Fuzzy Language in Action

Fuzzy Language in Action
Michael Franke
October 6, 2012
Main Idea
language use
constitutes
basic meaning
Main Idea
compositional semantic values
influences
language use
determines
constitutes
basic meaning
Main Points of Interest
1
optimal use of fuzzy language
2
vagueness
3
extreme-value use
4
use of borderline contradictions
Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Lewis-Style Signaling Games
ts ∈ T
m∈M
tr ∈ T
ts = tr
m
success
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Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Lewis-Style Signaling Games
ts ∈ T
m∈M
tr ∈ T
ts 6 = tr
m
failure
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Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Signaling Games for Similarity Maximizing
ts ∈ T
m∈M
tr ∈ T
success
∝
similarity(ts , tr )
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Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Signaling Games for Similarity Maximizing
success(ts, tr ) = 1 − |ts − tr |
(c.f. Jäger, 2007; Jäger and van Rooij, 2007)
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Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Utilities
Sim-Max Games
linear
Gaussian
9 / 37
Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Utilities
Lewis-Style Signaling Games
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Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Strategies
Sender
σ — | T | × | M | row-stochastic matrix
Receiver
ρ — | M | × | T | row-stochastic matrix
Example

0.2
0.5


σ = 0.1

0.0
1.0
| T | = 5 and | M | = 2

0.8
0.5


0.9

1.0
0.0
ρ=
0.1
0.5
0.8
0.1
0.1
0.0
0.0
0.3
0.0
0.1
!
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Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Example Simulation Run
12 / 37
Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Example Simulation Run
12 / 37
Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Example Simulation Run
12 / 37
Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Example Simulation Run
12 / 37
Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Example Simulation Run
12 / 37
Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Example Simulation Run
12 / 37
Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Example Simulation Run
12 / 37
Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Example Simulation Run
12 / 37
Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Example Simulation Run
12 / 37
Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Analytical Results
esss ⊂ “Voronoi languages”
• speaker strategy (≈ declarative meaning):
partitions states into cells
• receiver strategy (≈ imperative meaning):
“Bayesian estimators” of cells
Voronoi Language Example
(1 dimensional space, 2 messages)
R ( m1 )
|
•
{z
S ( m1 )
R ( m2 )
} |
•
{z
S ( m2 )
}
(Jäger et al., 2011)
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Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Solt and Gotzner (2012): Material
14 / 37
Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Solt and Gotzner (2012): Results
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Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Alxatib and Pelletier (2011): Material
x is tall
x is not tall
x is tall and not tall
x is neither tall nor not tall
True
False
Can’t tell
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Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Alxatib and Pelletier (2011): Results
P
not-P
both
neither
percentage true judgements
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
suspect
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Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Ripley (2011): Material
Disagree
• is near and it isn’t near .
• both is and isn’t near .
• neither is near nor isn’t near .
• neither is nor isn’t near .
Agree
1
2
3
4
5
6
7
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Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Ripley (2011): Results
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Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Shortcomings of Sim-Max Predictions
1
vagueness
2
extreme-value use
3
use of borderline contradictions
R ( m1 )
|
•
{z
S ( m1 )
R ( m2 )
} |
•
{z
S ( m2 )
}
20 / 37
Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Stochastic choice
Best Response

1

P(ai ) = | argk uk =maxj uj |
0
if ui = maxj uj
otherwise
Quantal Response
P(ai ) ∝ exp(λui )
Motivation
• real people are not perfect utility maximizers
• they make mistakes
sub-optimal choices
• still, high utility choices are more likely than low-utility ones
(Goeree et al., 2008)
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Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Quantal Response Equilibria in Sim-Max Games
6
1
·10−2
0.8
4
ρ(t | m)
σ(m | t)
0.6
0.4
2
0.2
0
0
0
0.2
0.4
0.6
t
0.8
1
0
0.2
0.4
0.6
0.8
1
t
(Franke et al., 2011)
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Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Where QRE goes wrong
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Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Noisy Best Response
Confusion Matrix
C — | T | × | T | row-stochastic matrix
Cij ∝ N (ti , sn ) is the probability of confusing j for i
Noisy Best Response
NBR(ρ) = C BR(ρ)
NBR(σ ) = BR(σ ) CT
Motivation
perceptual mistakes anti-proportional to similarity
perhaps also plausible: probability of suitable context
24 / 37
Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Noisy Response Equilibrium
25 / 37
Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Fuzzy Logical Semantics
Language L
• basic expressions {P} or {P, Q}
• degrees x ∈ R
• if A is a basic expression, then Ax is a formula
• if ϕ and ψ are formulas, then so are ¬ ϕ and ϕ ∧ ψ
Semantics
V (Ax) ∈ [0; 1]
V (¬ ϕ) = 1 − V ( ϕ)
V ( ϕ ∧ ψ) = min(V ( ϕ), V (ψ))
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Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Message Space
M ≈ {{ψ ∈ L | V ( ϕ) = V (ψ)} | ϕ ∈ L}
M1 = {P , ¬P , P ∧ ¬P}
M2 = {P , Q , ¬P , ¬Q , P ∧ Q , ¬P ∧ ¬Q , P ∧ ¬P , Q ∧ ¬Q}
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Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
The Dynamics
ρ0 =
0
1
...
...
1
0
!
σk = Extend(NBR(ρk−1 ))
ρk = NBR(σk )
Extend(sigma):
for i <= |M|:
if i is basic:
s_i = sigma_i
else:
s_i = normal fuzzy values based on use of
basic predicates as given in sigma
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Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Basic Results with P and Q
Pr = N (1/2, 1/4), noise level = .25
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Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Comparison with Solt and Gotzner (2012): Baseline
noise level = .25
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Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Comp. with Solt and Gotzner (2012): Left-Skewed
noise level = .25
31 / 37
Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Comp. with Solt and Gotzner (2012): Right-Skewed
noise level = .25
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Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Comparison with Alxatib and Pelletier (2011)
noise level = .25
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Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
Summary
compositional semantic values
influences
language use
determines
constitutes
basic meaning
Main Result
availability of complex forms shapes meaning of basic forms
34 / 37
Sim-Max Games
Experimental Data
Vagueness
Fuzzy → Action
To Do
1
charter dynamics
2
model fitting
3
more data
1
2
antonym pairs
unrelated properties
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References
Alxatib, Sam and Francis Jeffry Pelletier (2011). “The Psychology of
Vagueness: Borderline Cases and Contradictions”. In: Mind &
Language 26.3, pp. 287–326.
Franke, Michael et al. (2011). “Vagueness, Signaling & Bounded
Rationality”. In: JSAI-isAI 2010. Ed. by T. Onoda et al. Springer,
pp. 45–59.
Goeree, Jacob K. et al. (2008). “Quantal Response Equilibrium”. In:
The New Palgrave Dictionary of Economics. Ed. by Steven N. Durlauf
and Lawrence E. Blume. Basingstoke: Palgrave Macmillan.
Jäger, Gerhard (2007). “The Evolution of Convex Categories”. In:
Linguistics and Philosophy 30.5, pp. 551–564.
Jäger, Gerhard and Robert van Rooij (2007). “Language Structure:
Psychological and Social Constraints”. In: Synthese 159.1,
pp. 99–130.
Jäger, Gerhard et al. (2011). “Voronoi Languages: Equilibria in
Cheap-Talk Games with High-Dimensional Types and Few
Signals”. To appear in Games and Economic Behavior.
Ripley, David (2011). “Contradictions at the Borders”. In: Vagueness
in Communication. Ed. by Rick Nouwen et al. Springer, pp. 169–188.
Solt, Stephanie and Nicole Gotzner (2012). “Who Here is Tall?
Comparison Classes, Standards and Scales”. In: Pre-Proceedings of
the International Conference Linguistic Evidence, pp. 79–83.