Game Theory

Cryptocurrencies have opened a new field of opportunities for investors who want to diversify their portfolio. In 2019, staking was one of the most dynamic segments of the crypto market, with a growing number of blockchains with staking opportunities and an exploding number of companies offering a variety of staking services.

According to Staking Rewards, a sector data aggregator, the staking market capitalization in March 2020 amounted up to $10 billion, with more than $6 billion locked in staking. This trend is most likely to continue in 2020, despite cryptocurrency prices-driven volatility.

Staking cryptocurrencies allows crypto investors to earn a stable income, with little effort and sometimes even without the need to deposit their assets into a third party’s custody. Even with the existing regulatory uncertainties and high volatility of the crypto market, it offers an attractive option for investors to earn additional rewards on their holdings. At the same time, the mechanisms behind these earnings are often not clearly understood by investors. As a result, putting money in a ‘black box’ of staking can expose investors to higher risk and missed opportunities.

This article shows how applying game theory ideas can help understand the mechanisms behind staking and improve their basis for making investment decisions.


How staking for cryptocurrencies works

Staking cryptocurrencies is possible in the Proof-of-Stake (PoS) blockchains. The term ‘Proof-of-Stake’ is related to a variety of blockchains where consensus (in other words, decision on which blocks should be added to the blockchain) is achieved by a set of validators who commit (‘stake’) a number of blockchain tokens to this blockchain. In return, these validators earn a reward, usually in the form of more tokens. Examples of PoS blockchains include existing blockchains like Tezos and Cosmos, as well as highly anticipated ones such as Polkadot.

The blockchains expect validators to provide secure and readily available services, whenever required. For that, they need reliable and powerful servers which are running 24/7. Blockchains typically incentivize validators to avoid ‘bad’ or irresponsible behavior by reducing (‘slashing’) their stakes.

Not every owner of the tokens is ready to invest money and time into becoming a validator. In this case, token holders can become nominators by transferring their validation functions to the actors who are appropriately equipped for that. The transfer is usually accomplished by voting ‘with money’, i.e. delegating a nominator’s stake to the validators they trust.

In a PoS blockchain, a crypto investor can become either a validator, or a nominator. Becoming a validator requires some capital investment together with a commitment to a specific blockchain (each blockchain has specific technical and financial requirements), while as a nominator you only need to invest in cryptocurrency and delegate it to one or more selected validators.

Being a nominator allows more flexibility in asset allocation. An investor-nominator decides on a portfolio of cryptocurrencies, and then on validators to delegate to. These decisions are usually taken based on the performance of the blockchain and the validators. The validators’ performance can be predicted with the game theory tools.


Principles of game theory

Game theory offers a set of mathematical tools to analyze social and economic situations where rational actors interact with each other. Historically, the areas of game theory application range from military strategies (e.g. explaining the Cold War) to financial markets. Game theory can deal with uncertainties and probabilities, rewards and risks, and that is why it can be applied to so many real life situations.

A game assumes a number of players, each with a set of available strategies, or in other words decisions. A combination of chosen strategies of all players in the game results in a state with predefined payoffs to each player. The payoffs in different states are usually known to the players in advance.

Depending on the possibility of communication and agreement between the players, games can be noncooperative or cooperative. Imagine three generals before a big battle. If each general knows his own choices and the choices that are available to the other generals, but cannot communicate to them before taking a decision, this is a noncooperative game. If the generals of the allied armies can send messengers and agree on a common strategy, this is a cooperative game.

In addition to cooperative and noncooperative, games can also be simultaneous or sequential (depending on if the players move at the same time, or in a particular order), static or dynamic (a static game is played once, while dynamic game is repeated over time). Defining the type of the game is important when trying to find a solution for it.

In game theory, solving the game means to find an equilibrium (or show that the game does not have an equilibrium). Roughly speaking, an equilibrium is the most probable outcome of the game.

For noncooperative games, a desired solution is called Nash equilibrium. At the Nash equilibrium, none of the players have incentives to deviate from their current strategies to improve their payoff. Nash equilibrium does not mean that every player is getting the highest possible payoff, but just that it is the best situation given all other players’ strategies. Also, not all games have a Nash equilibrium, and in some cases, multiple equilibria exist.


Applying game theory to staking: Polkadot example

In PoS blockchains, validators and nominators are rational decision-makers trying to maximize their utilities, given a set of rules and other actors’ decision strategies. This is a classic game theory setting. The validators’ strategies include decisions on how much to stake, to run a single node or a multiple nodes, or any other parameters that are required by the blockchain design. Nominators decide on how much and which cryptocurrencies to stake and which validators to nominate. The payout to validators and nominators depends on their strategies, and a set of rules according to which validators are selected and rewards are distributed.

For illustration, consider the design of the Polkadot blockchain.


Polkadot is a project that is promising to provide a framework (or meta-protocol) for all other blockchains and crypto networks to interact without trusting each other with one of the co-founders of Ethereum, Dr. Gavin Wood, behind it. The information about staking design is taken from Polkadot Wiki, so the final design of the live blockchains can be different.


In the Polkadot blockchain, validators select their strategy by choosing a size of their validator node, or in other words – a stake in the blockchain. They also have an opportunity to divide their stake between two or more nodes, given that minimum stake condition is satisfied. The game of validators is noncooperative and dynamic.

Before the start of each period, a set of T validators is selected from all available candidates. After each period, the payoffs are distributed equally between T validators with stakes from s1 to sT, where sT is the lowest. The validator with the stake sT receives the highest return per DOT (DOTs are Polkadot tokens).

Based on the outcome of the first round, rational validators with stakes higher than sT are likely to reduce their stake sizes in the next period, by, for example, dividing the stakes among the nodes, or moving some of their assets to a different cause. If sT was higher than the minimum possible stake sMIN, required to qualify as a validator, even the smallest validators from the first round are likely to descale their nodes, to earn higher reward per DOT. This is repeated until sT is approaching sMIN as then the return per DOT is maximized. Indeed, when every validator is bidding sMIN, Nash equilibrium is achieved, meaning that no validator can improve their payoff by raising or lowering their stake size.

This analysis shows that the network is likely to converge in a decentralized state, where each validator has equal stakes. Decentralization is ensured due to the fact that rewards are distributed to the validators equally, and not proportionally to their stakes. The incentive is reinforced by the actions of the nominators, who are likely to support smaller validators to receive higher returns.

The illustration here is very simple. Introducing more factors into the game, such as possible irrationality of the validators and nominators, all complexities of the blockchain design, cost of running one or several validator-nodes, technical barriers, reputational factors, etc. can significantly complicate the solution, but also will provide a deeper insight into probable outcomes.



So, how does game theory help investors make better decisions?

On a wider scale, game theory is what lies in the foundation of decentralization in blockchain. Game theory operates in the realm of rational actors making choices, trying to maximize their utility, with a given set of rules. Blockchain is, in fact, a group of rational actors maximizing their reward, where rules are defined by the blockchain protocol. Designing a protocol with appropriate economic incentives allowed Satoshi Nakamoto to create Bitcoin and to revolutionize cryptocurrencies.

The economic incentives built in the protocol allow blockchains to function smoothly and efficiently over many years without the need for a centralized authority. Analyzing blockchain from the game theory perspective can help investors to understand the principles of blockchain, to look behind the numbers and see the ideas and structures.

From a more practical point of view, investors can use game theory tools to make tactical choices when allocating their assets. By modelling staking as a game, investors-validators can choose the optimal strategy to achieve the highest possible return on the stake. For example, in the Polkadot blockchain the optimal strategy for a validator is to run one or several validator nodes with the minimum possible stake.Investors-nominators can select validators to delegate their assets to according to their strategies and the expected equilibrium in the game of validators.

Ethereum 2 Is Now Live!

Learn more about the changes and how you can start staking on Ethereum 2 with Bitcoin Suisse.

Learn more