Boycotting elections is not uncommon: indeed, opposition parties have recently boycotted elections in Bangladesh and the main Islamist opposition party is calling for a boycott of upcoming elections in Algeria. The point of this post is, from a strategic standpoint, why would a party call for its supporters to not only not vote for it, but to not vote at all? 
I will discuss two theories that can justify election boycotts. The first concerns the opposition party’s strength, and the second concerns the ramifications of an election result being overturned. In general, the two are distinguished by whether the boycotting party expects to “win” the election or not.
In both cases, to make the stories succinct, suppose that some proposed “reform” is the main political issue and, without any loss of generality, let’s suppose that the ruling party is proposing the reform and the boycotting party opposes it. (This labeling doesn’t matter, but keeps the language simple.) I’ll start with “opposition party strength” explanation.
I’m So Popular…Nobody Showed Up. For the first explanation, suppose that the opposition party expects that it will lose the election—it suspects its supporters are outnumbered by the other party, and suppose that the ruling party will press ahead with the reform if it believes that (say) 60% of the citizens support the reform.
If the opposition party does not boycott, and the ruling party wins with (say) 62% of the vote, then the ruling party will proceed with the reform. The opposition party loses both electorally and in policy terms.
If the opposition party does boycott, then, while the ruling party will still win, the election result is less informative about the true latent support for the reform. In particular, as opposed to the baseline case—where abstention by a voter is more than likely due to indifference about (say) the reform—each “non-vote” might represent opposition to the reform. Thus, boycotting the election can lead to the ruling party being less certain about the underlying support for the reform and either modifying, or demurring from, the reform. 
Notice that this justification is based on signaling, and the logic is clearest when there is essentially no hope for the opposition party to win the election. If the opposition party might win (i.e., it has nearly the same number of supporters as the ruling party), then it must trade-off the potential increased probability of stymieing the reform (in the case of a loss) against the reduced probability of both winning office and stymieing reform. The second justification is more applicable when the opposition party suspects that the election won’t matter in any event.
It Didn’t Have to Play Out This Way. Sadly, election results are not sacrosanct. Suppose that the opposition party suspects that, if it wins the election, the ruling party might disregard the election result and impose the reform anyway. Such an undemocratic move might lead to various ancillary “bad” things unrest and/or a coup. To keep it simple just suppose that the opposition party prefers the reform to be implemented after being “ratified” (even in a boycotted election) rather than implemented against a contrary election result. In this case, because voting is costly, voting for the ruling party is otherwise distasteful, or for the purpose of recording an implicit score of (non-)support for the reform, the opposition might benefit from boycotting precisely when it suspects it might “win” the election. 
Elections Aren’t Just About Winning. Each of the arguments sketched out above rely on a key characteristic of elections: they aren’t the end of the game. Rather, an election results is always to some degree a signal about the electorate’s preferences about the issues being confronted at that time. Of course, the arguments also highlight how the proper interpretation of the “signal sent” by an election result need not be straightforward: as is usual, the fact that something might serve as a signal can infect the incentives of those sending it (in this case the opposition party) in counterinitutive ways (for example, see this point here).
More generally, the arguments provide two alternate routes to understand the legitimating power of participation. That is, many people understandably say that free, open, and active elections are a foundation of a healthy and legitimate government. This argument is often (to me, at least) based on the idea that people don’t want to participate in something that they don’t feel connected to and/or “served well by.” This isn’t a silly argument—it does have a self-enforcing quality that is reminiscent of equilibrium. But even if one is happy to accept that logic as “just so,” the next step is to examine incentives that logic provides to political actors in pursuit of policy and office.
With that, I leave you with this.
 Especially in multiparty systems (especially with nontrivial electoral “thresholds” for representation and/or public financing), there are clear reasons for a party to call for it supporters to vote for a different party. This phenomenon, known colloquially as strategic voting, is about coordination, and I will note it and set it to the side: strategic voting does not justify abstention unless there is some type of quorum/participation requirement.
 I’ll keep moving, but a moment’s thought suggests that, in some circumstances, this argument also suggests an incentive for the boycotting party to call for a boycott but send some of its supporters to the polls anyway. The details are a bit complicated, and such an incentive (or, “comparative static of the ruling party’s beliefs as a function of the actual turnout”) might not work out in equilibrium, because the ruling party’s inferences get complicated and depend upon what it knows/believes about the opposition party’s gambit in this regard. Nonetheless, it is a neat possibility. TO ME.
 One could embellish this argument quite easily (to account for smaller parties also subscribing to its logic) by having the opposition party increasingly dislike the reform being implemented over larger proportions of votes “against” the reform.