)
Asynchronous convergence to Nash equilibriumPayment and topological constraints
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
Q
pℓ(Q)
m > 0
b
b
0
b
0 1 2 k
−m
Q
pe(Q)
τ > 1
b
b
0
A
0 1 2 k
−Aτ
Q
ph(Q)
κ > 0
ε > κ+ 1
b
b
0
A
0 1 2 k
−Aκεκ
Q
pp(Q)
p > 1
q0 > k + p− 1
b
b
0
A
0 1 2 k
−Apq0
Sample stabilizing payment (inverse demand) functions
Assume that (Payment and topological constraints):
1. For all i ∈ C and j ∈ Vi, pij is a stabilizing payment function
2. For all j ∈ V , |Cj| ≤ 3 (i.e., no conveyor can have more than 3
outgoing links to cooperators)
3. For cooperator i ∈ C and j ∈ Vi, if j is a 3-conveyor (i.e.,
|Cj| = 3), then there must be some conveyor k ∈ Vi that is a
2-conveyor
Asynchronous convergence to Nash equilibriumOther example stable topologies
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
5
6
4
7
3
0
8 9
2
10
1
11
4
7
2
10
5
6
1
11
3
8 9
λ44
λ77
λ22
λ1010
λ55
λ66
λ11
λ1111
λ33
λ88 λ9
9
Rich yet stable task-processing network.
Asynchronous convergence to Nash equilibriumOther example stable topologies
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
5
6
4
7
3
0
8 9
2
10
1
11
4
7
2
10
5
6
1
11
3
8 9
λ44
λ77
λ22
λ1010
λ55
λ66
λ11
λ1111
λ33
λ88 λ9
9
Rich yet stable task-processing network.
“Pills” stabilize problematic areas by focussing attention
Asynchronous convergence to Nash equilibriumOther example stable topologies
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
5
6
4
7
3
0
8 9
2
10
1
11
4
7
2
10
5
6
1
11
3
8 9
λ44
λ77
λ22
λ1010
λ55
λ66
λ11
λ1111
λ33
λ88 λ9
9
Rich yet stable task-processing network.
“Pills” stabilize problematic areas by focussing attention
Future research direction: Stable network motifs
Asynchronous convergence to Nash equilibriumTotally asynchronous algorithm
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
Define T : [0, 1]n 7→ [0, 1]n by T (γ) , (T1(γ), T2(γ), . . . , Tn(γ)) where, foreach i ∈ C,
Ti(γ) , min1,max0, γi + σi∇iUi(γ)
(i.e., projected gradient ascent)
Asynchronous convergence to Nash equilibriumTotally asynchronous algorithm
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
Define T : [0, 1]n 7→ [0, 1]n by T (γ) , (T1(γ), T2(γ), . . . , Tn(γ)) where, foreach i ∈ C,
Ti(γ) , min1,max0, γi + σi∇iUi(γ)
(i.e., projected gradient ascent), where
1
σi
≥ 2|Vi|maxk∈Vi
|p′ik(0)|
for all γ ∈ [0, 1]n.
Asynchronous convergence to Nash equilibriumTotally asynchronous algorithm
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
Define T : [0, 1]n 7→ [0, 1]n by T (γ) , (T1(γ), T2(γ), . . . , Tn(γ)) where, foreach i ∈ C,
Ti(γ) , min1,max0, γi + σi∇iUi(γ)
(i.e., projected gradient ascent), where
1
σi
≥ 2|Vi|maxk∈Vi
|p′ik(0)|
for all γ ∈ [0, 1]n. If
minj∈Vi
|p′ij (|Cj |) | >
(
|Vi| −1
2
)
maxj∈Vi
|cij |, for all i ∈ C,
then the totally asynchronous distributed iteration (TADI) sequence γ(t)generated with mapping T and the outdated estimate sequence γi(t) for alli ∈ C each converge to the unique Nash equilibrium of the cooperation game.
Asynchronous convergence to Nash equilibriumTotally asynchronous algorithm
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
Define T : [0, 1]n 7→ [0, 1]n by T (γ) , (T1(γ), T2(γ), . . . , Tn(γ)) where, foreach i ∈ C,
Ti(γ) , min1,max0, γi + σi∇iUi(γ)
(i.e., projected gradient ascent), where
1
σi
≥ 2|Vi|maxk∈Vi
|p′ik(0)|
for all γ ∈ [0, 1]n. If (∝ Hamilton’s rule on networks)
Benefit︷ ︸︸ ︷
minj∈Vi
|p′ij (|Cj |) | >
Relatedness︷ ︸︸ ︷(
|Vi| −1
2
) Cost︷ ︸︸ ︷
maxj∈Vi
|cij |, for all i ∈ C,
then the totally asynchronous distributed iteration (TADI) sequence γ(t)generated with mapping T and the outdated estimate sequence γi(t) for alli ∈ C each converge to the unique Nash equilibrium of the cooperation game.
Asynchronous convergence to Nash equilibriumResults: cooperation by cyclic feedback
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
λ2 (encounters per second)
Optimal
coop
erationwillingn
ess
γ∗ ifori∈1,2,3
A B C
0
1
0 1 2 3 4
λ1 = 0.6
λ1
λ3 = 1.7
λ3
ba
b
bb
c
γ∗1
γ∗3
γ ∗2
A
λ2 < λ1 < λ3
γ∗2 > γ∗
1 > γ∗3
B
λ1 < λ2 < λ3
γ∗1 > γ∗
2 > γ∗3
C
λ1 < λ3 < λ2
γ∗1 > γ∗
3 > γ∗2
Simulation of AAV patrol scenario
Converges to predicted Nash equilibrium
Asynchronous convergence to Nash equilibriumResults: cooperation by cyclic feedback
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
λ2 (encounters per second)
Optimal
coop
erationwillingn
ess
γ∗ ifori∈1,2,3
A B C
0
1
0 1 2 3 4
λ1 = 0.6
λ1
λ3 = 1.7
λ3
ba
b
bb
c
γ∗1
γ∗3
γ ∗2
A
λ2 < λ1 < λ3
γ∗2 > γ∗
1 > γ∗3
B
λ1 < λ2 < λ3
γ∗1 > γ∗
2 > γ∗3
C
λ1 < λ3 < λ2
γ∗1 > γ∗
3 > γ∗2
Simulation of AAV patrol scenario
Converges to predicted Nash equilibrium
Increases in one encounter rate (e.g., λ2) cause equilibrium shift so
neighbors (e.g., 1 and 3) help more and agent (e.g., 2) helps less
Asynchronous convergence to Nash equilibriumResults: cooperation by cyclic feedback
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Background
Task-processingnetwork
Cooperation game
Asynchronousconvergence tocooperation
Results
Closing remarks
Engineering Serendipity Successes and New Investigations
λ2 (encounters per second)
Optimal
coop
erationwillingn
ess
γ∗ ifori∈1,2,3
A B C
0
1
0 1 2 3 4
λ1 = 0.6
λ1
λ3 = 1.7
λ3
ba
b
bb
c
γ∗1
γ∗3
γ ∗2
A
λ2 < λ1 < λ3
γ∗2 > γ∗
1 > γ∗3
B
λ1 < λ2 < λ3
γ∗1 > γ∗
2 > γ∗3
C
λ1 < λ3 < λ2
γ∗1 > γ∗
3 > γ∗2
Simulation of AAV patrol scenario
Converges to predicted Nash equilibrium
Increases in one encounter rate (e.g., λ2) cause equilibrium shift so
neighbors (e.g., 1 and 3) help more and agent (e.g., 2) helps less
Emergent cooperation due to cyclic feedback effects
Closing remarks
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Both biology and engineering are full of interesting complex systems
Closing remarks
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Both biology and engineering are full of interesting complex systems
Real-time implementations in one domain are intuitive and
cognitively simple behaviors in another
Closing remarks
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Both biology and engineering are full of interesting complex systems
Real-time implementations in one domain are intuitive and
cognitively simple behaviors in another
Homomorphisms are not always obvious and should not be
forced
Closing remarks
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Both biology and engineering are full of interesting complex systems
Real-time implementations in one domain are intuitive and
cognitively simple behaviors in another
Homomorphisms are not always obvious and should not be
forced
Unifying principles are more valuable than mimicry
Closing remarks
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Both biology and engineering are full of interesting complex systems
Real-time implementations in one domain are intuitive and
cognitively simple behaviors in another
Homomorphisms are not always obvious and should not be
forced
Unifying principles are more valuable than mimicry
Catalyze interdisciplinary collaboration
Closing remarks
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Both biology and engineering are full of interesting complex systems
Real-time implementations in one domain are intuitive and
cognitively simple behaviors in another
Homomorphisms are not always obvious and should not be
forced
Unifying principles are more valuable than mimicry
Catalyze interdisciplinary collaboration
Inject new ideas
Closing remarks
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
Both biology and engineering are full of interesting complex systems
Real-time implementations in one domain are intuitive and
cognitively simple behaviors in another
Homomorphisms are not always obvious and should not be
forced
Unifying principles are more valuable than mimicry
Catalyze interdisciplinary collaboration
Inject new ideas
Provides new avenues for careers after graduate school!
Thanks!
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Engineering Serendipity Successes and New Investigations
(bringing engineers and animals together)
Thank you!
Helpful People: Kevin Passino, Tom Waite, Ian Hamilton
Funding Sources:
Questions?
Further reading
Introduction
Solitary foraging: fromecology to engineeringand back
Cooperative taskprocessing
Closing remarks
Further reading
Engineering Serendipity Successes and New Investigations
Pavlic TP (2007) Optimal foraging theory revisited. Master’s thesis, The
Ohio State University, Columbus, OH. URL
http://www.ohiolink.edu/etd/view.cgi?acc_num=osu118193
Pavlic TP, Passino KM (2010a) Cooperative task processing. IEEE
Transactions on Automatic Control Submitted
Pavlic TP, Passino KM (2010b) The sunk-cost effect as an optimal
rate-maximizing behavior. Acta Biotheoretica Accepted pending
revisions
Pavlic TP, Passino KM (2010c) When rate maximization is impulsive.
Behavioral Ecology and Sociobiology doi:10.1007/s00265-010-0940-1,
in press