Bertrand-Edgeworth games under oligopoly with a complete characterization for the triopoly

The paper extends the analysis of price competition among capacity-constrained sellers beyond the cases of duopoly and symmetric oligopoly. We first provide some general results for the oligopoly and then focus on the triopoly, providing a complete characterization of the mixed strategy equilibrium of the price game. The region of the capacity space where the equilibrium is mixed is partitioned according to the features of the mixed strategy equilibrium arising in each subregion. Then computing the mixed strategy equilibrium becomes a quite simple task. The analysis reveals features of the mixed strategy equilibrium which do not arise in the duopoly (some of them have also been discovered by Hirata (2008)).Bertrand-Edgeworth; Price game; Oligopoly; Triopoly; Mixed strategy equilibrium

Statics and dynamics of selfish interactions in distributed service systems

We study a class of games which model the competition among agents to access some service provided by distributed service units and which exhibit congestion and frustration phenomena when service units have limited capacity. We propose a technique, based on the cavity method of statistical physics, to characterize the full spectrum of Nash equilibria of the game. The analysis reveals a large variety of equilibria, with very different statistical properties. Natural selfish dynamics, such as best-response, usually tend to large-utility equilibria, even though those of smaller utility are exponentially more numerous. Interestingly, the latter actually can be reached by selecting the initial conditions of the best-response dynamics close to the saturation limit of the service unit capacities. We also study a more realistic stochastic variant of the game by means of a simple and effective approximation of the average over the random parameters, showing that the properties of the average-case Nash equilibria are qualitatively similar to the deterministic ones.Comment: 30 pages, 10 figure

Derivation of a mathematical structure for market-based transmission augmentation in oligopoly electricity markets using multilevel programming

In this paper, we derive and evaluate a new mathematical structure for market-based augmentation of the transmission system. The closed-form mathematical structure can capture both the efficiency benefit and competition benefit of the transmission capacity. The Nash solution concept is employed to model the price-quantity game among GenCos. The multiple Nash equilibria of the game are located through a characterisation of the problem in terms of minima of the R function. The worst Nash equilibrium is used in the mechanism of transmission augmentation. The worst Nash equilibrium is defined as the one which maximises the social cost, total generation cost + total value of lost load. Thorough analysis of a simple three-node network is presented to clearly highlight the mechanism of the derived mathematical structure from different perspectives

Mean-Field Games of Finite-Fuel Capacity Expansion with Singular Controls

We study Nash equilibria for a sequence of symmetric $N$-player stochastic games of finite-fuel capacity expansion with singular controls and their mean-field game (MFG) counterpart. We construct a solution of the MFG via a simple iterative scheme that produces an optimal control in terms of a Skorokhod reflection at a (state-dependent) surface that splits the state space into action and inaction regions. We then show that a solution of the MFG of capacity expansion induces approximate Nash equilibria for the $N$-player games with approximation error $\varepsilon$ going to zero as $N$ tends to infinity. Our analysis relies entirely on probabilistic methods and extends the well-known connection between singular stochastic control and optimal stopping to a mean-field framework

On the Saddle-point Solution and the Large-Coalition Asymptotics of Fingerprinting Games

We study a fingerprinting game in which the number of colluders and the collusion channel are unknown. The encoder embeds fingerprints into a host sequence and provides the decoder with the capability to trace back pirated copies to the colluders. Fingerprinting capacity has recently been derived as the limit value of a sequence of maximin games with mutual information as their payoff functions. However, these games generally do not admit saddle-point solutions and are very hard to solve numerically. Here under the so-called Boneh-Shaw marking assumption, we reformulate the capacity as the value of a single two-person zero-sum game, and show that it is achieved by a saddle-point solution. If the maximal coalition size is k and the fingerprinting alphabet is binary, we show that capacity decays quadratically with k. Furthermore, we prove rigorously that the asymptotic capacity is 1/(k^2 2ln2) and we confirm our earlier conjecture that Tardos' choice of the arcsine distribution asymptotically maximizes the mutual information payoff function while the interleaving attack minimizes it. Along with the asymptotic behavior, numerical solutions to the game for small k are also presented.Comment: submitted to IEEE Trans. on Information Forensics and Securit

Capacities and Capacity-Achieving Decoders for Various Fingerprinting Games

Combining an information-theoretic approach to fingerprinting with a more constructive, statistical approach, we derive new results on the fingerprinting capacities for various informed settings, as well as new log-likelihood decoders with provable code lengths that asymptotically match these capacities. The simple decoder built against the interleaving attack is further shown to achieve the simple capacity for unknown attacks, and is argued to be an improved version of the recently proposed decoder of Oosterwijk et al. With this new universal decoder, cut-offs on the bias distribution function can finally be dismissed. Besides the application of these results to fingerprinting, a direct consequence of our results to group testing is that (i) a simple decoder asymptotically requires a factor 1.44 more tests to find defectives than a joint decoder, and (ii) the simple decoder presented in this paper provably achieves this bound.Comment: 13 pages, 2 figure

Small games and long memories promote cooperation

Complex social behaviors lie at the heart of many of the challenges facing evolutionary biology, sociology, economics, and beyond. For evolutionary biologists in particular the question is often how such behaviors can arise \textit{de novo} in a simple evolving system. How can group behaviors such as collective action, or decision making that accounts for memories of past experience, emerge and persist? Evolutionary game theory provides a framework for formalizing these questions and admitting them to rigorous study. Here we develop such a framework to study the evolution of sustained collective action in multi-player public-goods games, in which players have arbitrarily long memories of prior rounds of play and can react to their experience in an arbitrary way. To study this problem we construct a coordinate system for memory-$m$ strategies in iterated $n$-player games that permits us to characterize all the cooperative strategies that resist invasion by any mutant strategy, and thus stabilize cooperative behavior. We show that while larger games inevitably make cooperation harder to evolve, there nevertheless always exists a positive volume of strategies that stabilize cooperation provided the population size is large enough. We also show that, when games are small, longer-memory strategies make cooperation easier to evolve, by increasing the number of ways to stabilize cooperation. Finally we explore the co-evolution of behavior and memory capacity, and we find that longer-memory strategies tend to evolve in small games, which in turn drives the evolution of cooperation even when the benefits for cooperation are low