2-Player Nash and Nonsymmetric Bargaining Games: Algorithms and Structural Properties

The solution to a Nash or a nonsymmetric bargaining game is obtained by maximizing a concave function over a convex set, i.e., it is the solution to a convex program. We show that each 2-player game whose convex program has linear constraints, admits a rational solution and such a solution can be found in polynomial time using only an LP solver. If in addition, the game is succinct, i.e., the coefficients in its convex program are ``small'', then its solution can be found in strongly polynomial time. We also give a non-succinct linear game whose solution can be found in strongly polynomial time

On the Impact of Fair Best Response Dynamics

In this work we completely characterize how the frequency with which each player participates in the game dynamics affects the possibility of reaching efficient states, i.e., states with an approximation ratio within a constant factor from the price of anarchy, within a polynomially bounded number of best responses. We focus on the well known class of congestion games and we show that, if each player is allowed to play at least once and at most $\beta$ times any $T$ best responses, states with approximation ratio $O(\beta)$ times the price of anarchy are reached after $T \lceil \log \log n \rceil$ best responses, and that such a bound is essentially tight also after exponentially many ones. One important consequence of our result is that the fairness among players is a necessary and sufficient condition for guaranteeing a fast convergence to efficient states. This answers the important question of the maximum order of $\beta$ needed to fast obtain efficient states, left open by [9,10] and [3], in which fast convergence for constant $\beta$ and very slow convergence for $\beta=O(n)$ have been shown, respectively. Finally, we show that the structure of the game implicitly affects its performances. In particular, we show that in the symmetric setting, in which all players share the same set of strategies, the game always converges to an efficient state after a polynomial number of best responses, regardless of the frequency each player moves with

Braess's Paradox in Wireless Networks: The Danger of Improved Technology

When comparing new wireless technologies, it is common to consider the effect that they have on the capacity of the network (defined as the maximum number of simultaneously satisfiable links). For example, it has been shown that giving receivers the ability to do interference cancellation, or allowing transmitters to use power control, never decreases the capacity and can in certain cases increase it by $\Omega(\log (\Delta \cdot P_{\max}))$, where $\Delta$ is the ratio of the longest link length to the smallest transmitter-receiver distance and $P_{\max}$ is the maximum transmission power. But there is no reason to expect the optimal capacity to be realized in practice, particularly since maximizing the capacity is known to be NP-hard. In reality, we would expect links to behave as self-interested agents, and thus when introducing a new technology it makes more sense to compare the values reached at game-theoretic equilibria than the optimum values. In this paper we initiate this line of work by comparing various notions of equilibria (particularly Nash equilibria and no-regret behavior) when using a supposedly "better" technology. We show a version of Braess's Paradox for all of them: in certain networks, upgrading technology can actually make the equilibria \emph{worse}, despite an increase in the capacity. We construct instances where this decrease is a constant factor for power control, interference cancellation, and improvements in the SINR threshold ($\beta$), and is $\Omega(\log \Delta)$ when power control is combined with interference cancellation. However, we show that these examples are basically tight: the decrease is at most O(1) for power control, interference cancellation, and improved $\beta$, and is at most $O(\log \Delta)$ when power control is combined with interference cancellation

A Direct Reduction from k-Player to 2-Player Approximate Nash Equilibrium

We present a direct reduction from k-player games to 2-player games that preserves approximate Nash equilibrium. Previously, the computational equivalence of computing approximate Nash equilibrium in k-player and 2-player games was established via an indirect reduction. This included a sequence of works defining the complexity class PPAD, identifying complete problems for this class, showing that computing approximate Nash equilibrium for k-player games is in PPAD, and reducing a PPAD-complete problem to computing approximate Nash equilibrium for 2-player games. Our direct reduction makes no use of the concept of PPAD, thus eliminating some of the difficulties involved in following the known indirect reduction.Comment: 21 page

Greedy Selfish Network Creation

We introduce and analyze greedy equilibria (GE) for the well-known model of selfish network creation by Fabrikant et al.[PODC'03]. GE are interesting for two reasons: (1) they model outcomes found by agents which prefer smooth adaptations over radical strategy-changes, (2) GE are outcomes found by agents which do not have enough computational resources to play optimally. In the model of Fabrikant et al. agents correspond to Internet Service Providers which buy network links to improve their quality of network usage. It is known that computing a best response in this model is NP-hard. Hence, poly-time agents are likely not to play optimally. But how good are networks created by such agents? We answer this question for very simple agents. Quite surprisingly, naive greedy play suffices to create remarkably stable networks. Specifically, we show that in the SUM version, where agents attempt to minimize their average distance to all other agents, GE capture Nash equilibria (NE) on trees and that any GE is in 3-approximate NE on general networks. For the latter we also provide a lower bound of 3/2 on the approximation ratio. For the MAX version, where agents attempt to minimize their maximum distance, we show that any GE-star is in 2-approximate NE and any GE-tree having larger diameter is in 6/5-approximate NE. Both bounds are tight. We contrast these positive results by providing a linear lower bound on the approximation ratio for the MAX version on general networks in GE. This result implies a locality gap of $\Omega(n)$ for the metric min-max facility location problem, where n is the number of clients.Comment: 28 pages, 8 figures. An extended abstract of this work was accepted at WINE'1

The Influence of Coalition Formation on Idea Selection in Dispersed Teams: A Game Theoretic Approach

Sie, R. L. L., Bitter-Rijpkema, M., & Sloep, P. B. (2009). The Influence of Coalition Formation on Idea Selection in Dispersed Teams: A Game Theoretic Approach. In U. Cress, V. Dimitrova & M. Specht (Eds.), Learning in the Synergy of Multiple Disciplines. Proceedings of the Fourth European Conference on Technology-Enhanced Learning (EC-TEL 2009) (pp. 732-737). September, 29 - October, 2, 2009, Nice, France. Lecture Notes in Computer Science Vol. 5794. Berlin: Springer-Verlag.In an open innovation environment, organizational learning takes place by means of dispersed teams which expand their knowledge through collaborative idea generation. Research is often focused on finding ways to extend the set of ideas, while the main problem in our opinion is not the number of ideas that is generated, but a non-optimal set of ideas accepted during idea selection. When selecting ideas, coalitions form and their composition may influence the resulting set of accepted ideas. We expect that computing coalitional strength during idea selection will help in forming the right teams to have a grand coalition, or having a better allocation of accepted ideas, or neutralising factors that adversely influence the decision making process. Based on a literature survey, this paper proposes the application of the Shapley value and the nucleolus to compute coalitional strength in order to enhance the group decision making process during collaborative idea selection. This document does not represent the opinion of the European Union, and the European Union is not responsible for any use that might be made of its content.The idSpace project is partially supported/co-funded by the European Union under the Information and Communication Technologies (ICT) theme of the 7th Framework Programme for R&

Actinide complexation kinetics: rate and mechanism of dioxoneptunium (V) reaction with chlorophosphonazo III

Rates of complex formation and dissociation in NpO{sub 2}{sup +}- Chlorophosphonazo III (2,7-bis(4-chloro-2-phosphonobenzeneazo)-1,8- dihydroxynapthalene-3,6-disulfonic acid)(CLIII) were investigated by stopped-flow spectrophotometry. Also, limited studies were made of the rates of reaction of La{sup 3+}, Eu{sup 3+}, Dy{sup 3+}, and Fe{sup 3+} with CLIII. Rate determining step in each system is an intramolecular process, the NpO{sub 2}{sup +}-CLIII reaction proceeding by a first order approach to equilibrium in the acid range from 0.1 to 1.0 M. Complex formation occurs independent of acidity, while both acid dependent and independent dissociation pathways are observed. Activation parameters for the complex formation reaction are {Delta}H=46.2{+-}0.3 kJ/m and {Delta}S=7{+-} J/mK (I=1.0 M); these for the acid dependent and independent dissociation pathways are {Delta}H=38.8{+-}0.6 kJ/m, {Delta}S=-96{+-}18 J/mK, {Delta}H=70.0{+-} kJ/m, and {Delta}S=17{+-}1 J/mK, respectively. An isokinetic relationship is observed between the activation parameters for CLIII complex formation with NpO{sub 2}{sup +}, UO{sub 2}{sup 2+}, Th{sup 4+}, and Zr{sup 4+}. Rates of CLIII complex formation reactions for Fe{sup 3+}, Zr{sup 4+}, NpO{sub 2}{sup +}, UO{sub 2}{sup 2+}, Th{sup 4+}, La{sup 3+}, Eu{sup 3+}, and Dy{sup 3+} correlate with cation radius rather than charge/radius ratio

Statistical mechanics of random two-player games

Using methods from the statistical mechanics of disordered systems we analyze the properties of bimatrix games with random payoffs in the limit where the number of pure strategies of each player tends to infinity. We analytically calculate quantities such as the number of equilibrium points, the expected payoff, and the fraction of strategies played with non-zero probability as a function of the correlation between the payoff matrices of both players and compare the results with numerical simulations.Comment: 16 pages, 6 figures, for further information see http://itp.nat.uni-magdeburg.de/~jberg/games.htm