Yahtzee

Yahtzee is a dice-based minigame. Five whole numbers from 1 to 6 are randomly generated, and certain combinations will award the player credits. Users each get some Free Yahtzees in each Bot Restart, and can use Paid Yahtzees once the free ones have been used up which cost 10 credits each.

Users may use the command !yahtzee number to play Yahtzee, or the command !yahtzee all to use as many rolls as possible within Discord's character limitation. Note that this will also use a user's paid rolls, and can reduce their credits to negative. Users have some free rolls each Bot Cycle. Every Server Booster grants 1.5 free rolls, rounded up. After all free rolls have been used, each subsequent paid roll costs 10 credits. Each player has a limited number of paid rolls per Bot Cycle. This number starts at 150 and is increased by 150 for every Ephemera Level.

There are four types of combinations that will award credits.

1-Roll Yahtzee: All five numbers are the same. This awards 1989x the number shown on all dice.
4-of-a-Kind: Four of five numbers are the same. This awards 89x the number shown on the four dice.
Large Straight: The dice show a consecutive sequence of numbers when rearranged. This awards 4x of the sum of all numbers shown.
Full House: Three dice share one number, and the other two dice share another number. This awards 3x the sum of all numbers shown.
3-of-a-Kind: Three of five numbers are the same. This awards 1x the number shown on the 3 dice.
Small Straight: Four of the five dice show a consecutive sequence when rearranged. This awards 1x the sum of the numbers in the sequence.

The expected value of a free Yahtzee is around 17.8349 credits. The following calculations and explanations show the expected value of all the ways to get returns. The notation $^{n}\text{C}_{r}$ is the binomial coefficient, or the number of combinations that can be formed by drawing $r$ elements from a set with $n$ elements.

1-Roll Yahtzees can occur for any of the six numbers. There is 1 way to arrange 5 identical numbers.

$$E(\text{1-Roll Yahtzee})=\sum_{i=1}^{6}1989i\cdot\left(\cfrac{1}{6}\right)^{5}\approx5.3715$$

Four-of-a-Kinds can occur for any of the six numbers. There are 5 options for the last die as it cannot be a repeated number. There are 5 ways to position the non-repeated number among the five dice.

$$E(\text{4-of-a-Kind})=\sum_{i=1}^{6}89i\cdot5\cdot5\cdot\left(\cfrac{1}{6}\right)^{5}\approx6.0089$$

Large Straights have only two possible options, namely 12345 and 23456. There are $5!$ ways to arrange the five distinct numbers.

$$E(\text{Large Straight})=4\cdot\left(\sum_{i=1}^{5}i+\sum_{j=2}^{6}j\right)\cdot5!\cdot\left(\cfrac{1}{6}\right)^{5}\approx2.1605$$

Full Houses have 30 possible options as there are $6^2$ pairs of numbers that can be made from the 6 numbers and 6 of those pairs have a repeated number. There are $^{5}\text{C}_{2}$ ways to position the two numbers among the five dice.

$$E(\text{Full House})=3\left(\sum_{i=1}^{6}\sum_{j=1}^{6}\left(3i+2j\right)-5\sum_{k=1}^{6}k\right)\cdot^{5}\text{C}_{2}\cdot\left(\cfrac{1}{6}\right)^{5}\approx2.0255$$

3-of-a-Kinds can occur for any of the six numbers. There are 5 and 4 options for the fourth and fifth dice as neither can be a repeated number. There are $^{5}\text{C}_{2}$ ways to position the two non-repeated numbers among the five dice.

$$E(\text{3-of-a-Kind})=\sum_{i=1}^{6}i\cdot5\cdot4\cdot^{5}\text{C}_{2}\cdot\left(\cfrac{1}{6}\right)^{5}\approx0.5401$$

Small Straights have only three possible options, namely 1234, 2345, and 3456. For 1234 and 3456, the last number has 5 options, four of which are duplicates of numbers in the sequence. There are $^{5}\text{C}_{2}$ ways to position the two repeated numbers among the five dice and $3!$ ways to arrange the three distinct numbers. The last option that is not a duplicate has $5!$ ways to arrange the five distinct numbers. For 2345, the last number has 4 options, all of which are duplicates. Again, there are $^{5}\text{C}_{2}$ ways to position the two repeated numbers among the five dice and $3!$ ways to arrange the three distinct numbers.

$$E(\text{Small Straight})=\left(\left(\sum_{i=1}^{4}i+\sum_{j=3}^{6}j\right)\left(4\cdot^{5}\text{C}_{2}\cdot3!+5!\right)+\sum_{k=2}^{5}k\cdot\left(4\cdot^{5}\text{C}_{2}\cdot3!\right)\right)\cdot\left(\cfrac{1}{6}\right)^{5}\approx1.7284$$

Thus, the expected value of a free Yahtzee is given by the sum of these expected values, and the expected value of a paid Yahtzee is 22 less than that:

$$E(\text{Free Yahtzee})=\sum E(\text{Winning Combinations})\approx17.8349 \\ E(\text{Paid Yahtzee})=E(\text{Free Yahtzee})-22\approx-4.1651$$

Many thanks to Brent who corroborated this data through a spreadsheet. Given that the resulting value from these two different methods are identical up to at least 10 significant figures, the result is very likely accurate. View the spreadsheet here!

Note that the expected value of 1-Roll Yahtzees are very high despite their rarity, worth just over 30% of the total expected value. This implies that not getting any 1-Roll Yahtzees, which is a reasonable situation, will significantly decrease your returns. Similarly, getting slightly more 1-Roll Yahtzees than average will significantly increase your returns.

There is 1 Invocation that affects Yahtzees.

Discount is used to reduce the invoker's Paid Yahtzee cost. The command !discount consumes up to 13 Ephemera to reduce the cost of each Paid Yahtzee by the same number of credits, i.e. invoking Discount with 5 Ephemera decreases the cost of each Paid Yahtzee by 5 credits. If the invoker has 13 or less Ephemera, all will be consumed. If the invoker has 14 or more Ephemera, 13 will be consumed and the excess will remain on-hand. The effect of Discount is reset by Bot Restarts. Discount can only be used once per Bot Cycle.

Discount will misfire if the invoker has already used Discount during the Bot Cycle, consuming Ephemera as if it had been successfully used, i.e. all Ephemera on-hand or 13, whichever is lesser.

Since the expected value of a free Yahtzee is 17.8349, Discounting using 5 Ephemera will give positive returns on average. However, by discounting using the maximum of 13 Ephemera, the expected value becomes 8.8349. This makes Yahtzee a viable method for obtaining credits, given a user has sufficient Ephemera and a large enough number of paid rolls. The total expected returns on Yahtzees is given by:

$$\text{returns}=n(17.8349-c)=n(d-4.1649)$$

where $n$ is the number of rolls, $c$ is the cost per roll, and $d$ is the number of Discounts used.

Command	Description
!yahtzee number	Executes a specified number of Yahtzees. number is an optional parameter that refers to the number of rolls, defaults to 1.

1 Expected Value of paid Yahtzees:

$$\text{returns}=n(17.8349-c)=n(d-4.1649)$$

where $n$ is the number of rolls, $c$ is the cost per roll, and $d$ is the number of Discounts used.

Term	Definition
Free Yahtzee	Yahtzee rolls that do not require credits.
Paid Yahtzee	Yahtzee rolls that require credits.