Knowledge
The knowledge base of for the game 31 consists of a common knowledge part for the widow’s cards (the three face-up cards) There is common knowledge about the other players too, that they know the action the other player performed. This action is exchanging a hand card with the Widows’s cards, then both cards are revealed. The structure of this common knowledge is simple and the same for all agents that participate in the game.
The other knowledge involved are the cards that the players have in their hand. This knowledge is individual to the players, not common knowledge. There are too many cards in the game to infer which card another players has from what cards are in the game and the cards they have themselves. Each player makes a decision on their strategy based on the cards they have and the actions of the other players. A goal indicates what to do this round.
The other aspects of the game rely mostly on reasoning on the strategy of other players based on their actions. This is reasoning on intentions, where intentions can change depending on the progress of the game. The beliefs, desires and intentions (BDI) architecture is well suited for this. The architecture allows to distinguish between the possible choice a player has and what decisions he makes in that case. The beliefs are used to represent the estimations about other players intentions (based on their actions)
If a player knows that another player has a certain card, then it is also common knowledge among all the players.
Such that "η" is a random card from the card deck (CD). η∈CD
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Two players cannot have same cards at the same time.
Such that "η" is a random card from the card deck (CD). η∈CD
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If a player knows his own card, it does not indicate that it is a common knowledge among all players.
Such that "η" is a random card from the card deck (CD). η∈CD
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The states in the BDI architecture are the situations in a series of time points, connected and branching out in future states. The increase in possible states in the future is due to optional choices for actions about what occurs in the game. There are other belief-accessible worlds that involve possible actions of the other players. These can be used to model the actions in the game that modify the cards the players have. These actions are based on the intention of an agent, and therefore there can be guesses about the intention of other players. This deduction will be modeled in the simulation.
Representation of game model using the BDI architecture uses a time tree to describe the series of events in the world. Each turn in the game is a state (tx), the next turns of players are new states (tx+1). In their turn, the players take an action that changes the facts in the game. For each possible action there is one state in the next time point. The time points branch out quickly in further time points.
In a turn of the game 31 each player has an option to exchange one of his three cards with one of the three cards from the widows deck. The player can choose from nine possible options to exchange the card. Adding time points would multiply the number of states in each consecutive time point by 9. This would lead to a quickly expanding problem space.
Most of the possible card exchanges are not increasing a players chance win. A rational agent would not choose sub-optimal actions. This results in a subset of actual options, that may be chosen by the player. In order to keep the branches relevant, we choose to generate a world model that branches on the actions the agents actually may perform.
(Erik Dansen, Hirad Emami Alagha) * Axioms
1. Action 1 (A1): Dropping a high card and picking a lower card in the same suite
2. Action 2 (A2): Dropping a high card and picking a lower card in the different suite
3. Action 3 (A3): Dropping a low card and picking a higher card in the same suite
4. Action 4 (A4): Dropping a low card and picking a higher card in the different suite
The actions are a categorization that makes it easier to infer the intention of the agent. And is based value and type of the actual card exchange. In the next turn (time point) there is a new state for each action and each possible intention that follows from it. The same action can be from to several different intentions. The intentions are the facts that the other agents try to infer in the game. This knowledge can be used to block them, when they know that they try to obtain a specific suite of cards, or three of a kind.