Blockchain: A Coordination Mechanism

Blockchain technology is firmly established in the public awareness as a revolutionary new technology underpinning cryptocurrency. However, its potential applications can be found across sectors and industries in providing a novel way of producing coordination necessary to transact online, making it a timely invention in the age of progressing digitalization and increasing demands for efficiency and security of online transactions, and a promising research topic addressing the growing academic interest in the coordination aspect of the contract scholarship. The aim of this conceptual paper is to model blockchain as a coordination mechanism for online transactions. Three key aspects of coordination with blockchains are identified and examined – (1) producing consensus about the facts relevant to a transaction, (2) coding contracts, and (3) autonomously executing transactions. They are argued to be integral parts of the mechanism, jointly enabling blockchains to function as a complete mechanism of coordination for online transactions. The model is intended to inform debates on the prospects for the blockchain technology and can be further used to integrate coordination and contract scholarship. This work is licensed under a&nbsp;Creative Commons Attribution-NonCommercial 4.0 International License.</p

Resource Buying Games

In resource buying games a set of players jointly buys a subset of a finite resource set E (e.g., machines, edges, or nodes in a digraph). The cost of a resource e depends on the number (or load) of players using e, and has to be paid completely by the players before it becomes available. Each player i needs at least one set of a predefined family S_i in 2^E to be available. Thus, resource buying games can be seen as a variant of congestion games in which the load-dependent costs of the resources can be shared arbitrarily among the players. A strategy of player i in resource buying games is a tuple consisting of one of i's desired configurations S_i together with a payment vector p_i in R^E_+ indicating how much i is willing to contribute towards the purchase of the chosen resources. In this paper, we study the existence and computational complexity of pure Nash equilibria (PNE, for short) of resource buying games. In contrast to classical congestion games for which equilibria are guaranteed to exist, the existence of equilibria in resource buying games strongly depends on the underlying structure of the S_i's and the behavior of the cost functions. We show that for marginally non-increasing cost functions, matroids are exactly the right structure to consider, and that resource buying games with marginally non-decreasing cost functions always admit a PNE

On Existence and Properties of Approximate Pure Nash Equilibria in Bandwidth Allocation Games

In \emph{bandwidth allocation games} (BAGs), the strategy of a player consists of various demands on different resources. The player's utility is at most the sum of these demands, provided they are fully satisfied. Every resource has a limited capacity and if it is exceeded by the total demand, it has to be split between the players. Since these games generally do not have pure Nash equilibria, we consider approximate pure Nash equilibria, in which no player can improve her utility by more than some fixed factor $\alpha$ through unilateral strategy changes. There is a threshold $\alpha_\delta$ (where $\delta$ is a parameter that limits the demand of each player on a specific resource) such that $\alpha$-approximate pure Nash equilibria always exist for $\alpha \geq \alpha_\delta$, but not for $\alpha < \alpha_\delta$. We give both upper and lower bounds on this threshold $\alpha_\delta$ and show that the corresponding decision problem is ${\sf NP}$-hard. We also show that the $\alpha$-approximate price of anarchy for BAGs is $\alpha+1$. For a restricted version of the game, where demands of players only differ slightly from each other (e.g. symmetric games), we show that approximate Nash equilibria can be reached (and thus also be computed) in polynomial time using the best-response dynamic. Finally, we show that a broader class of utility-maximization games (which includes BAGs) converges quickly towards states whose social welfare is close to the optimum

Resource Competition on Integral Polymatroids

We study competitive resource allocation problems in which players distribute their demands integrally on a set of resources subject to player-specific submodular capacity constraints. Each player has to pay for each unit of demand a cost that is a nondecreasing and convex function of the total allocation of that resource. This general model of resource allocation generalizes both singleton congestion games with integer-splittable demands and matroid congestion games with player-specific costs. As our main result, we show that in such general resource allocation problems a pure Nash equilibrium is guaranteed to exist by giving a pseudo-polynomial algorithm computing a pure Nash equilibrium.Comment: 17 page

Network Topology and Equilibrium Existence in Weighted Network Congestion Games

Every finite noncooperative game can be presented as a weighted network congestion game, and also as a network congestion game with player-specific costs. In the first presentation, different players may contribute differently to congestion, and in the second, they are differently (negatively) affected by it. This paper shows that the topology of the underlying (undirected two-terminal) network provides information about the existence of pure-strategy Nash equilibrium in the game. For some networks, but not for others, every corresponding game has at least one such equilibrium. For the weighted presentation, a complete characterization of the networks with this property is given. The necessary and sufficient condition is that the network has at most three routes that do traverse any edge in opposite directions, or it consists of several such networks connected in series. The corresponding problem for player-specific costs remains open.Congestion games, network topology, existence of equilibrium

On the Convergence Time of the Best Response Dynamics in Player-specific Congestion Games

We study the convergence time of the best response dynamics in player-specific singleton congestion games. It is well known that this dynamics can cycle, although from every state a short sequence of best responses to a Nash equilibrium exists. Thus, the random best response dynamics, which selects the next player to play a best response uniformly at random, terminates in a Nash equilibrium with probability one. In this paper, we are interested in the expected number of best responses until the random best response dynamics terminates. As a first step towards this goal, we consider games in which each player can choose between only two resources. These games have a natural representation as (multi-)graphs by identifying nodes with resources and edges with players. For the class of games that can be represented as trees, we show that the best-response dynamics cannot cycle and that it terminates after O(n^2) steps where n denotes the number of resources. For the class of games represented as cycles, we show that the best response dynamics can cycle. However, we also show that the random best response dynamics terminates after O(n^2) steps in expectation. Additionally, we conjecture that in general player-specific singleton congestion games there exists no polynomial upper bound on the expected number of steps until the random best response dynamics terminates. We support our conjecture by presenting a family of games for which simulations indicate a super-polynomial convergence time

Strong Nash Equilibria in Games with the Lexicographical Improvement Property

We introduce a class of finite strategic games with the property that every deviation of a coalition of players that is profitable to each of its members strictly decreases the lexicographical order of a certain function defined on the set of strategy profiles. We call this property the Lexicographical Improvement Property (LIP) and show that it implies the existence of a generalized strong ordinal potential function. We use this characterization to derive existence, efficiency and fairness properties of strong Nash equilibria. We then study a class of games that generalizes congestion games with bottleneck objectives that we call bottleneck congestion games. We show that these games possess the LIP and thus the above mentioned properties. For bottleneck congestion games in networks, we identify cases in which the potential function associated with the LIP leads to polynomial time algorithms computing a strong Nash equilibrium. Finally, we investigate the LIP for infinite games. We show that the LIP does not imply the existence of a generalized strong ordinal potential, thus, the existence of SNE does not follow. Assuming that the function associated with the LIP is continuous, however, we prove existence of SNE. As a consequence, we prove that bottleneck congestion games with infinite strategy spaces and continuous cost functions possess a strong Nash equilibrium

Approximate Pure Nash Equilibria in Weighted Congestion Games

We study the existence of approximate pure Nash equilibria in weighted congestion games and develop techniques to obtain approximate potential functions that prove the existence of alpha-approximate pure Nash equilibria and the convergence of alpha-improvement steps. Specifically, we show how to obtain upper bounds for approximation factor alpha for a given class of cost functions. For example for concave cost functions the factor is at most 3/2, for quadratic cost functions it is at most 4/3, and for polynomial cost functions of maximal degree d it is at at most d + 1. For games with two players we obtain tight bounds which are as small as for example 1.054 in the case of quadratic cost functions