Anna Yaroslaski, Howard Community College

Mentored by: Loretta FitzGerald Tokoly, Ph.D.

Abstract

The developers of the board game Dungeon! suggest levels of play for each character. This project tested whether their recommendations are mathematically supported by determining the probabilities of defeating the monsters by every character at each dungeon level. Using these probabilities, I further calculated the average probability of successfully completing a room and the subsequent Expected Value in gold pieces to determine an ideal strategy for each character to win the game. Although elements of the suggested ranges were reasonable, I found that some variation on the suggested levels led to a possibly faster winning strategy.

Background of the game

The board game Dungeon was first published by Tactical Studies Rules, Incorporated (TSR) in 1975. In the original version there were four character classes: Elf, Hero, Superhero, and Wizard. In 1997, TSR was bought out by a company by the name of Wizards of the Coast LLC (WotC). A new version of the game was not released until 2012, this time under the Dungeons & Dragons (D&D) brand name. With this update, the characters were changed to match some of the class options of the latest edition of D&D at that time. There are four class types (Cleric, Fighter, Rogue, Wizard) with male and female versions of each for a total of eight possible characters. This paper will be using the rules of the second WotC release from 2014, which has the same character options as the 2012 release. In the rule book, the developers have included a table of levels they suggest as the “safest” for each class. The goal of this paper is to test those suggested level ranges to find out whether they are supported mathematically.

Introduction

My first Introduction to Dungeon was through the Table Top Gaming Club. One of the club members had brought in his copy of Classic Dungeon several times and I decided to try playing it. I ended up having so much fun playing that I bought my own copy. At the time I was also taking Discrete Structures Honors. The honors project for that course, was to give a presentation analyzing a board game, so I chose Dungeon.

The rules of Dungeon can be a little complex for people have never played the game before, so this section will mostly focus on the most basic elements. However, the calculations in this paper will focus on the fighter, rogue and cleric classes. The board has six levels representing the various floors of a dungeon, and a total of 80 rooms. In addition, all levels, with the exception of level one, contain two chambers. Level one contains only one chamber, the Great Hall, which acts as the starting place of the game. There are four basic character class types: Rogue, Fighter, Cleric and Wizard. All players, regardless of class, can move up to five spaces each turn. The goal of the game is to collect the amount of gold specified by your class and return to the Great Hall. To earn gold the players must defeat monsters. Each room contains one monster and each chamber contains three. The degree of difficulty of defeating a monster changes, depending on the level of the dungeon the player is on. To fight a monster the player rolls two six-sided dice and sums the results. Monsters have numbers that correspond to the sum the player must roll equal to or higher than in order to win the fight. If the player does not win the dice roll then they would roll again and refer to the “Monster Fights Back!” table (Table 1) to determine the damage inflicted by the monster. If the player is successful against the monster the player draws the top card from the treasure deck of that level.

Sum of Roll on 2 cube (6-sided) dice	Probability of outcome (Table 3)	Outcome of Monster Fighting Back
2-5	$10 / 36 \approx 27.8 %$	Miss! The monster misses you. You are not harmed and you lose no treasure.
6-7	$11 / 36 \approx 30.6 %$	Stunned. Drop 1 random treasure card.
8-10	$12 / 36 \approx 33.3 %$	Hurt. Drop 1 random treasure card and move your character back to Great Hall (Level 1).
11	$2 / 36 \approx 5.6 %$	Seriously Hurt. Randomly drop half your treasure and move your character back to Great Hall.
12	$1 / 36 \approx 2.8 %$	Crushed! Drop all your treasure cards. Remove your character from the board. Begin again.

Table 1: Monster fights back. Damage monster does by roll of two dice.

The player also would collect any treasure dropped by a previous player who had been defeated by a monster and dropped treasure on a previous turn.

The rogue is the weakest of the fighters, often having the highest number required to win the dice rolls. However, they have a better chance of finding secret doors. In order to open a secret door, the player rolls one die. While the other three classes must roll a five or six in order to spot the door, the rogue finds the door on a roll of three or better. The cleric has the same gold requirement as the rogue but will usually have a better chance of defeating monsters. The Fighter often has an even better chance of success than the cleric, but also requires more gold. The Wizard is the most complex of the four classes. They are able to cast spells and defeat monsters from outside of a room, with no risk of losing gold or dying. The wizard has two types of combat spells they can cast: fire and lightning. After defeating a monster, the wizard must use one space of movement to enter a room and collect the treasure. As they have a very low chance of ever losing treasure, they have to collect the most treasure. The game has levels it considers to be the “safest” for each class. This level breakdown, as well as the gold required for each class to win, is shown in the following table (Table 2).

Class	Suggested Level	Gold to Win
Rogue	1-3	10,000
Cleric	2-4	10,000
Fighter	3-5	20,000
Wizard	4-6	30,000

Table 2: Display of Class of Character, suggests playing levels and gold pieces required to win.

The purpose of this paper to determine the accuracy of the suggested level ranges. To do this we must look at the Expected Take per room by level, in the case that the player wins the dice roll against a monster, as well as the Expected Loss, if the player loses the dice roll. Due to the complexity of the wizard class, this paper will primarily focus on the other three classes and allow wizard to addressed in a future paper.

Theory of Calculations

To calculate the Expected Take, we first need to know the probability that a class will defeat each individual monster. In order to win the dice roll, the player must roll equal to or higher than the number specified on the monster card, so the total chance of success is the probability for rolling the given number plus the probabilities of any higher number. For example, the probability of rolling a six or higher is $\frac{26}{36}$ or about 72.2% (Table 3).

Table 3: Outcomes of rolling two 6-sided dice.

After finding the chance of success for each monster, we must determine the average probability that a player will beat any monster on a level. To determine this, we add up all of the probabilities of the individual monsters and divide by the total number monster cards for that level. Next, we must find the Expected Value of each room for any given level. To find this, we must sum all of the treasure cards on that level and divide by the number of cards. With these two values, we can then calculate the Expected Take (E_T) using the following formula where p is the average probability of success and v is the Expected Value.

$p \times v = E_{T}$

In order to find the Expected Loss, we need to know the average chance of losing, the Expected Value, and the probabilities of each outcome on the Monster Fights Back (Table 1). We can use the same Expected Value calculated in the last section. We will start by determining the probability of losing the dice roll. As the chance of losing and the chance of winning sum to one we can find the probability of losing (L) using the following formula where the probability of winning is W.

$1 - W = L$

The Monster Fights Back table has five outcomes (Table 1). For the purposes of this paper, all calculations were done with the players current treasure amount being half of their class’s total to win. The Expected Loss was then calculated using the formula below, where the variables $p_{0}, p_{1} p_{h a l f},$ and $p_{a l l}$ represent the probability of dropping each amount of treasure, M represents the total amount of treasure needed to win, $E_{L}$ represents Expected Loss, and the variables L and v continue to represent probability of losing and Expected Value respectively.

$L [(p_{0} \times 0) + (p_{1} \times v) + (p_{h a l f} \times \frac{1}{4} M) + (p_{h a l f} \times \frac{1}{2} M)] = E_{L}$

The last thing that needs to be calculated is the Net Expected Value for each level. To find this, we would subtract the Expected Loss from the Expected Take. The results for the three classes we are considering are shown in the tables below. Note “gold pieces” is abbreviated “gp”.

Rogue	Expected Value (gp)	Average % chance of success	Average % chance of failure	Expected Take (gp)	Expected Loss (gp)	Net Expected Value (gp)
Level 1	555	88.47	11.53	491	73	418
Level 2	916	78.5	21.5	719	186	533
Level 3	1640	38.9	61.1	638	810	-172
Level 4	2750	22.7	77.3	624	1579	-955
Level 5	4343	14.1	89.5	612	2622	-2010
Level 6	5666	0.4	99.6	23	3882	-3859

Table 4: Expected Value, Take, Loss, and Net Expected Value for the Rogue Player.

Cleric	Expected Value (gp)	Average % chance of success	Average % chance of failure	Expected Take (gp)	Expected Loss (gp)	Net Expected Value (gp)
Level 1	555	90.5	9.5	502	60	442
Level 2	916	77.8	22.2	713	192	521
Level 3	1640	42.7	57.3	700	759	-59
Level 4	2750	26.4	73.6	726	1498	-772
Level 5	4343	15.97	84.03	694	2565	-1871
Level 6	5666	3.2	96.8	181	3773	-3592

Table 5: Expected Value, Take, Loss, and Net Expected Value for the Cleric Player.

Fighter	Expected Value (gp)	Average % chance of success	Average % chance of failure	Expected Take (gp)	Expected Loss (gp)	Net Expected Value (gp)
Level 1	555	98.4	1.6	546	15	531
Level 2	916	93.4	6.6	856	75	781
Level 3	1640	69.7	30.3	1143	486	657
Level 4	2750	41.1	58.9	1130	1362	-232
Level 5	4343	30.8	69.2	1338	2304	-966
Level 6	5666	13.1	86.9	742	3629	-2887

Table 6: Expected Value, Take, Loss, and Net Expected Value for the Fighter Player.

Discussion of Strategy

The above tables show, that despite the suggested levels for rogue being levels 1-3 and 2-4 for cleric, as was shown in Table 2, the ideal level range for them is, realistically, levels 1 and 2. In fact, the ideal strategy for both the rogue and cleric would be to go straight to level 2, as that has the highest net Expected Value. The flaw with this strategy is the limited number of rooms on level 2, particularly when playing with multiple people. The player would eventually need to supplement with rooms, ideally from level 1, but no higher than level 3, or else the Expected Loss greatly outweighs the Expected Take. Similarly, the fighter has a positive net Expected Value until level four. After level 4, the Expected Loss becomes dramatically greater than the Expected Take. The ideal strategy for the fighters would be to gather as much gold as they can on levels 1-3 and attempt level four rooms only as necessary.

A Persistence Strategy

Due to the limited number of rooms on levels one and two, (9 on each level), there is a good chance that a rogue or cleric may need to attempt some level three rooms in order to get their 10,000 gold pieces. Since the chances of being defeated by a monster are greater on level three, we will also look at a strategy for success and treasure. This leads us to the final question of this paper: how worthwhile is replaying a room the player was previously defeated in? If the player loses a dice roll, they will either lose no treasure, one treasure, half of their treasure, or all of their treasure, but if the player returns to that room and wins, they pick up all dropped treasure in addition to the room’s original treasure. (as shown in Table 1). The primary reason to consider this strategy is the probability of rolling the number needed to defeat a monster increases geometrically. For example, if a player draws a monster that requires a roll of nine or better to defeat, they would have about a 72.2% chance of being defeated on one roll (as shown in Table 3). However, as they replay a room, the probability of their being defeated twice is less than a single defeat (about a 52.1% chance of being defeated, found by 0.7222). The probability of being defeated three times in a row is less again (0.7223, or about 37.6%). The probability of a successful roll in n number of consecutive turns can then be represented by the following formula where d is the probability of being defeated, and w is the probability of a successful roll:

$1 - d^{n} = w$

With each loss, the player also has less treasure to lose and more to gain. Assuming another player does not sneak in and defeat the room before the player can defeat the monster, this persistence is a way for the player to get all their dropped treasure. In order to determine the effectiveness of persistence, we must split the calculation into four parts, one for each of the four outcomes. Of the monster striking back (See Table 1). We will start by looking at the values for rogues and clerics at level three and then determine the values for fighters on level four, the fighter’s first level with a negative net Expected Value.

Calculations for the Persistence Strategy for Rogue and Cleric

The first outcome we will look at (a roll of 2-5) is when no treasure is dropped. There is about a 27.8% chance of this outcome. Because Since no treasure is dropped, the Expected Take of the room remains the same, but the probability of being defeated twice in a row is lower, as stated above. The next outcome (a roll of 6-7 or 8-10) is dropping one treasure. The player has about a 63.9% chance of dropping one treasure which will increase the Expected Value of that room by the value of the treasure dropped. Assuming the player has a mixture of treasures from levels one and two, the Expected Value of the dropped treasure will be the average of the Expected Values of the two (555 gold pieces for level one and 916 gold pieces for level two, we add these and divide by 2 to find their average). This is about 736 gold pieces. This means the new Expected Value of that room is 2,376 gold pieces, the 716 gold pieces dropped previously plus the stated Expected Value of a level three room (1,640 gold pieces as seen in tables 4 and 5). Using the equation from before ( ), with this new Expected Value, we find that the Expected Take is now 924 gold pieces (0.389×2376) for rogue and 1,015 (0.411×2376) gold pieces for cleric. The Expected Loss calculation would also change with this new Expected Value of dropped treasure, as the player’s total amount of treasure would have decreased by this value. The new equation, given the player had half of the 10,000 gold pieces total need to win before dropping one treasure can be represented by the following:

$L [(\frac{10}{36} \times 0) + (\frac{23}{36} \times 716) + (\frac{2}{36} \times (2500 - 716)) + (\frac{1}{36} \times (5000 - 716))] = E_{L}$

Using this new formula, we find the Expected Loss of level would then be 420 gold for the rogue (with a value of 0.611 for L) and 394 gold for the cleric (with a value of 0.573 for L).

The player has about a 5.6% chance of dropping half of their treasure. Again, assuming the player has half of the total treasure needed to win, the Expected Value of 1,640 gold pieces for level three would increase by 2,500 gold pieces to a total of 4,140 gold pieces. Using the same formula as the previous outcome gives new Expected Takes of 1,610 gold pieces and 1,768 gold pieces for rogues and clerics respectively. By modifying the Expected Loss formula to account for having lost half of their treasure with the L values from before we have the formula:

$L [(\frac{10}{36} \times 0) + (\frac{23}{36} \times 716) + (\frac{2}{36} \times \frac{2500}{2}) + (\frac{1}{36} \times \frac{5000}{2})] = E_{L}$

We see the Expected Loss would become 372 gold pieces for the rogue and 349 gold pieces for the cleric. The last outcome is that the player drops all of their gold. The new Expected Value of the room would be the gold dropped (an assumed 5,000) plus the stated Expected Value of 1,640 gold pieces for a total of 6,640 gold pieces. If we once again apply the formula we see the Expected Take for a rogue would be 2,583 gold pieces and 2,835 gold pieces for a cleric. Since in this scenario, they player would have just restarted with no treasure, the Expected Loss would be zero.

Calculations for the Persistence Strategy for Fighter

Fighters still have a positive net Expected Value on level three (as indicated in Table 6), so for them, we will do our calculations with the probabilities of success and failure on level four. Assuming the player has defeated rooms in levels one, two and three, the Expected Value of treasure dropped will be 1,037 gold pieces. Just as before, rolling 2-5 results in the “no treasure dropped” outcome, which will not change the calculations at (with the exception of the decreased chance of defeat in consecutive turns discussed before), so we will start with the outcome of dropping one random treasure. The new Expected Value of the room will be 3,787 gold pieces (the 1,037 gold pieces Expected Value of treasure dropped plus the 2,750 gold pieces Expected Value of a typical level four room). Once again, using the formula:

$p \times v = E_{T}$

gives us an Expected Take of 1,556 gold pieces. To find to new Expected Loss, we will be using the same formula as for rogue and cleric, with the adjustment of the fighter’s total gold needed to win, which is 20,000 gold pieces. This means the new Expected Loss is 667 gold pieces. For the outcome “of dropping half of the current total treasure,” the Expected Value of the room becomes 7,750 gold pieces, assuming the player is at half of their total needed to win. The Expected Take and Expected Loss become 3,185 gold pieces and 1,251 gold pieces respectively. Once again, for the outcome of “drop all treasure” outcome, the player has no treasure to lose, and therefore the Expected Loss is zero. The expected gain, on the other hand, is 5,240 gold pieces.

Rational of Persistence

For all classes, persistence seems to be a very valid strategy, as on consecutive turns the Expected Take is greater than the Expected Loss. However, there are a couple of caveats to this strategy. First, there is a less than 3% chance of rolling a 12, which results in character death. The player must then start over with a new character. As there are only two of each class, the number of times a player can die and return as the same class is limited. Also, this strategy can be time consuming. In particular, a roll of 8 -10, a player loses a turn on a roll of 11 or 12 on the monster fights back table requires the player to return to the Great Hall in the center of the board. With five spaces of movement each round, the trek back to the room a player was defeated in may take several turns. This is not a problem in a single player game, but in a multiplayer game. While player one is repeatedly playing a room or making their way back to the room, the other players may be defeating several monsters gaining gold and heading toward “home” or deviously, another player may enter the room Player One is repeatedly playing in order to defeat the monster before Player One returns, taking all of the treasure dropped before treasure before Player One can defeat the monster! The latter is especially likely if Player One is a rogue or cleric and the other player is a fighter, as fighters have better chance of success against monsters.

Conclusion

In testing the recommendations of the game designers, we found level 2 to be the most advantageous level for both the rogue and cleric, with relatively easy but small gains in gold pieces to be found on level 1. We found that especially with games involving more than a single rogue and/or cleric, it was necessary to venture onto level 3 and here when your character is defeated by a monster, persistence paid off. Although it was suggested by the designers, we saw no reason in games with 4 or fewer players for the cleric to go to level 4. With more than 4 players especially if this included 2 rogues and a second cleric, one cleric might have to move to level 4 simply to find rooms with treasure still in them!

For the fighter, who has to earn twice as many gold pieces as the rogue and cleric, the designers suggest playing on levels 3-5. Since their net Expected Value is highest at level 3, moving to this level as soon as possible is advantageous. We have explained above that persistence on level 4 also has a positive net Expected Value. As there are 16 rooms on level 3 and 18 more on level 4, the fighter is not as pressed to move to the higher levels unless three are many players in the game.

Acknowledgements

I would like to thank Dr. Mike Long for inspiring me to start this project and being an incredibly supportive teacher.

Contact: ayaroslaski@howardcc.edu

References

[1] Wizards of the Coast, "Dungeon! Rulebook (2012 version)," 2012. [Online]. Available: http://media.wizards.com/downloads/dnd/Dungeon!_Rulebook.pdf.

[2] Wizards of the Coast, "Webcitation," 04 10 2008. [Online]. Available: http://www.wizards.com/dnd/DnDArchives_History.asp.