The problem with this question is that the real world answer is probably different than any theoretical answer. Below is my stab at some "real world" nonsense.
1) Assuming that both Foo and Bar have the same market capitalization and no debt, their market caps would be co-integrated (over time, they wouldn't stray too far from each other).
2) Because of liquidity issues, the volatilities (uncertainties) of Bar I and Bar II would probably be higher than the volatility of Foo.
3) Because of voting rights, the volatility (uncertainty) of Bar I would be higher than Bar II.
4) Assuming that the difference in voting rights of Bar I and Bar II aren't too extreme, the co-integration of the market caps of Bar I with Bar II would probably be tighter than item 1) above.
If all of this is true, then a Market Maker would be more interested in making a market with the higher volatility Bar shares (more moving around leaves more room for monkeying around). Also, because of item 4) above, Bar pairs trades could be leveraged higher for the same ASSUMED risk.
The problem with all of this is, this is bait. What you think is going to happen is NOT what will happen. The MM might be better off with some oddball 3-way position and less leverage, or in a less "well defined" situation.
Edit 1 (05/19/2011) ======================================
If you ask 10 MM's, you'll probably get 10 different, but related schemes. The two extremes of these 10 methods might be heavily opposed. A major percentage of the MM's income is the bid/ask spread multiplied by volume. Some MM's might go for higher volume. Some go for higher bid/ask spreads. The bigger MM's might have a long term position inventory while the smaller MM's may clear their inventory by the end of the week (or even by the end of the day). The bottom line is, the bid/ask profits have to more than offset the costs of controlling risk, regardless of the underlying strategy.
In addition, because of the way the problem was described, any MM of Bar I/II will likely be a MM of Foo. Why? Is there a cheaper way to reduce inventory risk than another hedge (again, you're in the bid/ask business)?
A bigger consideration than all of the above is what kind of business is this? Is it a company that can easily explode on the upside, downside, both? How liquid are its options? Is there more than one way to "volatize" your position?
