Neil F. Johnson, Sean Gourley, Sehyo C. Choe, David Smith Pak Ming Hui Oxford University Chinese University of Hong Kongn.johnson@physics.ox.ac.ukMulti-Agent Populations & Networks:Competition for Limited ResourcestheoryCrowd-Anticrowd theory analytic, many-body description of dynamical correlationsnumerical simulationsBinary Agent Resource modelsreal worldbiologysociologySee also Financial Market Complexity (Oxford University Press, 2003)

updated history at time t +1 . . . . 1 0S history at time t. . . . 0 1agent memory m = 2action+1

action-1

historiesstrategiesSfdbe ca with resource level reward structure2 possible winning outcomesfdbe ca N-agent network

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updated history at time t +1 . . . . 1 0S history at time t. . . . 0 1agent memory m = 2action+1

action-1

historiesstrategiesSfdbe ca with resource level fdbe ca N-agent network

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Complex agent-based dynamic networke.g. B-A-R Binary Agent Resource system

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+ 2What affects the typical fluctuation size and hence efficient use of global resource? systems time evolutiontimeglobal resource levelL[t] = L

coin-tosscrowd - anticrowd pairs executeuncorrelatedrandom walks

sum of variances theory ~ numerical

.. also works for general B-A-R systemswalk step-size# of walksMinority Game no networkChallet and ZhangLarge crowds >> 0 wastage Wastage reduced by smaller crowds, using e.g. stochastic strategy choice mixed-ability population: large m and small m better for large m and small m agentstypical fluctuation size

Memory m

2m+1 > N.s

Crowd

size

large

medium

~ 1

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~ 0

Net crowd anticrowd pair size

large

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small

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# crowd - anticrowd pairs

~ 2m

inter-agentnetworkconnectionsKey to collective dynamics: Heterogeneity in strategies within population

single macrostate many microstatesstrategy space

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strategy scores strategy allocation matrix small m crowded regimeSee cond-mat/0306516no networkCrowd-Anticrowd theory typical fluctuation size

increasing K (

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( increasing K

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Basic Minority GameAnalytic Crowd-Anticrowd Theory vs. Numerical Simulation vs. mThermal Minority Game vs. MG: Challet and Zhang, Numerical results: Savit et al.Numerical results: Garrahan, Sherringtontypical fluctuation size

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Data

Theta

Standard Deviation

Sigma vs Theta. m=2, N=101, s=2, 10 runs each, 10000 initialisation time. 1000 timesteps.

2011.452342823941.23211.701333046

2010.937990658241.85311.1561775208

2010.034175834841.6411.2141227722

2011.801070513541.09212.380887031605.025

2011.855697261640.93113.11353492740.55.025

209.724122508442.16411.2935819626

2012.36182894340.95112.0363950729

2014.978630089538.72312.0985565186

208.727781744243.21612.1533098221014.46

209.732611351742.51512.25303840640.3753.615

20.05000000079.712122596342.68810.4344301224

20.050000000710.453720882141.63511.095425605811.1606251729011.16063

20.050000000713.072830561940.35211.85569763180.42.232126

20.050000000712.490196355740.61311.6240177155

20.050000000711.092837826841.48610.6346502304010.22710.933

20.050000000712.344082646240.6711.11727333070.3753.06812.73325

20.050000000710.739708624641.92911.4400157928

20.05000000078.416700115943.33510.9239749908

20.050000000710.4237063342.00310.6311531067

20.05000000079.971179690742.1610.6322088242

20.100000001510.063611492742.6910.30536174770.44.71

20.10000000158.571550120743.6999.65476608280.474.71

20.10000000159.398741210242.9489.9545192719

20.100000001510.518966139742.26910.6035480499

20.10000000158.169194313743.7378.024096489

20.10000000158.141147233543.6947.8896183968

20.10000000159.178193680642.8538.8257369995

20.10000000157.756556205244.1379.498925209

20.100000001510.169337019542.5849.9438714981

20.10000000159.881733079242.6768.9759407043

20.1500000067.197972687644.4897.6558947563

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20.1500000068.740503211943.4118.6527614594

20.1500000069.122731182342.9218.9332408905

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20.1500000068.461891254543.4189.5630645752

20.1500000068.864851131643.3448.4588537216

20.1500000067.550165892544.1527.8529996872

20.1500000069.079892314643.3349.5449399948

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20.2000000036.2243.6498.137858390820.2000000037.506553693244.4297.4593300819

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20.2000000037.001715791444.7746.816038608620.2000000036.672219909344.9717.15467643748.1055351273

20.2000000037.555798441644.3037.467988967920.2000000038.002814820143.9357.31675672537.0453983307

20.254.0944.7576.606594562520.2000000036.644482609645.0857.03109836588.0606435211

20.256.010418049145.0526.349344253520.2000000037.240540409844.5697.42189788827.9951812314

20.257.522206031744.8156.848714828520.2000000039.234352888743.1538.17900371557.0017157914

20.257.152283212744.6656.910768032120.2000000037.120372160244.6357.07871484767.5557984416

20.257.012358602744.6196.936688920.2000000037.325412047244.6437.5550384521

20.256.345807219745.6575.773780345920.2000000037.125923012444.6566.8447227478

20.256.353374329145.8866.221004009220.256.595111484745.1416.50541543966.9869748688

20.256.604666654844.9347.061967849720.256.687280442745.2686.22164487846.6947606245

20.256.191796051144.4167.25683164620.256.457297940845.3446.44643592837.0547692925

20.255.77168193145.8075.304504871420.256.591088107645.0896.24893569957.0937522737

20.3000000119645.6965.739013671920.257.018708019444.9036.41911602027.310913038

20.30000001193.719386217447.5573.597972869920.256.025471459345.535.58476305015.9987486183

20.30000001196.274740821145.3955.995621204420.254.095579784847.3053.76179575925.8357566824

20.30000001195.402516745746.1544.950710773520.256.010418049145.7355.65671348576.84169634

20.30000001195.109315522246.3675.176528453820.257.522206031744.4747.1391463287.6118289484

20.30000001195.75309465445.8595.422694206220.257.152283212744.816.84326314935.9214917094

20.30000001194.281276600447.1624.060286521920.257.012358602744.7837.2155389786

20.30000001195.13618344546.3554.96937942520.256.345807219745.4795.9583063126

20.30000001194.564080502246.8514.290552616120.256.353374329145.3786.0876870155

20.30000001195.287149889546.2285.0891351720.256.604666654845.1676.1576285362

20.3499999943.564646785747.6493.318047761920.256.191796051145.4445.9410123825

20.3499999944.515128612446.9444.148738861120.255.77168193145.8945.4761943817

20.3499999944.309939161547.0663.9248044491

20.3499999944.082482904647.2283.8751518726

20.3499999944.316437377247.0564.1162133217

20.3499999943.993864162847.263.7646846771

20.3499999943.827878348747.4523.4680650234

20.3499999945.008701137746.4634.7527279854

20.3499999945.10147282946.3934.6890945435

20.3499999945.13920338346.3924.8915257454

20.4000000063.815699091947.493.5871806145

20.4000000064.033766387847.2653.9443645477

20.4000000063.733622897447.5683.4688887596

20.4000000064.108147621147.2543.8531126976

20.4000000063.609574632747.6113.3803079128

20.4000000063.7550147.4793.5890445709

20.4000000064.298892936647.074.0588526726

20.4000000063.649152821747.4883.4593605995

20.4000000063.641601469347.6513.3699784279

20.4000000063.307860629247.8833.0911180973

20.45000001793.667349103847.5783.4445717335

20.45000001793.694949614447.5953.4980783463

20.45000001794.082237703147.1863.851051569

20.45000001793.190168832142.4895.2215027809

20.45000001793.675936957147.3173.5964541435

20.45000001794.02146393447.323.801006794

20.45000001794.471800196546.6424.4075722694

20.45000001793.741657386847.5173.4685122967

20.45000001793.712921462447.5563.4344334602

20.45000001793.517580613647.4933.7835769653

20.54.14018612447.2083.6917059422

20.53.75527656847.5673.509649992

20.54.537244382546.8214.3070673943

20.54.670418911946.5864.3154673576

20.53.921144651247.3563.6650247574

20.55.037596388946.5455.2526192665

20.55.154859122545.0554.8535575867

20.54.808754945346.2735.4529633522

20.54.022335040647.1934.3927121162

20.53.830884450847.3413.78166937834.38787005864.3222437143

2011.286424738741.43711.8231487274

2010.803458460841.36411.8470277786

2012.447969792440.93912.234703064

2013.292693319940.08312.2289390564

2011.55904890640.89210.9024324417

2012.47018466240.44411.8994188309

208.934754044242.7611.7409887314

2012.497487234940.84212.6774358749

209.618810816442.10110.2713088989

2011.485944755541.46112.7322511673

2012.514656072640.48310.8650932312

2010.835747016941.0213.4338359833

2013.681088120939.79312.7481536865

2013.097348607340.36711.8237257004

2010.35462303642.27912.1890010834

2014.041872174839.65711.9255285263

20.050000000710.620489385841.85210.933588028

20.05000000079.507386096642.86210.9006271362

20.050000000712.235346488940.62510.68060112

20.05000000078.986866181542.28511.1254901886

20.05000000079.826064906742.08410.2922744751

20.050000000713.988376413139.79910.4562015533

20.050000000710.297768886741.9919.3272209167

20.050000000712.277285193940.83212.2149114609

20.050000000711.685099124241.10510.7457675934

20.050000000711.386467918241.57210.7566709518

20.050000000710.310640586841.80310.1847162247

20.05000000079.408268465342.5311.4091625214

20.050000000710.137147614742.30310.3744792938

20.050000000713.171464895339.91211.5755138397

20.050000000712.673303145340.68211.4133815765

20.050000000710.819290968342.17211.2560110092

20.10000000159.801636193642.1549.9888839722

20.100000001510.090779148542.4629.9201688766

20.10000000159.851245964242.4129.5154037476

20.10000000158.718084939143.4359.8796567917

20.100000001511.903852389140.71511.5042982101

20.100000001511.732248107141.09210.0585975647

20.10000000159.987980764942.31410.6448564529

20.100000001510.387728572941.9119.5128974915

20.10000000159.429576709342.999.6817407608

20.100000001510.820308640741.92911.1436071396

20.100000001510.879998822441.6129.5818548203

20.10000000157.697288369844.1097.8477406502

20.100000001510.319471473541.88610.2342510223

20.10000000159.502699684942.7929.1073064804

20.10000000158.9449993343.5887.7024884224

20.10000000159.105707765743.2049.7578277588

20.1500000068.414321172943.5228.7386398315

20.1500000068.114977070443.9278.0381851196

20.1500000067.197416397844.7297.3996882439

20.1500000066.154256646845.5496.1522307396

20.1500000068.610176254643.6158.7582798004

20.1500000068.120156213843.6468.0592374802

20.1500000068.957971045143.368.3239822388

20.1500000068.870382386643.4038.9321737289

20.1500000069.123005493243.1258.0482654572

20.1500000068.865020506943.3628.1440954208

20.1500000067.187535089444.7817.2042975426

20.1500000068.938002455143.228.1159009933

20.1500000069.205473412143.0539.4639205933

20.1500000069.18457164842.9578.3363161087

20.15000000610.178192157542.2529.0680036545

20.15000000610.315687733442.4059.7610549927

20.2000000036.961426682944.8727.0382413864

20.2000000035.52045971645.9755.7950849533

20.2000000036.884213273344.8777.0114507675

20.2000000038.134504912443.8227.6949319839

20.2000000037.329988317144.5777.1378865242

20.2000000035.476128923645.9915.4528465271

20.2000000038.051262188244.0467.9481811523

20.2000000037.125642060544.9077.0738019943

20.2000000036.605651719145.0256.9926986694

20.2000000037.423416134644.437.0551638603

20.2000000037.210408448244.3667.7318472862

20.2000000039.014835953443.2067.9914922714

20.2000000038.073114341843.8647.5971570015

20.2000000036.605803254545.1116.1570901871

20.2000000038.920233773343.3738.2444601059

20.2000000036.293775624545.3016.5510692596

20.256.489384571145.2186.7355051041

20.255.787183381545.785.7914791107

20.256.750375364945.0976.8324108124

20.255.342147371546.245.5347003937

20.256.208908966245.456.1591701508

20.256.256368127145.4766.2435660362

20.255.388788271946.145.17258358

20.255.850318472245.7625.4265151024

20.256.554002971845.1536.3393788338

20.256.057945088845.6095.7443213463

20.257.201448525244.6546.6022996902

20.256.730772860845.1916.471092701

20.256.003419112645.615.8353133202

20.254.937750639946.564.5954031944

20.256.263563860345.4296.233830452

20.255.101767147646.424.8678178787

20.30000001195.165722087446.4634.9428601265

20.30000001195.293110334946.3535.0574178696

20.30000001195.532323798146.0755.4961004257

20.30000001195.763264284745.9225.3374881744

20.30000001194.289568789647.0484.2142000198

20.30000001196.122784373845.5235.5166106224

20.30000001194.446996264447.0454.4974293709

20.30000001196.941987553444.9996.7476935387

20.30000001194.171498699747.1833.9891982079

20.30000001195.537297324745.9585.4849176407

20.30000001196.149782095345.5365.6301388741

20.30000001195.454333116946.1355.1747932434

20.30000001195.663751237345.9155.3024377823

20.30000001194.774284636946.6844.4435706139

20.30000001194.667202886446.7464.86916399

20.30000001195.681133273945.9415.504650116

20.3499999944.420015445346.9794.2762932777

20.3499999945.165625197846.384.8721122742

20.3499999944.920997481846.5524.6668043137

20.3499999944.660227217446.8494.3479809761

20.3499999944.48755369246.8954.4347081184

20.3499999944.382602789147.0794.1320853233

20.3499999944.326860492547.0863.8697800636

20.3499999944.364062905747.0794.0263633728

20.3499999943.875179610947.4373.5934839249

20.3499999943.923569088847.4473.5010643005

20.3499999944.997296566446.484.6319522858

20.3499999944.449134162446.9424.1215758324

20.3499999945.185449728746.2724.8251366615

20.3499999943.698605099247.5783.4048323631

20.3499999943.855757625547.4923.4258949757

20.3499999944.499026699346.6024.796394825

20.4000000063.807426492647.5153.4833283424

20.4000000064.531173306346.8384.2341041565

20.4000000064.441590675446.9284.134829998

20.4000000063.991733199247.3593.6859641075

20.4000000063.801374810647.373.6589722633

20.4000000063.893476487647.3793.5907785892

20.4000000063.901053475447.1713.816075325

20.4000000063.792807058847.5323.5018589497

20.4000000063.758607072747.483.4288959503

20.4000000063.825131583647.4893.6356298923

20.4000000064.693295966346.7784.2446045876

20.4000000063.801638127547.3673.789347887

20.4000000064.015485540347.2913.8495719433

20.4000000064.01498693947.3483.7864565849

20.4000000064.027930513847.3273.6717426777

20.4000000063.755942905247.5323.45302248

Crowd-Anticrowd theory with network typical fluctuation size

m = 2

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

_984623702.unknown

_984623730.unknown

_984274705.unknown

Analytic Crowd-Anticrowd Theory vs. Numerical Simulation

e.g. 2 dimers, 4 monomersnumber of n-mers at given pProof that minimum in wastage can occur at low connectivity pCrowd-Anticrowd Theory at low p: n-mer gas

Details of Crowd-Anticrowd theory . . .

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Re-label strategies according to virtual-point ranking

score

time t

and hence

, are changing rapidly in time

but

isnt !!

highest-scoring strategy K=1

2nd highest-scoring strategy K=2

etc. .

_953171323.unknown

_953171366.unknown

_953178381.unknown

_953178538.unknown

_953235087.unknown

_953178497.unknown

_953178284.unknown

_953171345.unknown

_953171357.unknown

_953171333.unknown

_953171293.unknown

_953171303.unknown

_953170556.unknown

m1 = 3 m2 = 6

Neil F. Johnson, Sean Gourley, Sehyo C. Choe, David Smith Pak Ming Hui Oxford University Chinese University of Hong Kongn.johnson@physics.ox.ac.ukMulti-Agent Populations & Networks:Competition for Limited ResourcestheoryCrowd-Anticrowd theory analytic, many-body description of dynamical correlationsnumerical simulationsBinary Agent Resource modelsreal worldbiologysociologySee also Financial Market Complexity (Oxford University Press, 2003)