OddThinking

Introduction

Let’s play some dice games. They are all silly and hypothetical. They are toy games, designed to be analysed as a coding exercise for me, which I will discuss further later.

Typical casino games, like Baccarat, are asymmetric; i.e, each side of the game follows different rules. Sometimes, the croupier follows a strict predetermined strategy, like BlackJack.

The casino needs to work out what the optimal strategy is for a player, so they can price the game appropriately – at a point where it is slightly in the casino’s favour. That is, they need to know the expected value of the player’s optimal strategy.

(Casinos have close enough to infinite money, so we can ignore differing “marginal utility” of your last few dollars.)

Even though I’m not interested in actually gambling on these games, this approach maps exactly to what I want to figure out, so I will use a casino as an simple analogy.

So, suppose we’ve been hired by the research division of SomeBetOdd Casinos. They want an analysis of some new dice games they are thinking of introducing.

Each game involves the player rolling two fair six-sided dice. The dice are identical, but, one is red and one is blue, for ease of explanation.

For each game, think about what choices you would make to optimise your chances of winning, and how much you would be willing to pay to join such a game. I’d recommend stopping after each question and contemplating it (or even properly modelling it) before reading the later parts, but I understand that you won’t. That’s okay. This post is more to force me to understand my own model than to make an interesting puzzle for you.

Parts 2 and 3, including solutions, will be posted soon.

Game A

Rules: The player rolls two dice. The player is paid out the sum of the pips showing, in dollars.

What is the minimum the casino should charge to play?

Game B

Rules: The player rolls both dice. The player is then given three options:

• Sit: Keep the dice as is.
• Single Re-Roll: pick up one of the dice and re-roll it (only once) again.
• Double Re-Roll: pick up both of the dice and re-roll them (only once) again.

We’ll call this rule (where the player gets once chance to re-roll 0, 1 or 2 dice to maximise their total score) the “Basic Game”.

The player is paid out the sum of the pips showing at the end, in dollars.

What strategy should the player use? We’ll call this strategy the “Basic Strategy”.

How much should it cost to play?

Game C

Rules: A sign is posted that says “Beat a 7 to win”. The player plays the Basic Game. If the total score at the end is strictly greater than 7, the player receives a dollar. Otherwise they receive nothing.

What strategy should the player use? How much should it cost to play?

What if we tweaked the game to be “Beat a 9 to win.”?

Game D

So far, the player has had the only dice. Now, we add a croupier with their own pair of dice to add a second level of suspense to the game. The rest of the games involve a croupier who needs to be beaten. The croupier will always strictly follow a published procedure.

Rules: The player plays the Basic Game. Meanwhile, behind an opaque screen, the croupier simply rolls two dice (similar to the player in Game A). When the player is finished, the croupier’s score is revealed. If the player’s total score at the end is strictly greater than the croupier’s, the player receives a dollar. Otherwise they receive nothing.

What strategy should the player use? How much should it cost to play?

Game E

Rule: A sign is posted that says “Beat the Croupier by at least 2 to win”. It is played like Game D, but the player’s total score at the end must be strictly greater than the croupier’s score + 1.

What strategy should the player use? How much should it cost to play?

What if we increased the handicap by another point or two?

Game F

Rule: The player and the croupier both play the Basic Game. The croupier always plays the Basic Strategy. (Spoiler: Re-roll all 1s, 2s and 3s; keep all 4s, 5s and 6s.) The player does not see the croupier’s dice until the end. Again, the player must get a score strictly higher than the croupier to be paid out $1.

This improves the croupier’s chances of a better score. Does that affect the player’s strategy? How much should it cost to play?

Game G

Rule: The player and the croupier both play the Basic Game. The croupier always plays the Basic Strategy. The croupier always goes first, and the player is able to see their final total before attempting to beat it.

This surely affects the player’s strategy! How much should it cost to play?

Game H

Rule: The player and the croupier both play the Basic Game. The croupier always plays the Basic Strategy. The croupier always makes their first roll first. The player gets to see the first roll, but then the opaque screen is lowered, while the croupier makes their second roll (if any). The player finishes their game before the croupier’s final total is revealed.

Game I

Rule: The game is player like Game H. However, the croupier is no longer constrained to the Basic Strategy. Instead, the croupier will follow the optimum strategy for this game to minimise the amount the casino must play.

(That strategy will be published, so the player will know what it is.)

How is the croupier’s optimal strategy different from the Basic Strategy?