 We want to understand how democratic institutions should take into account that usually not everybody participates in an election. There are so many possibilities of how we could do it. The voting rule can be any specification of an outcome reform or status quo, depending on how many people vote for the reform, how many people vote for the status quo, how many people abstain. One common way of dealing with this is a majority rule with a quorum. That means that the reform is implemented if enough people of the entire electorate vote for the reform and in addition if there is a majority for the reform among those who do participate. But that's just one rule. Is it the best possible rule? We could do it in many other ways. For each of us it's individually costly to participate in an election. For example, in the EU referendum on the Daylight Savings time only 4 million people participated although hundreds of millions of people were eligible to participate. If so few people participate, then we don't get a very good representation of the preferences in the entire electorate. On the other hand, we save a lot on the participation costs. The question is how should we deal with that trade-off, how should we resolve it in the best possible way. We compare the voting rules and that means we need a prediction of what will happen in each voting rule. That prediction will depend on the prediction of what each individual voter will do. Each individual voter will think about what's the distribution of participation costs, benefits of the outcome in the population, what's my own participation cost. In a close election, I will participate even if my participation cost is high. I need a low participation cost in order to be willing to participate in an election if it's not close. The conditions for closeness in turn depend on which voting rule is in place. So all the strategizing will depend on that. For example, in the majority rule of the quorum, there are two different conditions. One is that the quorum is just about to be reached. The other condition is that there is just as many votes for reform as for status quo. The probability that these conditions are met depends on how many people will participate on the reform and on the status quo side. So rational voter will take all of that into account to make a decision optimally, assuming that voters on average have a correct expectation of what others are doing. We get a fixed point of behavior that we call a Nash equilibrium, assuming that a Nash equilibrium occurs in each voting rule. We can compare the various voting rules in terms of the expected utilities of average voters. The best voting rule is the one with the highest expected utility. So mathematically speaking, we have to solve an optimization problem under constraints. So that's what we will do. Actually I was surprised by our own findings. The common quorum rules that we all see everywhere around are not the best way of dealing with the participation problem. We found a new class of voting rules that we call the linear voting rules. They are as simple as quorum rules, just a bit different. Each linear rule is described by just three parameters, a weight on the reform votes, a weight on the status quo votes and the weight on the abstentions. According to such a rule, the reform will be implemented if and only if the weighted sum across votes and abstentions is positive. Let me give an example of a linear rule. You could say that if everybody participates in the referendum, then a simple majority is sufficient to make the reform go through. But then let's say the margin of victory that we would require for a reform should be proportional to the number of abstentions. For example, if only three quarters of the electorate participate, we would require 5%. For example, a margin of victory for the reform, then if only half of the electorate participates, we would require a 10% margin of victory. If only a quarter of the electorate participates, we would require a 15% margin of victory. So it's a much smoother way, a linear way of dealing with the abstentions. That's it. Linear rules are a better way of dealing with the abstentions than anything else and that's why we propose them. We do so many things in our modern societies by voting that it's extremely important to get the voting rules right and in particular to take into account the abstention problem in the best possible way. We have seen that the commonly used quorum rules are not the best way of dealing with this, but there are better ways. It can be very relevant to how well our voting works. You can even go to the Brexit example if we put just a little bit of weight on the abstentions favoring the status quo. In that example, we would immediately revert the election result of the Brexit election. We show the optimality of linear rules in one particular setting, the so-called private value setting, setting in which just the private interests of voters are relevant for the public decision, but there are many applications of voting that have a public element where the decision really affects us all in a way that is uncertain to all of us. For example, how much should we use antibiotics? How much should we use carbon dioxide? These settings in game theory we call interdependent values, analyzing the strategies of voters in such a setting is rather different. We would like to understand the properties of linear voting rules in such settings as well. That's one important topic for future research.