Introduction

In the game of 'What's My Number?' players are not allowed to ask questions. They will only answer questions.
The game we modeled is loosely based on the game Code 777 (Tricoda in The Netherlands). The goal of the game is to use the knowledge gained by the answers of other players to deduct the numbers you have.

Our game

In 'What's My Number?' three players draw two numbers at the start of the game from a pile of nine numbers. These range from 1 to 9. Players cannot see the numbers they drew but they are able to see which numbers the other players have drawn. This leaves five possible numbers to be on a player his own board in ten different combinations.

Each turn a player draws a question card from a pre-shuffled deck and answers the question. These answers provide the other two players with information on what their own board looks like from another players perspective. The questions we provided in the game are:

Do you see any consecutive numbers?
Is the sum of the numbers you see less than 16?
Is the sum of the numbers you see more than 20?
Do you see at least one board which has only even numbers?
Do you see at least one board which has only odd numbers?
Do you only see even numbers?
Do you only see odd numbers?
Do you see more even than odd numbers?
Do you see more odd than even numbers?
Do you see an equal amount of odd and even numbers?
Do you see at least one numbers of which modulo 3 returns 0?
Do you see at least two numbers of which modulo 3 returns 0?
Do you at least one number which is divisable by another number, not counting 1?
Do all boards you see have the same sum?
Do all boards you see have the same difference?
Do you see four consecutive numbers?
Do you see three consecutive numbers?
Do you see two consecutive numbers?
Do you see at least two consecutive numbers?
Do you see at least three consecutive numbers?

With the answering of a question the amount of possible combinations on a players board decreases and when only one possibility remains a player can declare and win. After each win, we restart the game.

Because there are twenty questions and three players we can reuse the pre-shuffled game stack by just resetting it after it has been emptied and letting the game continue. This way we make sure every player answers each question once. We implemented this because we some games would not have a winner after twenty turns

Strategies

There are two possible sources of information in the game:

Answers to questions. They exclude the possibility that certain numbers are on a player's board.
The fact that other players do not know their cards yet. Each combination of cards for which it hold that any player would have known her cards by know are then impossible.

In our research, we compare agents using only the first source of information to agents which use both sources of information. We hypothesize that games in which agents use both sources of information will take less turns on average for players to guess their numbers. Our hypothesis is that games where agents use the epistemological step will take less turns.

Kooi's Choice

As an additional feature we added an option where we let players choose the question they want to answer in the game. This was suggested by Barteld Kooi and therefore we named this variation of the game Kooi's Choice. With this option turned on players will try to minimize the information gain for other players by choosing to answer the question which releases the least amount of information. This is based on the amount of states made impossible by an answer to a question When a question is answered it is still removed from the stack so every question will be handled at least once. Just like in the original game the question stack is renewed when empty. Our first hypothesis about this feature is that it will take much longer for games to end. A consequence of this will be that there are more games that do not have a winner after 60 turns (3 times the question stack).

Kooi's choice is an interesting variation of the game because the amount of information other players gather by an answer differs a lot per question. To get an idea of the average information gain of each question, we simulated 7560 games and kept track of the number of states made impossible by the answer to each question. During these games, each question was posed approximately 3600 times. We have two distinct ways of determining the information gain:

Consider how many of the states considered possible by any of the players in the actual state become impossible.
Consider how many of the states considered possible by any of the players in any states in the model become impossible.

Figure 1 shows the mean information gain for our questions when looking at the full model. Figure 2 shows the mean information gain for our questions when only taking into account states considered possible in the actual state.

Figure 1: Mean information gain per question full model

Figure 2: Mean information gain per question true model

Figure 2: Mean information gain per question in true state