In their book "Radical Markets", the authors (Posner and Weyl) suggest a market for votes, where citizens can freely buy and sell their votes. But there is a catch: in order to have $$k$$ ballots, one must by $$k^2$$ votes.

Initially, the government gives each citizen 1 vote for free. Then, a free market opens, where citizens can buy and sell their votes. At the end, there is a referendun on some issue, where each person can put a "yes" or "no" ballot. A citizen who has, say, 9 votes, can put 3 ballots; a citizen with 16 votes can put 4 ballots; etc.

I am trying to understand how such a market will function, and specifically, what would be an equilibrium price for a single vote? For concreteness, suppose there are two tycoons: one wants the outcome to be "yes" and the other wants the outcome to be "no". Each tycoon $$i$$ gains $$v_i$$ if his favorite outcome is elected and $$0$$ otherwise. Can we compute the equilibrium price for a single vote, under some reasonable assumptions?

EDIT: Actually, I should have asked a more basic question first: how would the market function with linear voting (one vote gives one ballot) -- what would the equilibrium price be?

• Are the people indifferent, does the vote only affect the tycoons? Commented May 2 at 7:16
• @Giskard for the sake of simplicity, let's assume the other people are indifferent. Commented May 2 at 12:37
• Is the buying simultaneous, do they know each other's $v_i$? Could you try to define the game better? Commented May 2 at 12:40
• Clearly if tycoon A knows that tycoon B will buy more votes than they will since they value the outcome much more and can observe A's bought vote total before making a decision, they will not buy any. The exact market mechanism (game) is important. Commented May 2 at 12:49
• A game theory approach is also better than a general equilibrium approach, because the ballots have no inherent value; the number a player buys depends on how many they believe the other players will buy. Commented May 2 at 12:52