2,726 research outputs found
On a Stackelberg Subset Sum Game
This contribution deals with a two-level discrete decision problem, a
so-called Stackelberg strategic game: A Subset Sum setting is addressed with a
set of items with given integer weights. One distinguished player, the
leader, may alter the weights of the items in a given subset , and
a second player, the follower, selects a solution in order to
utilize a bounded resource in the best possible way. Finally, the leader
receives a payoff from those items of its subset that were included in the
overall solution , chosen by the follower. We assume that the follower
applies a publicly known, simple, heuristic algorithm to determine its solution
set, which avoids having to solve NP-hard problems.
Two variants of the problem are considered, depending on whether the leader
is able to control (i.e., change) the weights of its items (i) in the objective
function or (ii) in the bounded resource constraint. The leader's objective is
the maximization of the overall weight reduction, for the first variant, or the
maximization of the weight increase for the latter one. In both variants there
is a trade-off for each item between the contribution value to the leader's
objective and the chance of being included in the follower's solution set.
We analyze the leader's pricing problem for a natural greedy strategy of the
follower and discuss the complexity of the corresponding problems. We show that
setting the optimal weight values for the leader is, in general, NP-hard. It is
even NP-hard to provide a solution within a constant factor of the best
possible solution. Exact algorithms, based on dynamic programming and running
in pseudopolynomial time, are provided. The additional cases, in which the
follower faces a continuous (linear relaxation) version of the above problems,
are shown to be straightforward to solve.Comment: 13 pages, 1 figur
Playing Stackelberg Opinion Optimization with Randomized Algorithms for Combinatorial Strategies
From a perspective of designing or engineering for opinion formation games in
social networks, the "opinion maximization (or minimization)" problem has been
studied mainly for designing subset selecting algorithms. We furthermore define
a two-player zero-sum Stackelberg game of competitive opinion optimization by
letting the player under study as the first-mover minimize the sum of expressed
opinions by doing so-called "internal opinion design", knowing that the other
adversarial player as the follower is to maximize the same objective by also
conducting her own internal opinion design.
We propose for the min player to play the "follow-the-perturbed-leader"
algorithm in such Stackelberg game, obtaining losses depending on the other
adversarial player's play. Since our strategy of subset selection is
combinatorial in nature, the probabilities in a distribution over all the
strategies would be too many to be enumerated one by one. Thus, we design a
randomized algorithm to produce a (randomized) pure strategy. We show that the
strategy output by the randomized algorithm for the min player is essentially
an approximate equilibrium strategy against the other adversarial player
Stackelberg Network Pricing Games
We study a multi-player one-round game termed Stackelberg Network Pricing
Game, in which a leader can set prices for a subset of 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, and the problem is to find revenue maximizing prices. 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 -approximation for any . This can
be extended to provide a -approximation for the
general problem and followers. The latter result is essentially best
possible, as the problem is shown to be hard to approximate within
\mathcal{O(\log^\epsilon k + \log^\epsilon m). If followers have demands, the
single-price algorithm provides a -approximation, and the
problem is hard to approximate within \mathcal{O(m^\epsilon) for some
. 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. Our results
can be extended to provide constant-factor approximations for any constant
number of followers
Privatization in oligopoly : the impact of the shadow cost of public funds
The aim of this paper is to investigate the welfare eect of privatization in oligopoly when the government takes into account the distortionary eect of rising funds by taxation (shadow cost of public funds). We analyze the impact of the change in ownership not only on the objective function of the rms, but also on the timing of competition by endogenizing the determination of simultaneous (Nash-Cournot) versus sequential (Stackelberg) games. We show that, absent effciency gains, privatization never increases welfare. Moreover, even when large effciency gains are realized, an ineffcient public rm may be preferred
Mixed duopoly, privatization and the shadow costs of public funds
The purpose of this article is to investigate how the introduction of the shadow cost of public funds in the utilitarian measure of the economywide welfare affects the behavior of a welfare maximizer public firm in a mixed duopoly. We prove that when firms play simultaneously, the mixed-Nash equilibrium can dominate any Cournot equilibria implemented after a privatization, with or without efficiency gains. This can be true both in terms of welfare and of public firm's profit. When we consider endogenous timing, we show that either mixed- Nash, private leadership or both Stackelberg equilibria can result as subgameperfect Nash equilibria (SPNE). As a consequence, the sustainability of sequential equilibria enlarges the subspace of parameters such that the market performance with an inefficient public firm is better than the one implemented after a full-efficient privatization. Absent efficiency gains, privatization always lowers welfare
Privatization in oligopoly : the impact of the shadow cost of public funds
The aim of this paper is to investigate the welfare effect of privatization in oligopoly when the government takes into account the distortionary effect of raising funds by taxation (shadow cost of public funds). We analyze the impact of the change in ownership not only on the objective function of the firms, but also on the timing of competition by endogenizing the determination of simultaneous (Nash-Cournot) versus sequential (Stackelberg) games. We show that, absent efficiency gains, privatization never increases welfare. Moreover, even when large efficiency gains are realized, an inefficient public firm may be preferred
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