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Mathematical model of a political process

  1. Apr 5, 2009 #1
    I don't know if this is appropriate for this forum, but I have written a mathematical model of the recent political process and I would be interested in feedback on how I applied my math to an actual physical process. If you kick it back, a suggestion of where I should post it would be appreciated. Don't say lounge because I am not interested in the politics per say but rather in how I developed the model.

    Abstract:
    It is common knowledge that policy decisions affect how likely it is that a candidate will or will not be elected or reelected. A series of mathematical model are developed here to better quantify this effect. Asymmetries in the electoral system models create interesting side affects. After the models are developed, they are applied to the presidential elections of 2000 and 2008.

    How Electoral Asymmetries Can Magnify the Affect of Policy Decisions on the Outcome of an Election.

    A. The Models
    A policy decision will invariably lose a politician votes as well as gain him votes. This can be modeled in a simple manner as such:
    X₁(1-β) - X₁β-X₂(1-α) + X₂α = T for 0 <= β <= 1, 0 <= α <= 1 (1)
    Where
    X₁: Number of votes for Politician 1 in the absence of the policy decision.
    X₂: Number of votes for Politician 2 in the absence of the policy decision.
    β: Percentage of lost votes lost by politician 1 (gained by 2).
    α : Percentage of votes lost by politician 2 (gained by 1).
    T : If this number is positive, Politician 1 wins the election, 0 is a tie and negative is a lost for Politician 1 (a win for Politician 2).

    This model assumes that a vote lost by Politician A will be a gain by Politician B. It also assumes that the entire population cares about the policy decision under consideration, however a percentage (α or β) will actually change their vote with any one policy change. There are many considerations to be taken when voting and any single policy decision will normally only be the final tipping point for a small percentage of the voters.
    This is a rather simple model for what is actually going on under the hood. The final number of voters ending up in any one factor can be thought of as a function of each voter in the population as a whole and a movement variable (κ). The value of κ will vary with each voter. This is because each voter is affected by each policy decision by a different degree. Each voter has a decision variable (D). If the decision variable is less than 0, the voter will vote for candidate 1. If it is greater than 0, the voter will vote for candidate 2. If it is zero, the voter is torn between the candidates. Finally there is a random variable G with an inverse normal distribution centered on 0. The inverse normal is defined as such:
    P(1-Gaussian)
    P: Integer picked to represent the highest value G[κ] can obtain.
    Gaussian: A Gaussian distribution with μ=0 and varying between 0 and 1. The σ of the Gaussian must be determined empirically as is P.
    This random variable G determines how much D will move with any one decision. Kappa (κ) is positive or negative for each decision. It will determine the value of the random variable. G is centered around zero because that is the don’t care point. The user who doesn’t care about a policy decision will have a κ of zero and a G[κ] of zero. As kappa moves from zero either in a negative or positive direction its value increases slowly at first and then rapidly. A large κ represents a very emotional policy for the selected voter and therefore a large influence on the voter’s decision for or against a specific politician. So for each policy decision we have this recursive equation:
    D[n+1] = D[n]+ G(κ) (2)
    Where D[n+1] is the final decision after a policy change and D[n] is the decision before the policy change. If D changes sign due to G(κ) from negative to positive then it increases the size of β, if it changes sign from positive to negative then α increases.
    α = Dcpn/X₂ (3)
    β=Dcnp/X₁ (4)
    Where Dcpn is the total number of voters making the transition from positive D to negative D and Dcnp is the total number of voters making the transition from negative D to positive D.
    Finally X₁ and X₂ can be thought of as linear recursive equations:
    X₁ [n] = X₁[n-1](1-β) + X₂[n-1]α (5)
    And
    X₂[n] = X₁[n-1]β + X₂[n-1](1-α) (6)

    If equation 1 is rearranged, equation 7 can be arrived at:
    (X₁ +2 X₂ α) – (X₂ + 2 X₁β) = T (7)
    From this equation it can be seen that if X₂ >> X₁ (and if α and β are reasonably close in value), then the factor 2X₂α would have a larger effect on the outcome than the factor 2 X₁β. This would cause X₂ to shrink and X₁ to grow. However, if X₁ >> X₂, then X₁ would shrink and X₂ grow. Since α and β often tend to average out over time (reducing their effect), this would mean that the equation would tend to be stable around a value of T=0 for a large number of decisions. This would result in close elections where only a few percentage point would indicate a large movement.
    The basic stability of equation 7 can be explained by the symmetry of the equation. It will always push back when badly out of balance. However, it is not consistent with all observation. While many elections are close (T near 0), this is not universal. There are asymmetries that distort the symmetry of equations 1 and 2.
    There will be two asymmetries explored. The first is caused because not everyone has an interest in policy decisions. Their minds are made up and they quit paying attention. This is especially true in local government where as long as the expected services are maintained, many voters couldn’t care less how the city or county councils vote. This can be modeled by adding the population of indifferent voters (negative for voters opposed to candidate 1, positive for voters in favor of Candidate 1). These indifferent voters can be thought of having decisions variables (D in equation 2) of very large positive or negative values which are not influenced much by any one or even a group of policy decisions combined with very low κ for each policy decision. This can be modeled as the following:
    X₃-X₄+ X₅(1-β) - X₅β-X₆(1-α)+ X₆α = T (8)
    Where:
    X₃: The portion of X₁ who are not influenced by policy decisions.
    X₄: The portion of X₂ who are not influenced by policy decisions.
    X₅: The portion of X₁ who are influenced by policy decisions
    X₆: The portion of X₂ who are influenced by policy decisions
    X₁ = X₃ + X₅ and X₂ = X₄ + X₆
    X₅ and X₆ are modeled in the same manner as X₁ and X₂ in equation 1. X₃ and X₄ can be thought of as constants.
    Rearranging the above equation:
    (X₃+X₅ +2 X₆ α) – (X₄+X₆ + 2 X₅β) = T (9)
    Or:
    (X₁+2 X₆ α) – (X₂ + 2 X₅β) = T (10)
    If X₅ and X₆ remain proportional to the size of X₁ and X₂, then once again a stable point at T=0 would be expected for a large number of decisions if α and β are of more or less equal size (or at least average out). This would happen when X₅ and X₆ pull from the general population (X₁ and X₂) for each policy decision and the size of each X₅ and X₆ are dependent on the size of each X₁ and X₂.
    However, another possibility is that X₅ and X₆ instead of being proportional to X₁ and X₂, merely exchange population between themselves. This would happen if there is a group of activists (X₅ and X₆) with the rest of the population (X₃ and X₄) being indifferent or if a group has their minds made up (X₃ and X₄) and another group doesn’t (X₅ and X₆). Equation 5 can then be rewritten:
    (X₃- X₄) +(2 X₆ α+X₅-2 X₅β-X₆) = T (11)
    If (X₃- X₄) > (X₆ + X₅), then X₆ and X₅ make no difference. Then
    T = (X₃- X₄) + δ (12)
    Where δ is an irrelevant factor due to the size of (2 X₆ α+X₅-2 X₅β-X₆). In this case the voting is decided in advance by the relative sizes of X₃ and X₄, delta (δ) not being large enough to make any difference.
    If on the other hand, X₃ ~= X₄ << (2 X₆ α-2 X₅β), equation 11 should act in much the same manner as equation 1. There should be a stable point around 0 and the election should be close.
    Perhaps of greater interest is what will happen if there is a third-party candidate. In this case equation 1 becomes:
    θ+X₁(1-β) - ζX₁β-λ-X₂(1-α)+ ηX₂α = T (13)
    Where ζ and η are the factors that determine the percentage of voters lost from the system by going to the third party. θ and λ are the factors that determine voters gained by the system by coming back from the third party candidate. This assumes an irrelevant third party candidate. Interestingly enough, there really doesn’t need to be an actual third party candidate for this to take effect. This also models the loss of voters as they become discouraged and drop out or decide that the candidates are not so bad after all and come back into the electoral process.
    Combining with equation 8 results in this equation:
    X₃- X₄+ (Θ+ X₅(1-β)) -ζX₅β- (λ +X₆(1-α))+ ηX₆α = T (14)

    That is quite a mess and it is hard to see how it will shed any light on the electoral process. However, it will be seen in the next section that a few simplifying assumptions can result in equations that shed great light on recent elections.

    B. Using the Equations to Model Recent Elections

    In this section we will use equation 14 with simplifying assumptions to answer the following questions:
    1) What affect (if any) did Bill Clinton’s veto of the late term abortion bill have on the presidential election of 2000?
    2) How did the selection of Palin by McCain affect the outcome of the election of 2008?


    Question 1:
    First assume for the use of the equations that Gore is candidate 1 and Bush is candidate 2.
    Let’s start by looking at the converse. What if Clinton had signed the late term abortion bill? Since Gore was closely associated with Clinton, it is reasonable to have expected liberal voters to have abandoned Gore in favor of Nader. At the same time, some other moderate Republican leaning voters with decision variables (D) near zero would have been pushed over to Gore. Looking at equation 14, the following can be seen:
    1) Θ and λ would be 0 since no voters would be gained by the system. Gore would have only lost votes to Nader. It is very doubtful that any of Nader’s very liberal voters would have been impressed by Clinton signing the bill so they would have not moved back to support Gore. Of course, Bush would not have gained any of Nader’s supporters either because of Clinton’s decision.
    2) ζ would be 0. None of the voters abandoning Gore for this policy decision would have gone over to Bush. After all he is even more conservative on the abortion issue than Gore.
    3) η would be 1. It is doubtful that any voters near the decision variable point of 0 would have suddenly become extreme liberals (voted for Nader) just because Clinton signed the late term abortion bill. Therefore none of the voters lost by Bush would have gone over to Nader (all would have gone to Gore or η=1).
    With the above assumptions, equation 14 becomes the following:
    X₃-X₄+ X₅(1-β) -X₆(1-α)+ X₆α = T (15)
    Rearranging:
    X₃-X₄+(X₅+2αX₆) – (X₆+ βX₅) (16)
    What pops out in this equation is the 2x multiplier effect on α (votes gained by Gore). In other words, β (votes lost by Gore) could be almost twice as large as α and Gore would have still come out ahead assuming equal X₅ = X₆ and X₃ = X₄.
    Given the results of the 2000 election, it can be safely assumed that X₃ was more or less equal to X₄ in the state of Florida (near tie). Since the polls in Florida at the time showed strong support for the late term abortion bill, it is likely that α was larger than β.
    If the 2x α multiplier in equation 16 is combined with the probability that α was larger than β for the late term abortion decision and a large κ value due to abortion’s emotional power, it can be seen that a significant number of votes would have been gained by Gore if Clinton had signed the late term abortion ban, probably enough to throw the election to Gore in the 2000 election.
    Note: Keep in mind that with such a close election (X₃ ~= X₄), many other policy decision changes could have pushed the election one way or another.

    Question 2:

    In this case McCain will be assumed to be candidate 1 and Obama candidate 2.
    First unlike question 1, it must be stated that since X₃ >> X₄ in the 2008 presidential election, none of what I will say below would have thrown the election one way or another. It merely affected the size of Obama’s victory.
    In this case there is no significant third party candidate. Instead, McCain gained voters on his right (his base)by picking up discouraged voters and lost voters on his left to Obama because to the decision to pick Palin for his running mate.
    So what affect did Palin have on the election?
    Given the above, the following can be assumed:
    1) ζ can be assumed to be near 1. Almost all of the votes lost by McCain due to the Palin decision probably went to Obama. Since Palin is very conservative, it is unlikely that her choice discouraged very many conservative voters. Instead she pushed away the more moderate wing of the party – those that would tend to go to Obama.
    2) α can be assumed to be near 0. It can be assumed that very few moderate Obama supporters were influenced to vote for McCain because of the choice of the ultra conservative Palin.
    3) Θ would be assumed to be a significant number. Many conservatives were motivated to vote by Palin’s choice.
    4) λ would be much smaller than Θ, so it can be eliminated in this rough approximation.
    Given this, equation 14 can be rewritten:
    X₃- X₄+ (Θ+ X₅(1-β)) - X₅β -X₆ = T (17)
    Rewriting:
    X₃- X₄+ (θ+ X₅) – (2X₅β+X₆) = T (18)
    Once again, the 2x multiplier is seen in the β factor. This means that the loss of each moderate to Obama cost twice what was gained for each conservative gained from the discouraged pool. So how did it influence the election? Hard to say without hard numbers for θ and β, but the 2x multiplier would seem to indicate that it probably wasn’t a wise move. A choice of a more moderate running mate (smaller β at the expense of a smaller θ), might have been a smarter.

    C. Conclusions

    A number of mathematical models of the effect of policy decisions on elections were developed. From these models two interesting conclusions can be made:
    1) There is a stable point for the number of voters for and against candidates. If all voters are influenced by all policy decisions, then the stable point will often be at 50-50. In this case, a close election is expected. This is one reason why even a small differential in the popular vote can represent a large movement by the voters (for example take McCain and Obama in the 2008 election). Indifferent voters (ones whose minds are made up and will not change) can cause a shift in this stable point.
    2) Losing voters in the center costs more than losing them from a candidate’s base. As was seen in equations 16 and 18, there is a 2x multiplier effect in moving to the center. This happens because when one candidate gains votes at the center, his opponent loses the same number of votes, but when he gains votes from his base, his opponent is unaffected.

    Finally, these models could easily be converted into a computer program with possibly interesting results.
     
  2. jcsd
  3. May 18, 2009 #2
    Last edited by a moderator: Apr 24, 2017
  4. Aug 22, 2009 #3
    Quite impressive! I would contact a professor in the policy or political sciences. This could be publishable in a game theoretic journal in political science.
     
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