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By Martin Gairing, Rahul Savani

This publication constitutes the refereed lawsuits of the ninth overseas Symposium on Algorithmic online game conception, SAGT 2016, held in Liverpool, united kingdom, in September 2016.The 26 complete papers provided including 2 one-page abstracts have been conscientiously reviewed and chosen from sixty two submissions.
The permitted submissions hide numerous very important aspectsof algorithmic online game thought equivalent to computational facets of video games, congestion video games and networks, matching and vote casting, auctions and markets, and mechanism design.

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Extra info for Algorithmic Game Theory: 9th International Symposium, SAGT 2016, Liverpool, UK, September 19–21, 2016, Proceedings

Example text

We are ready to prove the main result of the section. Theorem 3. In any λp -Lipschitz game L that possess an equilibrium and any > 2 2 2 0, there is a k-uniform strategy profile, with k = 16M λ2 pγ that is an -equilibrium. Theorem 3 establishes the existence of a k-uniform approximate equilibrium, but this does not immediately give us our approximation algorithm. The obvious approach is to perform a brute force check of all k-uniform strategies, and then output the one that provides the best approximation.

XN ): brk (x) = argmax uk (α; x−k ) . e. a profile x∗ such that x∗k ∈ brk (x∗ ) for every player k. Definition 2 (Potential games and its generalizations). A game is an (exact) potential game [11] if it admits a function (called the potential) Φ : A → R such that for any player k and any unilateral deviation of k from strategy profile x to x : uk (α, x−k ) − uk (α , x−k ) = Φ(α, x−k ) − Φ(α , x−k ). A game is a generalized ordinal potential game [11] (or G-potential game for short) if there exists a potential function Φ : A → R such that, for any player k and any state x, uk (α, x−k ) > uk (α , x−k ) ⇒ Φ(α, x−k ) > Φ(α , x−k ).

2 Best Response Algorithm and Potential Games We consider a game with a finite number N of players, each with A strategies. def Definition 1 (N -player game). A game is a tuple G = G(N , A, u) with – a finite set of players N = {1, . . , N }; – a finite set Ak of pure strategies for each player k ∈ N . def The set of strategy profiles or states of the game is A = A1 ×A2 ×· · · AN . – The players’ payoff functions uk : A → R. We define the best response correspondence brk (x) as the set of all strategies def that maximizes the payoff for player k under profile x = (x1 , .

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