A Difficulty in Controlling Blockchain Mining Costs via Cryptopuzzle Difficulty

Blockchain systems often employ proof-of-work consensus protocols to validate and add transactions into hashchains. These protocols stimulate competition among miners in solving cryptopuzzles (e.g. SHA-256 hash computation in Bitcoin) in exchange for a monetary reward. Here, we model mining as an all-pay auction, where miners' computational efforts are interpreted as bids, and the allocation function is the probability of solving the cryptopuzzle in a single attempt with unit (normalized) computational capability. Such an allocation function captures how blockchain systems control the difficulty of the cryptopuzzle as a function of miners' computational abilities (bids). In an attempt to reduce mining costs, we investigate designing a mining auction mechanism which induces a logit equilibrium amongst the miners with choice distributions that are unilaterally decreasing with costs at each miner. We show it is impossible to design a lenient allocation function that does this. Specifically, we show that there exists no allocation function that discourages miners to bid higher costs at logit equilibrium, if the rate of change of difficulty with respect to each miner's cost is bounded by the inverse of the sum of costs at all the miners.Comment: 8 pages. This is a working draft and can potentially have errors. Any feedback will be greatly appreciated and will be acknowledged in the updated versio

Reward Sharing Schemes for Stake Pools

We introduce and study reward sharing schemes (RSS) that promote the fair formation of {\em stake pools}\ in collaborative projects that involve a large number of stakeholders such as the maintenance of a proof-of-stake (PoS) blockchain. Our mechanisms are parameterized by a target value for the desired number of pools. We show that by properly incentivizing participants, the desired number of stake pools is a Nash equilibrium arising from rational play. Our equilibria also exhibit an efficiency / security tradeoff via a parameter that calibrates between including pools with the smallest cost and providing protection against Sybil attacks, the setting where a single stakeholder creates a large number of pools in the hopes to dominate the collaborative project. We then describe how RSS can be deployed in the PoS setting, mitigating a number of potential deployment attacks and protocol deviations that include censoring transactions, performing Sybil attacks with the objective to control the majority of stake, lying about the actual cost and others. Finally, we experimentally demonstrate fast convergence to equilibria in dynamic environments where players react to each other's strategic moves over an indefinite period of interactive play. We also show how simple reward sharing schemes that are seemingly more "fair", perhaps counterintuitively, converge to centralized equilibria

SoK: Tools for Game Theoretic Models of Security for Cryptocurrencies

Cryptocurrencies have garnered much attention in recent years, both from the academic community and industry. One interesting aspect of cryptocurrencies is their explicit consideration of incentives at the protocol level. Understanding how to incorporate this into the models used to design cryptocurrencies has motivated a large body of work, yet many open problems still exist and current systems rarely deal with incentive related problems well. This issue arises due to the gap between Cryptography and Distributed Systems security, which deals with traditional security problems that ignore the explicit consideration of incentives, and Game Theory, which deals best with situations involving incentives. With this work, we aim to offer a systematization of the work that relates to this problem, considering papers that blend Game Theory with Cryptography or Distributed systems and discussing how they can be related. This gives an overview of the available tools, and we look at their (potential) use in practice, in the context of existing blockchain based systems that have been proposed or implemented

Game-Theoretic Randomness for Blockchain Games

In this paper, we consider the problem of generating fair randomness in a deterministic, multi-agent context (for instance, a decentralised game built on a blockchain). The existing state-of-the-art approaches are either susceptible to manipulation if the stakes are high enough, or they are not generally applicable (specifically for massive game worlds as opposed to games between a small set of players). We propose a novel method based on game theory: By allowing agents to bet on the outcomes of random events against the miners (who are ultimately responsible for the randomness), we are able to align the incentives so that the distribution of random events is skewed only slightly even if miners are trying to maximise their profit and engage in block withholding to cheat in games

Computational Aspects of Equilibria in Discrete Preference Games

We study the complexity of equilibrium computation in discrete preference games. These games were introduced by Chierichetti, Kleinberg, and Oren (EC '13, JCSS '18) to model decision-making by agents in a social network that choose a strategy from a finite, discrete set, balancing between their intrinsic preferences for the strategies and their desire to choose a strategy that is `similar' to their neighbours. There are thus two components: a social network with the agents as vertices, and a metric space of strategies. These games are potential games, and hence pure Nash equilibria exist. Since their introduction, a number of papers have studied various aspects of this model, including the social cost at equilibria, and arrival at a consensus. We show that in general, equilibrium computation in discrete preference games is PLS-complete, even in the simple case where each agent has a constant number of neighbours. If the edges in the social network are weighted, then the problem is PLS-complete even if each agent has a constant number of neighbours, the metric space has constant size, and every pair of strategies is at distance 1 or 2. Further, if the social network is directed, modelling asymmetric influence, an equilibrium may not even exist. On the positive side, we show that if the metric space is a tree metric, or is the product of path metrics, then the equilibrium can be computed in polynomial time

Rethinking Blockchain Security: Position Paper

Blockchain technology has become almost as famous for incidents involving security breaches as for its innovative potential. We shed light on the prevalence and nature of these incidents through a database structured using the STIX format. Apart from OPSEC-related incidents, we find that the nature of many incidents is specific to blockchain technology. Two categories stand out: smart contracts, and techno-economic protocol incentives. For smart contracts, we propose to use recent advances in software testing to find flaws before deployment. For protocols, we propose the PRESTO framework that allows us to compare different protocols within a five-dimensional framework.Comment: 8 pages, 1 figure, accepted for presentation as a regular paper at IEEE Blockchain 2018 (Halifax, Canada

The Space of Planar Soap Bubble Clusters

Soap bubbles and foams have been extensively studied by scientists, engineers, and mathematicians as models for organisms and materials, with applications ranging from extinguishing fires to mining to baking bread. Here we provide some basic results on the space of planar clusters of n bubbles of fixed topology. We show for example that such a space of clusters with positive second variation is an n-dimensional manifold, although the larger space without the positive second variation assumption can have singularities. Earlier work of Moukarzel showed how to realize a cluster as a generalized Voronoi partition, though not canonically.Comment: 10 pages, 3 figure

Jump-starting coordination in a stag hunt: Motivation, mechanisms, and their analysis

The stag hunt (or assurance game) is a simple game that has been used as a prototype of a variety of social coordination problems (ranging from the social contract to the adoption of technical standards). Players have the option to either use a superior cooperative strategy whose payoff depends on the other players' choices or use an inferior strategy whose payoff is independent of what other players do; the cooperative strategy may incur a loss if sufficiently many other players do not cooperate. Stag hunts have two (strict) pure Nash equilibria, namely, universal cooperation and universal defection (as well as a mixed equilibrium of low predictive value). Selection of the inferior (pure) equilibrium is called a coordination failure. In this paper, we present and analyze using game-theoretic techniques mechanisms aiming to avert coordination failures and incite instead selection of the superior equilibrium. Our analysis is based on the solution concepts of Nash equilibrium, dominance solvability, as well as a formalization of the notion of "incremental deployability," which is shown to be keenly relevant to the sink equilibrium.Comment: Some overlap with arXiv:1210.778

Homoclinic dynamics in a restricted four body problem

We describe a method for computing an atlas for the stable or unstable manifold attached to an equilibrium point, and implement the method for the saddle-focus libration points of the planar equilateral restricted four body problem. We employ the method at the maximally symmetric case of equal masses, where we compute atlases for both the stable and unstable manifolds. The resulting atlases are comprised of thousands of individual chart maps, with each chart represented by a two variable Taylor polynomial. Post-processing the atlas data yields approximate intersections of the invariant manifolds, which we refine via a shooting method for an appropriate two point boundary value problem. Finally we apply numerical continuation to the BVPs. This breaks the symmetries and leads to connecting orbits for some non-equal values of the primary masses

How Peer Effects Influence Energy Consumption

This paper analyzes the impact of peer effects on electricity consumption of a network of rational, utility-maximizing users. Users derive utility from consuming electricity as well as consuming less energy than their neighbors. However, a disutility is incurred for consuming more than their neighbors. To maximize the profit of the load-serving entity that provides electricity to such users, we develop a two-stage game-theoretic model, where the entity sets the prices in the first stage. In the second stage, consumers decide on their demand in response to the observed price set in the first stage so as to maximize their utility. To this end, we derive theoretical statements under which such peer effects reduce aggregate user consumption. Further, we obtain expressions for the resulting electricity consumption and profit of the load serving entity for the case of perfect price discrimination and a single price under complete information, and approximations under incomplete information. Simulations suggest that exposing only a selected subset of all users to peer effects maximizes the entity's profit.Comment: 9 pages, 4 figure