In praise of the double square root voting system for blockchain governance

In 2004, the scientific community and media was convinced that the voting procedure of the European Council of Ministers was unfair [1]. Many studies showed that the voting procedure formulated in the “Treaty of Nice”, which would come into effect on the 1. November of that year, would unfairly favor countries with big populations like Germany while being especially unfavorable for mid-sized countries like Spain and Poland. As a compromise, mathematicians from the Jagiellonian University in Krakow proposed to use the “double square root voting system” which, different from all previous voting procedures, should be based on game theoretical principles [2].

Similar to the EU, blockchain communities are a diverse group of stakeholders with individual interests and different levels of stake in the system. In both cases, stakeholders rely on an underlying shared infrastructure which all members benefit from. Amending this underlying infrastructure, be it the European law or a blockchain protocol can have a big impact on the individual stakeholders. As a result, the decision power to make changes to this infrastructure should be distributed fairly between the stakeholders.

But what does it mean to distribute decision power in a fair way? In the EU it means that a compromise has to be found between assigning decision power according to population size and at the same time giving a voice to all the smaller member states. In the blockchain space, assigning decision power according to the stake an actor has in the system, like e.g. the number of owned tokens or hashing power, is often considered as fair. But the widespread interest in systems like quadratic voting also shows that many would support a more egalitarian distribution of decision power, where having more vote weight becomes costly for an individual. The double square root voting system tries to avoid centralized decision power and at the same time ensures that the voting power of stakeholders actually equals their assigned vote weight.

A-priori voting power

The most important concept that has to be understood in order to grasp the double square root voting system, is that of a-priori voting power. A-priori voting power is a central concept in voting theory and describes the real voting power of players in a weighted binary voting game (e.g. deciding for or against a proposal). There are cases in which the voting weight does not equal the voting power of a player. This can be best explained using an example: Let’s imagine a simple weighted voting game with only two voters. One has 51% of the voting weight and the other 49%. Even though the voting weights are nearly identical, it is clear that if a majority (> 50%) is needed, the first voter has 100% of the voting power. Such a voter is called a dictator, as he or she possesses the entire decision power. The other voter is called a dummy, as there will never be a situation in which his or her vote will have an effect on the outcome.

There are different power indices that can be used to express the a-priori voting power of players of a weighted voting game. The relative Banzhaf index is the one that is most widely recognized in academia and best fits the case of blockchain governance. It can be used to compare the voting powers of players in one specific voting game. The relative Banzhaf index is calculated by counting the number of coalitions in which a player is critical, meaning that without this player the coalition would not be winning, and dividing this number by the total number of times voters are critical in the voting game [3].

This can also be better illustrated by an example: Imagine a voting game with the following parameters. The quota describes the vote weight that is needed for a coalition to win.

The first winning coalition of this voting game is {A, B, C} with a shared vote weight of 9. In this coalition, A and B are critical while C is not critical, as the shared vote weight of A and B is 7 which already meets the required quota. Accordingly, the second winning coalition is {A, B} with a shared vote weight of 7. In this coalition, both members are critical because one member alone would not reach the quota. As a result, A and B are both critical in the first and the second winning coalition and C is not critical in any winning coalition. Hence, the Banzhaf index for both A and B is 2/4=0.5 and for C 0/4=0. It becomes clear that A and B both have half of the decision power as the quota can only be reached if they agree with each other and form a coalition. This means that C is a dummy in this voting game and the real relative voting power of the players differs considerably from their relative vote weight.

It should be pointed out that a-priori voting power does not make any predictions about the probability of the occurrence of any outcome but only expresses the voting power of a player that is derived solely from the decision rule itself, meaning the voting power that a player has because of the quota that is required to win and the way the voting weights are distributed.

Taking the square root

As the name suggests, the double square root voting system is made up of two parts. First, in the spirit of quadratic voting, the square root of the value is taken which the vote weight should be based on (e.g. population or number of tokens etc.). Taking the square root has an aligning effect on the resulting vote weights. A country for example that has a population that is a 100x higher than that of another country or a tokenholder that has a 100x more tokens than another tokenholder only has vote weight that is 10x higher. And because of the gradient of the square root function, this effect increases with bigger differences in population or tokens ( sqrt(10,000)=100, sqrt(1,000,000)=1,000 etc. ). As a result, countries with a higher population or tokenholders with a larger number of tokens will still always have more vote weight than smaller countries or small-scale tokenholders but not to the same extent as the differences in population or tokens would suggest. Therefore, the principle (which can be very beneficial) that entities with a higher stake in the system should have more decision power is upheld while centralized decision power in the hands of a few big countries or crypto-whales is effectively prevented.

Taking the square root of the population in the case of the EU Council is actually based on the Penrose Method which says that the voting power of a citizen in the two-tier voting system decreases with 1/sqrt(population) and therefore the voting power of every citizen in the EU is the same if the vote weight of a member state is the square root of its population. This is of course not really applicable to blockchain governance, but the beneficial properties are the same nonetheless

Adjusting the quota

But as we have learned, the vote weight of a player in a voting game is not always a valid expression of this player’s voting power. In order to ensure that the assigned vote weight equals their voting power, in the second part of the double square root voting system, the quota is adjusted in a way that the vote weight of each player equals their relative Banzhaf index. This is achieved by using a mathematical approximation that produces the wanted quota as a function of the vote weight distribution of all participating voters. This quota, which depends on the number of voters and their vote weights, is always above 50% but approaches majority rule for a big number of voters and evenly distributed vote weights.

Making the vote weights equal the relative Banzhaf index has many beneficial properties. First and most importantly, if there is more than one voter, there can never exist a dictator because that would require a vote weight of 100%. If a voter for example has a vote weight of 80% the quota would adjust in a way that the relative Banzhaf index equals 80%. In simple majority rule (quota = 50%), such a voter would be a dictator. But because the quota is adjusted, this is not the case in the double square root voting system. In practice, this means that depending on the other vote weights, the resulting quota will be well beyond 80% to ensure that this voter actually only holds 80% of the voting power. Moreover, in contrast to other voting mechanisms, where the quota is often set more or less arbitrarily based on the “level of conservatism” that seems appropriate, here the quota is adjusted based on game theoretic principles, dynamically adapting to the respective situation. The quota is automatically higher in situations where a more conservative quota is appropriate (big differences in vote weight and/or low number of voters) and lower in situations where a less conservative quota is acceptable (evenly distributed vote weight and/or high number of voters). This is a very elegant solution to choosing an appropriate quota because it automatically creates a fair voting system without needing any additional rules.

You can play around and get a feeling for the voting system with an example of only 10 voters in this small example calculator:

https://docs.google.com/spreadsheets/d/16hG8AW8YEOfR4trli5TpZdM4_sJmqZasFFWG95gWTmA/edit#gid=0

The double square root voting system can also disincentivize different forms of manipulation. One important objection against the use of formal voting in blockchain governance is its vulnerability to manipulation and especially vote buying. Because of the basic characteristics of blockchains like transparency and the validation of every transaction, voting on the blockchain allows for very effective vote buying [4].

Vote buying is most efficient when the vote seller and buyer have certainty about their transaction. This is why all modern democracies have introduced the secret ballot: If the vote buyer or coercer has no way of knowing for sure what choice an individual makes in the voting booth, vote buying or coercion become unprofitable. Considering this, it is clear why blockchain voting is perfect for vote buying. Vote buyer and seller can have absolute certainty that the other party holds up their bargain and the exchange can even be automated in a smart contract.

The double square root voting system can disincentivize vote buying and coercion by introducing uncertainty at another level. Because the required quota is calculated from the vote weight distribution of all participating voters, until the end of the voting period it will be unclear what quota will be required e.g. for a proposal to pass. Only after the voting period is over and all information is known can the outcome be calculated. The uncertainty about voter participation, the decision of other voters, their vote weight and the resulting quota might deter vote buyers from engaging in such manipulation. If the effect that vote buying will have on a specific election is really hard to predict, its expected value becomes quite hard to calculate. As a result, similar to introducing uncertainty about the transaction, introducing substantial uncertainty about the effect that this transaction will have, can prevent or at least disincentivize vote buying.

Moreover, in blockchain governance, one can imagine many metrics like tokens or even “used gas” to potentially act as a basis of vote weight. These metrics can often be “artificially” increased for the sole reason of increasing voting power. In the double square root voting system, the effect that such an artificial increase will have is limited by design. Taking the square root and adjusting the quota makes increasing decision power for a malicious actor a lot more costly than in simple e.g. token weighted voting systems. Additionally, similar to manipulation through vote buying, the effect that such an artificial increase might have is very uncertain as explained above. The high cost and uncertainty about its effect might disincentivize many actors from engaging in such activities.

The only kind of manipulation that similar to many other voting systems can, unfortunately, be quite problematic for the double square root voting system is the Sybil attack, in which an actor assumes multiple identities to acquire more decision power. Even though the effects of a Sybil attack in the double square root voting system are a lot less severe than in a one-person-one-vote system, it can, depending on the setup, increase the decision power of an attacker significantly. A setup where a Sybil attack can for example have quite a big effect is when there is are very unevenly distributed vote weights. If e.g. an actor with 80% of the vote weight manages to split up into entities of 40% and 40% or even 20%, 20%, 20%, 20% by assuming multiple identities, it might be enough to make this actor a dictator. But there exist many setups in which a Sybil attack does not have such a severe effect. Whenever the vote weights become more unevenly distributed because of the Sybil attack for example, the quota increases, and the effects of the attack are diminished. However, because of its vulnerability to Sybil attacks, the double square root voting system requires some kind of identity scheme to reach its full potential.

It becomes clear, that the double square root voting system fulfills many requirements for becoming a valuable voting system for blockchain governance. The mechanism is relatively simple, it is moderately conservative, can be easily adapted to changing circumstances, all parameters can be easily verified, and it distributes the decision power fairly between the stakeholders. It can help to tackle different forms of manipulation and is based on game theoretical principles. Together with some form of identity system, it can be used to make collective decisions on-chain. It is easy to implement and less complex than quadratic voting and can therefore be used in blockchain governance today. But it can of course also be seen as a first step towards experimenting with even more complex formal voting systems.

Both parts of the system could also be applied individually. If aligning vote weights and taking the first step towards something resembling quadratic voting seems like a good idea but ever-changing quotas and “this whole voting power thing” are a little bit too much, only taking the square root to align vote weights could be an option. In contrast, if actors should have the voting power that resembles their stake in the system directly, without aligning vote weights through taking the square root, only the second part can be used to choose a meaningful quota.

In conclusion, I believe that people in the blockchain space should look into all the different ways that formal voting could be realized on the blockchain before ruling it out completely. At the same time, I encourage everyone that works on designing governance processes to look for answers outside of their usual circles and learn from other democratic institutions. Especially learning from their mistakes of course. The EU Council, for example, did not accept the proposal of the polish mathematicians and instead went with an option that was the result of political negotiations without considering any game theoretical principles.

[1] The Telegraf: The EU constitution is ‘unfair’, according to game theorists: https://www.telegraph.co.uk/news/science/science-news/3325470/The-EU-constitution-is-unfair-according-to-game-theorists.html

[2] Słomczynski, W., & Życzkowski, K. (2007). From a toy model to the double square root voting system. Homo Oeconomicus, 24(3-4): https://arxiv.org/pdf/physics/0701338.pdf

[3] Felsenthal, D. S., & Machover, M. (1998). The measurement of voting power: Edward Elgar Publishing: https://www.elgaronline.com/view/9781858988054.xml

[4] Hacking, Distributed: On-Chain Vote Buying and the Rise of Dark DAOs: http://hackingdistributed.com/2018/07/02/on-chain-vote-buying/

Enter the Blockchain Writing Contest

In 2004, the scientific community and media was convinced that the voting procedure of the European Council of Ministers was unfair [1]. Many studies showed that the voting procedure formulated in the “Treaty of Nice”, which would come into effect on the 1. November of that year, would unfairly favor countries with big populations like Germany while being especially unfavorable for mid-sized countries like Spain and Poland. As a compromise, mathematicians from the Jagiellonian University in Krakow proposed to use the “double square root voting system” which, different from all previous voting procedures, should be based on game theoretical principles [2].

Similar to the EU, blockchain communities are a diverse group of stakeholders with individual interests and different levels of stake in the system. In both cases, stakeholders rely on an underlying shared infrastructure which all members benefit from. Amending this underlying infrastructure, be it the European law or a blockchain protocol can have a big impact on the individual stakeholders. As a result, the decision power to make changes to this infrastructure should be distributed fairly between the stakeholders.

But what does it mean to distribute decision power in a fair way? In the EU it means that a compromise has to be found between assigning decision power according to population size and at the same time giving a voice to all the smaller member states. In the blockchain space, assigning decision power according to the stake an actor has in the system, like e.g. the number of owned tokens or hashing power, is often considered as fair. But the widespread interest in systems like quadratic voting also shows that many would support a more egalitarian distribution of decision power, where having more vote weight becomes costly for an individual. The double square root voting system tries to avoid centralized decision power and at the same time ensures that the voting power of stakeholders actually equals their assigned vote weight.

A-priori voting power

The most important concept that has to be understood in order to grasp the double square root voting system, is that of a-priori voting power. A-priori voting power is a central concept in voting theory and describes the real voting power of players in a weighted binary voting game (e.g. deciding for or against a proposal). There are cases in which the voting weight does not equal the voting power of a player. This can be best explained using an example: Let’s imagine a simple weighted voting game with only two voters. One has 51% of the voting weight and the other 49%. Even though the voting weights are nearly identical, it is clear that if a majority (> 50%) is needed, the first voter has 100% of the voting power. Such a voter is called a dictator, as he or she possesses the entire decision power. The other voter is called a dummy, as there will never be a situation in which his or her vote will have an effect on the outcome.

There are different power indices that can be used to express the a-priori voting power of players of a weighted voting game. The relative Banzhaf index is the one that is most widely recognized in academia and best fits the case of blockchain governance. It can be used to compare the voting powers of players in one specific voting game. The relative Banzhaf index is calculated by counting the number of coalitions in which a player is critical, meaning that without this player the coalition would not be winning, and dividing this number by the total number of times voters are critical in the voting game [3].

This can also be better illustrated by an example: Imagine a voting game with the following parameters. The quota describes the vote weight that is needed for a coalition to win.

The first winning coalition of this voting game is {A, B, C} with a shared vote weight of 9. In this coalition, A and B are critical while C is not critical, as the shared vote weight of A and B is 7 which already meets the required quota. Accordingly, the second winning coalition is {A, B} with a shared vote weight of 7. In this coalition, both members are critical because one member alone would not reach the quota. As a result, A and B are both critical in the first and the second winning coalition and C is not critical in any winning coalition. Hence, the Banzhaf index for both A and B is 2/4=0.5 and for C 0/4=0. It becomes clear that A and B both have half of the decision power as the quota can only be reached if they agree with each other and form a coalition. This means that C is a dummy in this voting game and the real relative voting power of the players differs considerably from their relative vote weight.

It should be pointed out that a-priori voting power does not make any predictions about the probability of the occurrence of any outcome but only expresses the voting power of a player that is derived solely from the decision rule itself, meaning the voting power that a player has because of the quota that is required to win and the way the voting weights are distributed.

Taking the square root

As the name suggests, the double square root voting system is made up of two parts. First, in the spirit of quadratic voting, the square root of the value is taken which the vote weight should be based on (e.g. population or number of tokens etc.). Taking the square root has an aligning effect on the resulting vote weights. A country for example that has a population that is a 100x higher than that of another country or a tokenholder that has a 100x more tokens than another tokenholder only has vote weight that is 10x higher. And because of the gradient of the square root function, this effect increases with bigger differences in population or tokens ( sqrt(10,000)=100, sqrt(1,000,000)=1,000 etc. ). As a result, countries with a higher population or tokenholders with a larger number of tokens will still always have more vote weight than smaller countries or small-scale tokenholders but not to the same extent as the differences in population or tokens would suggest. Therefore, the principle (which can be very beneficial) that entities with a higher stake in the system should have more decision power is upheld while centralized decision power in the hands of a few big countries or crypto-whales is effectively prevented.

Taking the square root of the population in the case of the EU Council is actually based on the Penrose Method which says that the voting power of a citizen in the two-tier voting system decreases with 1/sqrt(population) and therefore the voting power of every citizen in the EU is the same if the vote weight of a member state is the square root of its population. This is of course not really applicable to blockchain governance, but the beneficial properties are the same nonetheless

Adjusting the quota

But as we have learned, the vote weight of a player in a voting game is not always a valid expression of this player’s voting power. In order to ensure that the assigned vote weight equals their voting power, in the second part of the double square root voting system, the quota is adjusted in a way that the vote weight of each player equals their relative Banzhaf index. This is achieved by using a mathematical approximation that produces the wanted quota as a function of the vote weight distribution of all participating voters. This quota, which depends on the number of voters and their vote weights, is always above 50% but approaches majority rule for a big number of voters and evenly distributed vote weights.

Making the vote weights equal the relative Banzhaf index has many beneficial properties. First and most importantly, if there is more than one voter, there can never exist a dictator because that would require a vote weight of 100%. If a voter for example has a vote weight of 80% the quota would adjust in a way that the relative Banzhaf index equals 80%. In simple majority rule (quota = 50%), such a voter would be a dictator. But because the quota is adjusted, this is not the case in the double square root voting system. In practice, this means that depending on the other vote weights, the resulting quota will be well beyond 80% to ensure that this voter actually only holds 80% of the voting power. Moreover, in contrast to other voting mechanisms, where the quota is often set more or less arbitrarily based on the “level of conservatism” that seems appropriate, here the quota is adjusted based on game theoretic principles, dynamically adapting to the respective situation. The quota is automatically higher in situations where a more conservative quota is appropriate (big differences in vote weight and/or low number of voters) and lower in situations where a less conservative quota is acceptable (evenly distributed vote weight and/or high number of voters). This is a very elegant solution to choosing an appropriate quota because it automatically creates a fair voting system without needing any additional rules.

You can play around and get a feeling for the voting system with an example of only 10 voters in this small example calculator:

https://docs.google.com/spreadsheets/d/16hG8AW8YEOfR4trli5TpZdM4_sJmqZasFFWG95gWTmA/edit#gid=0

The double square root voting system can also disincentivize different forms of manipulation. One important objection against the use of formal voting in blockchain governance is its vulnerability to manipulation and especially vote buying. Because of the basic characteristics of blockchains like transparency and the validation of every transaction, voting on the blockchain allows for very effective vote buying [4].

Vote buying is most efficient when the vote seller and buyer have certainty about their transaction. This is why all modern democracies have introduced the secret ballot: If the vote buyer or coercer has no way of knowing for sure what choice an individual makes in the voting booth, vote buying or coercion become unprofitable. Considering this, it is clear why blockchain voting is perfect for vote buying. Vote buyer and seller can have absolute certainty that the other party holds up their bargain and the exchange can even be automated in a smart contract.

The double square root voting system can disincentivize vote buying and coercion by introducing uncertainty at another level. Because the required quota is calculated from the vote weight distribution of all participating voters, until the end of the voting period it will be unclear what quota will be required e.g. for a proposal to pass. Only after the voting period is over and all information is known can the outcome be calculated. The uncertainty about voter participation, the decision of other voters, their vote weight and the resulting quota might deter vote buyers from engaging in such manipulation. If the effect that vote buying will have on a specific election is really hard to predict, its expected value becomes quite hard to calculate. As a result, similar to introducing uncertainty about the transaction, introducing substantial uncertainty about the effect that this transaction will have, can prevent or at least disincentivize vote buying.

Moreover, in blockchain governance, one can imagine many metrics like tokens or even “used gas” to potentially act as a basis of vote weight. These metrics can often be “artificially” increased for the sole reason of increasing voting power. In the double square root voting system, the effect that such an artificial increase will have is limited by design. Taking the square root and adjusting the quota makes increasing decision power for a malicious actor a lot more costly than in simple e.g. token weighted voting systems. Additionally, similar to manipulation through vote buying, the effect that such an artificial increase might have is very uncertain as explained above. The high cost and uncertainty about its effect might disincentivize many actors from engaging in such activities.

The only kind of manipulation that similar to many other voting systems can, unfortunately, be quite problematic for the double square root voting system is the Sybil attack, in which an actor assumes multiple identities to acquire more decision power. Even though the effects of a Sybil attack in the double square root voting system are a lot less severe than in a one-person-one-vote system, it can, depending on the setup, increase the decision power of an attacker significantly. A setup where a Sybil attack can for example have quite a big effect is when there is are very unevenly distributed vote weights. If e.g. an actor with 80% of the vote weight manages to split up into entities of 40% and 40% or even 20%, 20%, 20%, 20% by assuming multiple identities, it might be enough to make this actor a dictator. But there exist many setups in which a Sybil attack does not have such a severe effect. Whenever the vote weights become more unevenly distributed because of the Sybil attack for example, the quota increases, and the effects of the attack are diminished. However, because of its vulnerability to Sybil attacks, the double square root voting system requires some kind of identity scheme to reach its full potential.

It becomes clear, that the double square root voting system fulfills many requirements for becoming a valuable voting system for blockchain governance. The mechanism is relatively simple, it is moderately conservative, can be easily adapted to changing circumstances, all parameters can be easily verified, and it distributes the decision power fairly between the stakeholders. It can help to tackle different forms of manipulation and is based on game theoretical principles. Together with some form of identity system, it can be used to make collective decisions on-chain. It is easy to implement and less complex than quadratic voting and can therefore be used in blockchain governance today. But it can of course also be seen as a first step towards experimenting with even more complex formal voting systems.

Both parts of the system could also be applied individually. If aligning vote weights and taking the first step towards something resembling quadratic voting seems like a good idea but ever-changing quotas and “this whole voting power thing” are a little bit too much, only taking the square root to align vote weights could be an option. In contrast, if actors should have the voting power that resembles their stake in the system directly, without aligning vote weights through taking the square root, only the second part can be used to choose a meaningful quota.

In conclusion, I believe that people in the blockchain space should look into all the different ways that formal voting could be realized on the blockchain before ruling it out completely. At the same time, I encourage everyone that works on designing governance processes to look for answers outside of their usual circles and learn from other democratic institutions. Especially learning from their mistakes of course. The EU Council, for example, did not accept the proposal of the polish mathematicians and instead went with an option that was the result of political negotiations without considering any game theoretical principles.

[1] The Telegraf: The EU constitution is ‘unfair’, according to game theorists: https://www.telegraph.co.uk/news/science/science-news/3325470/The-EU-constitution-is-unfair-according-to-game-theorists.html

[2] Słomczynski, W., & Życzkowski, K. (2007). From a toy model to the double square root voting system. Homo Oeconomicus, 24(3-4): https://arxiv.org/pdf/physics/0701338.pdf

[3] Felsenthal, D. S., & Machover, M. (1998). The measurement of voting power: Edward Elgar Publishing: https://www.elgaronline.com/view/9781858988054.xml

[4] Hacking, Distributed: On-Chain Vote Buying and the Rise of Dark DAOs: http://hackingdistributed.com/2018/07/02/on-chain-vote-buying/

In 2004, the scientific community and media was convinced that the voting procedure of the European Council of Ministers was unfair [1]. Many studies showed that the voting procedure formulated in the “Treaty of Nice”, which would come into effect on the 1. November of that year, would unfairly favor countries with big populations like Germany while being especially unfavorable for mid-sized countries like Spain and Poland. As a compromise, mathematicians from the Jagiellonian University in Krakow proposed to use the “double square root voting system” which, different from all previous voting procedures, should be based on game theoretical principles [2].

Similar to the EU, blockchain communities are a diverse group of stakeholders with individual interests and different levels of stake in the system. In both cases, stakeholders rely on an underlying shared infrastructure which all members benefit from. Amending this underlying infrastructure, be it the European law or a blockchain protocol can have a big impact on the individual stakeholders. As a result, the decision power to make changes to this infrastructure should be distributed fairly between the stakeholders.

But what does it mean to distribute decision power in a fair way? In the EU it means that a compromise has to be found between assigning decision power according to population size and at the same time giving a voice to all the smaller member states. In the blockchain space, assigning decision power according to the stake an actor has in the system, like e.g. the number of owned tokens or hashing power, is often considered as fair. But the widespread interest in systems like quadratic voting also shows that many would support a more egalitarian distribution of decision power, where having more vote weight becomes costly for an individual. The double square root voting system tries to avoid centralized decision power and at the same time ensures that the voting power of stakeholders actually equals their assigned vote weight.

A-priori voting power

The most important concept that has to be understood in order to grasp the double square root voting system, is that of a-priori voting power. A-priori voting power is a central concept in voting theory and describes the real voting power of players in a weighted binary voting game (e.g. deciding for or against a proposal). There are cases in which the voting weight does not equal the voting power of a player. This can be best explained using an example: Let’s imagine a simple weighted voting game with only two voters. One has 51% of the voting weight and the other 49%. Even though the voting weights are nearly identical, it is clear that if a majority (> 50%) is needed, the first voter has 100% of the voting power. Such a voter is called a dictator, as he or she possesses the entire decision power. The other voter is called a dummy, as there will never be a situation in which his or her vote will have an effect on the outcome.

There are different power indices that can be used to express the a-priori voting power of players of a weighted voting game. The relative Banzhaf index is the one that is most widely recognized in academia and best fits the case of blockchain governance. It can be used to compare the voting powers of players in one specific voting game. The relative Banzhaf index is calculated by counting the number of coalitions in which a player is critical, meaning that without this player the coalition would not be winning, and dividing this number by the total number of times voters are critical in the voting game [3].

This can also be better illustrated by an example: Imagine a voting game with the following parameters. The quota describes the vote weight that is needed for a coalition to win.

The first winning coalition of this voting game is {A, B, C} with a shared vote weight of 9. In this coalition, A and B are critical while C is not critical, as the shared vote weight of A and B is 7 which already meets the required quota. Accordingly, the second winning coalition is {A, B} with a shared vote weight of 7. In this coalition, both members are critical because one member alone would not reach the quota. As a result, A and B are both critical in the first and the second winning coalition and C is not critical in any winning coalition. Hence, the Banzhaf index for both A and B is 2/4=0.5 and for C 0/4=0. It becomes clear that A and B both have half of the decision power as the quota can only be reached if they agree with each other and form a coalition. This means that C is a dummy in this voting game and the real relative voting power of the players differs considerably from their relative vote weight.

It should be pointed out that a-priori voting power does not make any predictions about the probability of the occurrence of any outcome but only expresses the voting power of a player that is derived solely from the decision rule itself, meaning the voting power that a player has because of the quota that is required to win and the way the voting weights are distributed.

Taking the square root

As the name suggests, the double square root voting system is made up of two parts. First, in the spirit of quadratic voting, the square root of the value is taken which the vote weight should be based on (e.g. population or number of tokens etc.). Taking the square root has an aligning effect on the resulting vote weights. A country for example that has a population that is a 100x higher than that of another country or a tokenholder that has a 100x more tokens than another tokenholder only has vote weight that is 10x higher. And because of the gradient of the square root function, this effect increases with bigger differences in population or tokens ( sqrt(10,000)=100, sqrt(1,000,000)=1,000 etc. ). As a result, countries with a higher population or tokenholders with a larger number of tokens will still always have more vote weight than smaller countries or small-scale tokenholders but not to the same extent as the differences in population or tokens would suggest. Therefore, the principle (which can be very beneficial) that entities with a higher stake in the system should have more decision power is upheld while centralized decision power in the hands of a few big countries or crypto-whales is effectively prevented.

Taking the square root of the population in the case of the EU Council is actually based on the Penrose Method which says that the voting power of a citizen in the two-tier voting system decreases with 1/sqrt(population) and therefore the voting power of every citizen in the EU is the same if the vote weight of a member state is the square root of its population. This is of course not really applicable to blockchain governance, but the beneficial properties are the same nonetheless

Adjusting the quota

But as we have learned, the vote weight of a player in a voting game is not always a valid expression of this player’s voting power. In order to ensure that the assigned vote weight equals their voting power, in the second part of the double square root voting system, the quota is adjusted in a way that the vote weight of each player equals their relative Banzhaf index. This is achieved by using a mathematical approximation that produces the wanted quota as a function of the vote weight distribution of all participating voters. This quota, which depends on the number of voters and their vote weights, is always above 50% but approaches majority rule for a big number of voters and evenly distributed vote weights.

Making the vote weights equal the relative Banzhaf index has many beneficial properties. First and most importantly, if there is more than one voter, there can never exist a dictator because that would require a vote weight of 100%. If a voter for example has a vote weight of 80% the quota would adjust in a way that the relative Banzhaf index equals 80%. In simple majority rule (quota = 50%), such a voter would be a dictator. But because the quota is adjusted, this is not the case in the double square root voting system. In practice, this means that depending on the other vote weights, the resulting quota will be well beyond 80% to ensure that this voter actually only holds 80% of the voting power. Moreover, in contrast to other voting mechanisms, where the quota is often set more or less arbitrarily based on the “level of conservatism” that seems appropriate, here the quota is adjusted based on game theoretic principles, dynamically adapting to the respective situation. The quota is automatically higher in situations where a more conservative quota is appropriate (big differences in vote weight and/or low number of voters) and lower in situations where a less conservative quota is acceptable (evenly distributed vote weight and/or high number of voters). This is a very elegant solution to choosing an appropriate quota because it automatically creates a fair voting system without needing any additional rules.

You can play around and get a feeling for the voting system with an example of only 10 voters in this small example calculator:

https://docs.google.com/spreadsheets/d/16hG8AW8YEOfR4trli5TpZdM4_sJmqZasFFWG95gWTmA/edit#gid=0

The double square root voting system can also disincentivize different forms of manipulation. One important objection against the use of formal voting in blockchain governance is its vulnerability to manipulation and especially vote buying. Because of the basic characteristics of blockchains like transparency and the validation of every transaction, voting on the blockchain allows for very effective vote buying [4].

Vote buying is most efficient when the vote seller and buyer have certainty about their transaction. This is why all modern democracies have introduced the secret ballot: If the vote buyer or coercer has no way of knowing for sure what choice an individual makes in the voting booth, vote buying or coercion become unprofitable. Considering this, it is clear why blockchain voting is perfect for vote buying. Vote buyer and seller can have absolute certainty that the other party holds up their bargain and the exchange can even be automated in a smart contract.

The double square root voting system can disincentivize vote buying and coercion by introducing uncertainty at another level. Because the required quota is calculated from the vote weight distribution of all participating voters, until the end of the voting period it will be unclear what quota will be required e.g. for a proposal to pass. Only after the voting period is over and all information is known can the outcome be calculated. The uncertainty about voter participation, the decision of other voters, their vote weight and the resulting quota might deter vote buyers from engaging in such manipulation. If the effect that vote buying will have on a specific election is really hard to predict, its expected value becomes quite hard to calculate. As a result, similar to introducing uncertainty about the transaction, introducing substantial uncertainty about the effect that this transaction will have, can prevent or at least disincentivize vote buying.

Moreover, in blockchain governance, one can imagine many metrics like tokens or even “used gas” to potentially act as a basis of vote weight. These metrics can often be “artificially” increased for the sole reason of increasing voting power. In the double square root voting system, the effect that such an artificial increase will have is limited by design. Taking the square root and adjusting the quota makes increasing decision power for a malicious actor a lot more costly than in simple e.g. token weighted voting systems. Additionally, similar to manipulation through vote buying, the effect that such an artificial increase might have is very uncertain as explained above. The high cost and uncertainty about its effect might disincentivize many actors from engaging in such activities.

The only kind of manipulation that similar to many other voting systems can, unfortunately, be quite problematic for the double square root voting system is the Sybil attack, in which an actor assumes multiple identities to acquire more decision power. Even though the effects of a Sybil attack in the double square root voting system are a lot less severe than in a one-person-one-vote system, it can, depending on the setup, increase the decision power of an attacker significantly. A setup where a Sybil attack can for example have quite a big effect is when there is are very unevenly distributed vote weights. If e.g. an actor with 80% of the vote weight manages to split up into entities of 40% and 40% or even 20%, 20%, 20%, 20% by assuming multiple identities, it might be enough to make this actor a dictator. But there exist many setups in which a Sybil attack does not have such a severe effect. Whenever the vote weights become more unevenly distributed because of the Sybil attack for example, the quota increases, and the effects of the attack are diminished. However, because of its vulnerability to Sybil attacks, the double square root voting system requires some kind of identity scheme to reach its full potential.

It becomes clear, that the double square root voting system fulfills many requirements for becoming a valuable voting system for blockchain governance. The mechanism is relatively simple, it is moderately conservative, can be easily adapted to changing circumstances, all parameters can be easily verified, and it distributes the decision power fairly between the stakeholders. It can help to tackle different forms of manipulation and is based on game theoretical principles. Together with some form of identity system, it can be used to make collective decisions on-chain. It is easy to implement and less complex than quadratic voting and can therefore be used in blockchain governance today. But it can of course also be seen as a first step towards experimenting with even more complex formal voting systems.

Both parts of the system could also be applied individually. If aligning vote weights and taking the first step towards something resembling quadratic voting seems like a good idea but ever-changing quotas and “this whole voting power thing” are a little bit too much, only taking the square root to align vote weights could be an option. In contrast, if actors should have the voting power that resembles their stake in the system directly, without aligning vote weights through taking the square root, only the second part can be used to choose a meaningful quota.

In conclusion, I believe that people in the blockchain space should look into all the different ways that formal voting could be realized on the blockchain before ruling it out completely. At the same time, I encourage everyone that works on designing governance processes to look for answers outside of their usual circles and learn from other democratic institutions. Especially learning from their mistakes of course. The EU Council, for example, did not accept the proposal of the polish mathematicians and instead went with an option that was the result of political negotiations without considering any game theoretical principles.

[1] The Telegraf: The EU constitution is ‘unfair’, according to game theorists: https://www.telegraph.co.uk/news/science/science-news/3325470/The-EU-constitution-is-unfair-according-to-game-theorists.html

[2] Słomczynski, W., & Życzkowski, K. (2007). From a toy model to the double square root voting system. Homo Oeconomicus, 24(3-4): https://arxiv.org/pdf/physics/0701338.pdf

[3] Felsenthal, D. S., & Machover, M. (1998). The measurement of voting power: Edward Elgar Publishing: https://www.elgaronline.com/view/9781858988054.xml

[4] Hacking, Distributed: On-Chain Vote Buying and the Rise of Dark DAOs: http://hackingdistributed.com/2018/07/02/on-chain-vote-buying/

Enter the Blockchain Writing Contest

In 2004, the scientific community and media was convinced that the voting procedure of the European Council of Ministers was unfair [1]. Many studies showed that the voting procedure formulated in the “Treaty of Nice”, which would come into effect on the 1. November of that year, would unfairly favor countries with big populations like Germany while being especially unfavorable for mid-sized countries like Spain and Poland. As a compromise, mathematicians from the Jagiellonian University in Krakow proposed to use the “double square root voting system” which, different from all previous voting procedures, should be based on game theoretical principles [2].

Similar to the EU, blockchain communities are a diverse group of stakeholders with individual interests and different levels of stake in the system. In both cases, stakeholders rely on an underlying shared infrastructure which all members benefit from. Amending this underlying infrastructure, be it the European law or a blockchain protocol can have a big impact on the individual stakeholders. As a result, the decision power to make changes to this infrastructure should be distributed fairly between the stakeholders.

But what does it mean to distribute decision power in a fair way? In the EU it means that a compromise has to be found between assigning decision power according to population size and at the same time giving a voice to all the smaller member states. In the blockchain space, assigning decision power according to the stake an actor has in the system, like e.g. the number of owned tokens or hashing power, is often considered as fair. But the widespread interest in systems like quadratic voting also shows that many would support a more egalitarian distribution of decision power, where having more vote weight becomes costly for an individual. The double square root voting system tries to avoid centralized decision power and at the same time ensures that the voting power of stakeholders actually equals their assigned vote weight.

A-priori voting power

The most important concept that has to be understood in order to grasp the double square root voting system, is that of a-priori voting power. A-priori voting power is a central concept in voting theory and describes the real voting power of players in a weighted binary voting game (e.g. deciding for or against a proposal). There are cases in which the voting weight does not equal the voting power of a player. This can be best explained using an example: Let’s imagine a simple weighted voting game with only two voters. One has 51% of the voting weight and the other 49%. Even though the voting weights are nearly identical, it is clear that if a majority (> 50%) is needed, the first voter has 100% of the voting power. Such a voter is called a dictator, as he or she possesses the entire decision power. The other voter is called a dummy, as there will never be a situation in which his or her vote will have an effect on the outcome.

There are different power indices that can be used to express the a-priori voting power of players of a weighted voting game. The relative Banzhaf index is the one that is most widely recognized in academia and best fits the case of blockchain governance. It can be used to compare the voting powers of players in one specific voting game. The relative Banzhaf index is calculated by counting the number of coalitions in which a player is critical, meaning that without this player the coalition would not be winning, and dividing this number by the total number of times voters are critical in the voting game [3].

This can also be better illustrated by an example: Imagine a voting game with the following parameters. The quota describes the vote weight that is needed for a coalition to win.

The first winning coalition of this voting game is {A, B, C} with a shared vote weight of 9. In this coalition, A and B are critical while C is not critical, as the shared vote weight of A and B is 7 which already meets the required quota. Accordingly, the second winning coalition is {A, B} with a shared vote weight of 7. In this coalition, both members are critical because one member alone would not reach the quota. As a result, A and B are both critical in the first and the second winning coalition and C is not critical in any winning coalition. Hence, the Banzhaf index for both A and B is 2/4=0.5 and for C 0/4=0. It becomes clear that A and B both have half of the decision power as the quota can only be reached if they agree with each other and form a coalition. This means that C is a dummy in this voting game and the real relative voting power of the players differs considerably from their relative vote weight.

It should be pointed out that a-priori voting power does not make any predictions about the probability of the occurrence of any outcome but only expresses the voting power of a player that is derived solely from the decision rule itself, meaning the voting power that a player has because of the quota that is required to win and the way the voting weights are distributed.

Taking the square root

As the name suggests, the double square root voting system is made up of two parts. First, in the spirit of quadratic voting, the square root of the value is taken which the vote weight should be based on (e.g. population or number of tokens etc.). Taking the square root has an aligning effect on the resulting vote weights. A country for example that has a population that is a 100x higher than that of another country or a tokenholder that has a 100x more tokens than another tokenholder only has vote weight that is 10x higher. And because of the gradient of the square root function, this effect increases with bigger differences in population or tokens ( sqrt(10,000)=100, sqrt(1,000,000)=1,000 etc. ). As a result, countries with a higher population or tokenholders with a larger number of tokens will still always have more vote weight than smaller countries or small-scale tokenholders but not to the same extent as the differences in population or tokens would suggest. Therefore, the principle (which can be very beneficial) that entities with a higher stake in the system should have more decision power is upheld while centralized decision power in the hands of a few big countries or crypto-whales is effectively prevented.

Taking the square root of the population in the case of the EU Council is actually based on the Penrose Method which says that the voting power of a citizen in the two-tier voting system decreases with 1/sqrt(population) and therefore the voting power of every citizen in the EU is the same if the vote weight of a member state is the square root of its population. This is of course not really applicable to blockchain governance, but the beneficial properties are the same nonetheless

Adjusting the quota

But as we have learned, the vote weight of a player in a voting game is not always a valid expression of this player’s voting power. In order to ensure that the assigned vote weight equals their voting power, in the second part of the double square root voting system, the quota is adjusted in a way that the vote weight of each player equals their relative Banzhaf index. This is achieved by using a mathematical approximation that produces the wanted quota as a function of the vote weight distribution of all participating voters. This quota, which depends on the number of voters and their vote weights, is always above 50% but approaches majority rule for a big number of voters and evenly distributed vote weights.

Making the vote weights equal the relative Banzhaf index has many beneficial properties. First and most importantly, if there is more than one voter, there can never exist a dictator because that would require a vote weight of 100%. If a voter for example has a vote weight of 80% the quota would adjust in a way that the relative Banzhaf index equals 80%. In simple majority rule (quota = 50%), such a voter would be a dictator. But because the quota is adjusted, this is not the case in the double square root voting system. In practice, this means that depending on the other vote weights, the resulting quota will be well beyond 80% to ensure that this voter actually only holds 80% of the voting power. Moreover, in contrast to other voting mechanisms, where the quota is often set more or less arbitrarily based on the “level of conservatism” that seems appropriate, here the quota is adjusted based on game theoretic principles, dynamically adapting to the respective situation. The quota is automatically higher in situations where a more conservative quota is appropriate (big differences in vote weight and/or low number of voters) and lower in situations where a less conservative quota is acceptable (evenly distributed vote weight and/or high number of voters). This is a very elegant solution to choosing an appropriate quota because it automatically creates a fair voting system without needing any additional rules.

You can play around and get a feeling for the voting system with an example of only 10 voters in this small example calculator:

https://docs.google.com/spreadsheets/d/16hG8AW8YEOfR4trli5TpZdM4_sJmqZasFFWG95gWTmA/edit#gid=0

The double square root voting system can also disincentivize different forms of manipulation. One important objection against the use of formal voting in blockchain governance is its vulnerability to manipulation and especially vote buying. Because of the basic characteristics of blockchains like transparency and the validation of every transaction, voting on the blockchain allows for very effective vote buying [4].

Vote buying is most efficient when the vote seller and buyer have certainty about their transaction. This is why all modern democracies have introduced the secret ballot: If the vote buyer or coercer has no way of knowing for sure what choice an individual makes in the voting booth, vote buying or coercion become unprofitable. Considering this, it is clear why blockchain voting is perfect for vote buying. Vote buyer and seller can have absolute certainty that the other party holds up their bargain and the exchange can even be automated in a smart contract.

The double square root voting system can disincentivize vote buying and coercion by introducing uncertainty at another level. Because the required quota is calculated from the vote weight distribution of all participating voters, until the end of the voting period it will be unclear what quota will be required e.g. for a proposal to pass. Only after the voting period is over and all information is known can the outcome be calculated. The uncertainty about voter participation, the decision of other voters, their vote weight and the resulting quota might deter vote buyers from engaging in such manipulation. If the effect that vote buying will have on a specific election is really hard to predict, its expected value becomes quite hard to calculate. As a result, similar to introducing uncertainty about the transaction, introducing substantial uncertainty about the effect that this transaction will have, can prevent or at least disincentivize vote buying.

Moreover, in blockchain governance, one can imagine many metrics like tokens or even “used gas” to potentially act as a basis of vote weight. These metrics can often be “artificially” increased for the sole reason of increasing voting power. In the double square root voting system, the effect that such an artificial increase will have is limited by design. Taking the square root and adjusting the quota makes increasing decision power for a malicious actor a lot more costly than in simple e.g. token weighted voting systems. Additionally, similar to manipulation through vote buying, the effect that such an artificial increase might have is very uncertain as explained above. The high cost and uncertainty about its effect might disincentivize many actors from engaging in such activities.

The only kind of manipulation that similar to many other voting systems can, unfortunately, be quite problematic for the double square root voting system is the Sybil attack, in which an actor assumes multiple identities to acquire more decision power. Even though the effects of a Sybil attack in the double square root voting system are a lot less severe than in a one-person-one-vote system, it can, depending on the setup, increase the decision power of an attacker significantly. A setup where a Sybil attack can for example have quite a big effect is when there is are very unevenly distributed vote weights. If e.g. an actor with 80% of the vote weight manages to split up into entities of 40% and 40% or even 20%, 20%, 20%, 20% by assuming multiple identities, it might be enough to make this actor a dictator. But there exist many setups in which a Sybil attack does not have such a severe effect. Whenever the vote weights become more unevenly distributed because of the Sybil attack for example, the quota increases, and the effects of the attack are diminished. However, because of its vulnerability to Sybil attacks, the double square root voting system requires some kind of identity scheme to reach its full potential.

It becomes clear, that the double square root voting system fulfills many requirements for becoming a valuable voting system for blockchain governance. The mechanism is relatively simple, it is moderately conservative, can be easily adapted to changing circumstances, all parameters can be easily verified, and it distributes the decision power fairly between the stakeholders. It can help to tackle different forms of manipulation and is based on game theoretical principles. Together with some form of identity system, it can be used to make collective decisions on-chain. It is easy to implement and less complex than quadratic voting and can therefore be used in blockchain governance today. But it can of course also be seen as a first step towards experimenting with even more complex formal voting systems.

Both parts of the system could also be applied individually. If aligning vote weights and taking the first step towards something resembling quadratic voting seems like a good idea but ever-changing quotas and “this whole voting power thing” are a little bit too much, only taking the square root to align vote weights could be an option. In contrast, if actors should have the voting power that resembles their stake in the system directly, without aligning vote weights through taking the square root, only the second part can be used to choose a meaningful quota.

In conclusion, I believe that people in the blockchain space should look into all the different ways that formal voting could be realized on the blockchain before ruling it out completely. At the same time, I encourage everyone that works on designing governance processes to look for answers outside of their usual circles and learn from other democratic institutions. Especially learning from their mistakes of course. The EU Council, for example, did not accept the proposal of the polish mathematicians and instead went with an option that was the result of political negotiations without considering any game theoretical principles.

[1] The Telegraf: The EU constitution is ‘unfair’, according to game theorists: https://www.telegraph.co.uk/news/science/science-news/3325470/The-EU-constitution-is-unfair-according-to-game-theorists.html

[2] Słomczynski, W., & Życzkowski, K. (2007). From a toy model to the double square root voting system. Homo Oeconomicus, 24(3-4): https://arxiv.org/pdf/physics/0701338.pdf

[3] Felsenthal, D. S., & Machover, M. (1998). The measurement of voting power: Edward Elgar Publishing: https://www.elgaronline.com/view/9781858988054.xml

[4] Hacking, Distributed: On-Chain Vote Buying and the Rise of Dark DAOs: http://hackingdistributed.com/2018/07/02/on-chain-vote-buying/