Though the other answers are correct, in the sense that they make true and relevant statements, there are some subtle points of language design here that haven't been expressed yet. Many different factors contribute to the current design of the conditional operator.
First, it is desirable for as many expressions as possible to have an unambiguous type that can be determined solely from the contents of the expression. This is desirable for several reasons. For example: it makes building an IntelliSense engine much easier. You type x.M(some-expression.
and IntelliSense needs to be able to analyze some-expression, determine its type, and produce a dropdown BEFORE IntelliSense knows what method x.M refers to. IntelliSense cannot know what x.M refers to for sure if M is overloaded until it sees all the arguments, but you haven't typed in even the first argument yet.
Second, we prefer type information to flow "from inside to outside", because of precisely the scenario I just mentioned: overload resolution. Consider the following:
void M(object x) {}
void M(int x) {}
void M(string x) {}
...
M(b ? 1 : "hello");
What should this do? Should it call the object overload? Should it sometimes call the string overload and sometimes call the int overload? What if you had another overload, say M(IComparable x)
-- when do you pick it?
Things get very complicated when type information "flows both ways". Saying "I'm assigning this thing to a variable of type object, therefore the compiler should know that it's OK to choose object as the type" doesn't wash; it's often the case that we don't know the type of the variable you're assigning to because that's what we're in the process of attempting to figure out. Overload resolution is exactly the process of working out the types of the parameters, which are the variables to which you are assigning the arguments, from the types of the arguments. If the types of the arguments depend on the types to which they're being assigned, then we have a circularity in our reasoning.
Type information does "flow both ways" for lambda expressions; implementing that efficiently took me the better part of a year. I've written a long series of articles describing some of the difficulties in designing and implementing a compiler that can do analysis where type information flows into complex expressions based on the context in which the expression is possibly being used; part one is here:
http://blogs.msdn.com/ericlippert/archive/2007/01/10/lambda-expressions-vs-anonymous-methods-part-one.aspx
You might say "well, OK, I see why the fact that I'm assigning to object cannot be safely used by the compiler, and I see why it's necessary for the expression to have an unambiguous type, but why isn't the type of the expression object, since both int and string are convertible to object?" This brings me to my third point:
Third, one of the subtle but consistently-applied design principles of C# is "don't produce types by magic". When given a list of expressions from which we must determine a type, the type we determine is always in the list somewhere. We never magic up a new type and choose it for you; the type you get is always one that you gave us to choose from. If you say to find the best type in a set of types, we find the best type IN that set of types. In the set {int, string}, there is no best common type, the way there is in, say, "Animal, Turtle, Mammal, Wallaby". This design decision applies to the conditional operator, to type inference unification scenarios, to inference of implicitly typed array types, and so on.
The reason for this design decision is that it makes it easier for ordinary humans to work out what the compiler is going to do in any given situation where a best type must be determined; if you know that a type that is right there, staring you in the face, is going to be chosen then it is a lot easier to work out what is going to happen.
It also avoids us having to work out a lot of complex rules about what's the best common type of a set of types when there are conflicts. Suppose you have types {Foo, Bar}, where both classes implement IBlah, and both classes inherit from Baz. Which is the best common type, IBlah, that both implement, or Baz, that both extend? We don't want to have to answer this question; we want to avoid it entirely.
Finally, I note that the C# compiler actually gets the determination of the types subtly wrong in some obscure cases. My first article about that is here:
http://blogs.msdn.com/ericlippert/archive/2006/05/24/type-inference-woes-part-one.aspx
It's arguable that in fact the compiler does it right and the spec is wrong; the implementation design is in my opinion better than the spec'd design.
Anyway, that's just a few reasons for the design of this particular aspect of the ternary operator. There are other subtleties here, for instance, how the CLR verifier determines whether a given set of branching paths are guaranteed to leave the correct type on the stack in all possible paths. Discussing that in detail would take me rather far afield.