Communication Complexity of Cake Cutting

We study classic cake-cutting problems, but in discrete models rather than using infinite-precision real values, specifically, focusing on their communication complexity. Using general discrete simulations of classical infinite-precision protocols (Robertson-Webb and moving-knife), we roughly partition the various fair-allocation problems into 3 classes: "easy" (constant number of rounds of logarithmic many bits), "medium" (poly-logarithmic total communication), and "hard". Our main technical result concerns two of the "medium" problems (perfect allocation for 2 players and equitable allocation for any number of players) which we prove are not in the "easy" class. Our main open problem is to separate the "hard" from the "medium" classes.Comment: Added efficient communication protocol for the monotone crossing proble

Tit-for-Tat Dynamics and Market Volatility

We study the tit-for-tat dynamic in production markets, where each player can make a good given as input various amounts of goods in the system. In the tit-for-tat dynamic, each player allocates its good to its neighbors in fractions proportional to how much they contributed in its production in the last round. Tit-for-tat does not use money and was studied before in pure exchange settings. We study the phase transitions of this dynamic when the valuations are symmetric (i.e. each good has the same worth to everyone) by characterizing which players grow or vanish over time. We also study how the fractions of their investments evolve in the long term, showing that in the limit the players invest only on players with optimal production capacity

Nash Social Welfare Approximation for Strategic Agents

The fair division of resources is an important age-old problem that has led to a rich body of literature. At the center of this literature lies the question of whether there exist fair mechanisms despite strategic behavior of the agents. A fundamental objective function used for measuring fair outcomes is the Nash social welfare, defined as the geometric mean of the agent utilities. This objective function is maximized by widely known solution concepts such as Nash bargaining and the competitive equilibrium with equal incomes. In this work we focus on the question of (approximately) implementing the Nash social welfare. The starting point of our analysis is the Fisher market, a fundamental model of an economy, whose benchmark is precisely the (weighted) Nash social welfare. We begin by studying two extreme classes of valuations functions, namely perfect substitutes and perfect complements, and find that for perfect substitutes, the Fisher market mechanism has a constant approximation: at most 2 and at least e1e. However, for perfect complements, the Fisher market does not work well, its bound degrading linearly with the number of players. Strikingly, the Trading Post mechanism---an indirect market mechanism also known as the Shapley-Shubik game---has significantly better performance than the Fisher market on its own benchmark. Not only does Trading Post achieve an approximation of 2 for perfect substitutes, but this bound holds for all concave utilities and becomes arbitrarily close to optimal for Leontief utilities (perfect complements), where it reaches $(1+\epsilon)$ for every $\epsilon > 0$. Moreover, all the Nash equilibria of the Trading Post mechanism are pure for all concave utilities and satisfy an important notion of fairness known as proportionality

Multiplayer Bandit Learning, from Competition to Cooperation

The stochastic multi-armed bandit model captures the tradeoff between exploration and exploitation. We study the effects of competition and cooperation on this tradeoff. Suppose there are $k$ arms and two players, Alice and Bob. In every round, each player pulls an arm, receives the resulting reward, and observes the choice of the other player but not their reward. Alice's utility is $\Gamma_A + \lambda \Gamma_B$ (and similarly for Bob), where $\Gamma_A$ is Alice's total reward and $\lambda \in [-1, 1]$ is a cooperation parameter. At $\lambda = -1$ the players are competing in a zero-sum game, at $\lambda = 1$, they are fully cooperating, and at $\lambda = 0$, they are neutral: each player's utility is their own reward. The model is related to the economics literature on strategic experimentation, where usually players observe each other's rewards. With discount factor $\beta$, the Gittins index reduces the one-player problem to the comparison between a risky arm, with a prior $\mu$, and a predictable arm, with success probability $p$. The value of $p$ where the player is indifferent between the arms is the Gittins index $g = g(\mu,\beta) > m$, where $m$ is the mean of the risky arm. We show that competing players explore less than a single player: there is $p^* \in (m, g)$ so that for all $p > p^*$, the players stay at the predictable arm. However, the players are not myopic: they still explore for some $p > m$. On the other hand, cooperating players explore more than a single player. We also show that neutral players learn from each other, receiving strictly higher total rewards than they would playing alone, for all $p\in (p^*, g)$, where $p^*$ is the threshold from the competing case. Finally, we show that competing and neutral players eventually settle on the same arm in every Nash equilibrium, while this can fail for cooperating players.Comment: 41 pages, 5 figure

How to Charge Lightning: The Economics of Bitcoin Transaction Channels

Off-chain transaction channels represent one of the leading techniques to scale the transaction throughput in cryptocurrencies. However, the economic effect of transaction channels on the system has not been explored much until now. We study the economics of Bitcoin transaction channels, and present a framework for an economic analysis of the lightning network and its effect on transaction fees on the blockchain. Our framework allows us to reason about different patterns of demand for transactions and different topologies of the lightning network, and to derive the resulting fees for transacting both on and off the blockchain. Our initial results indicate that while the lightning network does allow for a substantially higher number of transactions to pass through the system, it does not necessarily provide higher fees to miners, and as a result may in fact lead to lower participation in mining within the system.Comment: An earlier version of the paper was presented at Scaling Bitcoin 201