Combinatorial Games under Auction Play

Lazarus, Andrew J.; Loeb, Daniel E.; Propp, James; Stromquist, Walter; Ullman, Daniel H.

doi:10.1006/game.1998.0676

Cited by 46 publications

(110 citation statements)

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“…Though we will not use this fact, it is interesting to point out that in the case where E is finite, there is a simple algorithm from [15] which calculates the value u of the game and proceeds as follows. Assuming the value u(v) is already calculated at some set V ′ ⊃ Y of vertices, find a path…”

Section: 3mentioning

confidence: 99%

Tug-of-war and the infinity Laplacian

Peres

Schramm

Sheffield⋆

et al. 2008

J. Amer. Math. Soc.

351

429

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We prove that every bounded Lipschitz function F on a subset Y of a length space X admits a tautest extension to X, i.e., a unique Lipschitz extension u for which Lip_U u = Lip_{boundary of U} u for all open subsets U of X that do not intersect Y. This was previously known only for bounded domains R^n, in which case u is infinity harmonic, that is, a viscosity solution to Delta_infty u = 0. We also prove the first general uniqueness results for Delta_infty u = g on bounded subsets of R^n (when g is uniformly continuous and bounded away from zero), and analogous results for bounded length spaces. The proofs rely on a new game-theoretic description of u. Let u^epsilon(x) be the value of the following two-player zero-sum game, called tug-of-war: fix x_0=x \in X minus Y. At the kth turn, the players toss a coin and the winner chooses an x_k with d(x_k, x_{k-1})< epsilon. The game ends when x_k is in Y, and player one's payoff is F(x_k) - (epsilon^2/2) sum_{i=0}^{k-1} g(x_i) We show that the u^\epsilon converge uniformly to u as epsilon tends to zero. Even for bounded domains in R^n, the game theoretic description of infinity-harmonic functions yields new intuition and estimates; for instance, we prove power law bounds for infinity-harmonic functions in the unit disk with boundary values supported in a delta-neighborhood of a Cantor set on the unit circle.Comment: 44 pages, 4 figure

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Section: 3mentioning

confidence: 99%

Tug-of-war and the infinity Laplacian

Peres

Schramm

Sheffield⋆

et al. 2008

J. Amer. Math. Soc.

351

429

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show abstract

“…Though we will not use this fact, it is interesting to point out that in the case where E is finite, there is a simple algorithm from [15] Vk …”

Section: Umentioning

confidence: 99%

Tug-of-War and the Infinity Laplacian

Peres¹,

Schramm²,

Sheffield⋆³

et al. 2011

Selected Works of Oded Schramm

100

233

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we later learned, tug-of-war games have been considered by Lazarus, Loeb, Propp and Ullman in [16] (see also [15]).Tug-of-war on a metric space is very natural and conceivably applicable (like differential game theory) to economic and political modeling.The intuition provided by thinking of strategies for tug-of-war yields new results even in the classical setting of domains in R n . For instance, in Section 4 we show that if u is infinity-harmonic in the unit disk and its boundary values are in [0, 1] and supported on a δ-neighborhood of the ternary Cantor set on the unit circle, then u(0) < δ β for some β > 0.Before precisely stating our main results, we need several definitions.1.2. Random turn games and values. We consider two-player, zero-sum randomturn games, which are defined by the following parameters: a set X of states of the game, two directed transition graphs E I , E II with vertex set X, a nonempty set Y ⊂ X of terminal states (a.k.a. absorbing states), a terminal payoff function F : Y → R, a running payoff function f : X Y → R, and an initial state x 0 ∈ X. Game play is as follows: a token is initially placed at position x 0 . At the k th step of the game, a fair coin is tossed, and the player who wins the toss may move the token to any x k for which (x k−1 , x k ) is a directed edge in her transition graph. The game ends the first time x k ∈ Y , and player I's payoff isPlayer I seeks to maximize this payoff, and since the game is zero-sum, player II seeks to minimize it.We will use the term tug-of-war (on the graph with edges E) to describe the game in which E := E I = E II (i.e., players have identical move options) and E is undirected (i.e., all moves are reversible). Generally, our results pertain only to the undirected setting. Occasionally, we will also mention some counterexamples showing that the corresponding results do not hold in the directed case.In the most conventional version of tug-of-war on a graph, Y is a union of "target sets" Y I and Y II , there is no running payoff (f = 0) and F is identically 1 on Y I , identically 0 on Y II . Players then try to "tug" the game token to their respective targets (and away from their opponent's targets) and the game ends when a target is reached.A strategy for a player is a way of choosing the player's next move as a function of all previously played moves and all previous coin tosses. It is a map from the set of partially played games to moves (or in the case of a random strategy, a probability distribution on moves). Normally, one would think of a good strategy as being Markovian, i.e., as a map from the current state to the next move, but it is useful to allow more general strategies that take into account the history.Given two strategies S I , S II , let F − (S I , S II ) and F + (S I , S II ) be the expected total payoff (including the running payoffs received) at the termination of the game -if the game terminates with probability one and this expectation exists in [−∞, ∞]; otherwise, let F − (S I , S II ) = −∞ and F + (S I , S II ) = +∞....

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“…In the unbiased case, using the absolutely minimizing Lipschitz property, there is a simple algorithm computing the discrete infinity harmonic extension on a finite graph, in polynomial time in the size of the graph [14,15]. The same idea can be used to approximate continuum solutions in R n [19].…”

Section: Explicit Biased Infinity Harmonic Functionsmentioning

confidence: 99%