8
U
ic1
- U
ic2
= -[(I
i
– C
1
)
2
-(I
i
– C
2
)
2
]
Scalar (adapted from Rabinowitz and Macdonald, 1995, 463)
U
ic1
- U
ic2
= [(I
i
* C
1
) -(I
i
* C
2
)]
The Euclidean and scalar representations now have more in common than they did in the single
candidate evaluation models above. In fact one is simply the other multiplied by a factor of two
as long as the candidates are symmetrically distributed around the center.
Proof:
Simplifying the Euclidean model when C
1
= -C
2
-[(I
i
– C
1
)
2
-(I
i
– C
2
)
2
] = -[(I
i
2
- 2I
i
C
1
+ C
1
2
) - (I
i
2
- 2I
i
C
2
+ C
2
2
)] = 2 [(I
i
* C
1
) -(I
i
* C
2
)]
Hence if C
1
= -C
2
, then Euclidean = 2*Scalar
The city block variant of the spatial model distinguishes itself from both the Euclidean
and the directional model when two candidates are compared simultaneously. Figure 2 shows
the utility curves for the models when candidates are compared simultaneously. The utility
dimension now has both a sign, indicating which candidate is most proximate, and a magnitude.
All three utility functions are equivalent when the candidates are located at the extremes
(Euclidean = 6*City Block = 2*Scalar, when candidates are at +/-3) and in the middle (all are
equal to zero when both candidates are at zero). As I show above, the Euclidean and scalar
models are equivalent when the candidates are symmetrically located around the center and both
are monotonic in any case. But as the candidates move toward the center, the city block curves
flatten in the tails.
***INSERT FIGURE 2 ABOUT HERE***
These utility functions would be sufficient for modeling purposes if one had data on
exactly which candidate respondents reported supporting with their campaign participation. The
Convention