Computing equilibria for integer programming games

The joint continuity of the expected payoff functions

Nash Equilibrium Seeking for Dynamic Systems with Non-Quadratic Payoffs

This paper introduces a bottleneck game with finite sets of commuters and departing time slots as an extension of congestion games of Konishi et al. (J Econ Theory 72:225–237, 1997a). After characterizing Nash equilibrium of the game, we provide sufficient conditions for which the equivalence between Nash and strong equilibria holds. Somewhat surprisingly, unlike in congestion games, a Nash equilibrium in pure strategies may often fail to exist, even when players are homogeneous. In contrast, when there is a continuum of atomless players, the existence of a Nash equilibrium and the equivalence between the set of Nash and strong equilibria hold as in congestion games (Konishi et al. 1997a).

We study the question of how long it takes players to reach a Nash equilibrium in uncoupled setups, where each player initially knows only his own payoff function. We derive lower bounds on the communication complexity of reaching a Nash equilibrium, i.e., on the number of bits that need to be transmitted, and thus also on the required number of steps. Specifically, we show lower bounds that are exponential in the number of players in each one of the following cases: (1) reaching a pure Nash equilibrium; (2) reaching a pure Nash equilibrium in a Bayesian setting; and (3) reaching a mixed Nash equilibrium. We then show that, in contrast, the communication complexity of reaching a correlated equilibrium is polynomial in the number of players.

A provisional transfer-based zero-sum tax/subsidy approach is characterized to stabilize a Nash equilibrium for the noncooperative dynamical systems with unknown sensitivity parameters. In the proposed approach, the system designer establishes a transfer graph between agents and modifies agents' payoff functions by collecting taxes from some agents and giving the same amount as subsidy to a neighbor corresponding agent(s) in the transfer graph. To achieve the zero-sum tax/subsidy approach with provisional transfer where there is no adjustment at the moment when the approach executes nor the moment when the Nash equilibrium is reached, some typical topologies of transfer graphs where the provisional transfer-based zero-sum tax/subsidy approach exists are presented.

Pure strategy Nash equilibria of large finite-player games and their relationship to non-atomic games

This paper provides a formal generalization of Nash equilibrium for games under Knightian uncertainty. The paper is devoted to counterparts of the results of Glycopantis and Muir (Econ Theory 13:743–751, l999, Econ Theory 16:239–244, 2000) for capacities. We prove that the expected payoff defined as the integral of a payoff function with respect to the tensor product of capacities on compact Hausdorff spaces of pure strategies is continuous if so is the payoff function. We prove also an approximation theorem for Nash equilibria when the expected utility payoff functions are defined on the space of capacities.

Cournot-model-based coordinated secondary voltage control considering reactive power coupling among multiple zones

We study the computation of equilibria of two-strategy anonymous games, via algorithms that may proceed via a sequence of adaptive queries to the game's payoff function, assumed to be unknown initially. The general topic we consider is query complexity, that is, how many queries are necessary or sufficient to compute an exact or approximate Nash equilibrium. We show that exact equilibria cannot be found via query-efficient algorithms. We also give an example of a 2-strategy, 3-player anonymous game that does not have any exact Nash equilibrium in rational numbers. Our main result is a new randomized query-efficient algorithm that finds a $$On^{-1/4}$$-approximate Nash equilibrium querying $$\tilde{O}n^{3/2}$$ payoffs and runs in time $$\tilde{O}n^{3/2}$$. This improves on the running time of pre-existing algorithms for approximate equilibria of anonymous games, and is the first one to obtain an inverse polynomial approximation in poly-time. We also show how this can be used to get an efficient PTAS. Furthermore, we prove that $$\varOmega n \log {n}$$ payoffs must be queried in order to find any $$\epsilon $$-well-supported Nash equilibrium, even by randomized algorithms.

In this paper, we consider last level cache (LLC) sharing problems in large-scale cloud networks with a fair payoff function. We formulate the problem as a strategic decision-making problem (i.e., a game). We examine the resource-sharing game with finite and infinite number of players. Exploiting the aggregate structure of the payoff functions, we show that the resource-sharing game has a Nash equilibrium in a wide range of return index. We show that the Nash equilibrium is not an evolutionarily stable strategy in the finite regime. Then, we introduce a myopic mean-field response where each player implements a mean-field-taking strategy. We show that such a mean-field-taking strategy is an evolutionarily stable strategy in both finite and infinite regime. We provide closed-form expression of the optimal pricing that gives an efficient resource-sharing policy. As the number of active players grows without bound, we show that the equilibrium strategy converges to a mean-field equilibrium, and the optimal prices for resources converge to the optimal price of the mean-field game. Then, we address the demand satisfaction problem for which a necessary and sufficient condition for satisfactory solutions is provided. In addition, a very fast mean-field learning algorithm is provided.

It is a main concern in applications of game theory to effectively select a Nash equilibrium. All Nash equilibria is often required to be computed for this selection process. However, it is well known that the problem of finding only one mixed-strategy Nash equilibrium is a PPAD-complete process. Therefore, it is very hard to find all Nash equilibrium for a certain problem by traditional methods. By exploiting the properties of multilinear terms in the payoff functions, this paper presents a good approximation of the multilinear terms and develops a mixed-integer linear programming for finding all mixed-strategy Nash equilibria. An example of this method will be given too.

ABSTRACTThe carbon tax is a cost‐efficient scheme to curb emissions, and it has been implemented in Australia, British Columbia, and other places worldwide. We aim to analyze its effect on dynamic pricing in a supply chain with multiple suppliers and one manufacturer. The profit‐maximizing manufacturer makes final products using raw materials from suppliers with heterogeneous prices and emission rates. A two‐stage game model is built over an infinite time horizon for this issue. In the first stage, suppliers face price‐dependent demand to set their prices and production rates under the constraint of inventory capacity. Then, in response to the carbon tax scheme, the manufacturer evaluates the procurement prices and emission rates of suppliers to control its emission volumes and sets the sales price of its product. This article predominately focuses on the optimal pricing strategies in a decentralized supply chain. The open‐loop equilibrium and Markovian Nash equilibrium for the dynamic pricing game models of both suppliers and the manufacturer are derived, respectively. The equilibrium prices of suppliers and the manufacturer can be solved based on both irreversible actions and real‐time states. These two types of equilibria can be regarded as the solutions of two different models in specific situations. To analyze the effect of sourcing diversity on pricing strategies and emissions control for the manufacturer, the more general equilibrium price for the manufacturer in an n‐suppliers oligopoly is studied. Numerical examples are presented to illustrate the equilibrium and its monotonicity with various parameter settings.

The existence of a pure Nash equilibrium in the two-player competitive diffusion game on graphs having chordality

About a symmetric three-players zero-sum game we will show the following results. 1. A modified version of Sion's minimax theorem with the coincidence of the maximin strategy and the minimax strategy are proved by the existence of a symmetric Nash equilibrium. 2. The existence of a symmetric Nash equilibrium is proved by the modified version of Sion's minimax theorem with the coincidence of the maximin strategy and the minimax strategy. Thus, they are equivalent. If a zero-sum game is asymmetric, maximin strategies and minimax strategies of players do not correspond to Nash equilibrium strategies. If it is symmetric, the maximin strategies and the minimax strategies constitute a Nash equilibrium. However, without the coincidence of the maximin strategy and the minimax strategy there may exist an asymmetric equilibrium in a symmetric three-players zero-sum game.

Real-world problems often have parameters that are uncertain during the optimization phase; stochastic optimization or stochastic programming is a key approach introduced by Beale and by Dantzig in the 1950s to address such uncertainty. Matching is a classical problem in combinatorial optimization. Modern stochastic versions of this problem model problems in kidney exchange, for instance. We improve upon the current-best approximation bound of 3.709 for stochastic matching due to Adamczyk et al. (in: Algorithms-ESA 2015, Springer, Berlin, 2015) to 3.224; we also present improvements on Bansal et al. (Algorithmica 63(4):733–762, 2012) for hypergraph matching and for relaxed versions of the problem. These results are obtained by improved analyses and/or algorithms for rounding linear-programming relaxations of these problems.