The recent parliamentary coalition chatter has generated three important conclusions. There will be no pre-election co-operation between the Liberals and NDP, the two parties will consider co-operation after the election if their seat totals allow them to form a government and they will not formally include the Bloc Québécois in any arrangement. Mix in some constitutional conventions and elementary game theory, and we have a road map to the next election.
Expect the Conservatives to run against the spectre of a red-orange coalition. "Vote for Michael Ignatieff and make Jack Layton finance minister." The Tories would do this anyway, based on the abortive 2008 coalition, and now the attacks will be even more plausible. Even though the Liberals and NDP deny that they would bring in the Bloc, the Conservatives can also attack them on that ground, since Mr. Ignatieff agreed to the 2008 deal with the BQ.
If either the Conservatives (improbable) or the Liberals (very improbable) win a majority of seats, there would be no issue, but the more likely outcomes all involve coalition strategies.
First (unlikely but possible), the Liberals might win a plurality of seats, i.e., fewer than a majority but more than any other party. In that case, Stephen Harper could attempt to stay on as Prime Minister, as Mackenzie King did in 1925 even though the Conservatives won more seats in that election. But such a course seems unlikely in light of recent precedents and statements by Mr. Harper himself about the importance of "winning the election" - i.e., getting the most seats - in order to form a government.
Another scenario (more plausible, according to all recent polls) is that the Conservatives may win a plurality of seats. In that case, Mr. Harper would certainly attempt to continue as Prime Minister, even if his party had lost seats and the Liberals had gained. He has every right to do so, as by convention he remains Prime Minister until he resigns, and he would have no reason to resign unless defeated in the House of Commons.
At that point, the crucial question becomes, how many seats do the Liberals and NDP hold together? If together, they have a majority (155 or more), their position will be dominant as long as they agree to work together, because they can defeat the government on a confidence vote and force the Prime Minister's resignation.
However, Mr. Harper would still have cards to play. As long as he remains Prime Minister, he has what game theorists call the first-mover advantage. For example, he does not have to ask the Governor-General to summon Parliament right away (Joe Clark waited four months after the 1979 election before meeting Parliament).
If he chose to play for time, Mr. Harper could try to break up the Liberal-NDP alliance by making a better offer to one or the other.
A Conservative-Liberal coalition makes intellectual sense because the two parties, under their current leaders, are close on major issues. But it violates the game-theoretic concept of a "minimum winning coalition." Coalitions are formed to benefit the included at the expense of the excluded, so you want to keep the magic circle as small as possible to maximize benefits per coalition member. A Conservative-Liberal coalition, representing more than 200 MPs and more than 60 per cent of the electorate, would be too large to be viable except in a national emergency.
A Conservative-NDP coalition would be the right size, but it would violate another concept of game theory, the "minimum connected winning coalition." In plain English, it's hard to form a coalition when the partners are far apart ideologically, and that certainly describes the Conservatives and NDP. They sometimes agree on specific legislation, but their world views are too far apart for continuing co-operation.
Finally, the Liberals and NDP together might win more seats than the Conservatives, but not a majority. In that case, the advantage remains with Mr. Harper's plurality. The red-orange coalition could form its own government only with the support of the BQ, and we saw how unpopular that was in 2008, when Canadians judged it a political non-starter.
The bottom line: Michael Ignatieff's best chance to become prime minister is for the Liberals to win more seats than the Conservatives, and his second-best chance is for the Liberals and NDP together to win an absolute majority of seats. Otherwise, Mr. Harper will continue in office.
Tom Flanagan is professor of political science at the University of Calgary and a former Conservative campaign manager.