Konrad Richter Presentation SWARM by oogACE

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									Agent Based Modeling in SWARM


Konrad Richter



Presentation Essex
November 24, 2004
AGENDA



         Introduction

             –   Example: Heatbugs



         Genetic Algorithms and Classifier Systems

             –   Example: The Santa Fe Artificial Stock Market




         The Auction Simulator

             –   Background
             –   Best Response in Auctions
             –   Simulation
             –   Conclusion and Outlook



                                                                 1
HeatBugs is a simple application that illustrates
the idea of agent based modeling
               Model Rules                                 Simulation Flow

    Setup:                                         Initialization
   • N Heatbugs in a 2 dimensional world
   • Each Heatbug has an individual optimal        • N Heatbugs are randomly
                                                   assigned a grid position, an
   temperature
                                                   optimal temperature and a heat
   • Each Heatbug produces an individual output
                                                   output
   of heat
   • Heat dissipates with a constant factor
    Updating rules:                                Simulation Flow:
   • If too cold, move to a warmer spot            • In each time step, each
   • If too warm, move to a colder spot            Heatbug:
   • If spot is occupied, try to move to an          – Measures heat at neighboring
   unoccupied spot                                    grid points
   • Small positive chance of moving to a random     – Moves to spot that minimizes
   spot                                               it‘s unhappiness (plus random
                                                      moves)
                                                   • Temperature of grid is updated



                                                                                      2
    OBJECT STRUCTURE OF HEATBUGS

 Heatbug                     Heatbug
BatchSwarm                ObserverSwarm




              Heatbug
             ModelSwarm




Heatspace                    Heatbug




                                          3
The behavior shows different regimes for different
model parameters
              Parameter Settings                           Characteristics

   Changing the number of Heatbugs:
   • 10 Heatbugs:
     – Slow cluster forming                       • Simulation shows a wide range
     – Discrete improvements of unhappiness       of different states that the model
   • 100 Heatbugs:                                can work in
     – All bugs clustered
     – Unhappiness decreases exponentially
   • 220 Heatubgs:
     – Interacting waves of agents
     – „Explosions“ as „holes“ in the heatspace
       become filled up
     – Unhappiness decreases exponentially        • Comparable Mathematical
   • 1000 Heatbugs:                               Analyses are usually nasty or
     – Strings of Heatbugs                        simply not solveable
     – Unhappiness shows kink
   • 3200 Heatbugs:
     – Complete random movements
   Changing diffusion rate: 0.9, 0.99, 0.995,
                                   0.999

                                                                                       4
Agent based modelling works bottom-up

          Agent Based Modelling                               Key Properties
                            Inanimate Objects
                                                      • Very simple rules can lead to
                                                      complex behavior
                                                      • Local interactions can lead to
                                                      collective behavior on scales that
                                                      the individual isn‘t aware of
                                                      (space or time)
                                                      • Model behavior is separated
                                                      into different regimes

         Schedule           Animate Agents            • Long-term behavior largely
                                                      independent of initial conditions
                                                      • Local solutions to simple
                                       Statistical
                                                      optimization problems („How do I
                                       Analysis
                                                      maximize my utility“) can lead to
                                                      pareto-optimal outcomes („How
 Model philospohy:
                                                      do I position the heatbugs in
 • Detailed definition of individual agent behavior   order to maximize their aggregate
 • Put agents into common world and analyse           utility?“)
 interactions

                                                                                           5
SWARM facilitates Agent Based Modelling

           Background                                                 Details

   SWARM                                   Features
   • Development started in 1994 at        • Library of Objective
   the Santa Fe Institute, New Mexico      C and Java routines, therefore
   • Goal:                                 provides all the flexibility and speed of
     – Facilitate agent based modelling    C
       for inexperienced programmers
                                           • Characteristics:
     – Set standard that allows easy
                                            – Easy manipulation of model specifications
       comparability of code
                                              due to object oriented structure (even online)
   • Free software, published under         – Comprehensive libraries for:
   the GNU open source licence                • Model online observation
   • Works on Windows, Unix, Linux            • Timing of updatings
   and Apple                                  • Random value distributions
   • Strong user community;                   • List handling
    – rich availability of tutorials and      • Spaces
      documentation                         – Reusability of code due to hierarchical
    – active newsgroups for design            program structure
      and programming questions
    – ongoing program development

                                                                                               6
AGENDA



         Introduction

             –   Example: Heatbugs



         Genetic Algorithms and Classifier Systems

             –   Example: The Santa Fe Artificial Stock Market




         The Auction Simulator

             –   Background
             –   Best Response in Auctions
             –   Simulation
             –   Conclusion and Outlook



                                                                 7
NATURE AS INSPIRATION FOR GAs



GAs mimic mechanisms of genetic evolution to obtain approximate solutions for optimization
problems

Nature                                        Computational Model

DNA: Genetic information is encoded in         Encode the potential problem solutions in
strings made up of 4 basic building blocks     BitStrings

Selection: The fitter an individual is, the    Use best solutions obtained so far as starting
more offspring it produces                     point to generate new solutions.

Sexual Reproduction: DNA of offspring is       Use crossover of BitStrings to generate new
created by merging DNA fragments of both       solutions
parents
Mutation: DNA is changed by radioactive        Randomly disturb obtained solutions to
and/or chemical influence                      explore solutions in the surrounding search
                                               space




                                                                                                8
GENETIC ALGORITHMS EXAMPLE (1/3)

Problem: Find argmax of 1- (x-0.5)2 on [0,1)
Solution: xopt=0.5
Question: How to implement with GAs?

Step 1: Generate random population* of BitStrings encoding values between 0 and 1

#   BitString                                 Encoded value                                      Fitness**

1    0 0 1 0 0 0 0 1                          0*0.5+0*0.25+1*0.125+...=0.1289                    1-(0.1289-0.5)2=0.8622


2    0 1 1 1 0 0 1 0                          0*0.5+1*0.25+1*0.125+...=0.4453                    1-(0.4453-0.5)2=0.997


3    1 0 0 1 0 1 1 0                          1*0.5+0*0.25+0*0.125+...=0.5859                    1-(0.4453-0.5)2=0.9926


4    1 1 0 1 1 0 0 1                          1*0.5+1*0.25+0*0.125+...=0.8477                    1-(0.8477-0.5)2=0.8791




     * In practice, the population consists of many more individuals. (N~102). The example is just illustrative
    ** In practice, fitness measures are used that distinguish sharper between the optimality of solutions.
                                                                                                                         9
GENETIC ALGORITHMS EXAMPLE (2/3)

Step 2: Generate Population of BitStrings at T+1 from Population at T by using genetic operators

           Step 2A: Select a BitString according to fitness
             #       Value                Fitness             Selection Probability
             1       0.1289               0.8622              0.8622/3.7309*=23.1%
             2       0.4453               0.997               0.997/3.7309*=26.7%
             3       0.5859               0.9926              0.9926/3.7309*=26.6%
             4       0.8477               0.8791              0.8791/3.7309*=23.6%
           Step 2B: With prob. Pco** select second string from remaining ones;
                    apply one-point crossover to obtain new string in population at T+1
                    With prob. 1-pco copy clone of selected string to population at T+1
           Assume that BitStrings # 2 and #3 were selected for crossover:
            Parent BitStrings                   Val                   Offspring BitString   Val

             0 1 1 1 0 0 1 0                    0.4453
                                                                        0 1 1 1 0 1 1 0 0.4609
             1 0 0 1 0 1 1 0                    0.5859

           Repeat step 2B till a new population has been generated

      * Sum of All Fitnesses is given by 0.8622+0.997+0.9926+0.8791=3.7309
     ** Typically, pco~0.8                                                                        10
GENETIC ALGORITHMS EXAMPLE (3/3)



           Step 2C: Browse through new population and flip each bit with pm*
           Original BitString                  Val      Mutated BitString      Val

            1 0 0 1 0 1 1 0                    0.5859   1 0 1 1 0 1 1 0        0.7109



 Step 3:     Repeat steps 2A-2C sufficiently often.*
             Eventually, the population converges to an optimum




    * Typically, pm~0.5%
   ** Typically for some 102-104 generations
                                                                                        11
HOLLAND'S SCHEMA THEOREM FOR GAs (1/2): BASIC DEFINITIONS



Entity             Symbol Description
Schema             H         Template made of 0, 1 and *; * denotes a don't care symbol
Instance of H      h         BitString that fits schema H
Order of H         O(H)      Number of fixed bits in schema H
Defining Length    L(H)      Number of symbols between the outermost fixed bits
of H

Example:

H          O(H)     L(H)     h1        h2        h3        h4
1**01    3         5         10001     11001     10101     11101

Building Block Hypothesis: The power of GAs relies on the selection and recombination
                           of ever fitter and more complex schemata over time. By
                           evaluating the fitness of N BitStrings, the GA implicitely evaluates
                           a much higher number of schemata.

Schema Theorems:             Describe statistically the time evolution of schemata in the
                             BitString population over time

                                                                                             12
HOLLAND'S SCHEMA THEOREM FOR GAs (2/2): THEOREM
                                                                                              pd


                                     f (H , t)                                L( H )      m( H , t ) f ( H , t )
     E[m( H , t  1)]  m( H , t )     
                                                 (1  pm ) O ( H ) (1  pco          (1                         ))
                                       f (t )                                 N 1                  _
                                                                                              M f (t )


                                     Selection      Mutation                        Crossover

Symbol              Interpretation
E(m,(H,t+1))        Expected number of BitStrings matching schema H at generation t+1
m(H,t)              Number of BitStrings matching schema H at generation t
f(H,t)              Mean fitness of instances of H at generation t
f(t)                Mean fitness of BitStrings in the population at generation t
pm                  Probability of mutation
O(H)                Order of H
L(H)                Defining Length of H
pco                 Probability of crossover
N                   Length of BitStrings in the population
M                   Population Size
pd                  Probability of Disruption if Crossover is applied



                                                                                                                      13
Classifiers are closely linked to the concept of a schema

             Descriptors                              Predictors


        State of the world: 42      Classifiers                            Predictions

                                      A   1 *     *    *   1 1 *       *      I
        Description Rules:                                                    II
                                      B   1 1 1 *          1 0 0 0
        1: Divisible by 2 5: >2
        2: Divisible by 3 6: >10
                                      C   1 1 0 *          *   *   *   *      III
        3: Divisible by 5 7: >20
        4: Divisible by 7 8: >50
                                      D   *   *   *    *   *   1 *     *      IV


        Descriptor of the World:    Reaction:

        1 1 0 1 1 1 1 0              Select one of the active classifiers
                                                 A, C or D
                                        Act based on it‘s prediction



                                                                                         14
The Santa Fe Artificial Stock Market simulates price prediction
via classifier systems
        Simulation Flow                           Updating of Price Expectation


     Investors place orders for           Market is characterized by a unique descriptor
     stock
                                           1 1 0 1 1 1 0 0
                                 (t )
     Market clearing price p is
                                (t )
     determined and dividend d              technical       fundamental
     is calculated according to
     AR(1)
                                          Bidders use classifiers to
     Investors update their ex-           predict future prices
                              ( t 1)
     pectations for ( p  d )         :
                                           1 *    *     *   *   *   0 *         (a,b)
      Choose predictors with
     probability proportional to            av50 ( p)  0 & & p! 4d
     accuracy
                                              ( p  d ) (t 1)  a( p  d ) (t )  b
     Investors decide on split
     between stock and riskless           • Accuracy and complexity determine
     asset according to                   fitness of classifiers
     riskaverse CAPM-portfolio            • GA evolves classifiers over time

                                                                                           15
   OBJECT STRUCTURE OF SANTA FE ARTIFICIAL STOCK MARKET

                ASM                                  ASM
             BatchSwarm                          ObserverSwarm



                               ASM
                                                    ModelParams
                            ModelSwarm



Specialist    Dividend    World      BFAgents          BFParams




                                     BFCasts




                                     BitVector



                                                                  16
The dynamics of the simulation shows
stylized facts of financial markets
              Simulation                         Key Observations

    Success of each prediction         Different market regimes depending on
    depends on all the other agent’s   frequency of GA updating:
    predictions (Keynesian             • Low frequency (1000-10000): market
    beautycontest)                     operates in RE regime
                                       • High frequency (100-1000): market operates
                                       in chaotic regime. Key characteristics:
                                         – Bubbles and Crashes
    Highly non linear feedback           – Clustered volatility of volumes and returns
    system                               – Crosscorrelation between volumes and
                                           returns
                                         – Lepto-kurtosis of return distribution
    Predictions can involve in
    permanent mutual adaptation        Bidders who can determine their GA updating
                                       rate endogenously fall into the complex regime
                                       (suboptimal: more risk, less earnings)
    Complex adaptive system with
    co-evolving ecology of
    heterogeneous traders              SF-ASM can serve as an explanation of
                                       stylized facts of real financial markets

                                                                                         17
WEB RESSOURCES FOR SWARM




Address                                     Material

http://wiki.swarm.org                       • Download of SWARM
                                            • Link to various tutorials and documentation
                                            • Example programs
http://sourceforge.net/projects/artstkmkt   • SWARM implementation of the Santa Fe
                                            artificial stock market

http://www.swarm.org/pubs.html              • Comprehensive overview of books and
                                            articles that deal with SWARM simulations

Konrad_richter@mckinsey.com                 • My e-mail address for further questions




                                                                                            18
AGENDA



         Introduction

             –   Example: Heatbugs



         Genetic Algorithms and Classifier Systems

             –   Example: The Santa Fe Artificial Stock Market




         The Auction Simulator

             –   Background
             –   Best Response in Auctions
             –   Simulation
             –   Conclusion and Outlook



                                                                 19
AGENDA - Auctions



             Background




             Best Response in Auctions




             Simulation




             Conclusion and Outlook




                                         20
This research aims at unifying two research areas that have so far been
largely separated
                                 Auction Theory under
        Auction Theory                                         Evolutionary Learning
                                 Evolutionary Learning

 • Assumptions about bidder     • Main Question:              • Assumptions about
 rationality:                     – How does accounting for    bidder rationality:
   – Each bidder is perfectly       bounded rationality        – Bidders observe
     rational and has all           change the predictions       their environment
     information about the          of auction theory          – Bidders react to
     auction setup              • Techniques:                    their environment
   – Each bidder knows that       – Complementary                by using simple
     each opponent is               assessment of best           updating rules for
     perfectly rational             response learning in         their strategies
   – Each bidder knows that         auctions by                – In particular: Best
     each opponent knows            mathematical analysis        response learning:
     that each of her               and agent based              Bidders use the
     opponents is perfectly         simulation                   strategy that would
     rational                                                    have generated
   – etc. ad infinitum                                           the highest payoff
                                                                 in the past




                                                                                       21
The theoretical investigation of auctions follows a standardized setup

                                                                Analytical
         Standardization of Auctions
                                                                Investigation
    Setup:                                                 Questions of Interest:
   • N Bidders with independent private values for
   the auctioned asset*
                                                           • Which Auction Format yields the
                                                           highest seller revenue
     – Values of all bidders drawn from same
       random distribution
                                                           • Which Auction Format has the
                                                           optimal allocative properties
     – Random distribution known to all bidders
   • Bidders decide on bid, put it into a sealed
   envelope and hand it over to the auctioneer
   • Auctioneer opens envelopes and assigns
   asset to highest bidder
   • Different Payment Mechanisms:
     – First Price (1PA): Winner pays her own bid          Usual Methodology:
     – Second & higher Price Auction: Winner pays
       2nd highest bid, 3rd highest bid, ...               • Calculation of NE Bidding
     – All Pay Auction: Each bidder pays her bid           Strategies
                                                           • Calculation of NE Bids and
                                                           Prices



    * Bidder   could have subpurchaser who takes the asset for a guaranteed price
                                                                                               22
The Revenue Equivalence Theorem holds for perfectly rational bidders

    Assumptions                            RET                        1st vs 2nd Price Auctions

 Standard                  Revenue Equivalence                 Explicitely for two bidders
 Framework:                Theorem:                            and values ~U(0,1):
 • Bidders have private      All auction formats where            1PA:
 independent values          • The item goes to the bidder who    • NE strategy:
 • Bidders' values are       submits the highest bid and          bidi  ( N  1) / N  vi  vi / 2
 drawn from the same         • The cost of submitting the lowest • Expected highest value:
                                                                   E[vi ]  N /(N 1)  2 / 3
                                                                       (1)
 distribution                feasible bid are the same
 • Bidders are risk          yield the same expected seller       •Expected Seller
 neutral                     revenue and result in the same       Revenue:
 • Bidders have no            allocation of goods.                     1/ 2  2 / 3  1/ 3
 budget constraint                                                     2PA:
 • Bidders are perfectly      If the seller posts no                   • NE strategy:
 rational and bid             reservation price, the final             bidi  vi
 according to their NE        allocation is pareto optimal.            • Expected second
 bidding functions                                                     highest value:
                                                                       E[vi ]  ( N 1) /(N 1)
                                                                            ( 2)

                                                                       • Expected Seller
                                                                       Revenue:
                                                                       1*1/ 3  1/ 3

                                                                                                      23
My research investigates whether the assumption of NE play
in auction theory is justifiable by evolutionary learning
  Revenue Equivalence Thm                Methodology                         This research

 Assumptions                       Main Idea:                     In accordance with RET
 • Private independent values      • Investigate                  • Private independent values
 • Identical value distributions   quantitatively the auxiliary   • Values ~U(0,1)
 • Risk neutrality                 game where each bidder         • Risk neutrality
 • No budget constraint            can observe each               • No budget constraint
 • Perfect rationality             submitted bid.                 • Evolutionary Learning
                                   • Use the results to arrive
                                   at conclusions about
                                   bounded rational bidding
                                   in sealed bid auctions.
                                                                  Auxiliary assumptions
                                                                  • 2 bidders
                                                                  • Linear bidding functions:
                                                                   bidi   i vi
                                                                  • Each bidder can observe
                                                                  all bids




                                                                                                24
The goal of the quantitative analysis is the characterization of the game
dynamics
              Rules of the Game                            Analysis of Game Dynamics

 The 2 Player Auction Game:                               Strategy Space:
 • Each bidder chooses a strategy  i                             j
 • Each bidder is assigned a private value vi ~ U (0,1)
 • Each bidder submits her bid bidi   i vi                                    NE-2PA
 • Each bidder observes her opponent's bid      jv j         1
 • The higher bidder receives a payoff of
   – in a 1PSBA: POi  (1   i )vi                                    NE-1PA
   – in a 2PSBA: POi  vi   j v j                        1/ 2
 • After R rounds, each bidder chooses a new
   strategy.
   – Myopic: Use strategy that would have                                                  i
                                                                      1/ 2  1
     maximized last round‘s payoff
   – Truncated Fictitious Play: Use strategy that         Questions in Dynamic Analysis:
     would have maximized last R rounds‘ payoff
   – Perfect Memory: Use strategy that would
     have maximized cumulated payoff since
     game start
 • Start from beginning                                                          Statistical
                                                            Convergence?
                                                                                Properties?


                                                                                                25
AGENDA - Auctions



             Background




             Best Response in Auctions




             Simulation




             Conclusion and Outlook




                                         26
Mathematical analysis shows that bidders in repeated first-price auctions
in general fail to coordinate on the NE
             Perfect Memory Best Response                Underbidding in 1PAs
             Function in First-Price Auctions

                                                         Estimation of
   0
     pBR
           ( 1 )                                        Opponent‘s Strategy:
                                                                    1 R   i v i
                                                         E[  i ]               
                                                                    R t 1 0.5
                                                          1 R bi
                                                          
 0.5                                                      R t 1 0.5


                                                          Perfect Memory Best
                                                          Response:
                                                         Slow convergence to the NE
                                                         from below
                                                          Truncated Fictitious
    0                                               1    Play:
                                                         Permanent Fluctuations
        0                     0.5               1
                                                         below the NE

                                                                                       27
Since bidders in second price auctions coordinate in the NE,
the Revenue Equivalence Theorem breaks down
         1PAs                Revenue Equivalence Revisited                2PAs

  • In general, no             First and second price           • Second price open
  convergence of best          auctions under best response     bid auctions under best
  response dynamics to         dynamics do in general not       response dynamics
  NE in first-price open       lead to the same expected        lead to NE bidding
  bid auctions                 seller revenue and final         either by upfront
  • First price sealed bid     allocation of goods              reasoning or by the
  auctions can be                                               best response
  expected even less to                                         dynamics
  converge to the NE                                            • Second price sealed
  since bidders have           The NE - and therefore a         bid auctions converge
  less information             pareto optimal allocation of     to the NE by the same
  available                    goods - is only reached in       reasons
                               second price auctions.
                               Any auction format that
                               incentivizes for value shading
                               leads to suboptimal allocation
                               of goods and to excess
                               volatility



                                                                                          28
AGENDA - Auctions



             Background




             Best Response in Auctions




             Simulation




             Conclusion and Outlook




                                         29
Double-checking the AS with the mathematical theory allows me to get
reliable computational results
      Approach                                     Program Flow
   Complementary            1. Initialization of 1 active and numStrategies-1 inactive
   Assessment:                 strategies
                            2. For numRounds auctions
                             1. Each bidder is assigned a random value vi
   Mathematical results
                             2. Each bidder submits her bid bidi   i vi according to her
   allow double-check of
                                  active strategy
        computational
                             3. The winner is determined
      simulation quality
                             4. Bidders update payoffs of active and passive strategies
                            3. Bidders activate a new strategy for next round’s play
                             1. Best Response: The best strategy is activated
                             2. Quantal Response: Payoff-proportional selection
                                  probability
       Quality-checked      4. If the GA is used:
       program allows        1. The best numElite strategies are kept unchanged for
    investigation of more         the next generation
       general setups        2. The best numParents strategies are taken as parents
                                  to create offspring
                             3. Bidders update their worst numStrategies-numParents
                                  strategies by using mutation and crossover


                                                                                             30
          OBJECT STRUCTURE OF THE AUCTION SIMULATOR

                                                             exper.
            ObserverSwarm                    ExperSwarm
                                                             setup



                                           model.
                              ModelSwarm
                                           setup



bidder.                                                      seller.
                Bidder                          Seller
 setup                                                       setup



bStrat.                                                      sStrat.
             BidderStrategy                 SellerStrategy
setup                                                        setup




             bStratElement                  sStratElement


                                                                       31
The determination of payoffs is the core of the Auction Simulator

       Get Bids and Asks                  Determine Winners                     Determine Payoffs


                                                                                 p1 :bPO1=0.28
                                                                                      sPO2=0.22
   B0:         B1:            B2:
  bv0:0.2     bv1:0.7       bv2:0.3      B1:            B2:           B0:
   0:0.4      1:0.8        2:0.9     b1:0.56    >   b2:0.27   >   b0:0.08     p2 :bPO1=0.265
  b0:0.08     b1:0.56       b2:0.27                                                  sPO2=0.235
                                                  Determine Price
                                                                                 p3 :bPO1=0.14
                                          e.g.,
                                          p1=(0.56+0.28)/2=0.42                  p4 :sPO2=0.08
                                          p2=(0.27+0.6)/2=0.435
                                          p3=0.56 && ai=0
                                          p4=0.28 && bi= 

   S0:         S1:           S2:
  sv0:0.6    sv1:0.5        sv2:0.2      S2:            S1:           S0:
  0:1/1.4   1 :1/1.2     2 :1/1..4   a2:0.28   <    a1 :0.6   <   a0: 0.84    Price Mechanism
  a0:0.84     a1:0.6        a2:0.28
                                                                                 determines
                                                                                 Strategy Updating




                                                                                                     32
Genetic Algorithms allow for the simulation of
truncated Fictitious Play with sufficiently high numRounds
           PO0 (  0 , 1  0.2)

                                                                                       Assumptions:
                                                                                       • numStrategies: 10
                                                                                       • numParents: 5
                                                                                       • numElite: 2




                                                                                                              0
Rank:             9     7                2        1          3       4        5    6     8          10

Choose rank 1 strategy as active for next round

Construct new strategy set
                                         4         5          3       2        1
Sel. Prob. for CO:                      15        15         15      15       15

Crossover: Parent-Pairings: (1,2) (2,4) (3,1) (2,1) (1,4) (3,5) (2,1) (1,3)
Mutation

New Population of Strategies

                                                                                                             33
Fixed Strategies are slower but yield correct results
on all time scales
            Simulation Technique                       Focus of Presentation
             Payoff in One Auction                   Variety of Possible Simulation
                                                     Setups:
                                                     • Myopic Best Response
                                                     •Perfect Memory Best
    0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0          Response
                                                     • Imperfect Memory Best
             Cumulated Payoff                        Response
                                                     • Truncated Fictitious Play
                        POcum1)  POcum  PO (t )
                          (t        (t )
                                                     • Quantal Response


                                                     Simulation
                                                     Assumptions in this
                                                     Presentation
    0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0          • Pure Bidder auctions
                                                     • Best Response Dynamics
             Fittest Strategy                        • First and second price auctions
    0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


                                                                                         34
The Auction Simulator allows for a wide range of
possible setups
                 Key Parameters                         Flexibility

   Model Setup:                                Saving of parameters in
   • numRounds                                 separate files outside the main
   • numBidders                                program allows easy changes
   • numSellers                                of the simulation setup
   • auctionType, priceDet
   • randomSeed
                                               SWARM allows for the fully
                                               automatic generation of series
   Bidder Setup:                               of simulation files with different
   • learningType (FS, GA)                     parameter settings
   • numStrategies
   • selectionType (BR, QR)
   •  (exponential parameter for QR)
   •  (memory strength)
   • fixedStrategyMarker, fixedStrategy
   • fixedValueMarker, fixedValue
   • valDistShape, valDet1, valDet2
   • numElite, numParents, crossoverPar,
   mutationProb, mutationType


                                                                                    35
SIMULATION RESULTS




            Double Checks against the Mathematical Theory




            Dynamic Simulation Results




                                                            36
The simulation of mBR and pBR against fixed strategies is in
accordance with the theory
       Analytic Prediction                         0mBR ( 1 )
                                                                     Simulation Result

 Myopic Best Response:
                                                 0.5
                    1  2 1 ln 1
   0mBR ( 1 )                    
                       4  2 1


 Perfect Memory Best
 Response:
                                                                                             1
                 1                                  0
          1 
  for :                                   0                       0.5        1
                 2                                 0
                                                     pBR
                                                        (   1   )
                    1 ( 81  1  9)
                   4/3    2/3        2     2/3
   0pBR ( 1 )  1                            
                       3( 81  1  9)
                                 2    1/ 3        0.5


                 1
          1 
  for : 
                 2
                   1
   0pBR ( 1 )  
                     
                   2
                                                                                             1
                                                    0
                                                        0                     0.5        1

                                                                                                  37
STRATEGY DISTRIBUTION UNDER MBR AGAINST1  0.3
                Simulated Distribution of   0




                Theoretical Prediction of the Distribution
      f (0 )




                        0.3
                                                         0



                                                              38
SIMULATION RESULTS




            Double Checks against the Mathematical Theory




            Dynamic Simulation Results




                                                            39
The predictions for the behavior of strategies under perfect memory best
response and truncated fictitious play are replicated correctly
              0       Perfect Memory Best Response
        0.5



                                    Slow convergence
                                   towards the NE from
                                          below

          0                                                         t
              0                    50.000                 100.000

              0             Truncated Fictitious Play

        0.5

                                       Permanent
                                   Fluctuations around
                                     mean below NE


          0                                                         t
              1.000                 1.050                  1.100           40
The simulation of myopic Best Response with uniform
value distributions is characterized by clustered volatility
                   Bidders‘ Strategies                                  Intuition


                                         •2 Bidders            General Trend:
                                         •Uniform              Convergence
         0 , 1                                               towards 0
                                         Values
                                                               Lowest Strategy:
       1
                                                               Lowest feasible
                                                               strategy constitutes
                                                               barrier for
                                                               convergence:
                                                               breakout probability:

                                                               P( 0   0 ) 
                                                                         *


                                                                     m inv1
       0                                                  t    P(               0 ) 
                                                                                  *
        1.000              1.050                  1.100                 v0
                                                                m in
                                                               2 0 *




                                                                                          41
                                                           Log (event
                                                           size)
            EFFECT OF VALUE DISTRIBUTION ON
               LOG-LOG-PLOT OF RETURNS                                  Log (event
                                                                        rank)
           N(100,0.001)                      N(100,0.01)
                                    1

10-1



                                  10-1


10-2


                                  10-2
       1                  24000          1                     24000

             N(100,1)                          U(0,100)
  1                               103



                                  102

10-1

                                   10


10-2                                1
       0                  24000          0                     24000


                                                                              42
                                                                 i , vi
In the ramp-up phase the strategies lock each other
in mutual oscillation                                                      t
                 Ramp-up: 50 TimeSteps                       Observations

            0                           v0   •2 Bidders
 0.03                      1.003              •Normal      Oscillations:
                                              Values       Strategies lock each
                                                           other in mutual
                               1                           oscillations



  0.0                      0.997
                                                           Oscillation Mean:
     0                50        0                50        Mean increases
                                                           steadily
            1                           v1
 0.03                      1.003



                               1
                                                           Oscillation
                                                           Amplitude:
                                                           Amplitude varies
                                                           over time
  0.0                      0.997
     0                50        0                50



                                                                                  43
                                                                       i , p
After the ramp-up, the time series of prices
is characterized by repeated crashes                                            t
                    200 TimeSteps                            Observations

            0
   1                                    •2 Bidders          Crashes:
                                        •Normal             Oscillations of
                                        Values              strategies are
                                                            interrupted by
                                                            simultaneous drops
                                         Price
                            1.01                            to lower levels;
                                                            Simultaneous
 0.97
     3900            4100                                   strategy drops lead
                                                            to price crashes
             1
   1                                                        Reason:
                             0.97
                                 3900                4100    i(t 1)   i(t )
                                                            if ( t )
                                                            vi  bti )
                                                                  (



 0.97
     3900            4100



                                                                                    44
Oscillations vanish as the memory strength is increased
                                                                                   f (t )
               memory Strength 0                      memory Strength 0.1
                                                      memory Strength 0.1
     1.0                                    1.0
                                                                                            t

     0.5                                    0.5




     0.0                                    0.0
        4500      4550     4600      4650      4500      4550    4600       4650
               memory Strength 0.5
               memory Strength 0.5                    memory Strength 0.9
     1.0                                    1.0




     0.5                                    0.5




     0.0                                    0.0
        4500      4550     4600      4650      4500      4550     4600      4650


                                                                                            45
                                                                    f (ret )
                  LEFT TAIL OF RETURN DISTRIBUTION
                                                                               ret
            memory Strength 0                      memory Strength 0.1
0.03%                                  0.03%



0.02%                                  0.02%



0.01%                                  0.01%



  0%                                     0%
    -100%                         0%       -100%                          0%

            memory Strength 0.5                    memory Strength 0.9
0.03%                                  0.03%



0.02%                                  0.02%



0.01%                                  0.01%



  0%                                     0%
    -100%                         0%       -100%                          0%


                                                                                     46
                                                                     0 ,  av
Truncated Fictitious Play leads to permanent fluctuations
of bidders‘ strategies                                                           r
                Time Series of Strategies                         Observations

                Individual                 •10 Bidders
      1                                     •Uniform      Individual Strategies:
                                            Values        • Permanent fluctuations
                                                          • Left- Skewed distribution
                                                          • Skewedness increases
                                                          for more bidders

    0.65
       18.500                           18.700

                  Average 
      1
                                                          Average Strategy:
                                                          • Permanent fluctuations
                                                          • Symmetric distribution


    0.65
       18.500                           18.700




                                                                                     47
                                                                        p,  av
The time series of prices and strategies is governed
by an AR(2) process                                                                  r
                Time Series                                       Properties

                Price
     84                                                AR(2)-process:
                                                       p (t )  78 .6 
                                                        0.285 ( p (t 1)  78 .6) 
                                                        0.009 ( p (t  2)  78 .6)   (t )

     72                                                 avt )  0.858 
                                                         (

      18.500                       18.550
                                                        0.344 (  avt 1)  0.858 ) 
                                                                   (

                 Average 
    0.94                                                0.03(  avt  2)  0.858 )   (t )
                                                                 (




                                                       Periods of fiercer
                                                       competition and higher
                                                       prices alternate with
    0.74                                               periods of lower
       18.500                      18.550              competition and lower
                                                       prices


                                                                                               48
AGENDA - Auctions



             Background




             Best Response in Auctions




             Simulation




             Conclusion and Outlook




                                         49
In a multitude of economic settings, a revision of Auction
Formats could significantly lower volatility
                 Auctions in the Economy                                                    Implications
 Economic Environment          S/D* Units Type Values
                                                                                       Value-shading auction
 Agricultural Products         S/D    S    1    A
                                                                                       designs used in many
 Procurement                            S         S      1       P                     economic settings

 Company Takovers                       S         S      1       A
 IPOs / privatization                   S         M      1       A
                                                                                       Replacing these auction
 Mining / Drilling rights               S         S      1       C
                                                                                       designs by truth-telling
 Land Conservation                      S         M      1       P                     formats could reduce
                                                                                       volatility
 Housing Markets                        D         S      1       A
 Wine Auctions                          S         S      2       A
                                                                                       However, caution about
 Art Auctions                           S         S      2       A                     additional constraints
 Treasury Bill Issuing                  S         M      D       C                     necessary (e.g.,
                                                                                       collusion)
 Stock and FX Markets                   D         M      U       C

 Electricity Markets                    D         M      U       C
    * Single or Double Auction; Single-Unit or Multi-Unit; 1PSBA, 2PSBA, Discriminatory or Uniform; Private Value,
      Affiliated Value or Common Value                                                                               50
The long-term goal of this research is to comprehensively investigate
auctions
                3-fold approach                          Final Goal

   Quantitative Analysis:
   Extend analytical investigation of best response
   and quantal response play to multi-unit double
   auctions


                                                          Comprehen-
   Simulations:
                                                         sive investiga-
   Extend Auction Simulator to simulate double-
                                                             tion of
   sided multi-unit auctions with affiliated values;
                                                            Auctions
   Incorporate profit maximizing investors



   Experiments:
   Estimate auction parameters by comparing
   statistics of experimental and empirical data sets
   with statistics of simulation runs in real auctions



                                                                           51
The next step is to extend the Auction Simulator
to nonlinear bidding strategies
                      Value-dependent Strategies       Extended Applications

                                                         Applications of Value-
                                                         dependent Strategies:
                                                         •Realistic simulation of
                                                         mutual adaptation of
    b(v)                         Current Simulation:     value-dependent bidding
                                 Linear Bidding          in repeated auctions
                                 Strategies              • Assessment of Nash
                                                         and Quantal Response
          Future Extension:                              equilibria for non-uniform
          Nonlinear Bidding                              value distributions
          Strategies                                     Further Extensions:
                                                         • Risk Aversion
                                                         • Correlated Values
                                                         • Asymmetric Value
                                                         Distributions
                                                   v     • Multi-unit Auctions
      0        0.25        0.5      0.75      1




                                                                                      52
In a later stage, the Auction Simulator shall simulate the effects of order
book redesign on the volatility and efficiency of financial markets
                         Model Structure                     Hypothesis

 Simulation of Financial Markets:
 • Integration of seller and buyer into one agent: Risk
                                                           The optimal orderbook
 averse investor maximizes utility under budget
                                                           design for financial
 constraints
                                                           markets is a dominant
 • Hierarchy of feedback loops between strategic
                                                           strategy format for
 bidding and updating of price expectations
                                                           multi-unit double
                                                           auctions with affiliated
            Price Prediction                               values:
                                                             – Reduces
           Bidding Strategies                                  occurence of
                                                               bubbles and
                                                               crashes
                                                             – Increases the
 In addition to the permanent mutual updating of price
                                                               efficiency of the
 predictions and bidding strategies, the simulation
                                                               capital allocation
 accounts for the mutual influence of bidding strategies
                                                             – Increases
 and price predictions
                                                               investors‘ wealth



                                                                                      53
Thank You




            54
   54

								
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