Towards Multi-Objective Game Theory - With Application to Go

A.B. Meijer and H. Koppelaar

Presented by Ling Zhao

University of Alberta

October 5, 2006

Introduction

Human players use multiple objectives (multi-goals) in strategic games.

Using multi-goals is not enough – need to know how to select right set of goals.

No theoretical work to perform tree search on conjunctions and disjunctions of goals yet.

Combinatorial Game Theory

A game consists of a set of independent subgames: G = G1 + G2 + … + Gn

G = GL | GR

Result of a game: W, L, KO, U (={W|L})More: W|W, L|L, L|W = ?KO: KO↑= {W | KO↓}, KO↓= {KO↑| L}{W|L} || {L|L}

ExampleW|W

KO↑

W|LL|W

Multi-goals

For independent games:G+H = {GL+H, G+HL | GR+H, G+HR}Previously Willmott use hierarchical planni

ng to deal with conjunctions of goals.This paper deals with disjunctions and co

mbinations of disjunctions and conjunctions of goals, and dependent games.

Multi-goals

A multi-goal is a logical expression of two or more ordinal-scaled objectives.

Logical: conjunction or disjunctions.Ordinal: values are partially orderedH = G1 and (G2 or G3)

Solving Multi-goals

Treat multi-goal as single goal: branching factor increased.

Divide and conquer approach for independent goals.

Logical Evaluation

W or g = WW and g = gL or g = gL and g = LU or U = UU and U = UW > U > U > U > LW > KO↑> KO↓> L

Example

G or H = {GL or H, G or HL | GR and H, G and HR} W|L or W|L = W | { W|L} = W | WL W|L and W|L = {W|L} | L = WL | L KO↑or KO↓= W | KO↑

Dependent Goals

G1= Connect(7, 8) = W|WG2 = Connect(8, 9) = W|LIf independent, then G1 and G2

= W|L.But for this case, G1 and G2 = L|

L

Definition of n-move

Sente: move in both W|L and WL | LL.Double threat by opponent: W|WL and W|

WL.A move is an n-move if it is one of n move

s in a row, which together achieve a better result if made consecutively.

2-move is a direct threat.

Definition of (n,m)-Dependent

Two games are (n,m)-dependent if n-moves of these two games overlap. Those moves are called (n,m)-moves.

Sente is (1,2)-move.Double threat is (2,2)-move.Effective (n,m)-dependent when opponent

has no counter move in both game simultaneously.

Example

In (m,n)-dependent games, friend can move G and/or H to GL and/or HL.

In (2,2)-dependent games for opponent, G = H = W|WL, then

G and H = {W|WL} | (WL and WL)

= {W|WL} | {WL|L}

Compute Solution and Threats

Proof-number search.Find proof trees.All moves and their neighbors in the proof

trees are recorded as threats.

Example

Conclusions

Define multi-goals in logical expression.Formalize sente and double threat.Simple algorithms to compute all solutions

and threats.

Future Work

Experiment with multiple goals in dependent games.

Experiment with ko.