I believe Adam Smith is saying that your description of “the rule” sounds like something which is true of any voting method. I don’t agree, but I must admit that I myself don’t understand what you’re describing in that paragraph. How is the game supposed to work? Is there a more direct explanation of the distribution, rather than just a characterization as a Nash equilibria?
FPTP would be if there weren’t more points awarded for winning by more votes.
Here is an example of an election.
3 prefer A > B > C
4 prefer B > C > A
5 prefer C > A > B
(note, this is a Condorcet cycle)
Now we construct the following payoff matrix for a zero sum game, where the number given is for the utility of the row player:
\ A B C
A 0 4 −6
B −4 0 2
C 6 −2 0
This is basically rock paper scissors, except that the A strategy wins twice as much when it wins as the C strategy does, and the B strategy wins 3 times as much as the C strategy does.
This game’s unique Nash equilibrium picks A 1⁄6 of the time, B 1⁄2 of the time, C 1⁄3 of the time. So this is the probability of the candidates being elected.
Ah, I see. What I was missing from this description:
What is the rule? Take a symmetric zero-sum game where each player picks a candidate, and someone wins if their candidate is preferred by the majority to the other, winning more points if they are preferred by a larger majority. This game’s Nash equilibrium is the distribution.
was understanding that we construct a two player game, not a game where the players are the voters (or even a game where the candidates are the players).
I believe Adam Smith is saying that your description of “the rule” sounds like something which is true of any voting method. I don’t agree, but I must admit that I myself don’t understand what you’re describing in that paragraph. How is the game supposed to work? Is there a more direct explanation of the distribution, rather than just a characterization as a Nash equilibria?
FPTP would be if there weren’t more points awarded for winning by more votes.
Here is an example of an election.
3 prefer A > B > C
4 prefer B > C > A
5 prefer C > A > B
(note, this is a Condorcet cycle)
Now we construct the following payoff matrix for a zero sum game, where the number given is for the utility of the row player:
\ A B C
A 0 4 −6
B −4 0 2
C 6 −2 0
This is basically rock paper scissors, except that the A strategy wins twice as much when it wins as the C strategy does, and the B strategy wins 3 times as much as the C strategy does.
This game’s unique Nash equilibrium picks A 1⁄6 of the time, B 1⁄2 of the time, C 1⁄3 of the time. So this is the probability of the candidates being elected.
Ah, I see. What I was missing from this description:
was understanding that we construct a two player game, not a game where the players are the voters (or even a game where the candidates are the players).