ESTIMATING THE PROBABILITY THAT THE GAME OF MONOPOLY NEVER ENDS. Eric J. Friedman Shane G. Henderson Thomas Byuen Germán Gutiérrez Gallardo

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1 Proceedigs of the 2009 Witer Simulatio Coferece M. D. Rossetti, R. R. Hill, B. Johasso, A. Duki, ad R. G. Igalls, eds. ESTIMATING THE PROBABILITY THAT THE GAME OF MONOPOLY NEVER ENDS Eric J. Friedma Shae G. Hederso Thomas Byue Germá Gutiérrez Gallardo School of Operatios Research ad Iformatio Egieerig Corell Uiversity Ithaca NY 14853, USA ABSTRACT We estimate the probability that the game of Moopoly betwee two players playig very simple strategies ever eds. Four differet estimators, based respectively o straightforward simulatio, a Browia motio approximatio, asymptotics for Markov chais, ad importace samplig all yield a estimate of approximately twelve percet. 1 INTRODUCTION Have you ever played a game of Moopoly ad, after a few hours, started to woder if the game would ever ed? We foud that o-edig games were a distict possibility while coductig simulatios to try to determie effective strategies for the game. So we aturally became curious: What is the probability that the game of Moopoly ever eds? By ever edig we mea over a ifiite horizo, ad ot just that the game lasts loger tha reasoable players might be willig to play. It is ot easy to estimate this probability usig stadard simulatio, sice oe ca ever be certai that a game that is still goig will ot evetually ed. The mechaism by which games ca go o forever has bee uderstood at least sice Lehma ad Walker (1975). Essetially, the game reaches a stage where players cotiue to accumulate cash idefiitely. Although there is a chace, at ay poit i time, that ay player could fall o a ru of bad luck ad lose their cash, this probability is small eough that the player s wealths simply grow to ifiity. There is o shortage of cash i the game, sice the rules of Moopoly (Fritzlei 2009) state that if the bak ever rus out of moey, oe ca simply prit more o extra sheets of paper. The probability α that the game goes o forever depeds o the strategies that the players adopt. We detail our assumptios about the strategies the players adopt i Sectio 2. For ow, suffice it to say that we use very simple strategies that ivolve buyig properties wheever possible ad buildig houses or hotels wheever possible, while still maitaiig a modest threshold of cash o had to deal with cotigecies. I this paper we give four differet simulatio-based estimators of α, assumig two-player games. The estimators are built from a combiatio of a extremely detailed simulatio model coded i Java, ad a approximatig Markov chai model. Surprisigly (we thik) there have bee few serious academic studies of the game of Moopoly. Our survey revealed a few relevat books, icludig Lehma ad Walker (1975) i which two (the) Corell udergraduates studied strategy usig what were, at the time, state of the art simulatio techiques playig may moopoly games. They also performed some aalysis based o Markov chais. Somewhat more recetly, Darziskis (1987) simulated may moopoly games to derive good strategies. More recet work was spurred by Stewart s aalysis i Scietific America (Stewart 1996a, Stewart 1996b), which describes the use of Markov chais i the aalysis. This led to several aalyses o websites (although such aalyses have o doubt occurred sice the game was itroduced). For example, oe computes accurate site-visitig probabilities for Markov chais (Collis 2009) ad aother applies these to develop a fiace-ispired aalysis of good strategies (Darlig 2009). Lastly, for referece, the rules ad other facts about moopoly ca be foud o Fritzlei (2009).

2 While may have cursed the legth of moopoly games, we have ot foud ay detailed aalyses of the game s legth, or ay studies that compute the probability that the game goes o forever. But surely, millios of players have thought that the game might ever ed. Sectio 2 describes the simulatio model ad the player strategies we adopt. The, i Sectio 3 we describe a simple estimator of a upper boud o α. Sectio 4 explais i more detail how games ca go forever, describig wealth plots as a coceptual basis for uderstadig how this arises. It also explais how we ca view games as cosistig of potetially two stages. Oe ca the estimate the probability α that the game goes o forever by coditioig o first reachig Stage 2 ad the state of the board upo reachig Stage 2. Sectio 5 gives a approximatio for the coditioal probability based o a Browia approximatio of Stage-2 dyamics of player wealths. The coditioal probability is istead estimated usig asymptotics for Markov Chais usig a stadard Markov chai model of Stage 2 dyamics i Sectio 6. I Sectio 7 we implemet a importace samplig estimator usig the Markov chai asymptotics. Sectio 8 rouds out the paper with discussio of the results ad a outlie of future research efforts. 2 THE SIMULATION MODEL AND PLAYER STRATEGIES We have developed a very detailed simulatio model of the game of Moopoly i Java. The simulatio model is object based, ad ca accommodate more tha two players, although we restrict attetio to the two-player game i this paper. To verify the model we have performed a large umber of detailed game traces, ad compared the simulatio model s predictios with available results o, e.g., frequecies of ladig o particular squares, average et cash ifluxes from the bak per tur, ad so forth. We have also compared the model s results with those of a Markov chai model that is detailed i Sectio 6 below. The Markov chai model eeds to have a small state space to allow computatios to be performed quickly. Accordigly, throughout the remaider of this paper, both the simulatio ad the Markov chai operate uder the followig assumptio. I the real game of Moopoly, a player s tur ca ivolve rollig the dice up to three times, depedig o the umber of doubles rolled. I our versio of the game, each player oly rolls the dice oce, ad doubles are treated exactly the same as ay o-double roll. Players must pay ret ad so forth whe rollig doubles, ad this is i accordace with the official rules of the game. Each step of the Markov chai correspods to oe roud, i.e., oe roll of the dice for each player. We will measure time i rouds rather tha turs i the remaider of this paper. Uder this assumptio, the 3 doubles i a row leads to jail rule is igored. The impact of this assumptio is that players go to jail less ofte tha i the real game. However, the impact seems slight, sice players roll 3 doubles i a row oly oce every 216 turs, or oce every 258 rolls. Our players play a commo, simple strategy. That strategy ca be approximately characterized as follows. 1. Players maitai a reserve threshold of cash that is easily varied but we set to the maximum of $200 ad the maximum ret o a property owed by aother player. 2. Players buy every property they lad o that is for sale, uless doig so would reduce their cash below the reserve threshold. 3. Players ever bid o properties that are up for auctio. 4. Players build o streets as log as they have the cash o had to do so, buildig o the more expesive properties whe they have a choice. Buildig is always doe evely, so that for example, if 3 properties form a Moopoly, the either 3 houses are purchased or oe are purchased. 5. Whe a double is throw while i jail, the player comes out ad advaces their toke by the double that was rolled, ad this completes their tur. (They do ot roll agai.) 6. A player does ot pay to get out of jail, uless he/she has rolled o-doubles o three successive attempts. After payig to get out of jail, the last of the three rolls is take as the player s roll. 7. Whe a player receives a Get out of Jail Free card, the card is immediately sold back to the bak for $ Players do ot trade properties. 9. Whe a player receives a Advace toke to your choice of railroad card, the player moves to the railroad they are curretly o or, if ot curretly o a railroad, moves to the oe immediately precedig them. This will result i them receivig $200 for passig Go uless they are o the first Commuity Chest square after Go. 3 A SIMPLE ESTIMATOR Our first estimator of the probability α that a game goes o forever is very atural ad simple. Simulate a large umber of games, keepig track of the fractio of games that are still goig after rouds have bee completed, for various values of

3 Figure 1: Estimated fractio of games still goig after rouds have bee completed. The solid lie gives the estimate ad the dashed lies represet (poitwise) 95% cofidece itervals.. The look for a horizotal asymptote o the graph. We took exactly this approach with our simulatio model, simulatig 3100 games. Abbreviated results are give i Figure 1. The curve i Figure 1 appears to asymptote to a value somewhere ear 10%. A 95% cofidece iterval for the fractio of games still goig after = turs is 0.12 ± This represets a upper boud o α. Ufortuately, we have o way of kowig whether the curve would drop to a eve lower level were we to simulate for loger. To obtai a better sese of the probability that games go o forever we eed a differet approach. To derive such a approach, it is helpful to try to uderstad exactly how games ca go o forever. 4 HOW GAMES GO ON FOREVER To gai a coceptual uderstadig of how the game ca go o forever, let W = (W (1),W (2)) be the wealth of the two players after rouds, i.e., after each player has rolled the dice times. Here, we measure wealth as the cash o had plus the value of all other properties, houses, ad hotels if they were to be sold back to the bak. (These are worth oe half of the face value i this case.) If either player s wealth ever becomes egative the they are bakrupt ad the game fiishes. If we plot the sequece of poits W 0,W 1,W 2,... (see Figure 2) the we see that the game goes o forever as log as the plotted poits stay withi the oegative quadrat. At first, the game cofiguratio chages rapidly, with properties beig bought ad houses built. Call this period Stage 1. Some games are completed withi Stage 1, but others cotiue beyod this trasiet stage, reachig a poit where the board cofiguratio has essetially solidified, with o further property exchages or house buildig uless oe player or other approaches bakruptcy. Let us call this secod period Stage 2. For our simulatios, we defied Stage 2 as beig the

4 Figure 2: Coceptual plot of the wealth process. first time that all properties are purchased ad both players have at least $5000 i cash. For games that reach Stage 2, oe ca compute the drift of the player wealths β = (β(1),β(2)), where the term drift meas the average chage i player wealth over a large umber of turs. If either drift is egative, the oe is essetially certai that oe of the players will go bakrupt evetually ad the game will ed. But if both drifts are positive, the there is a chace that the wealth process will escape to ifiity before hittig either axis, ad the game goes o forever. Figure 3 gives some examples of such plots for simulated games. Stage 1 is idicated by a cluster of poits ear the origi ad the, after reachig Stage 2, the wealth of the two players accumulates ad we see rays headig up ad to the right. Cosider how to estimate 1 α, the probability that the game eds i fiite time. This probability is simply the probability that the game eds i Stage 1, plus the probability that the game eds i fiite time after reachig Stage 2. The first probability is easily estimated, sice there is a clear time at which to stop the simulatios the miimum of the times whe oe player goes bakrupt ad the time at which Stage 2 is reached. It is the secod probability that is challegig to estimate. Suppose we have reached Stage 2, so that the board cofiguratio is fixed, ad let us thik of this poit as time zero. Let A(i) be the evet that Player i goes bakrupt, i = 1,2, i.e., the wealth of Player i hits 0. Here we assume that each player cotiues to play over a ifiite horizo, eve if the opposig player goes bakrupt. Coceptually, play cotiues after such a poit, with paymets goig to ad from a phatom player. We are iterested i P(A(1) A(2)), the probability that oe of the players goes bakrupt ad the game eds. This probability seems complicated to estimate, sice it ivolves a hittig problem i two dimesios. Boole s iequality bouds this probability by P(A(1)) +P(A(2)), ad so we ow have two hittig problems each i oe dimesio, which seems much more tractable. We expect that these bakruptcy probabilities are very small for the cases of iterest, ad that oe will ted to domiate the other, so the overestimatio implied through the use of Boole s iequality should be slight. We have suppressed the depedece of these probabilities o the iitial wealth w = (w(1), w(2)) of the two players, ad the iitial layout of the board (player locatios, property owership, house ad hotel developmet etc.). We will later wat to vary these parameters, so P(A(1)) P(A(2)) i geeral.

5 Figure 3: Plots of the wealth process for two simulated games. I both cases we see the liear tred of the curve that correspods to the drift β. 5 BROWNIAN MOTION The ext questio we must address is how to estimate P(A(1)), the probability that Player 1 goes bakrupt, where we agai suppress the iitial coditios correspodig to the board cofiguratio ad iitial player wealths upo reachig Stage 2. (The calculatios for Player 2 are idetical.) Let w be Player 1 s iitial wealth, ad W be Player 1 s wealth after rouds, with W 0 = w. The player goes bakrupt if there exists a 0 such that W 0. Lookig at Figure 3, it is atural to approximate ( W : 0) by a Browia motio startig at w i oe dimesio. Such a process is characterized by a drift γ, ad a variace parameter η 2. If γ < 0, the the Browia motio drifts to, so is certai to hit 0, ad so P(A(1)) = 1. Similarly, if the drift γ = 0 the the Browia motio hits ay poit o the real lie i fiite time, ad so P(A(1)) = 1. Whe γ > 0, the Browia motio drifts to + so is ot certai to hit 0, but istead hits 0 with probability exp( 2γ/η 2 ). (This is a stadard calculatio that ca be foud, e.g., i Ross 1996, Sectio 8.4.) We thus estimate γ ad η 2 from the path of ( W : 0) ad the estimate P(A(1)) by mi{exp( 2γ/η 2 ),1}. The drift γ is easily estimated as ( W W 0 )/ for some large. To estimate η we first remove the drift to get Y i = W i W 0 iγ for all i, ad the use o-overlappig batch meas with batches of size b: We first compute the k batch meas M j, j = 1,...,k, where M j is the average of the observatios Y ( j 1)b+1,...,Y jb, j = 1,...,k (assumig kb = ). We ca the estimate η 2 by b multiplied by the sample variace of M 1,M 2,...,M k. I our code we took b = 60 rouds ad k = 170 for a total rulegth of = after reachig Stage 2. This estimator of secod-stage bakruptcy probabilities is compared with two others i Sectio 8, oce we defie those two other estimators. The above steps are completed for each game that reaches Stage 2 for each player. Let us ow reuse some coutig idices to describe the overall estimator that Moopoly goes o forever. Defie U k = 1 if the kth game reaches Stage 2, ad U k = 0 otherwise. Also, let V k be zero if the kth game does ot reach Stage 2, ad otherwise let it be the miimum of 1 ad the sum of the Browia estimates of the game edig as outlied above. Our overall estimator that the game goes o forever is the 1 1 U k +U k V k ] = k=1[1 1 k=1 U k (1 V k ). This estimator is a sample average of i.i.d. observatios, so ca be aalyzed usig stadard cofidece-iterval methodology. A 95% cofidece iterval for the probability that games will go o forever, based o = 3100 replicatios is 0.12 ± This is the same (to two decimal places) as the cofidece iterval obtaied usig the brute-force strategy of simulatig for a very log time horizo, thereby reiforcig that result. This estimator is appealig i the sese that it attempts to model the ifiite-time behaviour of the wealth process. However, it suffers from the fact that each replicatio that reaches Stage 2 requires further estimatio of the drift ad volatility parameters of the approximatig Browia motio, ad this is computatioally expesive. Furthermore, the estimated

6 parameters are subject to the usual oise i simulatio estimates ad are combied i a oliear fuctio, ad oe might expect that this could lead to errors i the overall estimate of the probability that the game goes o forever. We ow tur to aother approach to estimatig this probability that addresses, to some extet, each of these difficulties. 6 MARKOV CHAINS It has bee oted may times that the game of Moopoly ca be modeled, at various levels of detail, by a Markov chai. See, for example, Stewart (1996a), Stewart (1996b) where the dice rolls are modeled as a Markov chai, as is the board motio of a sigle player, resultig i the steady-state probabilities of ladig o various squares. I this sectio we preset a simple Markov model of a 2-player Moopoly game. Our purpose is to apply Markov chai theory to obtai two further estimates of the probability that the game goes o forever, preseted i this ad the followig sectio. We rely heavily o the theory preseted i Lehtoe ad Nyrhie (1992) for rui probabilities i Markov processes. We use a Markov chai X = (X : 0), where each state trasitio represets a roll of the dice by both players. Here X = (X (1),X (2)), where X (i) is the curret locatio of Player i o the board. As is stadard, X (i) ca take oe of 3 values for i jail, which distiguish how may turs the player has bee preset i jail. So there are 4 states for jail: just visitig, ad i jail for j turs so far, j = 0,1,2. Let d = 42 2 = 1764 be the umber of states i the chai. (There is o state for the Go to Jail square.) Let B(x,i) deote a geeric radom paymet paid to Player i from the bak durig oe trasitio of the Markov chai startig from the state x. This quatity could be egative if the player makes a paymet to the bak, e.g., whe assessed for street repairs. Similarly, let R(x,i) deote ay ret or other paymet received by Player i from the other player i oe trasitio of the Markov chai startig from the state x. This quatity could also be egative. The amout of moey received by Player i (i = 1,2) over m steps of the Markov chai is the where m 1 M (i) = C (i), =0 C (i) = B (X,i) + R (X,i). If Player i s iitial wealth is w(i), the the probability that Player i evetually goes bakrupt is P(A(i)) = P( 0 : w(i) + M (i) 0), i.e., Player i goes bakrupt if his et worth hits 0. Suppose we fix the layout of the board i the sese that all properties are owed ad o further buildig will occur. The coditioal distributio of ( C (1), C (2)), coditioal o (X : 0), the depeds oly o X ad X +1, ad so ((X, M ) : 0) is a Markov-additive process. We ca therefore apply the theory of rui probabilities of Markov-additive processes i our cotext. To apply that theory it is coveiet to work with C (i) = C (i) ad M (i) = M (i), ad defie the probability of bakruptcy as P(A(i)) = P( 0 : M (i) w(i)), i.e., the chace that et outgoigs will equal or exceed iitial wealth. This bakruptcy probability depeds o the log-ru drift of M (i) for each i. To compute this drift, first let c(x,i) be the coditioal mea of C (i) coditioal o X = x. Next, ote that the chai X is irreducible ad aperiodic o a fiite state space, ad is therefore positive recurret. Let π deote its statioary distributio. The strog law of large umbers for Markov chais, e.g., Mey ad Tweedie (1993), Theorem , the esures that M (i) π(x)c(x, i) = µ(i) x as with probability 1. We refer to µ(i) as the drift of Player i s outgoigs. It correspods to the egative of the previously computed drift β(i), which is the drift of the wealth process.

7 Figure 4: Scatter plots of the estimated drift for a sigle player. The solid lie is the y = x lie. The expected drift o the horizotal access is computed usig steady-state calculatios from the Markov chai, whereas the vertical axis values are simulatio results. The left (right) plot gives results before (after) debuggig, where a bug i the simulatio ad a differece i modelig assumptios betwee the simulatio ad the Markov chai model geerated a couple of outlier poits ad a skewed slope of the poits, respectively. We have used these calculatios to help verify our simulatio code. I particular, we ca compute drifts of games that reach Stage 2 usig both the Markov model as above, ad the simulatio code, ad compare those drifts. See Figure 4. If µ(i) > 0 the Player i will evetually ru out of moey with probability 1, i.e., P(A(i)) = 1, so that the game will certaily ed. (Note that this probability is, as metioed earlier, coditioal o the iitial coditios of the chai, which we specify below.) If µ(i) = 0 the agai P(A(i)) = 1, i.e., whe the drift is 0, Player i is agai certai to go bakrupt. This follows from the fuctioal cetral limit theorem Mey ad Tweedie (1993), Theorem If µ(i) < 0 the the situatio is more complicated. Here M(i) = (M (i) : 0) has egative drift, so there is a probability, depedig o the iitial wealth of the players ad the iitial state of the chai, that Player i will go bakrupt. We eed to compute this probability. We apply a result due to Lehtoe ad Nyrhie (1992) for Markov-additive processes o discrete Markov chais. Lehtoe ad Nyrhie specialized existig large-deviatios theory for Markov-additive processes to develop asymptotics ad a simulatio estimator for the rui probability for Markov chais. The simulatio estimator has provably good performace (asymptotic optimality i the laguage of rare-evet simulatio) as the probability of rui becomes smaller. We ow specialize their developmet to the Moopoly cotext. I order to do so, we eed to compute the root of a certai equatio ivolvig Laplace trasforms. Defie ˆp xy (θ,i) = E[e θc (i) I(X +1 = y) X = x] to be the Laplace trasform (LT) of the oe-step outgoigs of Player i o the evet that the chai trasitios from x to y. This quatity does ot deped o, ad here x ad y each deote a pair of locatios o the Moopoly board. This quatity is 0 if trasitios from x to y have zero probability. The trasform is defied for all θ (, ), because C is bouded. Let ˆP(θ,i) be the d d matrix of LTs defied above, which is a o-egative matrix. The theory of such matrices implies that it has a eigevalue λ(θ,i) such that 1. λ(θ,i) is simple, real ad strictly positive, ad λ(θ,i) > λ for ay other eigevalue λ, ad 2. λ(θ,i) has a strictly positive right eigevector h(θ,i). Defie r(θ,i) = logλ(θ,i) (here ad elsewhere we use the atural logarithm uless otherwise specified). It is kow that r(,i) is strictly covex i θ, that r(0,i) = 0, ad that r (0,i) = µ i < 0. So r(,i) iitially dips below 0, ad ca have at most oe positive root θ (i) (so that r(θ (i),i) = 0 ad r (θ (i),i) > 0). The it is kow that lim w(i) 1 w(i) logp(a(i)) = θ (i). (1)

8 The logarithmic asymptotic result (1) suggests the approximatio P(A(i)) e θ (i)w(i), (2) although there are may other possibilities for the right-had side of (2) that also satisfy (1). So how ca we compute this approximatio? We first have to idetify θ (i). This ivolves formig the matrix ˆP(θ,i) for various θ, computig its maximum eigevalue, ad the seekig the positive value θ (i) for which that eigevalue = 1, i.e., r(θ (i),i) = 0. This is a root-fidig problem that ca be efficietly solved usig a biary search. We implemeted this approach i Java, computig the eigevalues usig the power method with 200 iteratios, which came to withi 0.002% of the eigevalues as computed usig MATLAB. The remaiig questio is how to compute ˆP(θ,i) for a give θ. Recall that ˆP(θ,i) is a d d matrix, where d = For a give value of θ we eed to fill i the values of this matrix. This is algorithmically the most difficult part of the calculatio. Our approach higes o the fact that we ca compute the LT of the total outgoigs of Player 1 due to Players 1 ad 2 each takig a sigle tur by computig the LT of the total outgoigs of Player 1 due to Player 1 s tur, the the LT of the total outgoigs of Player 1 due to Player 2 s tur separately, ad the multiplyig these LTs. (Recall that the LT of a sum of idepedet radom variables is the product of the idividual LTs.) We do ot provide further details here. So our estimator based o the asymptotic approximatio (2) is as follows. Recall that U k is the idicator that the game reaches Stage 2. Defie Ṽ k to be the miimum of 1 ad the sum of the probabilities (2) for the two players. The estimator is the 1 1 U k +U k Ṽ k ] = k=1[1 1 k=1 U k (1 Ṽ k ). This estimator is a sample average of i.i.d. observatios, so ca be aalyzed usig stadard cofidece-iterval methodology. A 95% cofidece iterval for the probability that games will go o forever based o = 3100 replicatios is 0.12 ± This agrees with the previous estimators. This approach is appealig i that it captures the ifiite-time behaviour of the wealth process, ad i cotrast to the Browia-motio estimator is relatively efficietly computed, ad coditioal o the state of the board at the start of Stage 2 cotais o simulatio oise. It suffers from a importat disadvatage, however. The approximatio (2) is oe of may possible approximatios that is cosistet with the logarithmic asymptotic (1) ad aother cosistet approximatio might yield very differet results. We ow address that cocer usig importace samplig. 7 IMPORTANCE SAMPLING I this sectio we preset the key ideas behid a importace samplig scheme for estimatig the probability that the game goes o forever, agai relyig heavily o the theory i Lehtoe ad Nyrhie (1992). If oe is familiar with that theory the our approach is coceptually exactly as they suggest. The difficulty really lies i the algorithmic implemetatio, because the discrete distributios ivolved have may poits of support. The overall simulatio procedure is as follows. We start by simulatig a game util either the game eds or Stage 2 is reached, whichever comes first, exactly as before. This is doe uder the usual probability distributio associated with the simulatio. Upo reachig Stage 2 we compute the drift of the player wealths as described i the previous sectio, ad if the drift of either player s wealth is ot strictly positive the the coditioal probability that the game eds is estimated to be 1. Otherwise, we eed to compute the coditioal probability that the game will ed, coditioal o the iitial cofiguratio at the oset of Stage 2. At this poit we chage the trasitio probabilities of the Markov chai (ad the correspodig paymets) so as to esure that both players will go bakrupt evetually with probability 1. This biases the probability estimates, so to correct the estimates we have to multiply by the likelihood ratio. Let p xy (c,i) be the probability, uder the usual probabilities that gover Moopoly, that from state x we trasitio to state y, ad i the process player i icurs outgoigs c. The importace samplig scheme chages these probabilities upo reachig Stage 2 to q xy (c,i) say. If we let τ(i) be the time at which player i goes bakrupt, the coditioal o reachig Stage 2 ad the cofiguratio of the board ad so forth at that time, ad takig the time of reachig Stage 2 to be time zero, the the coditioal probability that Player i goes bakrupt is EI(τ(i) < ) = Ẽ [ I(τ(i) < ) τ(i) k=1 ] p Xk 1,X k (C k (i),i), (3) q Xk 1,X k (C k (i),i)

9 Figure 5: The expected drift of the wealth process uder the stadard ad importace-samplig distributios. The effect of importace samplig is approximately to reverse the sig of the drift. The poits are broke dow ito games where there are o hotels, where oe player plays agaist a player with 2 hotels, where oe player ows 2 hotels ad is playig someoe with o hotels, ad whe there are more tha 2 hotels o the board. where Ẽ deotes the fact that we simulate the chai uder the ew dyamics q rather tha p i Stage 2. The theory i Lehtoe ad Nyrhie suggests that i order to estimate P(A(i)), we should simulate the chai uder the particular choice of trasitio probabilities q xy (c,i) = ˆp xy(θ )h(y) e θ c p xy (c) λ(θ )h(x) ˆp xy (θ ), where we write θ for θ (i) ad so forth for brevity. We have used this form of the expressio to show that if oe first adds over all possible values of c, the secod fractio sums to 1, ad the addig over all possible y the first fractio sums to 1, so that this is, ideed, a set of trasitio probabilities. However, these trasitio probabilities ca also be writte q xy (c,i) = eθ c p xy (c)h(y), (4) h(x) sice λ(θ ) = 1. This secod expressio better reflects how we geerate trasitios usig the ew probabilities. With this particular choice of importace-samplig trasitio kerel, I(τ(i) < ) = 1 a.s., ad so (3) simplifies to [ ] Ẽ e θ τ(i) k=1 C k(i) h(x T ), (5) h(x τ(i) ) ad this is our estimator of the coditioal probability that Player i goes bakrupt. As with may other estimators of rui probabilities i the light-tailed settig, we have see that the effect of importace samplig is approximately to reverse the sig of the drift. Figure 5 shows this effect, where we plot the expected drift as computed by the Markov chai model uder the origial probability dyamics versus the expected drift uder importace samplig for a umber of games.

10 Figure 6: Scatter plots of the estimated coditioal probability of games that have reached Stage 2 edig i fiite time. The three plots give, from left to right, scatter plots of the Browia estimator versus the estimator based o the asymptotic approximatio (2), the Browia estimator versus the importace-samplig estimator, ad the asymptotic-approximatio estimator versus the importace-samplig estimator. The full estimator of the probability that the game goes o forever is the as follows. As before, let U k be the idicator that Stage 2 is reached o the kth simulatio replicatio. Upo reachig Stage 2, we switch from the full Moopoly simulatio uder the usual probabilities to simulatios of the Markov chai uder the importace-samplig trasitio probabilities (oe for each player), ad simulate each path util the players go bakrupt. We ca the compute the quatity iside the expectatio i (5) as a sigle-replicatio estimator of the coditioal expectatio of bakruptcy for each player i = 1,2. Next, defie ˆV k to be the miimum of 1 ad the sum of these two estimators of the coditioal expectatios i Stage 2. The importace samplig estimator of the probability that the game goes o forever is the 1 1 U k +U k ˆV k ] = k=1[1 1 k=1 U k (1 ˆV k ). This estimator is a sample average of i.i.d. observatios, so ca be aalyzed usig stadard cofidece-iterval methodology. A 95% cofidece iterval for the probability that games will go o forever based o = 3100 replicatios is 0.12 ± COMPARISON, DISCUSSION AND FUTURE RESEARCH All four of our estimators yield cofidece itervals that suggest that the probability that the game goes o forever is close to 12%. This was the coclusio we obtaied usig the simplest estimator of all, i Sectio 3. However, we could ot be certai, based o that estimator aloe, that the probability would ot actually be sigificatly lower tha this value. The other three estimators were motivated by this issue ad the fact that we had fu computig them. Usig those estimators we cofirmed that, i fact, the potetial for the game edig after Stage 2 is reached is very slight. The three estimators based o Stage 2 calculatios are all biased due to the use of Boole s iequality to boud the probability that either player goes bakrupt by the sum of those probabilities. Beyod that bias, the Browia motio estimator ad the asymptotic-approximatio estimator both possess additioal bias, but the importace-samplig estimator does ot. It is iterestig to cosider how these three estimators compare. We coducted a small experimet where we idetified approximately 100 games that reach Stage 2, ad that have coditioal probabilities that the game will evetually ed that are i the rage of 0.01 through We the computed each of the estimators, ad produced scatter plots of their values i Figure 6. There is close agreemet betwee the asymptotic-approximatio estimator ad the importace-samplig estimator, but the Browia estimator differs. This differece does ot exhibit itself i our fial estimators of the probability that the game goes o forever sice the probability of reachig a poit i Stage 2 where these differeces occur is so small that it caot sigificatly ifluece the results. I future work we might see how strategies ca ifluece the legth of the game. Our assumptio that players do ot trade severely limits the possibility of developig properties, which i tur limits the probability that the game will ed. Whe players have hotels, or may houses, the variace i the player wealth process is much higher tha whe there is oly

11 limited developmet, ad so the chace that the game will ed would likely be higher. I that settig we would expect that the three estimators that are based o Stage 2 calculatios would provide a beefit beyod the simple estimator, ad that the probability that the game goes o forever would be smaller tha twelve percet. But primarily we hope to use the tools we have developed to this poit to idetify highly effective player strategies. We caot hope to approach the complexity of the strategies outlied i Lehma ad Walker (1975), sice those strategies ivolve extremely complicated ad subtle trades that go well beyod what we evisage codig. However, we ca certaily ivestigate questios such as whether oe should build through to hotels, or to halt developmet earlier so as to soak up houses. A importat questio is how to use some kid of automated learig process to idetify the best of a family of strategies. The reaso this is importat is ot just that we are ethusiasts ad see it that way, but also because Moopoly is a microcosm of may importat systems i real life, where sequetial decisios uder ucertaity are made i the face of competitio from other etities. True ethusiasts of estimatig the probability that the game goes o forever might also implemet importace-samplig estimators that do ot rely o the use of Boole s iequality, so as to obtai completely ubiased estimators of the probability that the game goes o forever. However, our results suggest that such a effort would have to idetify the probability extremely accurately i order to provide additioal iformatio beyod the cofidece itervals produced i this paper. ACKNOWLEDGMENTS This work was partially supported by Natioal Sciece Foudatio Grat Numbers CMMI , ITR ad CDI We would like to thak curret ad former studets Alex Aidu, Raghu Chadrasekara, Sea Choi, Deis Li, ad Ady Tuchma for simulatio programmig. REFERENCES Collis, T Probabilities i the game of Moopoly. < moopoly.shtml> [accessed Aug 4, 2009]. Darlig, T How to wi at Moopoly a surefire strategy. < [accessed Aug 4, 2009]. Darziskis, K Wiig Moopoly: A complete guide to property accumulatio, cash flow strategy, ad egotiatig techiques whe playig the best-sellig board game. HarperCollis. Fritzlei Moopoly/official rules. < Rules> [accessed Aug 4, 2009]. Lehma, J., ad J. Walker ways to wi at Moopoly. Dell Books. Lehtoe, T., ad H. Nyrhie O asymptotically efficiet simulatio of rui probabilities i a Markovia eviromet. Scadiavia Actuarial Joural: Mey, S. P., ad R. L. Tweedie Markov chais ad stochastic stability. Lodo: Spriger-Verlag. Ross, S. M Stochastic processes. 2d ed. New York: Wiley. Stewart, I. 1996a. How fair is Moopoly? Scietific America 274: Stewart, I. 1996b. Moopoly revisited. Scietific America 275: AUTHOR BIOGRAPHIES ERIC J. FRIEDMAN is a associate professor i the School of Operatios Research ad Iformatio Egieerig at Corell Uiversity. His mai research iterests are i game theory, computer etworks ad other problems arisig o the Iteret. He also has a strog iterest i combiatorial games ad their coectios to dyamical systems theory ad physics. He prefers dogs to cats. His web page ca be foud via < SHANE G. HENDERSON is a professor i the School of Operatios Research ad Iformatio Egieerig at Corell Uiversity. He is the simulatio area editor at Operatios Research, ad a associate editor for the ACM Trasactios o Modelig ad Computer Simulatio ad Operatios Research Letters. He co-edited the hadbook Simulatio as part of Elsevier s series of Hadbooks i Operatios Research ad Maagemet Sciece, ad also co-edited the Proceedigs of the 2007 Witer Simulatio Coferece. He likes cats but is allergic to them. His research iterests iclude discrete-evet simulatio ad simulatio optimizatio, ad he has worked for some time with emergecy services. His web page ca be foud via <

12 THOMAS BYUEN is a udergraduate i the School of Operatios Research ad Iformatio Egieerig. He ca be reached at <tb287@corell.edu>. GERMÁN GUTIÉRREZ GALLARDO is a Master of Egieerig studet i the School of Operatios Research ad Iformatio Egieerig at Corell Uiversity. He ca be reached at <gg92@corell.edu>.