- ...Dynamics2#2
- 2#2 Thanks to James Snyder, Morris Fiorina and
Charles Franklin for comments on early versions of the game model, to Graeme
Bailey of Cornell's Math Department for discussion in the early stages of
work on the empirical model, to Jasjeet Sekhon for discussion of some key
substantive points, and to Diana Richards for comments on several drafts. I
thank Daniel Kheel and Robert Houck for assistance. Kheel's work was
supported by an endowment from Jonathan R. Meigs. Houck's work was
supported in part by Theodore J. Lowi, the John L. Senior Professor of
American Institutions. Data were made available in part by the Cornell
Institute for Social and Economic Research and the Inter-University
Consortium for Political and Social Research. Computing to simulate the
differential equation system was supported in part by Cornell Information
Technologies. Thanks also to Jonathan Cowden, Jasjeet Sekhon and Gregory
Wawro for letting tron, praxis and yoknapatawpha help macht and tempter with
the statistical computing. The author bears sole responsibility for all
errors.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...service.
- The relationship between my concept of
challenger quality and Jacobson's (1990b) challenger quality variable that
measures past experience in office is not clear.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...constant.
- Fudenberg and Tirole (1991, 23-26) discuss the basic
Cournot adjustment process.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...wins.
- As is well
known, quadratic losses in the form of euclidean preferences are often used
in spatial models. Berger (1985) discusses pros and cons of various loss
functions.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...period.
- If 15#15, then the
solution to the first-order necessary condition for equilibrium,
16#16, is 17#17. For 18#18, this solution is a maximum: 19#19. Evaluated at 20#20, the contributor's expected rate of return
21#21 is 22#22.
Solving for 23#23 gives 24#24, so that 23#23 is
the one-period-ahead discount rate at interest rate 25#25. But
26#26 is
the present discounted value of a permanent future income of one unit per
period. Notice that the rate of return in Snyder's (1990, 1197 eq. 1)
contingent-claim formulation corresponds to 27#27. I.e.,
Snyder's concept of the value of favor sold by a candidate (here the
incumbent) corresponds to 11#11, while his concept of the investor
contribution to the candidate corresponds to 9#9.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...contributions.
- The sign of
*r*in 6#6 is negative: 40#40 where 41#41 is nonnegative because 42#42. This negative effect works against the incentive to increase*r*suggested by the term*p r*in*I*. But even here it is important to notice that 43#43 can be negative if*g*>0.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...point.
- If there are
multiple, stable Cournot-Nash equilibria then the players can be assumed to
choose one at random, but so far as I have been able to determine, there is
never more than one stable fixed point.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...argument.
- If the Cournot-Nash equilibrium point is not stable, then
the flows of system (1) do not converge to the point as the
initial errors go to zero. Given any initial deviation from the
Cournot-Nash equilibrium point, the flows wander away from the point.
Compare Fudenberg and Tirole (1991, 351-352).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...HREF="node3.html#eqsys">1).
*MACSYMA*(Symbolics 1991) was used to do 4th-order Runge-Kutta method numerical integration. All simulations were computed using 49#49.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...cycle
- A homoclinic cycle occurs when flows connect one or more saddle
points so as to create a circuit (Guckenheimer and Holmes 1986, 45). The
cycles in region V include three saddle points.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...HREF="node15.html#tabcgame1mix">2.
- One may verify by direct
calculation that neither the incumbent nor the challenger can gain by
unilaterally switching to one of the pure strategies of Table 1,
as long as 54#54 and 55#55. As noted in the
text, reasonable valuations are 53#53.
Because I have not computed payoffs for all (
*g*,*h*) pairs on a fine, bounded lattice (the computing demands are prohibitive), I cannot definitively assert that the equilibrium of Table 2 is unique (compare Fudenberg and Tirole 1991, 34-36), but nothing about the dynamics for the (*g*,*h*) pairs I have simulated would suggest otherwise. Except for (*g*,*h*) pairs with small positive values of*h*in region VI of Figure 1, for which the dynamics of system (1) are highly irregular, there would appear to be no barrier in principle to demonstrating uniqueness by applying the Dasgupta-Maskin theorem that Fudenberg and Tirole (1991, 487-489) review. The computing requirements would be immense, however.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...orbit.
- The orbit is not exactly closed because the values
*g*=.0425 and*h*=.487 are approximate.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...sink.
- For examples and pictures of spirals,
sources, sinks and saddle points, see Hirsch and Smale (1974, 90-96).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...spurious.
- Results produced using a generalized linear
model (McCullagh and Nelder 1989) would also be spurious.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...midterm.
- It may be that special
efforts by the opposition party to recruit strong challengers during the
presidential election year are sufficient to overcome the tendency to have
lower quality opposition party challengers then. Opposition party
politicians may think it would be especially valuable, for instance, to have
a robust legislative majority to support what they hope will be their new
President's early policy initiatives. But one would expect such recruitment
efforts to succeed only when the opposition party is widely believed to be
highly likely to capture the White House, which was not the case in 1984.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...1986)
- The key regression coefficient in Grier and Munger's (1986)
analysis has the correct sign for a conclusion that corporate PACs favored
Republican incumbents but is statistically insignificant.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...1986.
- The 4DH model
log-likelihood exhibits severe global nonconcavity. To find global optima,
I use
*GENOUD*(Mebane and Sekhon 1996; Sekhon and Mebane 1998), an improved version of Michalewicz's (1992) evolution program.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...flows.
- Data from Ohio are excluded from the analysis due to
difficulties encountered in trying to match parts of counties to parts of
congressional districts after 1984.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...person.
- Transfers in each category are district
totals estimated from raw data in Bureau of the Census (1984; 1986-91;
1991) using the procedure of Mebane (1993). ``Social welfare transfers''
includes transfers for public welfare, employment security, health and
hospitals and housing. Per capita amounts are computed first for each
county using population values from Bureau of Economic Analysis (1990) and
then allocated to districts in proportion to each county's share of the
district population, as derived from Bureau of the Census (1985; 1986).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
*P*. - Observation
counts for each type of PAC in 1984/85 (1986/87) are: corporate 162 (150),
except highways spending 92 (101); labor 114 (94), except highways spending
63 (67); non-connected 186 (167), except highways spending 103 (119).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...measuring.
- Vector field plots give no
information about the divergence, because the divergence is a function of
partial derivatives of the vectors rather than of the vectors themselves.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...113#113.
- Using
114#114 rather than
115#115 makes the plots easier to interpret by
reducing clutter.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...reason.
- If 57#57, then 122#122.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...plane,
- See the preceding discussion of embedding.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...increase.
- Let
164#164 be a bounded set of positive volume in the four-dimensional
space of system (1) at time
*t*=0. Let 165#165 be the set of points produced by starting a flow of system (1) at each point of 164#164 and running the system for time period*t*. The flows are said to have decreased (resp. increased) the volume of 164#164 if the volume of 165#165 is less (resp. greater) than that of 164#164.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...set.
- Weibull (1995) reviews applications of
Liouville's theorem to assess stability in multipopulation evolutionary game
models.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...112#112.
- The
order of time is ambiguous in the recovered dynamics, so that it is not
clear a priori whether 107#107 or 106#106 should be
used to estimate the vector field. For the results discussed in the text I
am using 106#106. Using 107#107 reverses all the
results for the divergence test in Table 3 and all the arrows
in Figures 4 and 5.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fri Oct 23 17:45:50 EDT 1998