CPS 296.1 Extensive-form gamesExtensive-form games with perfect informationBackward inductionA limitation of backward inductionConversion from extensive to normal formConverting the first game to normal formSubgame perfect equilibriumImperfect informationA poker-like gameSubgame perfection and imperfect informationSubgame perfection and imperfect information…Computing equilibria in the extensive formCommitmentCommitment as an extensive-form gameSlide 15Solving for the optimal mixed strategy to commit to [Conitzer & Sandholm 2006; see also: von Stengel & Zamir 2004, Letchford, Conitzer, Munagala 2009]VisualizationCPS 296.1Extensive-form gamesVincent Conitzer [email protected] games with perfect informationPlayer 1Player 2 Player 2Player 12, 4 5, 3 3, 21, 0 0, 1•Players do not move simultaneously•When moving, each player is aware of all the previous moves (perfect information)•A (pure) strategy for player i is a mapping from player i’s nodes to actionsLeaves of the tree show player 1’s utility first, then player 2’s utilityBackward inductionPlayer 1Player 2 Player 2Player 12, 4 5, 33, 21, 0 0, 1•When we know what will happen at each of a node’s children, we can decide the best action for the player who is moving at that nodeA limitation of backward inductionPlayer 1Player 2 Player 23, 2 2, 3 4, 1•If there are ties, then how they are broken affects what happens higher up in the tree•Multiple equilibria…0, 11/2.876551/2.12345Conversion from extensive to normal formPlayer 1Player 2 Player 23, 2 2, 3 4, 1•Nash equilibria of this normal-form game include (R, LL), (R, RL), (L, RR) + infinitely many mixed-strategy equilibria•In general, normal form can have exponentially many strategies0, 13, 2 3, 2 2, 3 2, 34, 1 0, 1 4, 1 0, 1LRLL LR RL RRLR = Left if 1 moves Left, Right if 1 moves Right; etc.Converting the first game to normal formPlayer 1Player 2 Player 2Player 12, 4 5, 3 3, 21, 0 0, 12, 4 2, 4 5, 3 5, 32, 4 2, 4 5, 3 5, 33, 2 1, 0 3, 2 1, 03, 2 0, 1 3, 2 0, 1LL LR RL RRLLRLRR•Pure-strategy Nash equilibria of this game are (LL, LR), (LR, LR), (RL, LL), (RR, LL)•But the only backward induction solution is (RL, LL)LRSubgame perfect equilibrium2, 4 2, 4 5, 3 5, 32, 4 2, 4 5, 3 5, 33, 2 1, 0 3, 2 1, 03, 2 0, 1 3, 2 0, 1Player 1Player 2 Player 2Player 12, 4 5, 3 3, 21, 0 0, 1LL LR RL RRLLRLRR•Each node in a (perfect-information) game tree, together with the remainder of the game after that node is reached, is called a subgame•A strategy profile is a subgame perfect equilibrium if it is an equilibrium for every subgame LR3, 2 1, 03, 2 0, 1*L *R*L*R1, 00, 1***L*R• (RR, LL) and (LR, LR) are not subgame perfect equilibria because (*R, **) is not an equilibrium• (LL, LR) is not subgame perfect because (*L, *R) is not an equilibrium•*R is not a credible threatImperfect information•Dotted lines indicate that a player cannot distinguish between two (or more) states–A set of states that are connected by dotted lines is called an information set•Reflected in the normal-form representationPlayer 1Player 2 Player 20, 0 -1, 1 1, -1 -5, -50, 0 -1, 11, -1 -5, -5L RLR•Any normal-form game can be transformed into an imperfect-information extensive-form game this wayA poker-like game1 gets King 1 gets Jackbet betstay staycall fold call fold call fold call fold“nature”player 1player 1player 2player 22 1 1 1 -2 -11 10, 0 0, 0 1, -1 1, -1.5, -.5 1.5, -1.5 0, 0 1, -1-.5, .5 -.5, .5 1, -1 1, -10, 0 1, -1 0, 0 1, -1cc cf fc ffbbsbssbs2/31/31/32/3Subgame perfection and imperfect informationPlayer 1Player 2 Player 21, -1 -1, 1 -1, 1 1, -1•How should we extend the notion of subgame perfection to games of imperfect information? •We cannot expect Player 2 to play Right after Player 1 plays Left, and Left after Player 1 plays Right, because of the information set•Let us say that a subtree is a subgame only if there are no information sets that connect the subtree to parts outside the subtreeSubgame perfection and imperfect information…Player 1Player 2 Player 24, 1 0, 0 5, 1 1, 0•One of the Nash equilibria is: (R, RR)•Also subgame perfect (the only subgames are the whole game, and the subgame after Player 1 moves Right)•But it is not reasonable to believe that Player 2 will move Right after Player 1 moves Left/Middle (not a credible threat)•There exist more sophisticated refinements of Nash equilibrium that rule out such behaviorPlayer 23, 2 2, 3Computing equilibria in the extensive form•Can just use normal-form representation–Misses issues of subgame perfection, etc.•Another problem: there are exponentially many pure strategies, so normal form is exponentially larger–Even given polynomial-time algorithms for normal form, time would still be exponential in the size of the extensive form•There are other techniques that reason directly over the extensive form and scale much better–E.g., using the sequence form of the gameCommitment•Consider the following (normal-form) game:2, 1 4, 01, 0 3, 1• How should this game be played?• Now suppose the game is played as follows:–Player 1 commits to playing one of the rows,–Player 2 observes the commitment and then chooses a column• What is the optimal strategy for player 1?• What if 1 can commit to a mixed strategy?Commitment as an extensive-form gamePlayer 1Player 2 Player 22, 1 4, 0 1, 0 3, 1•For the case of committing to a pure strategy:Up DownLeft Left RightRightCommitment as an extensive-form gamePlayer 1Player 22, 1 4, 0 1, 03, 1•For the case of committing to a mixed strategy:(1,0) (=Up)Left Left RightRight1.5, .5 3.5, .5Left Right(0,1) (=Down)(.5,.5)……•Infinite-size game; computationally impractical to reason with the extensive form hereSolving for the optimal mixed strategy to commit to[Conitzer & Sandholm 2006; see also: von Stengel & Zamir 2004, Letchford, Conitzer, Munagala 2009]•For every column t separately, we will solve separately for the best mixed row strategy (defined by ps) that induces player 2 to play t•maximize Σs ps u1(s, t) •subject to for any t’, Σs ps u2(s, t) ≥ Σs ps u2(s, t’) Σs ps = 1•(May be infeasible, e.g., if t is strictly dominated)•Pick the t that is best for player 1VisualizationL C RU0,1 1,0 0,0M4,0 0,1 0,0D0,0 1,0 1,1(1,0,0) = U(0,1,0) = M(0,0,1) =
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