Aviv Zohar

On the complexity of achieving proportional representation

We demonstrate that winner selection in two prominent proportional representation voting systems is a computationally intractable problem—implying that these systems are impractical when the assembly is large. On a different note, in settings where the size of the assembly is constant, we show that the problem can be solved in polynomial time.

Secure High-Rate Transaction Processing in Bitcoin

Bitcoin is a disruptive new crypto-currency based on a decentralized open-source protocol which has been gradually gaining momentum. Perhaps the most important question that will affect Bitcoin’s success, is whether or not it will be able to scale to support the high volume of transactions required from a global currency system. We investigate the implications of having a higher transaction throughput on Bitcoin’s security against double-spend attacks. We show that at high throughput, substantially weaker attackers are able to reverse payments they have made, even well after they were considered accepted by recipients. We address this security concern through the GHOST rule, a modification to the way Bitcoin nodes construct and re-organize the block chain, Bitcoin’s core distributed data-structure. GHOST has been adopted and a variant of it has been implemented as part of the Ethereum project, a second generation distributed applications platform.

Complexity of strategic behavior in multi-winner elections

Although recent years have seen a surge of interest in the computational aspects of social choice, no specific attention has previously been devoted to elections with multiple winners, e.g., elections of an assembly or committee. In this paper, we characterize the worst-case complexity of manipulation and control in the context of four prominent multiwinner voting systems, under different formulations of the strategic agents goal.

Inclusive Block Chain Protocols

Distributed cryptographic protocols such as Bitcoin and Ethereum use a data structure known as the block chain to synchronize a global log of events between nodes in their network. Blocks, which are batches of updates to the log, reference the parent they are extending, and thus form the structure of a chain. Previous research has shown that the mechanics of the block chain and block propagation are constrained: if blocks are created at a high rate compared to their propagation time in the network, many conflicting blocks are created and performance suffers greatly. As a result of the low block creation rate required to keep the system within safe parameters, transactions take long to securely confirm, and their throughput is greatly limited.

Optimal Selfish Mining Strategies in Bitcoin

The Bitcoin protocol requires nodes to quickly distribute newly created blocks. Strong nodes can, however, gain higher payoffs by withholding blocks they create and selectively postponing their publication. The existence of such selfish mining attacks was first reported by Eyal and Sirer, who have demonstrated a specific deviation from the standard protocol (a strategy that we name SM1).

Bitcoin: under the hood

I JUST WANT to report that I successfully traded 10,000 bitcoins for pizza,” wrote user laszlo on the Bitcoin forums in May 2010—reporting on what has been recognized as the first item in history to be purchased with bitcoins.a By the end of 2013, about five years after its initial launch, Bitcoin has exceeded everyone’s expectations as its value rose beyond the

Mechanisms for multi-level marketing

1,000 mark, making laszlo’s spent bitcoins worth millions of dollars. This meteoric rise in value has fueled many stories in the popular press and has turned a group of early enthusiasts into millionaires. Stories of Bitcoin’s mysterious creator, Satoshi Nakamoto, and of illegal markets hidden in the darknet have added to the hype. But what is Bitcoin’s “ innovation? Is the buzz surrounding the new cryptocurrency justified, or will it turn out to be a modern tulip mania? To truly evaluate Bitcoin’s novelty, its potential impact, and the challenges it faces, we must look past the hype and delve deeper into the details of the protocol. Bitcoin, a peer-to-peer digital cryptocurrency launched in 2009, has been slowly growing. Nakamoto described the protocol in a white paper published in late 2008 and released the software as an open source project, which has since been maintained by a large number of developers, most of them volunteers. Bitcoin’s network and its surrounding ecosystem have grown quite substantially since its initial release. Its dollar value, which most will admit is largely based on speculation on its future worth, has been extremely volatile. The currency had gone through several hype-driven bubbles and subsequent devaluations, attaining higher values each time. Bitcoin’s promise is mainly a result of the combination of features it bundles together: It is a purely digital currency allowing payments to be sent almost instantly over the Internet with extremely low fees. Like cash, it is nearly anonymous, and transactions are effectively irreversible once they are committed. Bitcoin addresses (the Bitcoin: Under the Hood

The learnability of voting rules

Multi-level marketing is a marketing approach that motivates its participants to promote a certain product among their friends. The popularity of this approach increases due to the accessibility of modern social networks, however, it existed in one form or the other long before the Internet age began (the infamous Pyramid scheme that dates back at least a century is in fact a special case of multi-level marketing). This paper lays foundations for the study of reward mechanisms in multi-level marketing within social networks. We provide a set of desired properties for such mechanisms and show that they are uniquely satisfied by geometric reward mechanisms. The resilience of mechanisms to false-name manipulations is also considered; while geometric reward mechanisms fail against such manipulations, we exhibit other mechanisms which are false-name-proof.

Mechanisms for information elicitation

Scoring rules and voting trees are two broad and concisely-representable classes of voting rules; scoring rules award points to alternatives according to their position in the preferences of the voters, while voting trees are iterative procedures that select an alternative based on pairwise comparisons. In this paper, we investigate the PAC-learnability of these classes of rules. We demonstrate that the class of scoring rules, as functions from preferences into alternatives, is efficiently learnable in the PAC model. With respect to voting trees, while in general a learning algorithm would require an exponential number of samples, we show that if the number of leaves is polynomial in the size of the set of alternatives, then a polynomial training set suffices. We apply these results in an emerging theory: automated design of voting rules by learning.

Asynchronous Best-Reply Dynamics

We study the computational aspects of information elicitation mechanisms in which a principal attempts to elicit the private information of other agents using a carefully selected payment scheme based on proper scoring rules. Scoring rules, like many other mechanisms set in a probabilistic environment, assume that all participating agents share some common belief about the underlying probability of events. In real-life situations however, the underlying distributions are not known precisely, and small differences in beliefs of agents about these distributions may alter their behavior under the prescribed mechanism. We examine two related models for the problem. The first model assumes that agents have a similar notion of the probabilities of events, and we show that this approach leads to efficient design algorithms that produce mechanisms which are robust to small changes in the beliefs of agents. In the second model we provide the designer with a more precise and discrete set of alternative beliefs that the seller of information may hold. We show that construction of an optimal mechanism in that case is a computationally hard problem, which is even hard to approximate up to any constant. For this model, we provide two very different exponential-time algorithms for the design problem that have different asymptotic running times. Each algorithm has a different set of cases for which it is most suitable. Finally, we examine elicitation mechanisms that elicit the confidence rating of the seller regarding its information.