Match Quality Graph There's a Forbes article about student-college match quality that contains an interesting graph based on a working paper by Eleanor Dillon and Jeff Smith. 
The description reads:

The chart below, which represents individuals who attended college in
  the early 2000s, details how well students in various ability
  quartiles (measured by a broad-ranging aptitude test) are matched to
  college quality quartiles. Perfect matching would place 25% of the
  student population in each circle along the diagonal, with no students
  in the other circles.
About 36 percent of students are appropriately "matched" to colleges
  based on ability. About 36 percent of students are appropriately
  “matched” to colleges based on ability. Students in both the top
  ability quartile and the top college quality quartile represent 11% of
  the overall student population, or 44% of all students in the top
  ability quartile. Overall, 36% of students attend a college in their
  corresponding quality quartile, and 77% attend a school within one
  quartile of their ability group.

Here's the Forbes graph:

I tried improving on this chart since I find circular areas hard to compare and I wanted to see not just the absolute percentage, but the marginals as well. Here's my attempt:

I did not bother to do the match shading (I am not sure there's a good metric), but I still find my graph unsatisfactory. It requires a lot of arithmetic to get insights out.
How would you display this data?
    cq   sa    pct  
     1    1   10.6  
     1    2    7.1  
     1    3    5.2  
     1    4      3  
     2    1    6.8  
     2    2    6.5  
     2    3    6.4  
     2    4    4.3  
     3    1      4  
     3    2    6.9  
     3    3    7.4  
     3    4    7.4  
     4    1    2.1  
     4    2    4.6  
     4    3    6.5  
     4    4   11.4  

 A: It seems to me it is worth noting that these are, in essence, agreement data.  We should use a plot designed for displaying and assessing such data.  The plot I'm most familiar with for this purpose is Bangdawala's agreement chart.  You can find it discussed here:  


*

*Bangdiwala, SI, & Shankar, V (2013).  The agreement chart.  BMC Medical Research Methodology, 13:97.  

*(See also Bangdiwala's B.)  


In R, you can create one with ?agreementplot in the vcd package.  (I know it can be done in SAS using the AGREE option in PROC FREQ, and I'm sure there are Stata macros for it as well.)  
library(vcd)
d = read.table(text="cq   sa    pct  
...  
4    4   11.4", header=T)
tab = xtabs(pct~cq+sa, d)

windows()
  agreementplot(tab)

## you can also get the Bangdiwala B agreement statistics: 
print(agreementplot(tab))
# $Bangdiwala
#           [,1]
# [1,] 0.1352742
# 
# $Bangdiwala_Weighted
#           [,1]
# [1,] 0.5426176
# 
# $weights
# [1] 1.0000000 0.8888889


Some things to note from this plot are:  


*

*The rectangles lie along the red diagonal.  This means that neither measure is systematically higher or lower than the other.  (That is, neither is a biased measure of the other.)  

*The heavy black rectangles are a fairly small proportion of the area of the outer rectangles, indicating that the matching of students to schools is far from perfect.  

*(The gray rectangles represent partial—'off by 1'—agreement.)  

A: I think the biggest weakness of the original is that the color intensity dominates our perception even though it is practically meaningless in that it duplicates information already represented by the positions. I imagine that leading to your dissatisfaction and search for alternatives.
Here is a version using color intensity for counts instead of using size for counts.

It does a decent job of showing the counts falling off from the diagonal and that quartiles 2 and 3 are not that different. Neither color not area is very easy to perceive accurately, but I switched from area to color for percents because it's more "glance-able" for pattern recognition. I used discrete colors instead of continuous colors to mask what I judged to be meaningless variations.
Looking at marginals, I'm finding it easier to see patterns in separate bar charts than in overlaid lines -- not sure why.


With some effort, it may work to append the bars charts to two edges of the heat map for a true "marginal" effect.
Lines do make it easier to think in terms of what happens when the X variable changes from over value to the next.

The data seems too coarse to go very far with a visualization.
A: I think the current charts show the data pretty well. The stacked bar chart has such nice progressions it is easier to follow along than most stacked bar charts. The original bubble chart shows that there is a reasonable correlation between the two (I calculated it at 0.36). 
One alternative is a dot plot/line chart.

One thing I like about this is the ability to de-trend, and then plot the same lines. (So you can see deviations from expected, as oppossed to simply the bivariate percentages.) I'm not sure what a reasonable model is though. A default model are the residuals from the cross-tab table, in this case it just replicates the original chart though.

It strikes me (both from the original bubble plot and this dot chart) that there is more binning at the extremes, but I'm not sure of a way off-hand to quantify that.
There are always more fancy things you could do (like a network graph that has the two quartile sets as nodes and shows weighted lines). But I think these examples are basically all you need.
