Multiplayer Knowledge Based Subset Construct

We study a multiplayer version of the traditional KBSC and implement it in Python 3.

Authors: August Jacobsson & Helmer Nylén

Supervisor: Dilian Gurov

Requirements

This library uses the following external programs, which need to be installed for the library to work.

• NetworkX, which can be installed via pip3 install networkx
• pydot, which can be installed via pip3 install pydot
• Graphviz, which can be downloaded from their website

The mkbsc package

Usage example - Wagon problem

An example of the code in main.py which applies the MKBSC to the two player triangular wagon problem.

#!/usr/bin/env python3
from mkbsc import MultiplayerGame, export

#states
L = [0, 1, 2]
#initial state
L0 = 0
#action alphabet
Sigma = (("w", "p"), ("w", "p"))
#action labeled transitions
Delta = [
    (0, ("p", "p"), 0), (0, ("w", "w"), 0),
    (0, ("w", "p"), 1), (0, ("p", "w"), 2),
    (1, ("p", "p"), 1), (1, ("w", "w"), 1),
    (1, ("w", "p"), 2), (1, ("p", "w"), 0),
    (2, ("p", "p"), 2), (2, ("w", "w"), 2),
    (2, ("w", "p"), 0), (2, ("p", "w"), 1)
]
#observation partitioning
Obs = [
    [[0, 1], [2]],
    [[0, 2], [1]]
]

#G is a MultiplayerGame-object, and so are GK and GK0
G = MultiplayerGame.create(L, L0, Sigma, Delta, Obs)
GK = G.KBSC()
GK0 = GK.project(0)

#export the GK game to ./pictures/GK.png
export(GK, "GK")

Behind the scenes

A brief summary is provided below. For full documentation and a tutorial, please refer to the user guide.

The package contains definitions for a multiplayer game structure, MultiplayerGame. The states in the game are defined by States, the transitions by Transitions and imperfect information is defined by an array of player-specific Partitionings, which are sets of Observations. The States contain a tuple of each of the players' knowledge, which can be accessed by State[player], where player is zero-indexed. When the first game is constructed it is sufficient to provide a single piece of knowledge for the states (usually an integer) which is considered to be the knowledge of all involved players.

Projection

Multiplayer game structures can be projected to study how an individual player experiences the game. MultiplayerGame.project(player) project the game onto player player.

KBSC

The KBSC is defined for both multi- and singleplayer games. Calling MultiplayerGame.KBSC() will yield a new MultiplayerGame. The knowledges in the states of the new game are sets of the states from the previous graph. This means that when iterating the construct (i.e. MultiplayerGame.KBSC().KBSC()...) the knowledge in the states of the resulting graph will form a sort of tree, where the leaves are the states of the original graph.

Rendering

The games can be written in the DOT language by calling MultiplayerGame.to_dot(), and will by default color-code observations for each player. The DOT representation can be written to a file and compiled by the dot command in Graphviz. This is all done automatically by calling mkbsc.export(game, filename), which saves and opens a PNG image.

Isomorphism

The isomorphism of two game graphs can be checked by calling MultiplayerGame.isomorphic(MultiplayerGame). The function can optionally also take the observations of each player into account.

Saving games

Games can be saved to disk with the function mkbsc.to_file(game, filename), and loaded with game = mkbsc.from_file(filename). For larger games, it is recommended to skip the validation when loading the game by passing the flag validate=False.