How to Simulate Elections with a Computer: A Multi-Agent System of Elections

• In Event: Poster Session: Political Methodology

Abstract

The paper shows how computer simulations of one-round majoritarian elections can be performed through a novel computational (iterative) Multi-Agent System of elections informed by canonical game-theoretical (analytical) models of voting.

First, I describe how models of strategic voting in plurality elections like Myerson and Weber (1993) can be implemented computationally, in discrete-time iterative simulations, with poll reports of out-of-equilibrium results (following Fey, 1997). For computational exactitude, all models are derived and implemented with both pivotal probabilities of breaking and making multi-way ties.

In the classical scenario where electors know the electorate size, I extend Palfrey’s (1989) Multinomial pivotal probabilities to also cover probabilities of making ties, and to cover multi-way ties. I offer an efficient algorithm to list all Multinomial pivotal scenarios, and derive the formula that counts those. For when electorate size is assumed to be unknown, that is in the frame of Myerson’s (1998, 2000) Poisson games, I derive novel Poisson pivotal probabilities for multi-candidate elections and an efficient algorithm for their calculation. I also derive Skellam pivotal probabilities that approximate those under some simplifying assumptions.

For all three types of pivotal probabilities, I offer Python code that calculates them in double and arbitrary precision, as well as detailed analysis of how each type of pivotal probability relates to each other and of the runtime performances of each of them and of their algorithms. Also, I show that following the analytical literature, except for really small electorates, the simulation converges to stable (nearly always Duvergerian) equilibria. Importantly, also as the electorate size grows, the equilibria election outcomes become increasingly identical regardless of type of pivotal probability used.

Second, I generalize the main derivations above to also cover elections where constituency magnitude is greater than 1, like in Cox (1994), but extending it to cover multi-way ties and to include pivotal probability of making a tie. I show that in its iterative discrete-time version (when path to equilibria is not abstracted away like in the analytical version of the model), the equilibrium type to which the simulation converges is a cycle that, when averaged, approximately recovers the equilibrium type described in by Cox (M+1 candidates sharing same amount of votes, others having zero).

Third, I show how including pivotal probabilities of making ties and also considering multi-way ties (differently from what analytical treatments usually did, since those were not needed in that framework), allows the extension of the models such that elector agents are also capable of strategic abstention. This means joining, for the first time in the literature, models of strategic voting and strategic abstention in multi-candidate elections. More specifically, I show how not considering the pivotal probability of making a tie or not considering multi-way ties can both lead to the degenerate result of zero turnout. Conversely, including them in the model prevents the problem (asymptotically) and then, a learning-based version of Palfrey and Rosenthal (1985), adapted from Demichelis and Dhillon (2010), can recover the equilibria of those work even with multi-candidate elections.

Fourth, to illustrate the usefulness of the computer simulations, I generate thousands of simulated election results and use those to study the accuracy of well known election forensics tools designed to detect fraud in real-life elections.

Lastly, I point how future steps of work aim at generalizing the model further to cover electoral rules that allow multiple weighted voting with magnitude 1 (e.g. Approval and Borda) and two-round elections.

(work supported by NSF award SES 1523355)

Author

• Fabricio Vasselai, University of Michigan