ABSTRACT (original article): We investigate the transient and steady-state dynamics of the Bennati-Dragulescu-Yakovenko money game in the presence of probabilistic cheaters, who can misrepresent their financial status by claiming to have no money. We derive the steady-state wealth distribution per player analytically, and show how the presence of hidden cheaters can be inferred from the relative variance of wealth per player. In scenarios with a finite number of cheaters amidst an infinite pool of honest players, we identify a critical probability of cheating at which the total wealth owned by the cheaters experiences a second-order discontinuity. Below this point, the transition probability to lose money is larger than the probability to gain; conversely, above this point, the direction is reversed. We further establish a threshold cheating probability at which cheaters collectively possess half of the total wealth in the game. Lastly, we provide bounds on the rate at which both cheaters and honest players can gain or lose wealth, contributing to a deeper understanding of deception in asset exchange models. CITATION (original article): Kristian Blom, Dmitrii E. Makarov, Aljaž Godec (2025), Phys. Rev. Research 7, 013279. https://doi.org/10.1103/PhysRevResearch.7.013279

Load MaTex package and set color functions


Monte-Carlo simulation for the money game with cheaters

This is the core code which is used to generate trajectories of the money game with honest players and cheaters.
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  • The parameter “Nplayers” sets the total number of players in the game.
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  • The parameter “Ncheaters” sets the total number of cheaters in the game. The number of honest players is then given by “Nplayers-Ncheaters”.
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  • The parameter “Nsteps” sets the total number of discrete steps taken in the simulation.
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  • The parameter “Nmoney” sets the total amount of money in the game.
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  • The parameter “bias” introduces an additional bias in the exchange between two players, which is not studied in the manuscript “Hallmarks of Deception in Asset-Exchange Models”. For “bias > 1/2$” it is more likely that a richer player receives money from a poorer player. For “bias < 1/2$” it is more likely that a poorer player receives money from a richer player. In all sections below we set “bias=1/2”.
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  • The parameter “qc” sets the cheating probability of the cheaters.
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  • The parameter “trackall” determines whether all money is tracked or only the money of a single player. If “trackall = True” the money of all players will be tracked. If “trackall = False” the money of the first player will be tracked. And when “Ncheaters > 0” this player will be a cheater. When one needs to generate a very large trajectory it can be useful to set “trackall = False” to prevent memory issues.
    • moneygame[Nplayers_,Ncheaters_,Nsteps_,Nmoney_,bias_,qc_,trackall_]:=​​Module{playerlist=Range[Nplayers],values=ConstantArray[0,Nplayers]},​​ ​​ (*Setmoneytransactionvalues*)​​ values[[1]]=1;​​ values[[2]]=-1;​​ ​​ (*Initializepath*)​​ Clear[X,Y];​​ path=Array[X,Nsteps];​​​​ (*Setinitialconditionofmoneyperplayer;westartfromauniformdistributionofmoneyperplayer*)​​ temp1=NmoneyNplayers;​​ temp2=Floor[temp1]; ​​ temp3=ConstantArray[temp2,Nplayers];​​ temp3[[Nplayers]]=Nmoney-(Nplayers-1)temp2;​​ If[trackall,X[1]=temp3,{X[1]=temp3[[1]],Y=temp3}];​​​​ (*Generaterandomwalkformoneygamewithcheaters*)​​ Fori=2,i<=Nsteps,i++,(*Selecttworandomplayerswhichwillinterchangemoney*)​​ players=RandomSample[playerlist],​​ (*Createtransactionlist*)​​ dm=Permute[values,players];​​ (*Determineamountofmoneypossessedbythetwoplayers*)​​ {N1,N2}=If[trackall,X[i-1][[players[[1;;2]]]],Y[[players[[1;;2]]]]],​​ (*Arewedealingwithcheatingplayers?Ifso,drawarandomnumberforthecheatingmove*)​​ If[players[[1]]<=Ncheaters,If[RandomReal[]≤qc,N1=0]],​​ If[players[[2]]<=Ncheaters,If[RandomReal[]≤qc,N2=0]],​​ (*Monte-Carlomove*)​​ IfN1==N2,If[N1!=0,If[trackall,X[i]=X[i-1]+dm,Y=Y+dm],If[trackall,X[i]=X[i-1],Y=Y]],​​ If[N2>N1,dm*=-1],​​ (*Thelinebelowcanbechangedtomakethegamesymmetricatm=0:If[RandomReal[]<=bias,X[i]=X[i-1]-dm,X[i]=X[i-1]]*)​​ IfN1==0||N2==0,If[trackall,X[i]=X[i-1]-dm,Y=Y-dm],​​ unfair=RandomReal[]<=bias,​​ If[unfair,If[trackall,X[i]=X[i-1]+dm,Y=Y+dm],If[trackall,X[i]=X[i-1]-dm,Y=Y-dm]],If[!trackall,X[i]=Y[[1]]];​​ (*Outputtrajectory*)​​ path​​

    Analytical expressions

    Here we list the analytical expressions for the wealth distribution which are derived and shown in the manuscript “Hallmarks of Deception in Asset-Exchange Models”.
    These expressions will be used in later sections to plot the wealth distribution per player, and the average money owned by the cheaters and honest players.
    Analytical expressions for the auxiliary functions; see Eq . (12)
    Delta0[m_,q_,phi_]:=1+2*m*q+(1/3)(2*m-phi*q)^2;​​Delta1[m_,q_,phi_]:=(2/27)*(2*m-phi*q)^3+(1/3)*(2*m-phi*q)(1+2*m*q)+phi*q;
    Steady - state probabilities at zero; see Eqs . (10) and (8), respectively
    p0h[m_,q_,phi_]:=2*Sqrt[Delta0[m,q,phi]/3]Cos[(1/3)ArcCos[-(3/2)(Delta1[m,q,phi]/Delta0[m,q,phi])Sqrt[3/Delta0[m,q,phi]]]]+(phi*q-2m)/3;​​p0c[m_,q_,phi_]:=(p0h[m,q,phi]-q)/(1-q);
    Steady - state probabilities; see Eqs . (11) and (9), respectively
    pmh[k_,m_,q_,phi_]:=If[k>0,2*p0h[m,q,phi]*((1-p0h[m,q,phi])/(1+p0h[m,q,phi]))^k,p0h[m,q,phi]];​​pmc[k_,m_,q_,phi_]:=If[k>0,2*((p0h[m,q,phi]-q)/(1-q^2))*((1-p0h[m,q,phi])*(1+q)/((1+p0h[m,q,phi])*(1-q)))^k,p0c[m,q,phi]];
    Steady - state probabilities in the large m limit; see Eqs . (16) and (17), respectively
    p0hlim[m_,A_,phi_]:=(1+2A+Sqrt[1+4A(1+A-2phi)])/(4m)​​p0clim[m_,A_,phi_]:=(1-2A+Sqrt[1+4A(1+A-2phi)])/(4m)​​pmhlim[k_,m_,A_,phi_]:=If[k>0,2*p0hlim[m,A,phi]Exp[-2p0hlim[m,A,phi]*k],p0hlim[m,A,phi]]​​pmclim[k_,m_,A_,phi_]:=If[k>0,2*p0clim[m,A,phi]Exp[-2p0clim[m,A,phi]*k],p0clim[m,A,phi]]
    Average money owned by a honest player or cheater; see Eqs . (26) and (25), respectively
    mh[m_,q_,phi_]:=(1-p0h[m,q,phi])(1+p0h[m,q,phi])/(2p0h[m,q,phi]);​​mc[m_,q_,phi_]:=(1-p0h[m,q,phi])(1+p0h[m,q,phi])/(2(p0h[m,q,phi]-q));
    Variance of total distribution
    varm[m_,q_,phi_]:=m*
    1
    p0h[m,q,phi]
    +
    1-
    2
    p0h[m,q,phi]
    p0h[m,q,phi]-q
    +
    -1+
    2
    p0h[m,q,phi]
    p0h[m,q,phi]-phiq
    -m^2;
    Asymptotic solutions to Eq . (B13)
    Collect[AsymptoticSolve[phi*(1-x)(1+x)/(2*x)+(1-phi)*(1-x)(1+x)/(2*(x-q))==m,x,{m,Infinity,2},Assumptions->{0<=phi<=1}],m]​​Collect[AsymptoticSolve[x^3+x^2(2*m-a*phi*m^(-B))-x(1+2*a*m^(1-B))+a*phi*m^(-B)==0/.{B->3/2},x,{m,Infinity,3},Assumptions->{0<=q<=1,0<=phi<=1,2/3<B<1}],m]​​FullSimplify[AsymptoticSolve[x^3+x^2(2*m-phi*q)-x(1+2*m*q)+phi*q==0,x,{m,Infinity,2},Assumptions->{0<=phi<=1,0<=q<=1}],Assumptions->{m>=0,0<=q<=1,0<=phi<=1}]
    Out[]=
    x
    phi
    2m
    +
    -phi+
    2
    phi
    4
    2
    m
    q
    ,xq+
    1-phi-
    2
    q
    +phi
    2
    q
    2m
    +
    phi-
    2
    phi
    -2
    2
    q
    +2phi
    2
    q
    +2
    4
    q
    -3phi
    4
    q
    +
    2
    phi
    4
    q
    4
    2
    m
    q
    ,x-2m-2
    q
    2
    -
    phiq
    2
    
    Out[]=
    x
    aphi
    3/2
    m
    +
    -2
    2
    a
    phi+2
    2
    a
    2
    phi
    2
    m
    +
    4
    3
    a
    phi-12
    3
    a
    2
    phi
    +8
    3
    a
    3
    phi
    5/2
    m
    if (aphi|a)∈
    ,x
    1
    2m
    +
    2a-2aphi
    2
    3/2
    m
    +
    4
    2
    a
    phi-4
    2
    a
    2
    phi
    2
    2
    m
    if (aphi|a)∈
    ,x
    -
    1
    2m
    -2m+
    -2a+2aphi
    2
    3/2
    m
    if (aphi|a)∈
    
    Out[]=
    x
    phi(-1+phi+2mq)
    4
    2
    m
    q
    if q>0
    ,x
    -((-1+phi)phi)-2m(-1+phi)q+2(-1+2
    2
    m
    +phi)
    2
    q
    +2m(-1+phi)
    3
    q
    +(-2+phi)(-1+phi)
    4
    q
    4
    2
    m
    q
    if q>0
    ,x
    -2m+(-1+phi)q if q>0
    

    Plot money game simulation (used in Fig1. a-c)

    Set constants
    Nplayers=10;​​Ncheaters=5;​​Nsteps=2500;​​Nmoney=100;​​bias=0.5;(*Thisparameterintroducesabiasintheexchangewhenitisnotequalto1/2*)​​qc=0.2;
    Plot trajectories
    ListPlotTranspose[moneygame[Nplayers,Ncheaters,Nsteps,Nmoney,bias,qc,True]],PlotRange->{{0,Nsteps},All},​​ Joined->True,Filling->False,​​ PlotStyle->{{Thickness[0.009]},{Thickness[0.009]}},​​ FrameTrue,FrameStyleDirective[Thick,Black],FrameLabelMaTeX/@{steps,money},​​ FrameTicks->{{Table[{x,MaTeX[x,"DisplayStyle"->False,FontSize->24]},{x,Range[0,Nmoney,25]}],None},{Table[{x,MaTeX[x,"DisplayStyle"->False,FontSize->24]},{x,Range[0,Nsteps,Nsteps/2]}],None}}
    Out[]=
    Set constants
    Vertex translation; this names the vertices as a tuple (i, j, k)
    Construct graph
    Animate random walk
    A video of the output of the above animation:

    Plot steady-state distribution of money game with 3 players (used in Fig. 1d-f)

    Set constants
    Run trajectory
    Vertex translation; this names the vertices as a tuple (i, j, k)
    Construct graph + edge list + vertex list
    Construct table with steady - state probabilities
    Find maximum occupancy probability to make color scheme more apparent
    Optional : Partition trajectory into consecutive jumps to determine steady - state current
    Optional : Construct table with steady - state fluxes
    Optional : Show steady - state fluxes by adding arrows
    Plot steady state distribution; if you want to also see fluxes change graph to graph2

    Plot steady state distribution of wealth (used for Fig. 3a-b)

    Set constants
    Run trajectory
    Plot histograms from numerical data
    Plot analytical results given by Eq . (17)

    Plot steady state distribution of wealth in the large m limit (used for Fig. 3c)

    Set constants
    Run trajectory
    Plot histograms from numerical data
    Plot analytical results given by Eq . (17)

    Plot average amount of money owned by the cheaters vs. the cheating probability qc (used in Fig. 4a-b)

    Set constants
    Determine threshold and critical cheating probability; See Eqs . (28) and (13), respectively
    Run trajectories; this part will take a while!
    Plot average money

    Plot average amount of money owned by the cheaters upon decreasing the fraction of cheaters (Fig. 4c)

    Set constants
    Set qcrit
    Plot average money owned by the cheaters; here we observe a second - order discontinuity

    Plot speed limits for gaining and losing money vs. number of steps (Fig. 5a)

    Set constants
    Set number of independent simulations; the larger this is, the longer it takes
    Initialize array containing amount of money owned by the cheaters per step
    Run simulations for various values of the cheating probability, and collect the money owned by the cheaters
    Normalize
    Compute upper bound given by Eq . 36 in the manuscript
    Plot results

    Plot speed limits for gaining and losing money vs. fraction of cheaters (Fig. 5b)

    Set constants
    Set number of independent simulations; the larger this is, the longer it takes
    Initialize array containing amount of money owned by the cheaters per step and per fraction of cheaters
    Compute average money for cheater
    Normalize
    Set theoretical bound
    Plot result

    CITE THIS NOTEBOOK

    Hallmarks of deception in asset-exchange models​
    by Kristian Blom, Dmitrii E. Makarov & Aljaž Godec
    Wolfram Community, STAFF PICKS, March 11, 2025
    ​https://community.wolfram.com/groups/-/m/t/3414673