Beet
Atlas Star
Posts: 28,914
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« on: February 26, 2006, 02:57:25 PM » |
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Oh fun, game theory.
Let O = {A, B, C, D} be a set of possible policy outcomes, S = {set of states}, and Di = {0, 1} be a decision set, where i represents state i.
Let F(x)i = Ui, where
(1) U is the total benefit gained by state i, measured in terms of the attention that it receives from presidential campaigns, and, by extension, from first-term presidents and political parties.
(2) F(x)i is a function with a domain as the set of possibly policy outcomes O.
Define A = Republican candidate is the winner, and the Republican candidate is highly attuned to the needs of the battleground states B = Republican candidate is the winner, and the Republican candidate is highly attuned to the needs of all the states C = Democratic candidate is the winner, and the Democratic candidate is highly attuned to the needs of all the states D = Democratic candidate is the winner, and the Democratic candidate is highly attuned to the needs of the battleground states
Suppose that there are 20 states i=1...5 have a function defined such that F(A) > F(B) > F(D) > F(C)
i=6...10 have F(x)i defined such that F(B) > F(A) > F(C) > F(D)
i=11...15 have F(x)i defined such that F(D) > F(C) > F(A) > F(B)
i=16...20 have F(x)i defined such that F(C) > F(D) > F(B) > F(A)
Now define Di = 0: state i DOES NOT award its votes to the popular vote winner. Di = 1: state i DOES award its votes to the popular vote winner.
Now define an aggregate state decision function
Sum(i=1...n=20) {G(X)i} = O{P(A),P(B),P(C),P(D)}
Where G(X)'s domain is defined over D {0, 1} and its range is a probability distribution of the set of outcomes. Since F(X)i takes domain of O, we can simply substitute F(X)i on the right-hand-side to obtain the aggregate state decision-->utility function
Sum(i=1...n=20) {G(X)i} = Ui
It is easy to see that P(A) and P(B) are increasing functions in Sum(i=11...n=20) {G(X)i}, in the sense that, if the Democratic candidate were to lose the popular vote but win the electoral vote, outcomes may be shifted from P(C) to P(B).
On the other hand, the Republican candidate's popular vote basis becomes evenly spread across the country, so Democratic states still prefer P(B) over P(A). The same goes for Republican states with Democratic Presidents for Sum(i=1...n=10) {G(X)i}.
Furthermore,the likelihood of the popular vote and electoral vote going in different directions, while possible (and strong in our current memory from 2000) is extremely small, it having occured only once in 112 years, and then under heavy dispute. There are well documented theoretic reasons for this that apply irrespective of how close an election is.
It is also easy to see that P(B) and P(C) are increasing functions in Sum(i=1...n=20) {G(X)i}, and they take on significant values as states that have more electoral votes choose D = 1. For example, if CALIFORNIA were to adopt D = 1, the Democratic candidate's guarantee-victory threshold under the criteria of winning the electoral vote but not the popular vote increases from 270 EVs to 325 EVs, causing a substantial shift in the Democratic candidate's calculus towards F(C) over F(D). The same goes with TEXAS on the Republican side, shifting his calculus from F(A) toward F(B). The probabilities here are NOT extremely small.
Since battleground states have a strict preference F(A) > F(B) and F(D) > F(C) and Sum(i) {G(X)i} is nonincreasing in utility otherwise, they have a clear preference for D = 0.
Safe states on the other hand are confronted with a conflicting set of factors. They strictly prefer F(B) > F(A) and F(C) > F(D), but D = 1 is nonincreasing over the partisan probabilities. If any safe state chooses D = 0 or D = 1, therefore, they are not maximizing their potential utility. The safe states' ideal would be all-states > battleground-states, yet at the same time without punishing their party relative to the other party.
I argue this is a classic collective action problem easily solved by contract theory. Suppose that New York and Texas, for example, both have roughly an equivalent number of electoral votes. The legislature of New York can then act unilaterally and pass a bill awarding its votes to the popular vote winner on the condition that Texas does the same, or on the condition that a list of named states does the same, or on the condition that states satisfying certain characteristics do the same. Texas then faces a strict preference with no risk to reciprocate by passing a similiar bill of its own, establishing a collective benefit for both states without sacrificing partisan preferences.
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