Filippo Radicchi
1.5
98
filrad.homelinux.org
1.4
1.3
1.2
filiradi@indiana.edu
105
0
10
β
102
0
10
102
105
Hunting98for numbers:
optimal strategies in
mental searches
Tracking animals
Movement patterns in nature
albatross
Viswanathan GM, et al., Nature 381, 413–415 (1996)
marine predators
Sims DW, et al., Nature 451, 1098–1102 (2008)
spider monkey
Fernandez G, et al., Beahv Ecol Sociobiol 55, 223–230 (2004)
honey bee
Reynolds AM,et al., J Exp Bio 210, 3763-3770 (2007)
Movement patterns in nature
Lévy flight
Humphries NE, et al., Nature 465, 1066–1069 (2010)
Shlesinger MF, Zaslavsky GM, Klafter J, Nature 363, 1–37
P (d)
d
µ
Lévy flights are optimal search strategies
Viswanathan GM, et al., Nature 401, 911-914 (1999)
What about humans?
107
105
104
years ago
102
101
Human mobility patterns
currency tracking
Brockmann D, et al., Nature 439, 462-465 (2006)
Human mobility patterns
mobile phone tracking
Gonzalez MC, et al., Nature 453, 779-782 (2008)
Lévy flights: a universal description for movement patterns of
animals and humans
from www.christineberrie.com
Problems
Results for animals are controversial
Journal of Animal Ecology
Volume 81, Issue 2, pages 432-442, 17 OCT 2011 DOI: 10.1111/j.1365-2656.2011.01914.x
http://onlinelibrary.wiley.com/doi/10.1111/j.1365-2656.2011.01914.x/full#f2
Problems
1) Data are aggregated over many “different”
individuals. Are we measuring heterogeneity at
the individual or at the population level?
2) Three orders of magnitude represent a
“luxury”. Is the scaling compatible with a power
law?
3) Do modern humans move in the space for
search purposes?
Lowest unique bid auctions
Lowest unique bid auctions
game rules
1
2
3
M-2
3
• high-valued goods are put up for auction V = 10 $
A
M = 101 $
101 $
not unique
not unique
unique and
lowest
not unique
unique but
not lowest
1¢
2¢
3¢
4¢
5¢
3 bids
2 bids
1 bid
M
105 $
• agents independently explore the bid space in search for the winning bid
• each bid costs c = 100 $
M-1
2 bids
1 bid
the winning bid is the lowest and unmatched bid
103 $
Lowest unique bid auctions
data sources
bid
8000
6000
A
Uniquebidhomes
agent 1632
auction 19
4000
change
2000
B
6000
4000
2000
0
100 200 300 400 500 600 700
t
Cumulative distribution
Patterns of bid space exploration
Using the model to fit the data
(Q )ji =
|i
j|
[1
(i
mj ( )
j)]
transition matrix
p (b1 , b2 , . . . , bT | ) = (Q )0,b1 (Q )b1 ,b2 (Q )b2 ,b3 · · · (Q )bT
fitting data by maximizing the likelihood
1 ,bT
Testing the validity of the model
1
P
(d) =
(Q )ij (d
i,j
|i
(Q )ij (d
j|)
d
i,j
goodness of fit
|i
j|)
Probability density
Patterns of bid space exploration
2.0
Uniquebidhomes
1.5
1.0
0.5
0.0
0.5
D
1.0
1.5
α
2.0
2.5
Patterns of bid space exploration
A
Probability density
10
B
0
10
Uniquebidhomes
Lowbids
Bidmadness
0
positive
negative
10
-2
10
-2
10
-4
10
-4
10
-6
10
-6
C10
10
0
10
1
10
2
10
3
10
4
D10
0
10
-2
10
10
T<10
10≤T<40
40≤T<200
T≥200
10
-2
-4
10
-4
-6
10
-6
10
0
10
1
10
2
10
3
10
0
10
1
10
2
10
3
10
4
10
4
0
10
Uniquebidhomes
Uniquebidhomes
4
change
first bid
t≤2
t≤5
t≥T-2
t=T-1
Uniquebidhomes
10
0
10
1
10
2
10
3
Modeling lowest unique bid auctions
single auction
0
1
p (i) = i
2
3
M-2
M-1
M
M
/m ( )
m( ) =
j
j=1
prob. that the agent bids on value i
N = Number of agents
T = Number of bids per agent
M
P ({n}) = N !
k=1
nk
[p (k)]
nk !
prob. to observe a configuration
{n} = (n1 , n2 , . . . , nk , . . . , nM )
M = Max bid value
= LF exponent of the population
= LF exponent of the agent
c = Fee value
V = Value of the good
Modeling lowest unique bid auctions
single auction
N 1
u (i) = P (ni = 1) = N p (i) [1
p (i)]
prob. that a bid on value i is unique
Focus on the agent with strategy
l
,
(v) = p (v) [1
N
p (v)]
[1
u (k)]
k<v
prob. that the agent places a unique and lowest bid on value v
M
w
,
=
M
l
,
(v)
v
,
=
vl
,
(v)
v=1
v=1
prob. that the agent wins the auction
average value of her winning bid
Modeling lowest unique bid auctions
multiple auctions
G = Total number of auctions
g = Total number of wins
P
,
(g) =
G
g
(w
,
g
) (1
w
,
G g
)
prob. that the agent wins g auctions
R
,
(I |g ) =
l
,
(v1 ) l
v1 +v2 +...+vg =I
prob. that the sum of her winning bids is I in g wins
I) /G
, (g) = (gV
economic return of the agent
r
,
(v2 ) · · · l
,
(vg )
Modeling lowest unique bid auctions
multiple auctions
R
(I) =
,
P
,
(g) R
,
(I |g )
g
prob. that the sum of her winning bids is I
In the limit
1
G
g = Gw
I= g v
,
,
=w
,
(V
,
v
,
sum of the winning bids
average number of wins
r
= Gw
,
v
economic return of the agent
,
)
r
,
>c
positive return
r
,
<c
negative return
Modeling lowest unique bid auctions
multiple bids
q
(1)
(i) = i
(Q )ji =
/m ( )
first bid
|i
j|
q
(i) =
T
(Q )ji q
(t 1)
(j) s(T ) (i) = 1
j=1
next bids
(T )
u ,
1
q (t) (i)
t=1
probability to visit a given site
(i) = N s
(T )
(i) 1
(T )
s
(i)
N 1
(T )
s
(i)
(T )
u ,
(v)
1
prob. that the agent makes on value i a unique bid
(T )
l ,
j)]
transition matrix
M
(t)
[1
(i
mj ( )
(T )
(v) = s
(v) 1
(T )
s
(v)
N
1
k<v
prob. that the agent makes on value i a lowest and unique bid
Model predictions
0
0
0
1.0 0.51.5 1.02.0 1.52.5 2.03.0 2.5
0.5 0.01.0 0.51.5 1.02.0 1.52.5 2.03.0 2.5 0.0
3.0 0.5 0.0
1.4
95
1.4
β
102
11
11
10
10
9
1.41.3
γ
1.51.4
γ
9
9
8
1.61.5
8
1.6
single bid, T = 1
9.5
1.5
9
1.6
1.2
1.31.2
9.5
1.4
1.2 90
1.5 1.2
1.3
1.5
94
12
β
1.3
1.3
12
10
105
9.5
10
102
9.5
3.0
10
β
9
11
11.5
10
12
1.4
1.4
γ
90
1.51.4
1.5
β
9
98
1.3
γ
1.41.3
94
11
11.5
12
1.31.2
95
98
1.5
1.1
1.21.1
98
1.4
95
1.5
1.3
98
1.4
1.2
1.3
1.2
1.1
1.1
1.2
β
106
102
1.4
0
10
β
1.5
102
0
10
102
1.3
98
102
0
10
105
0
10
1.3
105
β
98
1.2
106
β
11.5
95
110
D1.6110 D1.6
11
β
11.5
β
11
B 1.5 B 1.5
105
0
0.0
multiple bids, T = 10
parameters of real auctions
1.6
A
B
5.0
Evolutionary model for the auctions
e = 20
4.0
A Moran-like model
w (α)
At generation e=0, agents take their strategies form 3.0
an arbitrary pdf
(e)
1) they play the game.
g (0) ( )
2.0
2) a losing agent copies the strategy of the winner plus some random noise.
e=5
1.0
3) e -> e + 1/N. Go back to point 1.
e = 10
e=1
0.0
0
5.0
g
(0)
( ) = const.
1.0
w (α)
(e)
0.0
C
1.4
5
D
e = 20
g (0) ( λ)= =
const.
0.0
1.0
λ = 2.0
λ = 5.0
0.8
0
e=5
σ
e = 10
10
-1
10
-2
10
-3
e=3
0.4
e=2
0.2
e=1
e=1
0.0
0
4
10 0.6
2.0
1.0
α
3
0.5
3.0
(e)
w (α)
BC
1.5 1.2
e = 20
4.0
2
random mutations
2.0 1.4
<α>
A
no mutations
1
1
2
w
(e)
e = 20
α
3
4
5
0.0
0
10 0
1
2101
e
α
3
( ) = prob. thatDthe winning strategy is
2.0
4
10
2
5
0.0
A Moran-like model
10
0
2.0
1.0
e=5
e = 10
σ
(e)
w (α)
Evolutionary model for the auctions
0.5
3.0
λ = 0.0
λ = 2.0
λ = 5.0
10
-1
10
-2
10
-3
e=1
0.0
0
1.5
σ
w (α)
10
(e)
10
0
-1
10
-2
10
-3
D
e = 20
λ = 0.0
λ = 2.0
λ = 5.0
D
e=3
<α>
0.0
1.2
1.0
2
10
2
random mutations
2.0
λ = 0.0
λ = 2.0
λ = 5.0
0.8
0.6
e=2
0.4
0.2
1
10
e
1
2
α
3
1.5
0.5
e
10
1.0
0.2
4102
5
2.0
1.0
1
0.0
1.2
0.6
0.0
0
10
0.5
0.8
0.4
0
0
1.0
e=1
10
10
1.5
1.0
0.0
5
1.4
1.0
0.5
4
no mutations
1.2
<α>
α
3
<α>
C
2.0
2
σ
B
1
λ = 0.0
λ = 2.0
λ = 5.0
g
(0)
( )
0.0 0
10
10
e
1
Relative cost to winner
A 2.5
Agents bid rationally, but ...
C 2.5
Uniquebidhomes
Uniquebidhomes
2.0
2.0
1.5
1.5
1.0
1.0
0.5
0.5
0.0 3
10
Relative income of auctioneer
Bidmadness
B 6.0
10
4
10
5
10
6
10
7
5.0
4.0
4.0
3.0
3.0
2.0
2.0
1.0
1.0
0.0 3
10
10
4
10
5
10
6
10
10
D6.0
Uniquebidhomes
5.0
0.0 3
10
7
4
10
5
10
6
4
10
5
10
6
Bidmadness
0.0 3
10
10
Value of good
r
s, s
<w
s, s
V
V
<Tc
N +1
the economic return is always negative
Do humans search as animals?
107
105
104
years ago
102
101
We are running some experiments...
http://cgi.soic.indiana.edu/~i601levy/index.php
Preliminary results
Thanks
L.A.N. Amaral, Northwestern University
A. Baronchelli, City University London
References
Evolution of optimal Lévy-flight strategies in human mental searches !
F. Radicchi and A. Baronchelli !
Phys. Rev. E 85, 061121 (2012)
Rationality, irrationality and escalating behavior in lowest unique bid auctions !
F. Radicchi, A. Baronchelli and L. A. N. Amaral !
PloS ONE 7, e29910 (2012)
Lévy flights in human behavior and cognition!
A. Baronchelli and F. Radicchi!
Chaos Soliton. Fract. 56, 101-105 (2013)!