You could use the Metropolis-Hastings algorithm (or perhaps Gibbs sampling) to sample your sequence, as long as you can calculate the desired probability for the entire sequence (or a function that is proportional to the probability). As I understand you, you wish to let the probability of $a$ coming before $b$ be set to the odds $a\mathop{:}b$. If all such decisions were independent, you could simply multiply the probabilities (derived from the odds) over all the pairs—but of course they're not. This might be a good enough approximation of what you're after, though, depending on what the application is.
Anyway, once you have your probability function $f(s)$ (defined over the entire sequence $s$), an application of the Metropolis algorithm (a special case of Metropolis-Hastings) would be something like this:
- Start with an arbitrary ordering $s$.
- Generate a different ordering, $s'$, for example by randomly swapping two elements.
- Compute the acceptance ratio $\alpha = f(s')/f(s)$.
- If $\alpha \geq 1$, accept $s'$, that is, use it instead of $s$ for the next iteration. Go to 2.
- Otherwise, accept $s'$ with a probability $\alpha$. Go to 2.
And then you let this run for “a while”—exactly how long would depend on the length of your sequence (though the number of iterations should be on the order of the autocorrelation of the Markov process; you can probably find more details on this in a good book on Monte Carlo methods).
