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Logistic Constraints on 3D Termite Construction

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					Leeds University
Business School


   Termite Construction
   and
   Agent-Based Simulation
   Dan Ladley,
   Leeds University Business School and School of Computing




                               danl@comp.leeds.ac.uk
                         www.comp.leeds.ac.uk/danl
Leeds University
Business School

Social Insects

Social insects such as termites, ants and bees successfully
accomplish many complex tasks through cooperation.


These include:
Locating food sources
Building nests
Dividing labour
Brood Sorting
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Computing Applications


Insects have evolved solutions to challenging distributed
coordination problems which have been successfully adapted
to real world systems.


Locating food sources -> Shortest path algorithms
Building nests -> Nano-technology, Space Exploration
Dividing labour   -> Task Allocation problems
Brood Sorting     -> Graph partitioning, data analysis
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Termite nest formation


Many individual termites participate in the construction of
termite nests. Due to the large size of the next relative to
individual termites and the number of individuals involved this
is a difficult coordination problem.


The most common ways of coordination are:
Blueprint                         Leader
Plan                              Template
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Stigmergy

The above methods do not work for termites instead they employ
stigmergy. Cues in the environment encourage termites to make certain
behaviours which in turn effect the environment effecting future behaviours.


Termites respond to many environmental cues. These include:
• Pheromones
  • Cement, Queen, Trail
• Temperature
• Air Movements
• Humidity
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Structures Formed

Domes
Pillars
Walls
Entrances
Tunnels
Air conditioning
Fungus farms
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Previous Model


Demonstrated the existence of pillars, chambers, galleries and
covered paths
No consideration of logistic factors or inactive material




E. Bonabeau, G. Theraulaz, J-L. Deneubourg, N. Franks, O. Rafelsberger, J-L. Joly, S. Blanco. A
model for the emergence of pillars, walls and royal chambers in termite mounds. Philosophical
Transactions of the Royal Society of London, Series B, 353:1561-1576, 1998.
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Agent Based Model

Three dimensional discrete world
Populated by a finite number of „termites‟
Three pheromone types
• Cement – given off by recently placed material
• Trail – given off by moving termites
• Queen – given off by stationary queen
Diffusion through finite volume method
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Agent Movement


May move to any adjacent location as long as
• There is no building material present
• The new location is adjacent to material
Movement influenced by cement pheromone
Roulette wheel selection based on pheromone gradients
Random Movement with probability 1/Gradient
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Agent Building Behaviour


Probability of building when queen pheromone level lies in a
particular range
Crude physics
Newly placed material gives off cement pheromone
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               Chambers
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 Recruitment
                              1.8

                              1.6
        Deposits per worker



                              1.4

                              1.2

                               1

                              0.8

                              0.6

                              0.4

                              0.2

                               0
                                    10   20   40   80   160   320   640   1280

                                                   Workers
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               Tunnels
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               Flared Tunnels
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               Narrow Tunnels
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Dome Entrances


Currently no entrance in chambers


New class of “Worker” termites go to and from the queen


Deposit inhibitory trail pheromone
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               Entrances
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                   Targets
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Pros and Cons of this model


Reproduces results seen in nature
Importance of logistic constraints
Applications in real situations – space exploration, nano-tech…


Simplistic movement strategy
Artefacts due to tessellation of world
No accounting for castes of termites
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Agent-based modelling is employed in other fields, in particular
it is key to current research in epidemiology, transport studies
and defence.


Many fields investigate problems involving many interacting
individuals engaging in potentially complex and changing
relationships which are frequently difficult to analyse with more
traditional techniques.
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Agent Based Models


Allow the investigation of:
Heterogeneous individuals
Bounded rationality
Complex relationships
The time path or dynamics of a system
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Agent-Based Models

These models have draw backs:


They do not provide proofs only demonstrations of sufficiency
There are typically many ways to model any given situation
Parameters, parameters and more parameters
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A Game:


It‟s January 1926 you have £1 to invest


If you invested it in US Treasury bills, one of the safest bets
around, and reinvested all of the proceeds how much would
you have now?


                          £14
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If you invested it in the S&P 500 index (the stock market), a
much riskier bet, how much would you have now?




                        £1370
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Now suppose that each month you were able to divine which
would do better and invested everything in that, how much
would you have?



           £2,296,183,456
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Motivation


In order to predict what is going on in financial market it is vital
to separate the effect of the market mechanism and individual
behaviour.


The order book market mechanism is employed (with
variations) in the majority of the worlds major financial
institutions.
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Order book markets


Similar to a continuous double auction


Traders submit orders to the market
• Market Orders execute immediately at the best available price for the
  specified quantity
• Limit Orders are added to the order book at the specified quantity and
  price


Trade results in limit orders being removed from the book
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Example order book
   Buy                    Sell
   Order                  Order




                                                  10

      10    20       10            20    30       10 10

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

                          Price
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Example order book
   Buy                      Sell        Best Ask
   Order                    Order


                 Best Bid


                                                   10

      10    20        10       Spread   20   30    10 10

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

                            Price
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Understanding order book markets


Analytical work - Difficult to maintain analytical tractability


Empirical and experimental work - Difficult to separate trader
strategy from the effect of the market mechanism


Simulation work – how should the traders agents behave?
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Solution - Zero Intelligence
         Observed =   Effect of   +    Effect of
         Behaviour     Trader          Market
                      Strategy        Mechanism



Traders modelled to behave randomly, consequently
any effects observed in the data are due to the
market mechanism. Those not observed are then
dependant on individual behaviour.
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Agent-Based Model

100 traders each initially allocated 50 units to either buy or
sell with reservation prices stepped between 0 and 100


Each time step one trader selected at random to submit an
order for a random number of units at a random price drawn
from a uniform integer distribution constrained by the limit
prices of the traders units


With a set probability new traders enter and leave the market
each time step
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Orders classified into 12 types based on aggressiveness
(Biais et al. 1995)



            Buy Orders                     Sell Orders

        1   Market larger quantity    7    Market larger quantity

        2   Market equal quantity     8    Market equal quantity

        3   Market smaller quantity   9    Market smaller quantity

        4   Limit between quotes      10   Limit between quotes

        5   Limit at quote            11   Limit at quote

        6   Limit below Quote         12   Limit below Quote
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Order Book Mechanism
   Sell     Buy
   Order    Order

                    6    5       4   1,2,3        10

      10    20          10           20      30   10 10

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

                             Price
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  From\To   1   2   3   4   5   6   7   8   9   10   11   12
  1
  2
  3
  4
  5
  6
  7
  8
  9
  10
  11
  12
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Also predicts:
• Details of the bid ask spread
• Intra-book spreads
• Quantities available at the quotes
• Effect of changes of the tick size


Importance of the tips of the order book (Griffith et al. 2000
etc.)


Correlation between price movements and order book shape
(Huang & Stoll 1994, Parlour 1998 etc.)
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Conclusions


Much of the order dynamics typically observed in markets can
be explained as a consequence of the order book market
mechanism


In many cases trader strategy may not be the dominant force
in observed market behaviour


However this is only half of the story we still need to
understand the strategies employed by traders
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Model as before, except…


The agents are now trading a financial asset (e.g. a stock in a company)
and money


They are paid dividends and interest and must consume a fraction of their
wealth each time step


They are subject to margin constraints a limit on the amount of money a
trader may borrow to some fraction of there net-worth


And the traders have strategy…
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Genetic Programs


Programs are provided with the 8 input parameters
(information about the market)


Two outputs, the quantity and price are returned
      Quantity – Rounded to Integer Values
      Price – Rounded to [0,1] then mapped to [10000,20000]


Three registers for variable manipulation are provided
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Genetic Program Example
Instruction Program

1         R0 = 2

2         R 1 = ps

3         R0 = R0 * R1

4         R 1 = R 1 – pb

5         Return R0

Results   2ps
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Genetic Programming Tournaments


One Tournament per trading period


4 Individuals selected at random


Fitness equal to net worth


2 Least fit individuals have their strategies replaced
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Genetic Programming Mutation
  Instruction   Program          Instruction   Program

  1             R0 = 2           1             R0 = 2

  2             R1 = ps          2             R 1 = ps

  3             R0 = R 0 * R1    3             R 0 = R0 * R1

  4             R 1 = R 1 – pb   4             R0 = R0/5

  5             Return R0        5             Return R0

  Results       2ps              Results       2ps/5
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    Genetic Programming Recombination
         Program 1      Program 2                 Program 1        Program 2
1        R 0 = pb       R0 = 2             1      R 0 = pb         R0 = 2
2        R 1 = ps       R1 = pb            2      R 1 = ps         R 1 = pb
3        R0 = R0 * 5 R0 = R0/R1            3      R0 = R0/R1       R0 = R0 * 5
4        R 1 = R 1 – ps R 1 = R 1 - 1      4      R1 = R1 - 1      R 1 = R 1 – ps
5        Return R0      R0 = min(R0,R1)    5      R0 = min(R0,R1) Return R0
6                       Return R0          6      Return R0
Result   5pb            Min(2/ pb, pb-1)   Result Min(pb /ps, ps-1) 10
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Analysis of Margin Constraints


Vary β from 0 to 1 in increments of 0.1


β = 0 corresponds to no buying on margin


β =1 corresponds to having no restriction on capacity to buy
(unrealistic)
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Average Bankruptcy Size
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Wealth Distributions
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Conclusions


There exists an optimal level of market regulation reducing
bankruptcy


Traders strategies depend heavily on the level of borrowing
allowed


Agent-based models can provide insights into these systems
unachievable with other techniques.

				
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