Algorithms
COMP4141 Games
Probabilistic Theory of Computation
Lecture 14 Games
Kai Engelhardt
CSE, UNSW (and NICTA)
Revision: 1.2 of Date: 2007/02/01 08:51:09 UTC
(Credits: D Dill, R v Glabbeek, M Sipser, W Thomas, T Wilke)
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Games
Probabilistic Algorithms
Games
Game theory nowadays plays an important role in economics, politics, and computer science. Lets consider two-player turn-based games such as Tic-Tac-Toe, Checkers, Chess, Go, Geography Game, etc. only. A position is a comprehensive description of the game state. Example is such a description for Tic-Tac-Toe (assuming that the player placing always moves rst). Histories of moves are always usable as positions, however, they may be not concise enough for practical purposes.
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Games
Probabilistic Algorithms
Graphs and Winning Strategies
In general, the possible positions of (nite) games form directed graphs where the edges are sometimes labelled with the player who can move from the position at the source of the edge to the position at the destination. When we use histories as positions then graphs become trees, odd level moves belong to one player whereas even level moves belong to the other. The leaves in such a tree describe nal positions of the game. These leaves are marked depending on who wins the game when it ends in this position. A player has a winning strategy in a position if, by choosing only his moves from that position onwards, he can force a win, regardless of the other players moves.
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Games
Probabilistic Algorithms
Connection to Logic
The question does player E have a winning strategy in position p of game G can often be re-phrased as a quantied Boolean formula m1 (r1 (m2 (. . . ))) where the mi and ri range over player E s, resp. As moves and expresses that E wins.
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Games
Probabilistic Algorithms
Formula Game
Lets consider a simple formula game. Let be a Boolean formula over propositions x1 , . . . , xk . In move i a player chooses a truth value for xi . Player E takes the odd turns and player A takes the even turns. Player E wins if the formula turns out true, otherwise A wins. Player E has a winning strategy i the quantied Boolean formula x1 (x2 (xr (. . . ))) is true.
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Games
Probabilistic Algorithms
Formula Game cont.
Since it is pretty much the same as QBF it follows that { | E has a winning strategy for } is also PSPACE-complete.
Sipser puts the quantiers into but this doesnt a make dierence because we assume an order on the propositions and from that the quantier sequence follows.
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Games
Probabilistic Algorithms
Geography
Sipser writes that kids play a game in which one starts with a city name and the next kid always has to come up with a new city name that starts with the last letter of the previous one. The kid who cannot name another city in this sequence loses. We could re-phrase this as a graph problem by making all the known city names nodes and have edges from a name wa to bv i a = b. Or we could generalise this even further to an arbitrary directed graph and forget about the city names. GG = { G , b | Kid 1 has a w. s. in G starting at b }
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Games
Probabilistic Algorithms
Geography cont.
Theorem GG is PSPACE-complete. Proof of GG PSPACE. That GG is in PSPACE follows from having a polynomial space TM M which decides GG. On input (V , E ), b :
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Reject if b has no successor nodes in (V , E ) because kid 1 loses. Remove b from (V , E ): let V1 = V \ {b} and E1 = E (V1 V1 ). For each b1 such that (b, b1 ) E run M on (V1 , E1 ), b1 . If all of these recursive calls accept, then kid 2 has a winning strategy in the game so reject, otherwise accept.
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Games
Probabilistic Algorithms
Geography cont.
Proof of GG is PSPACE-hard. by reduction from QBF.
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Games
Probabilistic Algorithms
Real Games
Real games such as Chess and Go have xed board sizes and thus nite numbers of p...