1. 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

2. 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.

3. Combinatorial Game Theory A game consists of a set of independent subgames: G = G1 + G2 + … + Gn G = G L | G R 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}

4. Example W|W KO↑ W|L L|W

5. Multi-goals For independent games: G+H = {G L +H, G+H L | G R +H, G+H R } 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.

6. 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)

7. Solving Multi-goals Treat multi-goal as single goal: branching factor increased. Divide and conquer approach for independent goals.

8. 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 > U > U > U > L W > KO↑> KO↓> L

9. Example G or H = {G L or H, G or H L | G R and H, G and H R } 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↑

10. Dependent Goals G1= Connect(7, 8) = W|W G2 = Connect(8, 9) = W|L If independent, then G1 and G2 = W|L. But for this case, G1 and G2 = L|L

11. 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 moves in a row, which together achieve a better result if made consecutively. 2-move is a direct threat.

12. 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.

13. Example In (m,n)-dependent games, friend can move G and/or H to G L and/or H L. In (2,2)-dependent games for opponent, G = H = W|WL, then G and H = {W|WL} | (WL and WL) = {W|WL} | {WL|L}

14. Compute Solution and Threats Proof-number search. Find proof trees. All moves and their neighbors in the proof trees are recorded as threats.

15. Example

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

17. Future Work Experiment with multiple goals in dependent games. Experiment with ko.