Game Theory: Sharing, Stability and Strategic Behavior

1
(No Transcript)
2
Simple Foraging for Simple Foragers

• Frank Thuijsman
• joint work with
• Bezalel Peleg, Mor Amitai, Avi Shmida

3
Outline
4
Outline

• Two approaches that explain certain
• observations of foraging behavior
• The Ideal Free Distribution
• The Matching Law
• Risk Aversity

5
The Ideal Free Distribution

• Stephen Fretwell Henry Lucas (1970)
• Individual foragers will distribute themselves
  over various patches proportional to the amounts
  of resources available in each.

6
The Ideal Free Distribution

• Many foragers
• For example if patch A contains twice as much
  food as patch B, then there will be twice as many
  individuals foraging in patch A as in patch B.

7
The Matching Law

• Richard Herrnstein (1961)
• The organism allocates its behavior over various
  activities in proportion to the value derived
  from each activity.

8
The Matching Law

• Single forager
• For example if the probability of finding food
  in patch A is twice as much as in patch B, then
  the foraging individual will visit patch A twice
  as often as patch B

9
Simplified Model
Two patches
One or more bees
Yellow
Blue
?
p
q
y
b
Nectar quantities
Nectar probabilities
10
Only Yellow
11
And Blue
12
No Other Colors
13
Yellow and Blue Patches
14
IFD and Simplified Model
Yellow
Blue
two patches
y
b
nectar quantities
nY
nB
numbers of bees
IFD
nY / nB
y / b
15
Matching Law and Simplified Model
Yellow
Blue
two patches
p
q
nectar probabilities
nY
nB
visits by one bee
nY / nB
p / q
Matching Law
16
How to choose where to go?
Alone
17
How to choose where to go?
or with others
18
How to choose where to go?
bzzz, bzzz,
No Communication !
19
How to choose where to go?
e-sampling orfailures strategy!
20
The Critical Level cl(t)

• cl(t1) acl(t) (1- a)r(t)
• 0 lt a lt 1
• r(t) reward at time t 1, 2, 3,
• cl(1) 0

21
The e-Sampling Strategy

• Start by choosing a color at random
• At each following stage, with probability
• e sample other color
• 1 - e stay at same color.
• If sample at least as good,
• then stay at new color,
• otherwise return
• immediately.

e gt 0
22
IFD, e-Sampling, Assumptions

• reward at Y 0 or 1 with average y/nY
• reward at B 0 or 1 with average b/nB
• no nectar accumulation
• e very small only one bee sampling
• At sampling cl is y/nY or b/nB

23
e-Sampling gives IFD

• Proof
• Let P(nY, nB) y(1 1/2 1/3 1/nY)
  - b(1 1/2 1/3 1/nB)
• If bee moves from Y to B,
• then we go from (nY, nB) to (nY - 1, nB 1)
• and
• P(nY - 1, nB 1) - P(nY, nB)
• b/(nB 1) - y/nY gt 0

24
e-Sampling gives IFD

• So P is increasing at each move,
• until it reaches a maximum
• At maximum
• b/(nB 1) lt y/nY and y/(nY 1) lt b/nB
• Therefore
• y/nY b/nB
• and so
• y/b nY/nB

25
ML, e-Sampling, Assumptions

• Bernoulli flowers reward 1 or 0
• with probability p and 1-p resp. at Y
• with probability q and 1-q resp. at B
• no nectar accumulation
• e gt 0 small
• at sampling cl is p or q

26
ML, e-Sampling, Movements
e
Y1
B2
1- e
1- p
q
p
Markov chain
1- q
B1
Y2
1- e
e
nY/nB (p qe)/ (q pe) p/q
27
The Failures Strategy A(r,s)

• Start by choosing a color at random
• Next
• Leave Y after r consecutive failures
• Leave B after s consecutive failures

28
ML, Failures, Assumptions

• Bernoulli flowers reward 1 or 0
• with probability p and 1-p resp. at Y
• with probability q and 1-q resp. at B
• no nectar accumulation
• e gt 0 small
• Failure reward 0

29
The Failures Strategy A(3,2)
30
The Failures Strategy A(3,2)
31
ML and Failures Strategy A(3,2)
Now nY/nB p/q if and only if
32
ML and Failures Strategy A(r,s)
Generally nY/nB p/q if and only if
This equality holds for many pairs of reals (r,
s)
33
ML and Failures Strategy A(r,s)
If 0 lt d lt p lt q lt 1 d, and M is such that (1
d)2 lt 4d (1 dM), then there are 1 lt r, s lt
M such that A(r,s) matches (p, q)
34
ML and Failures Strategy A(fY,fB)
e.g. If 0 lt 0.18 lt p lt q lt 0.82, then there
are 1 lt r, s lt 3 such that A(r,s) matches (p, q)
35
ML and Failures Strategy A(r,s)
If p lt q lt 1 p, then there is x gt 1 such that
A(x, x) matches (p, q) Proof Ratio of visits Y
to B for A(x, x) is
It is bigger than p/q for x 1, while it goes to
0 as x goes to infinity
36
IFD 1 and Failures Strategy A(r,s)

• Assumptions
• Field of Bernoulli flowers p on Y, q on B
• Finite population of identical A(r,s) bees
• Each individual matches (p,q)
• Then IFD will appear

37
IFD 2 and Failures Strategy A(r,s)

• Assumptions
• continuum of A(r,s) bees
• total nectar supplies y and b
• certain critical levels at Y and B

38
IFD 2 and Failures Strategy A(r,s)

• If y gt b and ys gt br, then there exist
  probabilities p and q and related critical levels
  on Y and B such that
• i.e. IFD will appear

39
Learning
40
Attitude Towards Risk
2
1
3
2
2
2
?
41
Attitude Towards Risk
Assuming normal distributions If the critical
level is less than the mean, then any
probability matching forager will favour higher
variance
42
Attitude Towards Risk
Assuming distributions like below If many
flowers empty or very low nectar quantities, then
any probability matching forager will favour
higher variance
43
Concluding Remarks

• A(r,s) focussed on statics of stable situation
  no dynamic procedure to reach it
• e-sampling does not really depend on e
• e-sampling requires staying in same color for
  long time
• Field data support failures behavior
• Simple Foraging?
• The Truth is in the Field

44
Questions
?
frank_at_math.unimaas.nl
F. Thuijsman, B. Peleg, M. Amitai, A. Shmida
(1995) Automata, matching and foraging behaviour
of bees. Journal of Theoretical Biology 175,
301-316.