<p>In their book &quot;Radical Markets&quot;, 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 <span class="math-container">$k$</span> ballots, one must by <span class="math-container">$k^2$</span> votes.</p>
<p>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 &quot;yes&quot; or &quot;no&quot; ballot. A citizen who has, say, 9 votes, can put 3 ballots; a citizen with 16 votes can put 4 ballots; etc.</p>
<p>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 &quot;yes&quot; and the other wants the outcome to be &quot;no&quot;. Each tycoon <span class="math-container">$i$</span> gains <span class="math-container">$v_i$</span> if his favorite outcome is elected and <span class="math-container">$0$</span> otherwise. Can we compute the equilibrium price for a single vote, under some reasonable assumptions?</p>
<p>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?</p>

CC BY-SA 4.0
2024-05-02T07:16:59.303Z
Are the people indifferent, does the vote only affect the tycoons?
Giskard
CC BY-SA 4.0
2024-05-02T12:37:45.503Z
@Giskard for the sake of simplicity, let's assume the other people are indifferent.
Erel Segal-Halevi
CC BY-SA 4.0
2024-05-02T12:40:15.010Z
Is the buying simultaneous, do they know each other's $v_i$? Could you try to define the game better?
Giskard
CC BY-SA 4.0
2024-05-02T12:49:53.833Z
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.
Giskard
CC BY-SA 4.0
2024-05-02T12:52:06.323Z
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.
Giskard
CC BY-SA 4.0
2024-05-02T13:01:02.100Z
Re your edit: in the linear case what you are looking at is an all-pay auction with different valuations. Timing and knowledge/information structure determines what happens in equilibrium.
Giskard
CC BY-SA 4.0
2024-05-02T13:37:14.483Z
@Giskard Consider the linear case, and suppose tycoon A does not know how much exactly tycoon B is going to buy, but has some probability distribution. Then, each vote increases the chances of tycoon A to win, so the value of each vote for A is the increase in expected gains. Does this make sense?
Erel Segal-Halevi
CC BY-SA 4.0
2024-05-02T14:00:59.257Z
As I wrote before, the linear case is a simple all-pay auction which has abundant literature [E.g.. Krishna]. In fact, what you wrote is what happens in the equilibrium of the game where they know the valuations and have to make a bid simultaneously.
Giskard
CC BY-SA 4.0
2024-05-02T14:17:42.750Z
Note that there is no 'equilibrium price', there are bids of stochastic size.
Giskard
CC BY-SA 4.0
2024-05-02T14:21:08.223Z
In case you assume there are many more people than votes the tycoons are willing to buy, and you model the votes as a continuum, there is no difference between the linear and quadratic cases, as the number of votes does not appear in the payoff function beyond what is spent on them, so it is enough to focus on how much will each tycoon spend, with the larger spender winning.
Giskard
CC BY-SA 4.0
2024-05-07T06:30:36.770Z
@Giskard this is not exactly an all-pay auction, since in an all-pay auction, the spending directly determines the probability of winning, and here there is another variable, which is the price per ticket. For example, if there is only one citizen (besides the two tycoons), then I think the market will be more similar to a winner-pays auction, where the highest-valued tycoon will buy the vote and pay (in equilibrium) the second price.
Erel Segal-Halevi
CC BY-SA 4.0
2024-05-07T09:12:58.650Z
Yes, if the number of voters is low the setup is different from the all-pay auction. Hence I think you should *go to your question and edit in all your assumptions and much more information about the model structure*. Otherwise there probably 20 different interpretations with as many answers.
Giskard
<general-equilibrium><competitive-equilibrium><voting>
Economics of tradeable votes
Erel Segal-Halevi
# Economics of tradeable votes

(Asked by: username_0 on 2024-05-02)

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?

**Tags:** general-equilibrium, competitive-equilibrium, voting

> **Comments:**
> **username_1**: Are the people indifferent, does the vote only affect the tycoons?  
> **username_0**: @username_1 for the sake of simplicity, let's assume the other people are indifferent.  
> **username_1**: Is the buying simultaneous, do they know each other's $v_i$? Could you try to define the game better?  
> **username_1**: 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.  
> **username_1**: 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.  
> **username_1**: Re your edit: in the linear case what you are looking at is an all-pay auction with different valuations. Timing and knowledge/information structure determines what happens in equilibrium.  
> **username_0**: @username_1 Consider the linear case, and suppose tycoon A does not know how much exactly tycoon B is going to buy, but has some probability distribution. Then, each vote increases the chances of tycoon A to win, so the value of each vote for A is the increase in expected gains. Does this make sense?  
> **username_1**: As I wrote before, the linear case is a simple all-pay auction which has abundant literature [E.g.. Krishna]. In fact, what you wrote is what happens in the equilibrium of the game where they know the valuations and have to make a bid simultaneously.  
> **username_1**: Note that there is no 'equilibrium price', there are bids of stochastic size.  
> **username_1**: In case you assume there are many more people than votes the tycoons are willing to buy, and you model the votes as a continuum, there is no difference between the linear and quadratic cases, as the number of votes does not appear in the payoff function beyond what is spent on them, so it is enough to focus on how much will each tycoon spend, with the larger spender winning.  
> **username_0**: @username_1 this is not exactly an all-pay auction, since in an all-pay auction, the spending directly determines the probability of winning, and here there is another variable, which is the price per ticket. For example, if there is only one citizen (besides the two tycoons), then I think the market will be more similar to a winner-pays auction, where the highest-valued tycoon will buy the vote and pay (in equilibrium) the second price.  
> **username_1**: Yes, if the number of voters is low the setup is different from the all-pay auction. Hence I think you should *go to your question and edit in all your assumptions and much more information about the model structure*. Otherwise there probably 20 different interpretations with as many answers.  

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## 0 answers

