One used to think the world was an inconvenient place for those who liked their reality in black and white. Not any more. Voters everywhere are increasingly adopting entrenched positions. And yet, a simple game theory model shows that every democracy needs a little bit of disloyalty to survive. 

The game is adapted from a paper by Beatriz Magaloni in the American Journal Of Political Science and is played between three parties—the ruling party, a principal opposition party, and a third party. The model examines the incentives of the ruling party to carry out electoral fraud, with fraud being defined as the stealing of votes from the principal opposition party. The third party in a political contest is generally reviled as a spoiler. But the analysis shows, under some conditions, it may play an important role in ensuring free and fair elections.

While all parties have perfect information about the real nature of the election, voters do not know if fraud has taken place. They draw their conclusions from the decisions of the opposition parties to oppose or accept the election results. Given that there are two opposition parties, voters will have different responses when one opposes the election versus when both oppose. Successful opposition is often a matter of effective coordination and the model captures that by dividing the supporters of a party into diehards who never leave (these days referred to by the curious appellation of bhakts), and a floating population consisting of radicals and moderates. Radicals are convinced fraud has taken place if even one opposition party objects to the election. Moderates are only willing to accept that fraud has taken place if both opposition parties object. 

There are complicated rules about when radicals and moderates will abandon their party and who they will go to. For example, radicals will leave one opposition party for the other when their original party does not object to the election but the other one does. Think of these as people inclined to believe that the world is corrupt. Moderates, on the other hand, will leave one opposition party for the other if their original party is the only one opposing the election. These are people who prefer to believe that systems are in good order and disapprove of hasty actions based on “insufficient evidence". 

The game is played over time. The ruling party makes its decision based on the short-term gains from fraud versus the long-term losses from the possible defection of its floating voters. The opposition parties have to assess the shift of votes resulting from their decision to oppose or accept the election given what the other parties are doing.

Note that since the voters do not know whether fraud has actually taken place or not, given a set of actions by the opposition parties (example, principal opposition party objects and third party accepts), the pay-offs are the same with or without fraud with the sole difference being that with fraud, some votes are transferred from the principal opposition party to the ruling party. It seems obvious that the ruling party will choose fraud every time, a rather gloomy prognostication for democracy. 

And, indeed, this is what the analysis almost always yields. After observing the manner in which the election is conducted, when the opposition parties move simultaneously, or when the principal opposition party moves before the third party, there is no distribution of voters between moderates and radicals across the three parties that could result in an equilibrium in which elections are fair. However, when the third party is called upon to declare the elections fair or fraudulent before the principal opposition party does so, it is possible, under certain conditions, to get an equilibrium path of play in which the ruling party conducts free elections, and both the opposition parties accept the verdict. 

The enabling conditions show that democracy requires a reasonably large floating population of voters to survive. Further, there should be an appropriate distribution between moderates and radicals. For instance, the moderate voters supporting the principal opposition party should outnumber the radicals in the other two parties taken together to ensure the principal opposition party does not choose to reject the election after the third party has accepted. On the other hand, the moderates in the third party should not be much higher than the radicals in the principal opposition party, otherwise the principal opposition party would want to accept an election that the third party has opposed in order to get the votes of the moderates of the third party. This distribution of voters ensures a minimum level of coordination among the opposition parties. We also require that the floating population of supporters of the ruling party be high enough to incentivize both opposition parties to dislodge the floating population from the ruling party. Finally, the floating population should be higher than the gains to the ruling party from rigging elections. 

The third party is able to bring about a good equilibrium with fair elections because it is called upon to adjudicate on the election before the principal opposition party. Its standing stems from the fact that, unlike the principal opposition party, it does not gain or lose any votes on account of fraud. Neither does it gain the floating population of the ruling party if both opposition parties reject the election. However, a fair election is not the only outcome possible, even if the conditions on voter types are met, indicating that the task of protecting our democracy is indeed serious business, one that our Election Commission has performed successfully for a number of years. 

The political polarization index developed by Marina Azzimonti shows a steep increase in polarization since the Great Recession of 2008. The hardening of voter positions poses a far greater threat to a democracy than the new technologies for tallying votes. 

With inputs from Srijit Basu.

Rohit Prasad is a professor at MDI, Gurgaon, and author of Blood Red River. Game Sutra is a fortnightly column based on game theory.