Decentralization is a core element in the design of permissionless blockchains. To participate in the blockchain consensus mechanisms, prospective network nodes – also called miners – need to provide proof of some costly resource. This resource may be computational power in protocols with Proof of Work (PoW) selection mechanisms, [28, 15], or coins of the native cryptocurrency in Proof of Stake (PoS) selection mechanisms, [8, 5]. Under default conditions, the selection is proportional to miners’ resources and hence, it depends on their actual distribution. An integral assumption in the security philosophy of permissionless blockchains is that the network of mining nodes remains “sufficiently” decentralized and distributed. In the extreme case, sufficiently means that no single entity holds or more of the resources but in practice much more fragmentation may be desired to safeguard the safety properties of the underlying protocol [11, 22, 3].
With this in mind, the picture illustrated in Table 1 is disconcerting.
|Entity (Pool)||Blocks (%)||Entity (Pool)||Blocks (%)|
Table 1 shows the distribution of blocks among miners in the two largest111In terms of market capitalization, cf. coinmarketcap.com. cryptocurrencies, Bitcoin and Ethereum, and indicates that the desired assumption of a highly decentralized (and distributed) network is currently not satisfied. As can be seen, the vast majority of mining resources is concentrated in a small number of “major” nodes or mining pools
in which individual miners join forces to reduce the variance of their payments[14, 34]. The rest is scattered among a large number of minor and individually insignificant miners. The discrepancy between the intended distribution and the concentration of resources that is observed in practice raises some questions. What is the actual power of such pools or major miners to influence the evolution of the blockchain? Does this distribution create incentives for mergers and formation of coalitions (cartels) that will seize control of the majority of resources and manipulate the blockchain [26, 24]? What strategic considerations arise and what are their implications on blockchain governance and long-term sustainability?
Similar questions have been examined by conventional economics in the context of corporate governance. To study interactions between shareholders with various degrees of power in particular,  developed the model of Oceanic Games. These are games featuring a mixture of few large players (shareholders) and a continuum of infinitesimal players, called the ocean, each of which holds an insignificant fraction of corporate shares. The resemblance with blockchain mining – with shares corresponding to units of mining resources – is apparent. Our goal in this paper is to explore incentives in blockchain mining from the perspective of Oceanic Games and complement existing studies that focus on safety and security related issues [29, 13, 9].
The central idea in the literature of Oceanic Games is the measurement of a value for each entity and for each unit of resource given the distribution of resources among shareholders. The concept of value is considered as a powerful tool in the theory of decision making [2, 33, 31] and . For instance, if a miner holds of the total resources, then each of her units is worth much more than if she holds only of the total resources, since in the former case, the entirety of her shares gives her absolute control over the blockchain. Similar, but maybe less obvious considerations, arise also in intermediate cases. If a miner holds of the resources and a second miner holds of the resources, then both miner’s resources value higher than in the case in which the first miner only holds of the resources, since in the former case, the two miners may collude and jointly seize control of the blockchain.
Motivated by these considerations, we adapt the model of Oceanic Games from  on blockchain mining. Our aim is to measure the value of mining resources per miner and per unit of resource as a strategic component in the process of power gain and coalition formation between mining nodes. With this approach, we shift our attention from safety attacks and equilibration models, [20, 12], to the understanding of incentives related to the distribution and acquisition of protocol resources. The analysis of these issues is relevant to the broader subjects of long term sustainability and blockchain governance .
Based on the above, our contribution in the present paper can be summarized in the following points
We model instances of blockchain mining as Oceanic Games: the discrete set of large players corresponds to the large mining nodes, typically mining pools, and the continuum of infinitesimal oceanic players to the remaining, individual miners, cf. Figure 1.
Conveniently, the resulting model does not depend on the underlying selection mechanism (PoW, PoS or similar) or consensus protocol and hence can be used for the study of resource acquisition, strategic interactions, coalition formations (mergers) and governance related issues in a broad spectrum of permissionless blockchains [16, 9, 30, 21, 18, 4, 10]. We extend an example of  to illustrate the defined concepts in blockchain context.
The application of existing results uncovers incentives for the formation of mergers between miners. Starting from an initial distribution in which the oceanic players control the majority of resources, we use simulations to show that this holds in two instances: first, in the formation – crystallization – of a coalition out of the ocean and second, in the exogenous acquisition of additional resources by a group of individual miners who, nevertheless, have the ability to coordinate their actions (collude). In both cases, the value of the miners’ resources is higher when they act as a single entity rather than individual, oceanic players. This result provides an alternative perspective to the observed centralization in cryptocurrency mining, cf. Table 1.
Further numerical simulations demonstrate that the above conclusions do not hold in the whole range of parameters. Instead, the dynamics of coalition formations and entry barriers are shown to depend on the current distribution of mining resources among major miners and the ocean.
Finally, we use this model to raise issues relevant to the design of future cryptocurrencies and formulate prospective research questions.
In general, the present paper can be seen as a first step towards the application of the Oceanic Game concept in blockchain mining. Beyond some first insight, the extent to which this model can provide further results in the issues of (de)centralization, blockchain governance and long-term sustainability is yet to be fully understood.
The rest of the paper is structured as follows. In Section 2 we present the model of Oceanic Games and give an example to illustrate the defined notions. Revelant results from  and their application in blockchain settings are shown in Section 3 along with numerical simulations. In Section 4, we raise related issues and research questions and discuss limitations of the current approach. Section 5 concludes the paper.
2 The Model: Oceanic Games on the Blockchain
The current model adjusts the notation and terminology of  in a standard blockchain setting.
The miners are the physical entities that participate in the block proposal and creation process. The term is used here in the broadest sense and depending on the underlying protocol and selection mechanism, it may refer to “conventional” miners as in PoW, , or to virtual miners as in PoS or other alternative forms . The set of miners consists of two distinctive components
A finite, discrete set of major miners or mining pools.
An interval of infinitesimal miners. We refer to as the ocean and to miners in , as oceanic players. We only consider subsets of the ocean , e.g. , and not individual oceanic players.
To participate in the distributed consensus, each miner needs to provide proof of some designated, costly resource. This may be a physical or digital asset such as computational power in PoW or native coins in PoS mechanisms, respectively. To describe these resources, we use following notation
A set of real numbers , where denotes the amount of resources of miner . For any subset , we will write to denote the total resources of miners in .
A positive constant which denotes the total resources of the ocean . Accordingly, any subset controls of resources where .
Based on the above, the total protocol resources are equal to . While resources change over time, in the present analysis, we will focus on a single period or a static setting and hence, unless indicated otherwise, our notation is independent of the time . Resources may be expressed as absolute numbers or percentages but this will be made explicitly clear from the context.
- Blockhain Oceanic Games:
Given the above, a blockchain oceanic game is defined by a majority quota, , using the symbol222The notation is common in the literature of weighted voting games, see [31, 32] and  for a more related application. Also, in most cases, we will be interested in or but the current model applies to any of interest.
with the following interpretation: a coalition of miners with and wins in the game , if and only if its total resources are larger than or equal to , i.e., if
- Addition of resources:
Given an oceanic game , we want to study the situation in which new entities acquire resources and enter the protocol. For this, we will use the notation with
and . In words, results from by the addition of a new major player with exogenous resources . Similar notation can be used to denote the formation of a new entity crystallizing out of the ocean. In this case, for some , and .
The first core functionality of the present model is to calculate a value for each major miner and one value for the entirety of the oceanic players, also referred to as the oceanic value. Each miner’s value depends on that miner’s share of resources and on the total distribution of the remaining resources among the rest of major and oceanic miners. To define the miner’s values and the oceanic value , let. For each , let , where if and otherwise (indicator function). Then, the value of miner is defined by
and the oceanic value by . Intuitively, the value
is the probability that minerwill be the crucial entity to turn a random coalition of miners from losing (total resources of the coalition without are less than ) to winning (total resources of the coalition with are equal to or greater than )333For more details and the probabilistic derivation of these values, we refer to ..
- Value-per-unit of resource:
The second functionality of this model is to determine the value per-unit of resource or power ratio, , for each player , which is defined by
Similarly, the value per oceanic unit of resource or oceanic power ratio, , is equal to .
2.1 An Example: Why values and not Shares?
We illustrate the above with the help of an example adapted from [27, Section 6]. We consider a mining situation with two major mining entities or pools, , and the simple majority quota represented by the following game , where . The 0.5 or quota corresponds to cntrol of the majority of protocol resources and hence, of the blockchain as a whole. In this game a coalition wins, if , i.e., if it occupies or more of the protocol resources444Due to continuity properties, there is no difference between using the quota or symbolically, the quota, as is common in the related literature [28, 11, 22]..
All possible resource configurations are illustrated in Figure 2. The horizontal and vertical axes represent miner ’s and miner ’s fraction of the resources, respectively. Their possible combinations are divided in inner regions, . Region contains all configurations for which the combined resources of both major miner are less than , i.e., . In this case, the majority of resources is controlled by oceanic players. However, the ocean is not actually “in control”, since, by assumption, there is no coherence nor organizational structure between oceanic players. The explanation of regions and is similar and is briefly given in the legend of Figure 2.
Using (1), the value of the first major miner is given by
with for and . The value of Miner is analogous and the oceanic value is simply equal to .
The interpretation of the values in the extreme regions and is straightforward. In , miner controls more than of the resources and hence, has absolute power over the blockchain. This implies that her value is equal to and consequently, the value for both miner and the oceanic miners is . Region is similar. The interesting cases arise whenever , i.e., when the major miners and the ocean, each control less than of the resources, or , i.e., when the resources controlled by the ocean account for more than half of the total resources. This case is also referred to as the interior case in the original paper. Some instantiations in regions and are presented in Table 2.
|Resources %||Values %||Ratios|
|: Interior Game||40||9||51||65||4||31||1.62||0.42||0.62|
|: Balance of Power||35||20||45||44||11||44||1.27||0.56||0.99|
An indicative observation – which does not aim to an exhaustive analysis of the above measurements – is that the values and the ratios unveil disparities between shares and actual influence or power of the participating entities. For example, there are instances, as in the -configuration (first row in ), in which a major miners’ ratio is larger than the ratio of oceanic players. This imbalance generates a motive for oceanic players to merge with that miner to increase the power of their individual resources. Equivalently, the large miner has an increased influence to attract resources from the ocean. The picture is totally different in the -configuration (third row of ), in which the competition between the major miners raises the value of resources owned by the ocenic players. Both cases can be contrasted to the stability in the -configuration, in which all ratios are approximately equal to .
Yet, as argued in , the interpretation of values should be done with caution and only in addition to complementary analytical tools. This is because values do not take into account qualitative factors such as ethical commitments, operational constraints or other kinds of incentives.
3 Individual Mining is not Stable
A direct outcome of applying the model of Oceanic Games in the blockchain context is the next result due to . Both parts of Section 3 make critical use of the assumption that the majority of mining resources is controlled by oceanic players. Their proof relies on a recursion in the number of major miners and can be found in . Here, we will focus on the interpretation of Section 3 and its application in blockchain context.
[] Let with be a blockchain oceanic game, such that , i.e., such that the majority of mining resources is controlled by individual (oceanic) miners. Then
The value of any major player in is given by
where and is the number of major miners in . The oceanic value is equal to .
If for some , and are the values in , then
or equivalently, .
- Interpretation of Section 3.
Statement (a) of Section 3 is an analytical result which yields the exact formula to compute the values of the major miners and the ocean. Its usefuleness will become aparent in the applications. Statement (b) carries more intuition. It states that the value-per-unit of resource of a miner entering is equal to the oceanic value-per-unit of resource in . One possible interpretation, also supported by , is that this provides a stability argument in favor of decentralization, in the sense that there is no incentive for the formation of a “cartel” or a mining pool, provided that the size of the ocean is big enough, i.e., provided that the ocean controls the majority of the resources555This statement actually holds for any quota and not only for as formulated here..
However, as we will see in the following applications, this picture is misleading and decentralization is actually not stable. In practice, the oceanic value per unit of resource in can go below the value per unit of resource of the crystallizing or newly entering entity. Hence, given that a set of miners can coordinate their actions, then it may be beneficial for them to either crystallize out of the ocean or to acquire exogenous resources and form in both cases a single mining entity.
3.1 Applications of Section 3
The above interpretations of Section 3 are illustrated via the simulation of two representative scenarios. In both cases, we assess the stability of initial distributions of mining resources, in which the majority of resources is controlled by the ocean. This is achieved by comparing the oceanic value per unit of resource to the value per unit of resource of the same miners when acting as single entity.
- I. Crystallization out of the ocean:
In the first scenario, we consider an instance of the blockchain oceanic game in which all resources are initially controlled by oceanic players. This is described by the game and . Then, we simulate a gradual formation of a single mining entity by the process of crystallization out of the ocean. This is captured by a sequence of games (instances) with and . For each instance, we calculate the value per unit of resource of the single entity that is forming out of the ocean and compare it with the value per unit of oceanic resource. The results are shown in Figure 3.
It is apparent that is higher than even for arbitrarily low values of and that the difference is increasing in the percentage of crystallized resources. This uncovers a motive for coalition formations and merging between miners, even if the initial distribution is perfectly decentralized. Further simulations (not shown here) demonstrate that the same picture continues to hold even if , as long as and no single miner in holds a percentage close to . If there exists a “large” miner with, e.g., , then the oceanic players may be disincentivized to collude. However, this is only a semblance of stability, since in this case, oceanic miners have an incentive to merge with the “large” miner.
- II. Acquisition of exogenous resources:
In the second scenario, we consider miners who are acquiring exogenous resources to enter the mining process. We assume that these miners can either enter the ocean and mine individually or collude and form a single mining entity. We want to compare the value per unit of resource in these two cases. Formally, we denote the current distribution of resources by with and the total mining resources of the new entities by . We want to compare
in the game to
in the game .
The game describes the instance in which the entering miners merge in a single mining entity and the game the instance in which the entering miners become part of the ocean and mine individually. We assume that initially, the majority of resources is controlled by oceanic players and that there exist two other major mining entities. It turns out that the share of mining resources of the other major entities influences the incentives of the entering miners. To see this, we consider two cases.
- Case 1:
Let , so that and for any . As shown in Figure 4, in this case, both major miners are not large enough to create entry barriers for the third entity and for any .
In agreement with Section 3(b), the value per unit of resource, , of the new miners when they enter as a single entity is equal to the oceanic value in the initial game (red line). According to  this implies that “there is no incentive for a new entity to form”. However, this only says half the truth. As we can see by the blue line, if the newly entering miners enter the ocean as individual miners, then their value per unit of resource will be lower compared to the case in which they collude. Hence, given that a group of entering miners are capable to coordinate their actions, then they are better off if they enter as a single entity than as oceanic players.
- Case 2:
Let , so that and for any . As shown in Figure 5, in this case, the presence of major miner seems to create a disincentive for a forming coalition and for any such that .
The resulting picture shows that we cannot generalize the outcome of the previous case. In particular, we conclude that whether the entering miners have an incentive to form a single entity or to join the ocean as individual miners, may depend on the actual distribution of resources among the existing major miners and the ocean. However, this is only a semblance of stability, stemming from an already centralized initial distibution (). In this case, oceanic miners actually have a stronger incentive to merge with miner instead of forming a new entity.
The previous simulations create an inconclusive picture. In general, the incentives for miners to merge seem to depend on the current distribution of resources. Since blockchain mining is a dynamical system that evolves over time, they suggest that even if the blockchain starts from a sufficiently decentralized point, then it is unlikely to remain decentralized also in the future or equivalently that decentralization creates a negative feedback loop, [36, 19]. The dynamics of the coalition formation process and the entry barriers resemble these of conventional economic markets of either perfect or oligopolistic competition. These findings provide an alternative perspective to cryptocurrency mining along with , and suggest the need for further research in this direction.
4 General Issues, Research Perspectives & Limitations
The application of the oceanic-game model in blockchain mining opens new research perspectives but also has its own limitations. Beyond the insight from existing results, a complete model needs to account for the additional challenges and address the questions that are specific to the blockchain context. In the following discussion, we raise such relevant issues, discuss their connection and research possibilities via the current model and identify potential limitations.
- Cryptocurrencies as Resources:
The difference between PoW and PoS in terms of their resources – computational power versus native coins – has a direct impact on both the mining process and the value of the underlying cryptocurrency. When coins are used as mining resources (PoS protocols), their value depends on their distribution among existing miners, their availability for prospective miners and the returns (profits) from mining. This in contrast to PoW protocols, in which the price of the resources – e.g., hardware and electricity – is not tied to the price of the underlying cryptocurrency.
- Resource Acquisition & Entry Barriers:
The above suggest that the nature of protocol resources may also generate different entry barriers. In PoS, the acquisition of protocol resources, i.e., coins, by prospective nodes depends on the willingness of current owners to exchange their coins and the way that new coins are minted. Different configurations may lead to high entry barriers and centralization. In PoW, computational power is essentially unlimited and acquisition of additional resources is independent of the underlying cryptocurrency. This implies lower entry barriers but also more frequent changes in the configuration (distribution) of resources among miners.
In particular, the current cost of acquiring enough computational power to control the majority in the Bitcoin and Ethereum PoW-blockchains is estimated at 1.5 billion US Dollars. This amount is well within the budget of several physical or legal entities worldwide. Moreover, it is independent of the value of the underlying cryptocurrency and depends only on the size of the network and the hardware and electricity costs. With this in mind, it is natural to ask: how stable are PoW blockchains against arbitrary authorities able to acquire the majority of resources? How relevant are these questions to the current distribution of resources and how do they translate in the PoS setting?
In this context, further work on blockchain oceanic games can aid the community to raise and study questions about investment in cryptocurrencies. When viewed not only as assets but also as means to gain power in the mining process and the governance of a blockchain, cryptocurrencies fit to the current perspective and their mechanics can be better understood.
- Mathematical Modelling:
From a mathematical perspective, oceanic games bridge the gap between atomic and non-atomic congestion games . Yet, the use of values instead of equlibria to study real settings has its own limitations [27, 32]. This is mainly due to the probabilistic derivation of values, which ignores qualitative aspects such as ethical commitments, preferences or any other motives of the participating agents. However, despite these limitations, if properly interpreted, values can become a powerful tool in the analysis of strategic interactions. In an immediate direction, they can be used to rethink the notion of blockchain fairness or equitability, which is currently based on the theoretically tentative premise that one unit of resource – one vote also implies fairness [13, 35].
In this paper, we employed the concept of Oceanic Games , to model and study strategic interactions in blockchain mining. Oceanic Games have been used in conventional economics to analyze decision making in corporate settings with a small number of major players – shareholders or here, mining pools – and a continuum of minor, individually insignificant players, called the ocean. This stream of literature focuses on the measurement of the value per miner and per unit of resource for each miner given a distribution of resources. Values are then interpreted as strategic components in decisions related to resource acquisition, mergers and coalition formations and offer an alternative perspective to the common equilibration models.
An immediate implication of existing results was that given a sufficiently large initial distribution of resources, there are incentives both for active and for newly entering miners to merge (form cartels or coalitions) and act as single entities. These observations provide an alternative justification of the observed centralization and concentration of power in the mining process of the major cryptocurrencies. Contrary to common perceptions, they amount to the existence of a negative feedback loop in terms of decentralization as a core ingredient in permissionless blockchain philosophy, , and reveal the need for futher research in this direction. In a general discussion, we identified critical issues related to resource acquisition, entry barriers and centralization risks in blockchain mining and formulated relevant questions that may be answered by further exploration of the present model. These findings can be placed in the broader context of governance and long-term sustainability for pemissionless blockchains.
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