Knowledge Games
Hans van Ditmarsch
Abstract:
In the board game Cluedo the players have to determine what the murder cards on
the table are, by reasoning about their own cards, and about what they get to
know of other players' cards in game actions. Typical for Cluedo is that a
finite number of cards is dealt over a finite number of players, that the
players can only see their own cards, and that cards do not change hands. We
call these games knowledge games. A knowledge game that exemplifies most of the
general features that we want to study, is the game for three players each
holding one card. We describe game states, game actions, and their consequences.
A deal of cards is a function from cards to players. Two deals are
indistinguishable for a player if they agree on his cards. The set of relevant
deals is the set of deals that are publicly indistinguishable from the actual
deal of cards. These are the deals that players still have to consider while
reasoning about card possession. In the initial state of the game, the cards
have been dealt but the players have not acquired knowledge about other
players' cards. The initial state of the game is represented by a pointed
multiagent $S5$ model on the set of deals where all players hold the same
number of cards as in the actual deal. The point of the model is the actual
deal. Accessibility is indistinguishability of deals. Propositional atoms
describe that a player holds a card. Other knowledge game states are also $S5$,
but agents may know more.
The properties of the agents in the initial state of a knowledge game are
described by the theory $kgames = \{ deals, seedontknow \}$. $Deals$ is the
disjunction of atomic state descriptions $\delta_{d'}$ of relevant deals $d'$.
It expresses that exactly one of the relevant card deals must be actually the
case. $Seedontknow$ expresses that a player considers deals, if and only if
they correspond to what he knows about his own cards. Various agent properties
follow from $kgames$, such as that every player knows his own cards and that
every player holds a fixed number of cards, and different ways to describe
ignorance. By a bisimulation proof we show that $kgames$ (uniquely) describes
the model $I_d$ for the initial state of a knowledge game. We also characterize
the knowledge game state where the cards have been dealt but where the players
have not yet turned their cards. A theory $prekgames$ describes the model
$preI_d$ for that state. State descriptions that are computed with standard
methods for finite multiagent $S5$ models, are equivalent to our results.
Game actions in knowledge games consist of card requests and responses to those
requests, plus some other moves, such as announcing, or guessing, that you have
won. A typical game action is that of a player showing a card (only) to another
player. The remaining players see that a card is being shown, but cannot see
which card. Questions and answers are combined in one format for game actions:
a quintuple consisting of the requesting player, the question, the answering
player, the answer, and the `publicity': what other players get to know about
the answer. The question is a set of possible answers. The actual answer is one
of those. Each possible answer is a set of worlds of the current game state.
Not any set of worlds: an answer must be the union of classes of the partition
of that state for the responding player. The question must cover the state:
each world must be contained in a possible answer. All desirable constraints on
`publicity', such as that the answering player controls the response, can be
realized by formalizing it as a function from players to partitions of the
question. The game action format also describes other conceivable game actions,
sometimes by the trick of having a player respond to his own question. A game
action is executable in a game state if the answer contains the point of that
state. A game action minus the roles of the requesting and responding player
corresponds to a pointed $S5$ frame on the set of possible answers: a game
action frame. The next game state is a restriction of the direct product of the
current game state and a game action frame.
This game action format is purely semantic: game actions are defined for a
given game state. We introduce a general action language to describe game
actions, and a corresponding notion of interpretation that we call local
interpretation. The language $L^\Box_n$ for dynamic epistemic logic contains
dynamic modal operators for, what we call, knowledge action types and knowledge
actions. The basic programming constructs in the action language are test,
sequence, choice, learning and local choice. The first four define the class of
knowledge action types $KT$. From an action type we construct a knowledge
action of that type by the operation of local choice. $KA$ is the class of
knowledge actions. In the knowledge game for three players and three cards
where player 1 holds red, player 2 holds white, and player 3 holds blue, the
game action of 1 showing his red card to 2 is described by the knowledge action
$L_{123} (!L_{12} ?r_1 \union L_{12} ?w_1 \union L_{12} ?b_1)$. This stands
for: 1 and 2 learn that 1 holds red, and 1, 2 and 3 learn that either 1 and 2
learn that 1 holds red, or that 1 and 2 learn that 1 holds white, or that 1 and
2 learn that 1 holds blue.
The local interpretation of a knowledge action type is a relation between
multiagent $S5$ models and their worlds. The `learning' operator is interpreted
as follows: given a set of models that is the result of executing knowledge
action type $\tau$ in a given model $M$, the result of executing `group $A$ of
agents learn $\tau$' is the direct sum of that set of models plus access added
for the agents `only in $A$': an agent cannot distinguish worlds in two
different models in the direct sum if he does not occur in those models and if
he could not distinguish their origins in $M$ under the interpretation of
$\tau$. The local interpretation of a knowledge action in a state $(M,w)$ is
derived from the interpretation of that action's type $\tau$ by choosing,
determined by local choice, one model image in the interpretation of $\tau$ in
$M$ and one world image for $w$ in that model image. It can be seen as a
constraint on the state transformations induced by $\tau$. Bisimilarity of
models and states is preserved after execution of knowledge types and actions.
Game actions are described by knowledge actions. To establish that
correspondence, we define another notion of interpretation, called product
interpretation. The knowledge action type frame of a knowledge action type is
defined on the set of actions of that type. Access on this frame is determined
by the structure of actions. The frame of a knowledge action is a pointed
knowledge action type frame. The state transformation induced by that frame is
the product interpretation of a knowledge action in an $S5$ state. The notions
of local interpretation and product interpretation are the same, up to
bisimilarity of states. They are not the same if the local interpretation of a
knowledge action type in a model consists of models for different groups. A
knowledge action describes a game action for a given game state, if there is an
isomorphism between the game action frame for that game action and the
knowledge action frame for that knowledge action. The isomorphism relates
actions of the same type to possible answers to the question of the game
action, such that the precondition of an action is satisfied in the worlds that
the answer consists of. We suggest a procedure describe that constructs from a
game action a knowledge action that describes it.
Apart from game actions we can describe many other communicative acts in
$L^\Box_n$. We give examples. We describe suspicion, and we describe sequences
of calls over a network. We compare our research to other recent work in the
area of dynamic epistemics and multiagent systems.
We have shown that all game states and all game actions in knowledge games can
be formally described in a logical language. Given this formal description of
games, we can start to think about optimal strategies for winning them. There
are some formidable obstacles to overcome here. It is unclear what the
individual preferences of a player are among the different questions he can
ask. This requires a comparison of the partition refinements, i.e. the new game
states, created by the possible answers to those questions. It also requires a
(recursive) analysis of the questions that the next player can ask given those
refinements. Even when we have individual preferences for all players, it is
unclear what the mixed strategy equilibria are for such an imperfect
information game. Finally someone may be able to answer the question, what is
the value of Cluedo?