= Pivotal Politics Model =

The '''Pivotal Politics Model''' is a social choice theoretic model seeking to explain why (a) gridlock is common in a modern democratic institution and (b) gridlock is broken by large majorities rather than by margins.

<<TableOfContents>>

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== Formulation ==

Suppose that...
 1. legislators have preferred policy positions that are all linearly related
 2. the status quo policy position (''q'') also exists in this linear space
 3. the only policy position that will ever be brought to a vote is that of the median legislator (''m'')
    * The obvious weak point of the theory, but it can be rationalized in limited contexts. Competition in an open amendment process, for example.
 4. If median legislator's preferred policy positions ''is'' the status quo, nothing would be brought to a vote.
 5. between any two given policy positions, a legislator will always vote in a way that supports the policy position differing least from their own preference
    * a.k.a. rationality

If a simple majority voting rule is used by the legislature, the outcome policy position will be ''m''.

If a filibuster rule is introduced, whereby a policy position must instead pass a cloture vote with some greater threshold, then a filibuster pivot (''f'') is added. Some observations:
 * While a filibuster pivot legislator could exist on both sides of the median legislator, the relations are transitive and linear. Positions could be negated to cause a reversal of ordering, and the relations would be the same.
   * Given this, let's just say ''m'' <= ''f''
 * If the filibuster pivot position is identical to the status quo, the cloture vote will never pass. ''q'' is the outcome.
 * Category 1: If the linear relation is ''m'' < ''q'' <= ''f'', the cloture vote will never pass. ''q'' is the outcome.
 * Category 2: If the linear relation is ''q'' < ''m'' <= ''f'', the cloture vote will certainly pass and ''m'' is the outcome.
   * This model disavows the existence of parties, but given their existence, can this theoretical category really exist? Functionally the majority party would shifting policy from a preferred status quo towards the minority party's preference.
 * Category 3: If the linear relation is ''m'' <= ''f'' < ''q'', the legislator at the filibuster pivot point has the deciding vote. Whichever policy position they prefer will be the outcome.
 * Would the median legislator make concessions towards the filibuster pivot, to ensure the cloture vote passes?
   * Certainly not if the status quo is preferable to a concessionary policy position. So (1) is inexorably in gridlock.
   * Very possible in (3). The actual legislative outcome can be somewhere in [''m'',''f'']. If there is no possible proposal, the filibuster pivot point must be very close to the status quo.

Going by this logic, policy change is only possible if the status quo is far removed from the median legislator's preferred policy position. Enough that a supermajority up to and including the filibuster pivot legislator supports the proposal (either as-is or given some concessions).

If a veto rule is introduced, whereby a non-legislator can halt a legislative outcome, then a veto power pivot position (''p'') is added. All assumptions made about legislators and their policy positions are also made of this veto power. Some observations:
 * If the veto pivot position is identical to filibuster pivot or median legislator positions, there should be no veto. Actors will identical preferences have already demonstrated preference for the proposal.
 * If the veto pivot position is identical to the status quo, there should be a veto, because no legislative proposal can be preferred over the status quo.
 * If the status quo dominates, there can be no veto process.
 * The only theoretical categories that matter are (2) and (3) if and only if the filibuster pivot isn't very close to the status quo.
   * ...bearing in mind that (3) may also see the median legislator making concessions towards the filibuster pivot legislator.
   * So consider (2) specifically:
     * Category I: If the linear relation is ''p'' <= ''q'' < ''m'' <= ''f'', the policy will be vetoed. The outcome is ''q''.
     * Category II: If the linear relation is ''q'' < ''m'' <= ''p'' <=> ''f'' (i.e. the relation between ''f'' and ''p'' do not matter), the policy will not be vetoed. The outcome is ''m''.
     * Category III: If the linear relation is ''q'' < ''p'' <= ''m'' <= ''f'', the veto power has the deciding 'vote'. Whichever policy position they prefer will be the outcome.
     * Would the median legislator make concessions towards the veto pivot, to ensure that a policy is not vetoed?
       * Certainly not if the status quo is preferable to a concessionary policy position. So (I) is inexorably vetoed.
       * Very possible in (III). The actual policy outcome can be somewhere in [''p'',''m'']. If there is no possible proposal, the veto power pivot point must be very close to the status quo.
   * So consider (3) specifically:
     * Category A: If the linear relation is ''p'' <=> ''m'' <= ''f'' < ''q'', the policy will not be vetoed. The outcome is the legislative outcome (i.e. somewhere in [''m'',''f'']).
     * Category B: If the linear relation is ''m'' <= ''f'' < ''p'' < ''q'', the veto power has the deciding 'vote'. Whichever policy position they prefer will be the outcome.
     * Category C: If the linear relation is ''m'' <= ''f'' < ''q'' <= ''p'', the policy will be vetoed. The outcome is ''q''.
     * The median legislator may have already made concessions toward the filibuster pivot to pass the cloture vote. Would they make further concessions towards the veto pivot, to ensure that a policy is not vetoed?
       * Certainly not if the status quo is preferable to a concessionary policy position. So (C) is inexorably vetoed.
       * Very possible in (B). The actual policy outcome can now be re-stated be somewhere in [''m'',''p'']. If there is no possible proposal, the veto power pivot point must be very close to the status quo.

This logic simply amends the above conclusion to say that the veto power's approval is also required for policy change.

If a veto override rule is introduced, whereby the veto power can be overcome with some even greater threshold in the legislature, then an override pivot position (''v'') is added. Some observations:
 * The only theoretical categories that matter are (I), (III) if and only if the veto power pivot ''is'' very close to the status quo, (B) if and only if the veto power pivot ''is'' very close to the status quo, and (C).
   * Among the categories that matter, the filibuster pivot was ineffectual. The policy outcomes are always either ''m'' or a concessionary policy position somewhere in [''m'',''p''].
     * Therefore ''f'' can be discarded from the model if veto override pivots are in play.
   * For all the same reasons that we say ''m'' <= ''f'' is true, so too is ''m'' <= ''v'' true
   * So consider (I) specifically:
     * The linear relation must be ''p'' <= ''q'' < ''m'' <= ''v'', and the veto will be overriden. The outcome is ''m''.
   * So consider (III) specifically:
     * The linear relation must be ''q'' < ''p'' <= ''m'' <= ''v'', and the veto will be overriden. The median legislator will no longer offer concessions to the veto power's position. The outcome is re-stated as ''m''.
   * So consider (B) specifically:
     * If the linear relation is ''m'' <= ''v'' <= ''p'' < ''q'', the veto override pivot legislator has the deciding vote. Whichever policy position they prefer will be the outcome.
     * If the linear relation is ''m'' < ''p'' < ''q'' <= ''v'', the veto will not be overriden. The outcome is ''q''.
   * So consider (C) specifically:
     * If the linear relation is ''m'' <= ''v'' < ''q'' <= ''p'', the veto override pivot legislator has the deciding vote. Whichever policy position they prefer will be the outcome.
     * If the linear relation is ''m'' < ''q'' <= ''v'' <=> ''p'', the veto will not be overriden. The outcome is ''q''.
   * The median legislator may have already made concessions toward the filibuster pivot or veto pivot. Would they make further concessions to the veto override pivot?
     * Concessions became entirely unnecessary in (III).
     * In (B), the median legislator can make concessions to either the veto power pivot or the veto override pivot.


Altogether now, policy change is only possible if the status quo is far removed from the median legislator's preferred policy position. Enough that a supermajority up to and including the filibuster pivot legislator supports the proposal (either as-is or given some concessions). If the veto power does not support a proposal, the supermajority needs to be up to and including the veto override pivot legislator.
 * Proposals must favor the filibuster pivot legislator. It may be that the median legislator's policy position is already preferable to the status quo for this pivot legislator, or it may be that concessions were made between the two.
 * If the veto power disapproves of the proposal, proposals must favor the veto override pivot legislator.
 * There should only be concessions made to the veto power pivot if their support can be bargained for more 'cheaply' than the veto override pivot legislator.

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== History ==

This model was developed by Keith Krehbiel.



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