Gambler's Ruin

Use a discrete Markov process to model a repeated gambling game

In statistics, "gambler's ruin" is the fact that a gambler playing a game with negative expected value long enough will eventually go bankrupt.

A gambler, starting with 40 chips, places a one-chip bet at each step, with a winning probability of p=0.4 and a goal of winning 70 units before going broke. The states here are the integers 1 through 71, representing the gambler's wealth plus one:

In[1]:=

GamblersRuin[p_,n_]:=SparseArray[{{1,1}1,{n+1,n+1}1,{i_,j_}/;1<i<n+1&&ji+1p,{i_,j_}/;1<i<n+1&&ji-11-p},{n+1,n+1}]

In[2]:=

gamblerwealth=DiscreteMarkovProcess[40,GamblersRuin[0.4,70]];

Simulate the process:

In[3]:=

SeedRandom[2345];ListLinePlot{#〚1〛,#〚2〛-1}&/@RandomFunction[gamblerwealth,{0,150}]["Path"],



Out[3]=

Find the expected number of times the gambler has

1≤n≤6

units:

In[4]:=

Transpose[{MarkovProcessProperties[gamblerwealth,"TransientClasses"]〚1〛-1,MarkovProcessProperties[gamblerwealth,"TransientVisitMean"]}]〚1;;6〛//TableForm

Out[4]//TableForm=

1	1.66666
2	2.77777
3	3.51851
4	4.01233
5	4.34155
6	4.56103

Compare the above result to the one determined from a simulation:

In[5]:=

data=RandomFunction[gamblerwealth,{0,10^3},10^4,Method{Automatic,"StoppingFunction"Function[{len,pos},len<1001&&pos≠1&&pos≠71]}]["ValueList"];

In[6]:=

Table[Mean[Count[#,i]&/@data],{i,2,7}]//N

Out[6]=

{1.6508,2.73,3.4422,3.9235,4.2653,4.496}

Find the expected time until the gambler reaches his goal or goes broke:

In[7]:=

Mean[FirstPassageTimeDistribution[gamblerwealth,{1,71}]]

Out[7]=

194.999

Plot the wealth trajectories of three simulated gamblers over time and compare with this expected time of play:

In[8]:=

SeedRandom[122];ListLinePlotTable[{#〚1〛,#〚2〛-1}&/@RandomFunction[gamblerwealth,{0,250}]["Path"],3],



Out[8]=

Find the average total number of states visited before the gambler reaches his goal or goes broke:

In[9]:=

MarkovProcessProperties[gamblerwealth,"TransientTotalVisitMean"]

Out[9]=

40.9999

Verify the answer using simulation:

In[10]:=

data=DeleteDuplicates/@RandomFunction[gamblerwealth,{0,10^3},10^5]["ValueList"];

In[11]:=

Mean[Length[#]-1&/@data]//N

Out[11]=

40.9939

Find the probability of winning via two different methods:

In[12]:=

PDF[gamblerwealth[Infinity],71]

Out[12]=

3.47673×

-6

In[13]:=

Probability[x[Infinity]71,xgamblerwealth]

Out[13]=

3.47673×

-6

Note that as the gambler's goal is increased for fixed

p<1/2

, this probability tends to zero, demonstrating the concept of gambler's ruin.

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Gambler's Ruin

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Gambler's Ruin

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