We study a two-sided network investment game consisting of two sets of players, called providers and users. The game is set in two stages. In the first stage, providers aim to maximize their profit by investing in bandwidth of cloud computing services. The investments of the providers yield a set of usable services for the users. In the second stage, each user wants to process a task and therefore selects a bundle of services so as to minimize the total processing time. We assume the total processing time to be separable over the chosen services and the processing time of each service to depend on the utilization of the service and the installed bandwidth. We provide insights on how competition between providers affects the total costs of the users and show that every game on a series-parallel graph can be reduced to an equivalent single edge game when analyzing the set of subgame perfect Nash equilibria.
Leader-follower models as the previous ones are also known as Stackelberg games @cite_20 . Recently, lots of attention has been placed on Stackelberg pricing games. Starting with Labb ' e et al @cite_34 , who considered the problem in which a leader sets prices on a subset of edges and the follower chooses a shortest path, several authors generalized to different combinatorial problems for the follower. For example, @cite_32 studied the Stackelberg minimum spanning tree problem, and Bil o el al. @cite_10 and later Cabello @cite_11 investigated the Stackelberg shortest path tree problem. @cite_15 and @cite_8 considered the power of single-price strategies. B " o @cite_22 extended the analysis of this simple algorithm beyond the combinatorial setting.
@cite_22 @cite_8 @cite_32 @cite_15 @cite_34 @cite_10 @cite_20 @cite_11
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We study a multi-player one-round game termed Stackelberg Network Pricing Game, in which a leader can set prices for a subset of m priceable edges in a graph. The other edges have a fixed cost. Based on the leader’s decision one or more followers optimize a polynomial-time solvable combinatorial minimization problem and choose a minimum cost solution satisfying their requirements based on the fixed costs and the leader’s prices. The leader receives as revenue the total amount of prices paid by the followers for priceable edges in their solutions. Our model extends several known pricing problems, including single-minded and unit-demand pricing, as well as Stackelberg pricing for certain follower problems like shortest path or minimum spanning tree. Our first main result is a tight analysis of a single-price algorithm for the single follower game, which provides a (1+e)log m-approximation. This can be extended to provide a (1+e)(log k+log m)-approximation for the general problem and k followers. The problem is also shown to be hard to approximate within @math for some e>0. If followers have demands, the single-price algorithm provides an @math -approximation, and the problem is hard to approximate within @math for some e>0. Our second main result is a polynomial time algorithm for revenue maximization in the special case of Stackelberg bipartite vertex-cover, which is based on non-trivial max-flow and LP-duality techniques. This approach can be extended to provide constant-factor approximations for any constant number of followers. How do we build multiagent algorithms for agent interactions with human adversaries? Stackelberg games are natural models for many important applications that involve human interaction, such as oligopolistic markets and security domains. In Stackelberg games, one player, the leader, commits to a strategy and the follower makes their decision with knowledge of the leader's commitment. Existing algorithms for Stackelberg games efficiently find optimal solutions (leader strategy), but they critically assume that the follower plays optimally. Unfortunately, in real-world applications, agents face human followers (adversaries) who --- because of their bounded rationality and limited observation of the leader strategy --- may deviate from their expected optimal response. Not taking into account these likely deviations when dealing with human adversaries can cause an unacceptable degradation in the leader's reward, particularly in security applications where these algorithms have seen real-world deployment. To address this crucial problem, this paper introduces three new mixed-integer linear programs (MILPs) for Stackelberg games to consider human adversaries, incorporating: (i) novel anchoring theories on human perception of probability distributions and (ii) robustness approaches for MILPs to address human imprecision. Since these new approaches consider human adversaries, traditional proofs of correctness or optimality are insufficient; instead, it is necessary to rely on empirical validation. To that end, this paper considers two settings based on real deployed security systems, and compares 6 different approaches (three new with three previous approaches), in 4 different observability conditions, involving 98 human subjects playing 1360 games in total. The final conclusion was that a model which incorporates both the ideas of robustness and anchoring achieves statistically significant better rewards and also maintains equivalent or faster solution speeds compared to existing approaches. We consider a one-round two-player network pricing game, the Stackelberg Minimum Spanning Tree game or StackMST. The game is played on a graph (representing a network), whose edges are colored either red or blue, and where the red edges have a given fixed cost (representing the competitor’s prices). The first player chooses an assignment of prices to the blue edges, and the second player then buys the cheapest possible minimum spanning tree, using any combination of red and blue edges. The goal of the first player is to maximize the total price of purchased blue edges. This game is the minimum spanning tree analog of the well-studied Stackelberg shortest-path game. We analyze the complexity and approximability of the first player’s best strategy in StackMST. In particular, we prove that the problem is APX-hard even if there are only two different red costs, and give an approximation algorithm whose approximation ratio is at most min k,1+ln b,1+ln W , where k is the number of distinct red costs, b is the number of blue edges, and W is the maximum ratio between red costs. We also give a natural integer linear programming formulation of the problem, and show that the integrality gap of the fractional relaxation asymptotically matches the approximation guarantee of our algorithm. Let a communication network be modelled by a directed graphG = (V,E) of n nodes and m edges. We consider aone-round two-player network pricing game, the Stackelberg ShortestPaths Tree (StackSPT) game. This is played on G, by assuming thatedges in E are partitioned into two sets: a set E F of edges with afixed positive real weight, and a set E P of edges that should bepriced by one of the two players (the leader). Given adistinguished node r ∈ V, the StackSPT game isthen as follows: the leader prices the edges in E P in such a waythat he will maximize his revenue, knowing that the other player(the follower) will build a shortest paths tree of G rooted at r,say S(r), by running a publicly available algorithm. Quitenaturally, for each edge selected in the solution, theleader’s revenue is assumed to be equal to the loaded priceof an edge, namely the product of the edge price times the numberof paths from r in S(r) that use it. First, we show that theproblem of maximizing the leader’s revenue is NP-hard as soonas |E P | = Θ(n). Then, in search of aneffective method for solving the problem when the size of E P isconstant, we focus on the basic case in which |E P| = 2, and we provide an efficient O(n 2 logn) timealgorithm. Afterwards, we generalize the approach to the case |E P| = k, and we show that it can be solved inpolynomial time whenever k = O(1).