Rolling Dice?! Flipping a coin?! Randomly excluding voters?! To attain a fair vote?!
Here's a REAL set of answers that provide deterministic results and that start with the assumptions of the OP's conditions. Refer to the Addendum below if you need to understand how.
Some "fair" voting procedures that are deterministic:
You can skip to the section labeled with A,B,C etc...
In fairness, as a disenfranchised new member of the community (meaning that I can't vote), I would like to ask that those with the vote privilege please leave my vote count at zero if you likewise believe that my answer is of no value. Please leave a well-reasoned argument that is of value instead. I do edit my posts. Thank you...
Much of the work by the philosopher, Alexis de Tocqueville, could be quoted and then paraphrased to sum the very problem you describe in your question: 'The rule of the 51 percent majority is the tyranny and oppression over the 49 percent minority.' This is especially true, in the case of YES or NO, all or nothing, when essentially the other half of voters are given neither equally nor pragmatically viable alternative benefits or consolations to substitute with which they may be indifferent (as in a basket of goods A or B for them) to be satiated. (In America, by the way, this could even be as bad as a split-haired 49.99% vs. 50.01% popular vote.) They, the lesser half, cannot be ignored as they do not disappear. By little stretch of the imagination this sets the stage for a very Pareto Inefficient outcome.
In your comment, you say "there is a chance that one person will always make the decisions although all other people disagree with him." As you originally alluded, the contrapositive is also applicable: 'Many people may make decisions although one person may disagree.' This is a challenge for new thinking when there already exists conventional thought.
What you're asking is 'For the most optimal outcome, how do you balance votes when there's a group within the total set of voters whose unchanging vote makes one majority outcome more probable than another (thereby even making the voting process itself superfluous.)'
There are several things that can be done. These solutions can be applied to remedying loaded coin/dice tosses mixed in with legitimate coin/die tosses or people who vote with bias:
A. IGNORE/REMOVE THE VOTES THAT NEVER CHANGE.
If a subgroup always votes the same way, then the existence of them having a rationale is questionable. A vote is, in contrast to a random toss, assumed to be a discrimination between choices based on information. But voters can behave irrational in their choice. They may, without further consideration, always select one brand instead of another when faced between substitutes that have different labels but equal content. Perhaps they do this to minimize the risk of trying new. They operate on no, low, or old information. At any rate they skew the total vote by acting as a bias coefficient. 'Bias coefficient' means that the choice is completely inelastic. There are no other options or outcomes. This can mean that, because they neither question nor reason, they have no constructive voting input other than to skew the results. The solution: Simply ignore the never changing votes. Subtract the coefficient on the graph and bring the starting point back to zero. Conduct the real vote: Count the remaining votes as 100%, i.e., the votes which can sway either way depending on exogenous factors (as opposed to an inherent, endogenous bias.)
B1. WEIGHT THE VOTES AND DECIDE ON A DIFFERENT MAJORITY FRACTION.
An inelastic vote bias gives the remaining voters who vote in line with the bias an unfair advantage over the other side of the YES/NO coin. This is a handicap for the other side. It takes less input for the former side to outvote the latter side - perhaps even when the majority of the latter votes are the actual thinking/rational voters who do constructively weigh YES vs. NO in decision-making. As you know, various sports employ handicaps to equate units of input, e.g., effort, on both sides. Q: How did David beat Goliath? A: By using an equalizer, i.e., a slingshot!
Also, pick a tie-breaker that is culturally tolerable. The US Congress uses the simplest fraction, 2/3 majority to represent a scenario where 2 out of 3 discrete/indivisible people would commit decisively one way versus another. In the 2/3 example, for the purpose of inclusion, redefine the inelastic subgroup as having 1/3 weight. The remaining voters can represent the other 2/3 of the vote. Multiply each vote of the remaining votes by some factor that makes their vote count numerically 2/3 the size of the first subgroup.
For example, the inelastic/biased group is made up of 90 voters or 40% of all voters. The remaining number of voters is therefore 90*60%/40% = 135 voters. Multiply the 135 elastic voters by a factor that gives them a 2/3 decision weight, i.e., 135*x=90*2 --> x = 180/(135) --> x = 4/3. In this example, the vote of each elastic voting person (who can be YES or NO) is equal to a 4/3 biased vote. This is actually a variation on A. The drawback is that the required majority might not be attained. The benefit is it makes the minority contingent smaller.
Let's say that there is yet another subgroup within the elastic, changeable subgroup that does not have an equal probability of voting YES or NO. It may be partially biased. Let's say members in this variable subgroup may have a 2/3 probability of voting one way versus another. Again, find out the number of this special subgroup that has an uneven probability versus the number of those who have an even probability and multiply each side by factors that give each group, e.g., an equal 50/50 voting weight. For simplicity, one half has 2/3 probability of voting one way; and the second half has a 1/2 probability of voting either way. Multiply the first side votes by 3/2 and the second side votes by 2/1 to make the influential weight of both sides 1:1 again. If the numbers of either elastic subgroup are uneven then do the additional simple math for which you can refer above in B1.
C. INCREASE THE VOTING SAMPLE SIZE AND APPLY B.
Imagine a room of only 2 voters: one stubborn, hard-headed person and one changeable,open-minded person. The outcome is either unanimous or 50/50 ambiguous. Increase your sample size! The problem is that neither of them and very likely at least one of them will not trust the newcomers especially after the winning/losing votes are tallied.
D. MAKE PEOPLE ACCOUNTABLE FOR VOTING CONSEQUENCES. [...this is my favorite one...]
Hindsight is 20/20 but risk can truly sharpen one's focus. Involve risk management as part of the vote. Let the voters reap the fruit of their vote but also let them enjoy or suffer its taste. In this instance, voters would have to be registered/identifiable. Those voters who win get the benefits (and costs) of their vote (most fairly in proportion to the size of their vote.) If 67% of voters got to decide on how to use a budget, let them enjoy 67% of that budget toward their decision. Those voters who lose do not get to share in that benefit (or cost.) However, if the majority vote makes a bad decision, they must pay for it - not those who didn't vote for it. Most primates if not animals do not like to break even when budgeting input in exchange for output but the fear of loss is indeed greater than hope of gain. The perception of risks for voting vs. not voting can radically change voting behavior and motivate voters to acquire better information, not vote, or participate more actively thereby changing the voting sample (toward a more honest/constructive/informed turnout & vote) as well.
Fairness rules can be created and do exist (right here!) to balance a voting sample that contains biased subgroups in achieving a fair and deterministic outcome in both YES/NO votes or those that involve further complexity.
Hope these suggestions help, Erel!
A lengthy list of quotes by de Tocqueville:
[Originally a response to a comment below asking for clarification. Important for inclusion but too long for intro.]
An election is a decision. A vote is a decision. The difference between either is the word used for "decision" & fraction criteria for finalization. A decision represents a probability. The probable decision is the sum of all probabilities divided by the total number of probabilities. Therefore, without complete/perfect info, a priori, an election decision is a probability; a single vote decision is a probability. Before casting a vote, a voter conducts an election with themselves. Each issue may be comprised of sub-issues, all representing probabilities, each with weight.
The probable voting decision that a voter makes is the sum of all sub-decision probabilities (each multiplied by their weight of importance - analogous to individuals voting) divided by the total weight of sub-decisions. Taking issues, sub-decisions, etc., to infinity, using the formula, gives us the probability of a vote when total issues are taken to infinity. The same applies to a voter having infinite elections with themselves or conducting infinite elections. Whether a group given probability=1 for their preference gets their way (at infinity) depends on the required majority.
If the probability of the group's decision at infinity is greater than the required majority vote, then the group will have their way at infinity. The answer to the question above takes this as a starting assumption, a mutually understood given, and then offers solutions for a "fair decision rule" that's interpreted to mean an 'equitable voting outcome that is balanced, that is "fair."' Sources of error it addresses are groupsizes/weights and that, even when taken to infinity, voters do not work with the same sets of info at the sub-decision levels to come to the 'same vote decisions'*.
*In the above, the "same vote decision" implies that the vote of one voter is identical in scope and thus equivalent to another voter's vote if it encompasses an identical number of issues/sub-issues, identical sets of information, identical cost/benefit analyses, and identical degrees of consideration with all other things being equal. Votes are not the same if all that which goes into each vote (for each voter) is different from one vote to the next which therefore creates a probable bias in the process leading up to the voting decision and thus the vote...
Case in point: Five people standing in line at a ballot box for an infinite number of elections to vote YES/NO that covers an infinite number of issues. They're living in an episode of Rod Serling's Twilight Zone where it just keeps repeating with more and more issues added to infinity. The first two people eternally read the newspaper everyday and do much research, consultation, & contemplation over weeks/years in order to decide YES/NO. The probability of the first group's vote either way, given variable tastes, is more variable. The other three people who like the same style and, who stubbornly refuse to change, base their decisions on comparing bumper stickers, 5-second slogans, and operate - with no exception - on the bias that marketing image & affiliation is everything and that the final decision is one about image & affiliation. The probability of the second group's vote given the same never changing preferences (and for the purpose of illustration) is 1 or very close to it. How to establish a fair vote? Refer to all of the above...