Towards Multi-Objective Game Theory - With Application to Go

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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
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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.

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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 (WL)
• More WW, LL, LW ?
• KO KO? W KO?, KO? KO? L
• WL LL

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Example
WW
KO?
WL
LW
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Multi-goals

• For independent games
• GH GLH, GHL GRH, GHR
• Previously Willmott use hierarchical planning to
  deal with conjunctions of goals.
• This paper deals with disjunctions and
  combinations of disjunctions and conjunctions of
  goals, and dependent games.

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Multi-goals

• A multi-goal is a logical expression of two or
  more ordinal-scaled objectives.
• Logical conjunction or disjunctions.
• Ordinal values are partially ordered
• H G1 and (G2 or G3)

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Solving Multi-goals

• Treat multi-goal as single goal branching factor
  increased.
• Divide and conquer approach for independent
  goals.

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Logical Evaluation

• W or g W
• W and g g
• L or g g
• L and g L
• U or U U
• U and U U
• W gt U gt U gt U gt L
• W gt KO?gt KO?gt L

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Example

• G or H GL or H, G or HL GR and H, G and HR
• WL or WL W WL W WL
• WL and WL WL L WL L
• KO?or KO? W KO?

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Dependent Goals

• G1 Connect(7, 8) WW
• G2 Connect(8, 9) WL
• If independent, then G1 and G2 WL.
• But for this case, G1 and G2 LL

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Definition of n-move

• Sente move in both WL and WL LL.
• Double threat by opponent WWL and WWL.
• A move is an n-move if it is one of n moves in a
  row, which together achieve a better result if
  made consecutively.
• 2-move is a direct threat.

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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.

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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
  WWL, then
• G and H WWL (WL and WL)
• WWL WLL

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Compute Solution and Threats

• Proof-number search.
• Find proof trees.
• All moves and their neighbors in the proof trees
  are recorded as threats.

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Example
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Conclusions

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

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Future Work

• Experiment with multiple goals in dependent
  games.
• Experiment with ko.