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http://www.unc.edu/~rhwiley
A Signal-Detection Equilibrium
R. Haven Wiley
Department of Biology
University of North Carolina
Chapel Hill, NC 27599-3280, USA
in the Evolution of Communication
07 08 22
Animal Behavior Society, August 19, 2008
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Reverberation and attenuation of acoustic signals
Alerting components in signals
Adaptations to minimize degradation
Adaptations to utilize degradation (ranging)
Limitations on transmission of information
Limitations on receivers’ choices
Partitioning of signal space
Noise and acoustic communication
Richards and Wiley 1980
Richards 1981(2), Whitehead 1989, Naguib 1995-1997, Wiley and Godard 1996
Richards 1981(1)
Wiley and Richards 1978, 1982, Wiley 1991
Schroeder and Wiley 1983, Godard 1993-94, Mackin 2005, Wiley 2005 … and others
Wollerman 1999, Wollerman and Wiley 2002(1)
Wollerman and Wiley 2002(2), Luther 2007
http://www.unc.edu/~rhwiley
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How should noise affect
the evolution of communication?
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Signal Detection Theory/Decision Theory
provides the tools for an answer
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receiver performance evolves
noise and errors -- pervasive
signals evolve -- current view
signalers evolve -- signal-detection view
signaler-receiver equilibrium
history
news
new view of the evolution of communication
addendum (1) for what a signal is
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receiver performance evolves
noise and errors -- pervasive
signals evolve -- current view
signalers evolve -- signal-detection view
signaler-receiver equilibrium
history
news
new view of the evolution of communication
addendum (1) for what a signal is
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noise is pervasive
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CD -- correct detection (signal, response)
MD -- missed detection (signal but no response)
FA -- false alarm (response but no signal)
CR -- correct rejection (no signal, no response)
receiver's behavior has four possible outcomes . . .
when signals come with noise (all real communication) . . .
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receptors do not completely separate signals from noise
p rob
abili
ty
receptor activitylow high
N N+S
01
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only option is to set a criterion (threshold) for response
prob
abili
ty
receptor activitylow high
threshold for responseN N+S
01 receiver’s
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prob
abili
ty
receptor activitylow high
threshold for responseN N+S
01 receiver’s
receiver's threshold has in 4 possible outcomes
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N
prob
abili
ty
receptor activitylow high
N + S
threshold
P(CD|signal)
each outcome has a probability set by the threshold and S/N
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receiver's decision
signal
present
absent
response no response
CORRECTDETECTION
MISSEDDETECTION
FALSEALARM
CORRECTREJECTION
the 4 outcomes are mutually exclusive and exhaustive
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receiver's decision
signal
present
absent
response no response
CORRECTDETECTION
MISSEDDETECTION
FALSEALARM
CORRECTREJECTION
2 of the possible outcomes are errors
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the 2 kinds of errors are not independent
receivers cannot simultaneously minimize MD and FA
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raising the threshold decreases FA, but increases MD
lowering the threshold has the opposite effects
prob
abili
ty
receptor activitylow high
threshold for response
N N+S
01
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the receiver faces an inevitable trade-off
N
Pro
ba
bili
ty
Receptor activityLow High
N + S
threshold for response
1
1
P(FA)
P(C
D)
00
ROC ...ReceiverOperatingCharacteristic
the ROC -- p(CD) = f( p(FA) ) -- describes this trade-off
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N
Pro
ba
bili
ty
Receptor activityLow High
N + S
threshold for response
1
1
P(FA)
P(C
D)
00
ROC ...ReceiverOperatingCharacteristic
receivers should evolve optimal thresholds
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utility (overall payoff) of a threshold for a receiver
(1) location of the threshold
(2) probability that a signal occurs
(3-6) payoffs for each of the four possible outcomes
depends on . . .
when receiver samples its input
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receiver's optimal threshold falls along a continuum
adaptive fastidiousness . . .
adaptive gullability . . .
decreases MD but allows more FA
low threshold for response . . .
high threshold for response . . .decreases FA but allows more MD
Wiley 1994, 2000, 2006
between two extremes . . .
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P(CD) = P(FA) + s(j - m) - j + U(1 - s) (j - a)
s(h - m)
P(FA)
P(CD)
00
if we let U (which affects the y-intercept) vary, we can findthe largest value of U possible for these conditions ...
E(U) = P(signal) * P(CD | signal) * U(CD) +
P(signal) * {1 - P(CD | signal)} * U(MD) +
{1 - P(signal)} * P(FA | no signal) * U(FA) +
{1 - P(signal)} * {1 - P(FA | signal)} * U(CR)
For any value of E(U) = U, we can rearrange this equation
to obtain an indifference curve ...
P(CD) = (1 - s) (j - a)
s(h - m)P(FA) + s(j - m) - j + U
s = P(signal)
h, m, a, j = U(CD), U(MD), U(FA), U(CR)
Wiley 1994, 2000, 2006
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(1 - s) (j - a)
s(h - m)
the slope is low when ...
s and h-m are high ... and j-a is low
for a receiver listening for true alarm calls
when some individuals occasionally give false alarm calls
h >> 0 (avoiding predation)
m << 0 (exposure to predation)
j > 0 and a < 0 (a little time gained or lost feeding)
(1 - s) (j - a)
s(h - m)
this value for the slope is high when ...
j-a is high ... and s and h-m are low
for a receiver searching for cryptic prey
or subtly discriminable objects such as optimal mates
j and m > 0 (prospects for future success devalued by low cost of additional search)
h > j and m (optimal mate or appropriate prey)
a < 0 (suboptimal mate or inappropriate prey)
when search costs are low ...
alarm calls
cryptic prey
or
optimal mates
Wiley 1994, 2000, 2006
MD is costly
FA is costly
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most previous treatments of receivers' errors . . .
just added variance to receivers' responses
prevailing paradigm for evolution of communication
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most previous treatments of receivers' errors . . .
just substitute average payoffs for fixed payoffs
just added variance to receivers' responses
the result . . .
prevailing paradigm for evolution of communication
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prevailing paradigm for evolution of communication
receivers "must get what they want" . . .
so signaling would not pay
receiver is assumed to benefit from responding (on average) . . .
and no communication would occur
otherwise receiver would evolve not to respond
Grafen 1990b
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Grafen (1990a,b) somewhat similar to Zahavi (1975)
reliable (on average) signaling occurs
if and only if
signals are costly and condition-dependent (revealing)
prevailing paradigm for evolution of communication
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Grafen (1990a,b) somewhat similar to Zahavi (1975)
reliable (on average) signaling occurs
if and only if
signals are costly and condition-dependent (revealing)
prevailing paradigm for evolution of communication
because each individual has
a condition-dependent optimal level of signaling
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prevailing paradigm for evolution of communication
benefit from response to signal (fecundity)
survival of signaler
survival * fecundity
optimal level of signaling
surv
ival
or
bene
fit f
rom
res
pons
e
signal size
sign
aler
's
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prevailing paradigm for evolution of communication
signal size
benefit from response to signal (fecundity)
survival of signaler
survival * fecundity
optimal levels of signaling
surv
ival
or
bene
fit f
rom
res
pons
e
sign
aler
's
individuals differ in
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receiver is largely out of the picture after a few initial assumptions
prevailing paradigm for evolution of communication
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signal-detection approach differs from the prevailing approach
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receivers' optimal performance depends on . . . noise . . . but also the signal’s properties . . .
signal-detection paradigm
which depend on the signaler’s effort
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receivers' optimal performance depends on . . . noise . . . but also the signal’s properties . . .
signaler’s optimal effort depends on . . .
which depends on receiver's performance
signal-detection paradigm
signal's cost . . . but also its benefit . . .
which depend on the signaler’s effort
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signalers evolve in response to behavior of receivers
receivers evolve in response to behavior of signalers
and
signal-detection paradigm
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is there a joint optimum in communication . . .
a signal-detection equilibrium?
signal-detection paradigm
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from receiver's perspective . . .
higher signal level favors higher optimal thresholds
because p(FA) decreases for any p(MD)
but as signal level increases . . .
higher thresholds have diminishing benefits
signal-detection equilibrium: basics
because p(CD) approaches 1.0
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from signaler's perspective . . .
marginal benefits . . . a decreasing function of effortmarginal costs . . . constant or increasing function of effort
these benefits have diminishing returns
higher optimal thresholds of intended receivers result in . . .
and consequently greater effort
also results in greater receiver-dependent benefit
higher signal level requires . . .
greater exaggeration of signals . . .
higher probability of responses by receivers
because p(CD) approaches 1.0
signal-detection equilibrium: basics
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if signalers adjust exaggeration of signals
and receivers adjust thresholds
result might be a signaler-receiver equilibrium
exaggeration of signals by signalers reaches an optimum
at which receiver performance also reaches an optimum
(based on current receivers' thresholds)
(based on current exaggeration of signals)
signal-detection equilibrium: basics
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signaler's optimal signal level snlevel* =
snlevel | max[ uS(
signal-detection equilibrium: doing the math
uS, signaler’s utility
sS, signaler’s survival
snlevel, signal level in relation to noise
th, receiver’s threshold (th*, optimal)
brS, signaler’s benefit from a response
ps, probability of a signal
th*(snlevel),
brS,
sS(snlevel),
ps) ]
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signaler's optimal signal level snlevel* =
snlevel | max[ uS(sS(snlevel), th*(snlevel), brS, ps) ]
receiver's optimal threshold th* =
th | max[ uR(p[outcomes](th), payoff[outcomes], snlevel*, ps) ]
signal-detection equilibrium: doing the math
uR, receiver’s utility
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signaler's optimal signal level snlevel* =
snlevel | max[ uS(sS(snlevel), th*(snlevel), brS, ps) ]
receiver's optimal threshold th* =
th | max[ uR(p[outcomes](th), payoff[outcomes], snlevel*, ps) ]
signal-detection equilibrium: doing the math
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signaler's optimal signal level snlevel* =
snlevel | max[ uS(sS(snlevel), th*(snlevel), brS, ps) ]
receiver's optimal threshold th* =
th | max[ uR(p[outcomes](th), payoff[outcomes], snlevel*, ps) ]
signal-detection equilibrium: doing the math
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define the receiver's and signaler's utilities
find their maxima
by finding the root of ∂(utility)/∂(signal level) = 0
signal-detection equilibrium: doing the math
in Mathematica . . .
across all possible signal levels
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uR = receiver's utility . . .
signal-detection equilibrium: doing the math
defined earlier !
psR = p(receiver encounters a signal)
receiver's payoffs
dR
mR
fR
rR
Gaussian PDF's for noise (N) and signal + noise (SN)mean | N = 0, sd | N = 1.0
mean | SN = signal level, sd | SN = sd | N = 1.0
some constants and functions . . .
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uR = receiver's utility …
signal-detection equilibrium: doing the math
defined earlier !
psR = p(receiver encounters a signal)
receiver's payoffs
dR
mR
fR
rR
normal PDF's for noise alone (N) and signal+noise (SN)mean | N = 0, sd | N = 1.0
mean | SN = signal level, sd | SN = sd | N = 1.0
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uR = receiver's utility …
signal-detection equilibrium: doing the math
defined earlier !
psR = p(receiver encounters a signal)
receiver's payoffs
dR
mR
fR
rR
normal PDF's for noise alone (N) and signal+noise (SN)mean | N = 0, sd | N = 1.0
mean | SN = signal level, sd | SN = sd | N = 1.0
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uR = receiver's utility …
signal-detection equilibrium: doing the math
defined earlier !
psR = p(receiver encounters a signal)
receiver's payoffs
dR
mR
fR
rR
normal PDF's for noise alone (N) and signal+noise (SN)mean | N = 0, sd | N = 1.0
mean | SN = signal level, sd | SN = sd | N = 1.0
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uR = receiver's utility …
signal-detection equilibrium: doing the math
defined earlier !
psR = p(receiver encounters a signal)
receiver's payoffs for mate-choice task . . .
dR = 2.0
mR = 0.9
fR = 0.1
rR = 1.0
false alarm is bad news!
normal PDF's for noise alone (N) and signal+noise (SN)mean | N = 0, sd | N = 1.0
mean | SN = signal level, sd | SN = sd | N = 1.0
see addendum (2) for an alarm-call task
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signaler's signal level
signaler’s effort
signal-detection equilibrium: doing the math
signal exaggeration
cost
survival
oc
oc
oc
oc
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signaler's signal level
signaler’s effort
signal-detection equilibrium: doing the math
signal exaggeration
cost
survival
oc
oc
oc
oc
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signaler's signal level
signaler’s effort
signal-detection equilibrium: doing the math
signal exaggeration
cost
survival
oc
oc
oc
oc
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survival relative to maximal survival (without signaling)
signal-detection equilibrium: doing the math
signaler's cost = sS / maxsS
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(signaling effort)
(signal exaggeration)
signal level
survival
intercept = maximal survival of signaler (maxsS)
slope = marginal cost of exaggeration (mcX)
Signal-Detection Equilibrium: Doing the Math
sS
signaler's
signaler's cost = sS / maxsS
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any response outcome counts . . .
CD, correct detection of signal by receiver
FA, response when signaler present . . . despite absence of signal !
probability(response by receiver)
benefit (fecundity) for signaler from a response
Signal-Detection Equilibrium: Doing the Math
signaler’s benefit from receiver brS =
X
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any response outcome counts . . .
CD, correct detection of signal by receiver
FA, response when signaler present . . . despite absence of signal !
benefit (fecundity) for signaler from a response
signal-detection equilibrium: doing the math
signaler’s benefit from receiver brS =
probability(response by receiver) X
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survivalrelativeSurvival = sS/maxsS
when signal occurs maxsS cancels
when signal does not occur there is no cost
(p[signal] + p[no signal])
psS pe sS brS (1 - psS) pe maxsS brS
(relativeSurvival = 1)
p[encounter] maxsS relativeSurvival brS =
signal-detection equilibrium: doing the math
signaler's utility uS =
* fecundity =
+
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to calculate signaler's utility uS
for a particular signal level snlevel . . .
first calculate optimal threshold for receiver th*
for this snlevel
then calculate utility for signaler uS
for this snlevel and this th*
signal-detection equilibrium: doing the math
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optimal utility for the signaler
across all possible signal levels
use FindRoot( )
for a set of conditions (constants) . . .
pdR, pmR, pfR, pjR
maxsS, mcX, brS (payoffs for signaler)
(payoffs for receiver)
psS, peR (signaling rate, encounter probability)
Gracias, Mathematica!
signal-detection equilibrium: doing the math
to calculate
∂ uS / ∂ snlevel
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a signaler has an
optimal level of signal exaggeration (effort, cost)
signal-detection equilibrium: doing the math
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signaler’s utility uSdepends on exaggeration (meanSN) and receiver’s threshold
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signaler’s
exaggeration (meanSN)
utility (uS)
threshold = 4
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exaggeration (meanSN)
threshold = 8
signaler’sutility (uS)
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optimal signal exaggeration
as a function of receiver’s threshold
threshold
optimalexaggeration
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receiver’s threshold
as a function of optimal signal exaggeration
optimal exaggeration
threshold
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thresholdoptimal
exaggeration
receiver’s optimal threshold
as a function of exaggeration
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threshold
exaggeration
signaler’s optimum
receiver’s optimum
signal detection equilibrium
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signal-detection equilibrium: doing the math
both receiver and signaler thus have
optima for performance
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perhaps a general conclusion . . .
a joint optimum
any adapting signaler-receiver system
in a particular situation
evolves toward
for signaler efficiency and receiver performance
signal-detection equilibrium: conclusions
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just as in costly condition-dependent signal CCDS theory,
signalers evolve an optimal effort
just as in signal-detection SD theory,
receivers evolve an optimal performance
signal-detection equilibrium: conclusions
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signals do not reach maximal efficiency
(further exaggeration would improve
responses to not reach maximal performance
(further exaggeration would reduce errors)
probability of intended response)
signal-detection equilibrium: conclusions
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evolution does not result in perfect communication
signalers will not
receivers will not
always produce signals with the intended effect
always avoid errors
signal-detection equilibrium: conclusions
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signalers are always subject to . . .
unresponsive and unintended receivers
receivers with different constraints on performance
can always evolve responses to exploit signalers
(eavesdroppers, rivals)
signal-detection equilibrium: conclusions
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receivers are always subject to deception
they do not always "get what they want"
signalers with different constraints on signaling
can always evolve signals to exploit receivers
deception is an exception to the rule of "continuity in everything"
signal-detection equilibrium: conclusions
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Krebs and Dawkins (1985) were correct ...
manipulation is an unavoidable consequence of communication
but reliability is the predominant feature of communication
signal-detection equilibrium: conclusions
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avoids some weaknesses of CCDS theory
(1) predictions that reliable signaling requirescosts for signalers and benefits for receivers
are not strong predictions
cost-free signals are difficult to imagine
all signals have some costs
otherwise responses cannot evolve by selectionall responses have benefits on average
Signal Detection Equilibrium theory
alternative hypotheses . . .
*communication is non-adaptive . . . evolves randomly . . . rather than by selection
are indistinguishable from the null hypothesis*
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(2) CCDS theory predicts misleading costs for signalers
because CCDS theory does not take into account
adaptive adjustments by receivers
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predicts optimal costs for signalers that take into account adaptive adjustments by receivers
Signal Detection Equilibrium theory
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predicts the amount and the direction of signal exaggeration
exaggeration of signals should
Signal Detection Equilibrium theory
increase performance of intended receivers
decrease performance of unintended receivers
because noise varies across environments
signals should adapt to the environment
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signals should evolve to minimize costsfor any increase in performance of intended receivers
signalers should evolve to maximize efficiency of signaling
increased costs are only incidental
Signal Detection Equilibrium theory
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focuses on efficiency of signals
efficiency of signaling =
focus solely on the costs of signals is misdirected
Signal Detection Equilibrium theory
( benefits from signaling * survival of signaler )
( benefits without signaling * survival without signaling )
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predicts possibilities for manipulation
by unresponsive and unintended receivers
(subject to different constraints on performance in comparison to intended receivers)
by deceptive signalers
(subject to different constraints on signaling
in comparison to preferred signalers)
Signal Detection Equilibrium theory
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evolution of communication is more complicated than
"costs of signals" and "benefits to receivers"
Signal Detection Equilibrium theory
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previously neglected variables include . . .
probabilities and payoffs of all four outcomes for receivers
probabilities of signals (when receiver is attending)
payoffs for signaling as a function of effort
probabilities of responses
by unintended receiversby intended receivers
from each category of receiver
all of these measures …in different environments
Signal Detection Equilibrium theory
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neglected variables include . . .
probabilities and payoffs of all four outcomes for receivers
probabilities of signals (when receiver is attending)
payoffs for signaling as a function of effort
probabilities of responses
by unintended receiversby intended receivers
from each category of receiver
all of these measures …in different environments
Signal Detection Equilibrium theory
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neglected variables include . . .
probabilities and payoffs of all four outcomes for receivers
probabilities of signals (when receiver is attending)
payoffs for signaling as a function of effort
probabilities of responses
by unintended receiversby intended receivers
from each category of receiver
all of these measures …in different environments
Signal Detection Equilibrium theory
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neglected variables include . . .
probabilities and payoffs of all four outcomes for receivers
probabilities of signals (when receiver is attending)
payoffs for signaling as a function of effort
probabilities of responses
by unintended receiversby intended receivers
from each category of receiver
all of these measures …in different environments
Signal Detection Equilibrium theory
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neglected variables include . . .
probabilities and payoffs of all four outcomes for receivers
probabilities of signals (when receiver is attending)
payoffs for signaling as a function of effort
probabilities of responses
by unintended receiversby intended receivers
from each category of receiver
all of these measures …in different environments
Signal Detection Equilibrium theory
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stay tuned !
mathematical exploration of the joint optima ofreceivers’ performance and signalers’ effort
has just begun!
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addendum (1)
what is a signal?
the essential feature of signalsdetermines the essential features of communication
Signal Detection Equilibrium theory
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a signal is a pattern of energy (or matter)
that elicits a response from a receiver
but does not provide all of the power for the response
Signal Detection Equilibrium theory
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a receiver is a mechanism that . . .
associates signals with responses
it requires . . . transducers, gates, amplifiers, effectors
for instance . . . receptors, cns, musculo-skeletal system
Signal Detection Equilibrium theory
because a signal does not provide all of the power for a response . . .
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receiver is in control -- at least of the response
but receiver is also exposedbecause receiver uses low-power signals for decisions
receiver is inevitably subject to constraints on detecting and discriminating signals in noise
. . . open to possibilities of signals from unexpected signalers
. . . inherently open to the possibility of deception
Signal Detection Equilibrium theory
because a signal does not provide all of the power for a response . . .
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signaler is also in control -- at least of the signal
but signaler is also exposed
signaler is inevitably subject to constraints
. . . open to possibilities of unresponsive/unintended receivers
. . . inherently open to the possibility of eavesdropping
because the signaler relies on receivers' power and decisions
on directing signals to intended receivers
Signal Detection Equilibrium theory
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no need for a distinction between signal and index
insufficient power of a signal is enough
no need for a specialized communicative function for a signal
predictable responses by a receiver are enough
definition of a signal applies to all communication (all signaling)
electronic, organismal, cellular, molecular
Signal Detection Equilibrium theory
nevertheless . . .
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evolving (living) signalers and receivershave a special property . . .
both signaling and receiving should evolve
to maximize the utility of each
caveat -- we should not expect all organisms to have reached an adaptive optimum
maladaptive (nonadaptive) behavior can result from
Signal Detection Equilibrium theory
conditions changing faster than adaptationgenetic driftmigration from populations in other environmentsgenetic or developmental constraints
(valleys in the adaptive landscape)
because SDE theory makes quantitative predications about the direction and level of exaggeration of signals
and the performance of receivers . . .the adaptedness of communication becomes an empirical question
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the biological/psychological questions are ...
how should signaling and receiving evolve
so signalers and receivers reach optimal performance?
what features should adapted communication have
because evolution of optimal performance is evolutionary adaptation …
in different environmental conditions?
Signal Detection Equilibrium theory
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addendum (2)
Signal Detection Equilibrium theory
optima for receiver’s performance and signaler’s effort
receiver's payoffs for alarm calls . . .
dR = 2.0
mR = 0.1
fR = 0.9
rR = 1.0
missed detection is bad news!
for alarm calls