Table of Links
Abstract and 1. Introduction
1.1 Our Approach
1.2 Our Results & Roadmap
1.3 Related Work
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Model and Warmup and 2.1 Blockchain Model
2.2 The Miner
2.3 Game Model
2.4 Warm Up: The Greedy Allocation Function
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The Deterministic Case and 3.1 Deterministic Upper Bound
3.2 The Immediacy-Biased Class Of Allocation Function
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The Randomized Case
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Discussion and References
- A. Missing Proofs for Sections 2, 3
- B. Missing Proofs for Section 4
- C. Glossary
5 Discussion
Present Bias. It is interesting to note that although we incorporate miners’ present biases into our model in the form of the miner ratio α, it does not appear explicitly in the results. Technically, this is because for all the upper bounds we can ignore the first transaction in building our adversarial sequences. For the greedy and ℓ-immediacy-biased algorithms, the case analyses consist of either 1-1 comparisons within the same step, or of chains where the algorithm has the earliest transaction at least as high as OPT. In the analysis of RMIXλ, it is because the potential function has value 0 at the first step. On a deeper level, this is because all our algorithms are variations on the greedy algorithm, and have a bias towards allocating heavy transactions earlier compared to algorithms that plan ahead. This is another aspect of designing good competitive ratio algorithms for present-biased agents that may complicate the generalization of results from the packet scheduling literature.
Memorylessness. We find that a simple memoryless algorithm achieves the deterministic upper bound in the semi-myopic case. We provide a tight characterization of this optimality, and so that algorithm cannot achieve optimality with larger values of λ. We know that for the undiscounted case, i.e., when λ = 1, there is an optimal deterministic algorithm [VCJS], although it is not memoryless. One interesting direction is to see whether this algorithm can be extended to lower λ values.
Heterogeneous Choice of Allocation Rules by Miners. Our assumption throughout the paper is that all miners follow a given allocation protocol. This is essential to the analysis: a miner assumes that even when they do not mine a certain block, and thus do not extract revenue at this step, the same allocation rule is still followed. This assumption is somewhat bolstered by our results, that find that the optimal semi-myopic algorithm does not depend on α, as long as α > 0. Thus, as long as all miners have α > 0, it seems reasonable that all will follow the same optimal rule, even if we do not implement ways of ‘punishing’ miners who deviate from the rule. However, if some of the miners are ‘atomic’ (have α = 0), then their optimal allocation rule is the greedy allocation, and then all bets are off w.r.t. what may be an equilibrium of the allocation rule. The same is true for the cases which are not semi-myopic.
Semi-myopic discounts. Most blockchains rely on mechanisms that have a relatively slow blockrate, e.g., Bitcoin’s mechanism has an average of one block per 10 minutes. Thus, the discount factors of the semi-myopic range may seem excessive. However, we believe that there are settings, such as long-term project management, where a company decides which projects to commit its resources to on a time-scale of months / years, that are perfectly aligned with our framework and the semi-myopic regime in particular. Moreover, there are alternative interpretations to the discount factor that make the semi-myopic range more lucrative: One example is where clients, on top of having a strict deadline after which they leave the system, may also become impatient and leave following a geometric (memoryless) process. Another interpretation, which is perhaps very relevant in the blockchain setting, is exactly the one we discussed regarding heterogeneous miners: λ can be interpreted as a rule-of-thumb discount that stems from the probability of having miners with different allocation rules decide the next block. On a technical note, we remark that we expect the level of complexity of algorithms that go beyond the semi-myopic range to be higher than our optimal ℓ-immediacy-biased algorithm. In a sense, aggressive discounts eliminate longer pathological examples, and so enable such a simple algorithm to be optimal.
General time-dependent models. While multiplicative discounting over time is a natural model which is very common in the economic literature, other time-dependent models can be of interest. It is interesting to consider the most general case, where a transaction can specify a perblock fee, and these may not even be monotonically decreasing, for example, if this is a delayed transaction. It is also interesting to consider personalized discount factors instead of a uniform global discount factor, that represent differential patience between clients/transactions.
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Authors:
(1) Yotam Gafni, Weizmann Institute ([email protected]);
(2) Aviv Yaish, The Hebrew University, Jerusalem ([email protected]).
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This paper is available on arxiv under CC BY 4.0 DEED license.
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