<Game> Dice Rolling

Dice rolling is a two-player game where both players have five 6-sided dices. Before the game start, the dice values are randomly generated and hidden from the other player. If every dice holds a distinct value then they must be rolled again until there are at least two dices of the same value. The game starts with one player asserting a dice value and its appearance number indicating at least how many dices of that value are there among altogether ten dices. Then, the other player has two choices. He can either choose to raise or to showdown.

1. If he chooses to raise, then he can either raise the value with the same number or raise the number with any value. 

2. If he chooses to showdown, then both players will unveil their dices and determine who's the winner. If the number of the last assertion is larger than or equal to the truth, the asserter wins, otherwise, the caller wins.

An additional rule is usually applied in order to make the game more complicated and interesting. The counting is different when it comes to dice value 1, which can be counted as any other value as long as the appearance number of 1 has not been asserted, and once it is asserted, 1 can only be counted as itself.

By definition, this is a probability game with imperfect but complete information.

 

Rules of thumb (to be reorganized)

The game strategy is different depending on the order of the assertion.

If you are to make the first assertion, the most common strategies are straight bluffing and telling the truth, since whether your assertion is close to or away from the truth depends more on the information you have rather than the information your opponent has. If you are bluffing, make your opponent raise on the same value, and if you are telling the truth, make your opponent call for the showdown. In order to make your opponent do as you wish, you need to disguise your true intention in the following assertions. Make it looks like that you are telling the truth while you are actually bluffing or vice versa.

The expectation of the appearance number of a certain value among five dices is 5 * 1/6 = 5/6 ≈ 1 if one is counted as itself, otherwise, the expectation should be 5 * 2/6 = 10/6 = 1.667 ≈ 2.

 

To identify a bluffing, check how many dices of that value you have. If you have many, then your opponent is most likely bluffing, otherwise, he is most likely telling the truth. To counterattack a bluffing, call for the showdown. If you think he is telling the truth, change to another value of which you have many, otherwise, raise on the same value.

 

Probability table (1 counted as itself):

At least zero appearance: 100.00%

At least one appearance: 16.67%

At least two appearance: 3.33%

At least three appearance: 0.67%

At least four appearance: 0.13%

At least five appearance: 0.03%

 

Probability table for a value other than 1 (1 counted as any value):

At least zero appearance: 100.00%

At least one appearance: 33.33%

At least two appearance: 11.11%

At least three appearance: 3.70%

At least four appearance: 1.23%

At least five appearance: 0.41%