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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 bi 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 AGAINST1 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 bti ) ( 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