CHUTES AND LADDERS MICHAEL HOCHMAN

Transcription

1 CHUTES AND LADDERS MICHAEL HOCHMAN Abstract. This paper discusses ad the uses the theory of Markov chais to aalyze ad develop a theory of the board game Chutes ad Ladders. Further, a method of usig computer programmig to simulate Chutes ad Ladders is discussed, as are various experimets with the rules ad layout of the stadard game. Cotets. Itroductio 2. The Rules of Chutes ad Ladders 2 3. Overview of Markov Chais 3 4. Theory of Chutes ad Ladders 6 5. The Programmig 8 6. Miscellay Sides o the Die The Effects of Addig a Ladder (or a Chute) Startig Somewhere Besides Square Zero Ackowledgmets 2 Refereces 2. Itroductio The game Chutes ad Ladders, also referred to as Sakes ad Ladders, is a simple game that a child ca play. Despite this the mathematics of this game - specifically with regards to Markov chais - are quite iterestig. This paper is dedicated to explorig these mathematics. First, we shall explai the rules of Chutes ad Ladders. The, we shall briefly discuss the theory of Markov chais i geeral, ad the focus o absorbig Markov chais. The, we shall build a theory of Chutes ad Ladders, usig, i part, the theory of Markov chais. The, we will discuss how to use computer programmig to model Chutes ad Ladders. Next, we shall perform a few experimets o the ordiary rules ad layout of Chutes ad Ladders ad derive couterituitive results.

2 2 MICHAEL HOCHMAN 2. The Rules of Chutes ad Ladders Figure. The Stadard Board Chutes ad Ladders is played o a 00 square board game. There ca be as may players as desired; however, sice the actios of the players are idepedet from the actios of the other players, we shall oly cosider oe player. The player begis off of the board, at a figurative square zero. The player the rolls a six-sided die, ad advaces the umber of spaces show o the die. For example, if the player is at positio 8, ad rolls a 5, they would advace to positio 3. The game is fiished whe the player lads o square 00. There are two exceptios to the rule of movemet. The first is that if the player, after advacig, lads o a chutes (slide) or a ladder, they slide dow or climb up them, respectively. For example, if the player, o their first tur, rolls a 4, the player advaces to square 4, ad the climbs to square 4, o the same tur. Thus, if the players is o square 77, ad rolls a 3, the player is fiished, as he advaces to 80, ad the climb to 00. The secod is that the player must lad exactly o square 00 to wi. If the player rolls a die that would advace them beyod square 00, they stay at the same place. For example, a player at square 96 must roll exactly a 4 to wi. A roll of 5 would make the player stay put at positio 96. Do ote that these rules could easily apply to a board of ay size, with ay size die, ad with chutes ad ladders i ay positio.

3 CHUTES AND LADDERS 3 3. Overview of Markov Chais Imagie a frog, sittig o a lilypad. Surroudig the frog are other lilypads. At give itervals, the frog either moves from the lilypad to aother, or simply stays put, with fixed probabilities. Furthermore, the frog has o memory: the frog s set of probabilities from movig from a give lilypad to aother stay the same, regardless of the frog s previous movemets. That is, if the frog has a probability p of movig from lilypad i to j at oe istace, the probability of movig from lilypad i to lilypad j is the same regardless of the frog s previous jumps. The frog s behavior ca be modeled by a Markov Chai. Formally, a Markov Chai ca be described as a set of states, s, s 2,..., s, ad a set of probabilities p ij of trasitioig from oe state to aother. I terms of the frog aalogy, the states are the lilypads. p ij are the probabilities of trasitioig to state j give you are at state i. For example, p 24 is the probability of movig to state 4 give you are at state 2. These probabilities form a matrix called a trasitio matrix. Note that for all i, j, p ij always exists - although it may be zero. We will discuss a few examples of Markov chais ad trasitios matrices. *I go out for dier each ight. However, I am quite picky. I oly eat hamburgers, pizza, cereal, ad foie gras. Normally, I pick oe of the four at radom. There are three exceptios, however: () I ever eat pizza the ight after I eat hamburgers. I this case, I pick from the other three at radom. (2) The ight after eatig foie gras, I have a particular cravig for cereal, ad a distaste for hamburgers: I thus choose cereal 50 percet of the time, hamburgers 0 percet of the time, ad pizza ad foie gras each 20 percet of the time. (3) The ight after eatig cereal, I always eat cereal. The trasitio matrix ca be represeted as: H P C F H P C F 0 5 *I am o a moorail system. There are three statios that form a loop, ad each trai stops at each statio. I flip a coi ad decide to either go oe stop clockwise or oe stop couterclockwise. The trasitio matrix is: *Goig back to the frog example, imagie that there are 4 lilypads, yet the frog oly stays o the lilypad which it is curretly o. The trasitio matrix is: Notice that this is the idetity matrix.

4 4 MICHAEL HOCHMAN Furthermore, ote that all trasitio matrices are square. This is because each state always has a trasitio probability to every other state (icludig itself), eve if this probability is 0. Chutes ad Ladders ca be thought of as beig modeled by a Markov chai. Each square o the board, through 00, is a state, as is the startig positio, the figurative square zero. We will cosider the trasitio matrix of Chutes ad Ladders i the ext sectio. There are may differet types of Markov chais. This paper shall focus o oe specific type, absorbig Markov chais. First, we defie a absorptio state: Defiitio 3.. A absorptio state is oe i which, whe etered, it is impossible to leave. That is, p ii =. A state that is ot a absorptio state is called a trasiet state. Defiitio 3.2. A absorbig Markov chai is a Markov chai with absorptio states ad with the property that it is possible to trasitio from ay state to a absorbig state i a fiite umber of trasitios. I the above examples, the first ad third examples were absorbig Markov chais. I the first example, cereal was the absorbig state, while i the third example, all of the states were absorbig states. Furthermore, for early ay cofiguratio, Chutes ad Ladders is a absorbig Markov Chai. The fial square is the absorbig state, ad it is possible to reach the fial square from ay give square (icludig figurative square zero). Some exceptios to this rule are that if there are d or more chutes directly before the fial square, where d is the umber of sides o the dice, or if there is some loop of chutes ad ladders. Deotig P as the trasitio matrix, we derive the probability of goig from state i to state j i precisely steps. Propositio 3.3. The probability of goig from state i to state j i precisely steps is p () ij, the i, j-th etry of P. Proof. The probability of goig from state i to state j i two steps is the sum of the probability of goig from step i to step, the from step to step j, the probability of goig from step i to step 2, the from step 2 to step j, ad so o. Thus, lettig P be a w w matrix, p (2) ij = p i p j + p i2 p 2j p iw p wj = w p ir p rj This parallels the defiitio of matrix multiplicatio. That is, it is evidet that p (2) ij is the i, j-th etry of P 2 = P P. This proves the propositio for the = 2 case; the proofs for greater follow similarly. Propositio 3.4. The probability of beig i a absorbig state approaches, ad the probability of beig i a o-absorbig state approaches 0, correspodigly, as the umber of steps is icreased. Proof. Defie m j to be the miimum umber of steps required to reach a absorbig state from state j, ad t j to be the probability of ot reachig a absorbig state from state j i m j steps. t j <. Defie m to be the greatest of the m j, ad t to be the greatest of the t j. Thus, the probability of ot beig absorbed i m is r=

5 CHUTES AND LADDERS 5 less tha or equal to t <, the probability of ot beig absorbed i 2m is less tha of equal to t 2. Sice t is less tha zero, as the umber of steps icreases, the probability of ot beig i a absorbig state approaches 0. For a absorbig Markov chai, we ca defie a submatrix Q of P as the trasitio matrix betwee o-absorbig states. That is, Q is P, but with the rows ad colums correspodig to absorbig states removed. For example, i the dier sceario described above H P F H Q = P 4 F 0 Remark 3.5. By Propositio 3.3 ad Propositio 3.4, we see that as approaches ifiity, Q approaches 0. Propositio 3.6. (I Q) exists. Proof. Take x such that (I Q)x = 0. Note that this is possible, as the trivial solutio x = 0 works. Distributig ad subtractig yields x = Qx. This implies that x = Q x by iteratio. By Remark 3.5, we have that Q approaches 0 as approaches ifiity, which, due to the way Q x is costructed by iteratio, implies that x = 0. Thus, the oly solutio is the trivial solutio, which meas that (I Q) is ivertible. Propositio 3.7. (I Q) = I + Q + Q 2 + Q Proof (I Q)(I + Q + Q 2 + Q Q ) = I Q + (I Q) (I Q)(I + Q + Q 2 + Q Q ) = (I Q) (I Q + ) I + Q + Q 2 + Q Q = (I Q) (I Q + ) I + Q + Q 2 + Q = (I Q) The fial step is true as approaches ifiity per Remark 3.5. Defiitio 3.8. Now, we shall let N = (I Q). ij refers to the i, jth etry of N. We call N the fudametal matrix for P. Propositio 3.9. Give that we start i state i, ij is the expected umber of times that state j is reached. Remark 3.0. This icludes the startig state, so ii, as if we start i a give state, we must be i that state at least oce. Proof. We defie X (l) ij as a radom variable that is equal to if we are i state j after l steps startig from positio i, ad 0 otherwise. From our previous work, we see that P (X (l) ij = ) = q (l) ij. Thus, P (X ij = 0) = q (l) ij. Note that q(l) ij is the i, jth etry of Q l, ad ij is the i, jth etry of N. Note that this holds for l = 0, as we ca state Q 0 = I. Further, E(X (l) ij ) = P (X(l) ij = ) = q (l) ij, as 0 X(l) ij. Therefore, E(X (0) ij + X () ij + X (2) ij X (l) ij ) = q(0) ij + q () ij + q (2) ij + q (l) ij

6 6 MICHAEL HOCHMAN Lettig l approach ifiity, E( l=0 X (l) ij ) = l=0 q (l) ij = ij Remark 3.. A atural questio is, give we start i state i, how much time will it take util a absorbig state is reached. By our defiitio of N, the solutio is to sum the etries of row i i N. This is because ij is the umber of times a particular o-absorptive state will be reached, so to get the umber of times util absorptio, we simply eed to sum these values, j ij. 4. Theory of Chutes ad Ladders Now that we have discussed the termiology of Markov chais, it is time to apply them to Chutes ad Ladders. As metioed above, Chutes ad Ladders ca be represeted as a absorbig Markov chai, with the fial square as the oly absorbig state. Now, the job is to determie the trasitio matrix. Remark 4.. We shall deote a game of Chutes ad Ladders by C(k,, M), where k is the umber of squares o the board (ot icludig the figurative square zero), is the umber of sides o the dice, ad M is the set of chutes ad ladders. A chute is deoted as, for example, (5,2), which meas there is a chute from square 5 to square 2. Ladders are deoted similarly. Remark 4.2. The stadard board, described i Sectio 2, shall be deoted as T. Specifically, T = C(00, 6, {(, 38), (4, 4), (9, 3), (2, 42), (28, 84), (36, 44), (5, 67), (7, 9), (80, 00), (6, 6), (47, 26), (49, ), (56, 53), (62, 9), (64, 60), (87, 24), (93, 73), (95, 75), (98, 78)}). For the time beig, we shall cosider C(k,, Ø), give 2 ad < k. What would this trasitio matrix be? Remark 4.3. Sice figurative square zero is a state, the first row i the trasitio matrix is the row represetig state 0. However, for ease of otatio, we shall idex our matrix to begi at row 0. That is, row 0 represets square 0, row represets square, ad so o. Take some i N {0} such that + i k. i is some square o the board, icludig the figurative square zero. Rollig the dice will yield a probability of of advacig, 2,..., squares. Therefore, give beig i square i, after the ext trasitio, there is a chace of beig i square i+, a chace of beig i square i + 2 ad so o util there is a chace of beig i square i +. The ith row of the trasitio matrix thus looks like: ( 0... i i + i i + i + + i... k i ) However, the issue is slightly differet for i withi squares of k. Recall that a player must lad precisely o k. If the player rolls somethig that would place them beyod k, they stay put. For example, i T, if a player is o square 97 ad rolls a 4, they remai o 97. This meas that our trasitio matrix over C(k,, Ø),

7 CHUTES AND LADDERS 7 for the fial rows (icludig row k) will be differet from the remaiig rows discussed above. Row k will look as such: ( 0... k 3 k 2 k k k ) This is because it is a absorbig state. Row k will look as such: ( 0... k 3 k 2 k k ) k This is because there is a probability of rollig a o a -sided dice, ad a probability of = of ot rollig a oe ad stayig put. For similar reasos, row k 2 looks as such: 0... k 3 k 2 k k ( 2 k This patter cotiues util row k, which is simply a ordiary row. Now, we shall add i ladder (or chute) from square f to square g. Now, aytime we roll the die that would result i goig to square f, we move istead to square g. We therefore eed to alter our trasitio matrix accordigly. Remark 4.4. The same algorithm applies regardless if f < g (a ladder) or f > g (a chute). The probability of goig to square f is ow 0, ad the probability of goig to square g is ow what used to be the probability of goig to square f plus the origial probability of goig to square g, which is i most cases 0. The trasitio matrix would look like: f f + f f f... g... k f f f f f This still leaves the questio of what row f looks like. The aswer to this questio is that there is a i the gth colum, ad 0 elsewhere. This is because if a players is o square f, they move with probability to square g. This is quite irrelevat for a computer simulatio, as it is impossible to start a tur o square f, but is ecessary for accurately calculatig the fudametal matrix. Remark 4.5. This algorithm still works i the edge cases, such as where f ad g are close together, or g is ear k. Remark 4.6. The same algorithm ca be used to add i as may chutes ad ladders as we wish. For example, we could use this algorithm to geerate T. Sice oly the square k is a absorbig state, the Q matrix is simply the trasitio matrix, with the row k ad colum k removed. Note that Q is a k k matrix, as the trasitio matrix is a (k + ) (k + ) matrix, due to the figurative square zero. )

8 8 MICHAEL HOCHMAN At this poit, we ca had over the Q matrix to the computer, which, provided k is ot very large, ca calculate N, ad give us the expected umber of turs, completig the problem. Remark 4.7. The computer package Jama for Java is capable of calculatig iverses quickly ad efficietly; attemptig to calculate a matrix s iverse, as is ecessary for T, is quite challegig to do by had. However, Jama ca do it i uder oe secod. Remark 4.8. For T, the expected umber of turs eeded is approximately The Programmig Programmig was used to determie the aswer experimetally. The programmig was doe i the Java laguage. The program had a umber of methods. *The first method was to create a trasitio matrix of the form C(k,, Ø). The values of k ad were iputted from the user, ad the trasitio matrix was costructed as described i sectio 3. *The secod method added the chutes ad ladders. The user iputs the coordiates of the chutes ad the ladders, ad the trasitio matrix was altered accordigly, by the algorithm described i the sectio above. Remark 5.. For coveiece, there exists a method that combied the two methods above to automatically create T without iput from the user. * The third method ra through oe iteratio of the game chutes ad ladders. The player started at figurative square zero, correspodig to the 0th row i the matrix (usig the covetio described above). The, a radom umber is geerated correspodig to the dice roll, ad the player moves accordig to the established trasitio matrix, icludig usig the chutes ad ladders. Meawhile, it is recorded that the die has bee rolled oce. The process is repeated util the player reaches k exactly. The, the umber of dice rolls eeded is output. * The fourth method ra the third method multiple times i order to reap the beefits of the law of large umbers. The user iputs the umber of times they wish to third method to ru. The average of these rus is the output. For example, ruig the fourth method, with iput,000,000 o T 5 times yields: , , , , These are remarkably close to the value predicted by the theoretical methods described i sectio 4. The actual details are a little bit techical, ad thus best described i a footote. The radom umber geerated was ot betwee, say, ad. Istead, a radom umber was geerated betwee 0 ad. Each etry i the row correspodig to the space the player is curretly i was assiged a area accordig to it s value. Sice the sum of the values o a give row is equal to, the square moved to correspoded to which area it fell ito. For example, if the player was o the ith square, ad row i of the trasitio matrix looked like, ( 0... i i + i + 2 i + 3 i i k ) i the if the radom variable was betwee 0 ad.25, the the player would move to square i +, if the value of the radom variable was betwee.25 ad.5, the player would move to square i + 2, if the value of the radom variable was betwee.5 ad.75, the player would move to square i + 4, ad if the value of the radom variable was betwee.75 ad, the player would move to square i + 5. This represets a ladder betwee square i + 3 ad square i + 5.

9 CHUTES AND LADDERS 9 6. Miscellay I this sectio, we shall focus o three differet experimets, which shall yield somewhat couterituitive results. These couterituitive results are largely due to some of the large ladders ad chutes o the board. 6.. Sides o the Die. O T, a 6-sided die is used. Here, we shall modify T by chagig the umber of sides of the die. We would expect that, as we icrease the umber of sides o the die, the expected umber of rolls eeded to complete the game decreases, for each roll has a higher expected value. Usig the theoretical methods described i Sectio 4 2 we have established the followig table, with represetig the umber of sides ad e represetig the expected umber of rolls eeded to complete the game. e This table correspods roughly to a ituitive uderstadig: i geeral, as the umber of sides of the die icreases, the umber of turs it takes to complete the game icreases. However, there is a exceptio. Rollig a 2-sided die yields a e value of approximately 60.8, yet rollig a 3-sided yields a e value of That is, it would take loger to play a game of Chutes ad Ladders with a 3-sided die tha with a 2-sided die. This is likely due to the ladder from square to 38, a large ladder: There is a higher probability of ladig o this ladder immediately with a two-sided rather tha 3-sided die. Beyod this, however, icreasig the umber of die fuctios as expected The Effects of Addig a Ladder (or a Chute). This subsectio shall address the idea of addig oe additioal ladder or chute to T. It would seem likely that addig a ladder would decrease the umber of rolls expected to fiish the game, while addig a chute would correspodigly icrease the umber of times. However, there are some circumstaces whe a o-ituitive effect would occur. Lookig at the board earlier i this paper, it is clear that a ladder from square 27 2 The empirical methods described i Sectio 5 would likely work as well. However, whe usig such methods, there is a small but existig probability of error. Because the matrix that eeds to be iverted i order to calculate the precise value is small (00 00), it ca be iverted quickly i Java usig Jama. If we were dealig with larger matrices, the empirical method would probably be better.

10 0 MICHAEL HOCHMAN to 29 would probably icrease the expected umber of rolls eeded, for if a player is lucky eough to lad o square 27 ad advace to square 29, they would miss the opportuity to lad o a much bigger ladder, that from square 28 to 84. Similarly, a chute from square 29 to 27 would probably decrease the expected umber of rolls eeded. I fact, these guesses is correct. The e value, calculated usig the theoretical methods described i sectio 4, for T with a ladder added from square 27 to square 29 is approximately early oe roll loger tha T, ad the e value for T with a chute added from square 29 to 27 is approximately 38.05, more tha oe roll shorter tha T. However, these are fairly trivial examples that do ot, i geeral, couteract our otio that addig ladders decreases the expected umber of rolls eeded while addig chutes icreases the umber of expected rolls. For oe, these examples were oly of legth two, ad were situated immediately over a very importat ladder. Therefore, to see if our ituitio is correct, we should check for log ladders. For the remaider of this sectio, we shall oly cosider ladders of legth 0 or more. 3 First, however, a defiitio. Defiitio 6.. A o-ladder square is oe which is either the startig or fiishig poit of ay existig chute or ladder. We shall oly permit our additioal chutes ad ladders to go from o-ladder squares to o-ladder squares. 4 Usig the theoretical method (combied with a pair of ested for-loops to allow us to check all possible feasible ladders (or chutes) of legth 0 or greater), we fid that there are 5496 feasible ladders (or chutes) of legth 0 or greater, ad that 254 of them chage the expected value agaist 254 our ituitio = 4.6%. Although a small percetage, 4.6% is ot isigificat either. This meas that of all ladders ad chutes that ca be added oto the board 3 0 is mostly a arbitrary, reasoably-large umber 4 If this were ot the case, there would be errors. There are a few possible cases: () A additioal ladder (or chute) goes from square i to square k, ad there already exists a ladder (or chute) from square i to square j. This makes it so that whe a player lads o square i, there is o defied square to go to. (2) A additioal ladder (or chute) goes from square i to square k, ad there already exists a ladder (or chute) from square j to square i. Thus, if a player were to lad o square j, they would jump to square i, the immediately jump to square k. Thus, we are both addig a ladder (or chute) from square i to square k, ad, i essece, modifyig the existig ladder (or chute) from square j to square i to a ladder (or chute) from square j to square k. However, we solely wished to add a ladder. (3) A additioal ladder (or chute) goes from square i to square k, ad there already exists a ladder (or chute) from square k to square j. For the same reasos discussed i (2), this does ot work. (4) A additioal ladder (or chute) goes from square i to square k, ad there already exists a ladder (or chute) from square j to square k. This would make the Q matrix look like: 0... k k k i j Thus, the Q matrix is sigular, so (I Q) is sigular, so N = (I Q) does ot exist, which makes our theoretical method for calculatig the expected umber of turs eeded ot work.

11 CHUTES AND LADDERS ad are at least of legth 0, early i 20 defies our ituitio about their effect upo the expected value. Remark 6.2. Most of these are clustered ear either the (28,84) or (80,00) ladder. This is because the (28,84) ladder is quite large, ad the (80,00) ladder allows the player to lad precisely o the fiishig square Startig Somewhere Besides Square Zero. A rule that we have followed so far is that the player begis at the figurative square zero. What happes if we chage this rule? Recallig Remark 3., we have a method for calculatig the expected umber of rolls eeded to complete the game give a specified startig positio. Usig this, we ca see, for example, that startig at square 2 has a expected value of , ad that startig at square 5 has a expected value of , both of which are larger tha the expected value of startig at the figurative square zero. That is, it would be preferable to start at figurative square zero tha to start at either square 2 or square 5, a couterituitive result. What causes this? There are ladders goig from square to square 38, ad from square 4 to square 4. If a player is o square 2, they miss the opportuity to use the -38 ladder, ad if a player is o square 5, they miss the opportuity to use them both. This aalysis shows that it is ofte better to be just before a ladder rather tha just after, eve though there are fewer squares util the fiishig square, ad there is o guarateed advacemet oto the ladder. Remark 6.3. There is ot a similar aalysis for chutes, because this aalysis would show that it is better to be just after a chute rather tha before it, which already correspods to our itutitio. However, as we move further away from figurative square zero, this effect is ot strog eough to icrease the expected value above the expected value of startig at zero. For example, startig at square 29 - just after the massive (28,84) ladder - has a expected umber of rolls of Thus, aother way to see if uituitive results occur is to compare specific squares to the square precedig them. Because the expected value is ot precisely defied from startig at the bottom of a ladder, we shall oly cosider cases ot as such. 5 Ruig this algorithm from the computer, we fid that it is preferable to start at the prior square (or the prior o-ladder square, if ecessary) for the followig squares o T: 2, 5, 0, 23, 24, 25, 26, 27, 29, 4, 43, 52, 57 66, 67, 72, 73, 75, 76, 77, 78, 79, 8, 89, 92 There are a umber of thigs apparet from this list. First ad foremost is the sheer quatity of such uituitive squares, 25 i total. Cosiderig that we are ot coutig a umber of squares, this amouts to over 3 of the eligible squares. Clearly, our ituitio is off-kilter. Further, the placemet of these squares is quite iterestig. These squares show how importat the (28,84) ad (80,00) ladders are; it is but slightly more probably to lad o these latters beig, say, 5 rather tha 4 squares before the ladder (recallig that this is over multiple turs), ad the fact that eve 8 squares out, 5 For techological reasos, we shall also exclude squares at the top of ladders ad chutes, ad the bottom of chutes.

12 2 MICHAEL HOCHMAN for the (80,00) ladder ad 4 for the (28,84) oe, shows precisely how importat they are. Ackowledgmets. I would like to particularly thak my graduate-studet metor for the REU program, Be Seeger, for his woderful assistace with this paper. Refereces [] Altheo, Kig, ad Schillig. How Log Is a Game of Sakes ad Ladders? The Mathematical Gazette, Vol. 77, No. 478 (Mar., 993), pp [2] Gristead ad Sell. Itroductio to Probability (Chapter ) 7 6 A good overview of the topic. Used i Sectio 4. Was the source of the experimet i sectio Used extesively i Sectio 3; may of the proofs rely, to varyig degrees, o this source.