Gary Johnson, Voting Third-Party, and Game Theory
Libertarian presidential candidate Gary Johnson is looking to shake up the 2016 election, as he currently rivals Ross Perot for the largest third-party campaign since 1992. Johnson, a former Republican governor of New Mexico, seeks to appeal to voters down the aisle as a balanced, yet experienced candidate. Coupled with his vice presidential pick, former governor of Massachusetts Bill Weld, the two claim they are a voice of reason in an unusually hectic election season. As former Republican governors of blue states, the ticket seeks to appeal to a fiscally conservative and socially liberal platform that is becoming increasingly popular with millennials (Politico). According to a recent Quinnipiac poll, 62% of voters want to see Gary Johnson in the debates (Quinnipiac). His main weakness besides name recognition is that many voters see a third-party vote as a waste. Why vote Johnson when you could have a (slightly more) tangible effect on the election by voting for your pick of the lesser of two evils?
Alternative PAC, one of the main super PACs for the Johnson/Weld campaign, created the website Balance Rebellion (http://www.balancedrebellion.com/) in an effort to organize these disgruntled Republican and Democrat voters behind the Libertarian ticket. By matching disaffected #NeverTrump voters with similarly disenchanted #NeverHillary voters, the site reassures voters that their vote for Johnson is not a wasted. In a one-to-one match up between Clinton and Trump, users are asked whom they would choose with no other option. Upon being matched, both voters are urged to pledge to vote for Johnson instead of Trump or Clinton. The intuition behind this is such that the “Never X” voters will double the vote for Johnson, instead of cancelling each other out by voting for a mainstream ticket. This increases the perceived utility of a Johnson vote, especially if that is where one’s true policy principles lie.
But how useful is this tactic? By taking advantage of the matching system, devious pro-Clinton or pro-Trump voters can take away a vote from their opponent candidate by breaking the pledge to vote for Johnson on the site. If one breaks the pledge, then not only are they preventing another voter from cancelling their own vote, but they are also increasing the utility of their own vote. If they both keep the pledge and vote for Johnson, they receive some utility for taking away a vote from their non-preferred main ticket candidate. However, if both voters lie, then they go back to the pre-matchmaking condition of cancelling each other’s vote, an undesirable outcome. I break this down to the following game:
| Never Hillary Voter | |||
| Keep Pledge | Break Pledge | ||
| Never Trump Voter | Keep Pledge | 1, 1 | 0, 2 |
| Break Pledge | 2, 0 | -1, -1 |
In this game, the pure strategy Nash Equilibriums (PSNE) are the payoffs (2, 0) and (0, 2) where one voter lies and the other keeps their pledge. A mixed strategy Nash Equilibrium (MSNE) can also be calculated of (p* = ½, q* = ½).
Payoff to Never Trump voter:
Payoff to keep pledge = payoff to break pledge
1q + 0(1-q) = 2q + (-1)(1-q)
q = 2q -1 + q
-2q = -1
q* = ½
Payoff to Never Hillary voter:
Payoff to keep pledge = payoff to lie
1p + 0(1-p) = 2p + (-1)(1-p)
p = 2p -1 + p
-2p = -1
p* = ½
Thus, the probability in mixed strategy of breaking the pledge is fifty-fifty. With game theory in hand, one would be hard pressed to say that Alternative PAC’s matchmaking website offers users anything more than peace of mind for their third-party vote.
That being said, as of this blog post, the Commission on Presidential Debates (CPD) ruled that no third-party candidates made the 15% poll cut for the first debate (NPR). Although Johnson has been within the margin of error in important polls at 13%, and is as high as 19% in some states, the CPD chose to shut him out (Quinnipiac). In comparison, Ross Perot made it into the first debate in 1992 polling at just 8%, and ended with 19% of the general vote (Reason).
Although voting third-party may seem even grimmer under a game theoretic analysis, and although Gary might have not made it to the podium, it still begs the question: are you in?
For a more in-depth commentary on Gary Johnson and game theory see:
http://www.politico.com/magazine/story/2016/09/gary-johnson-swing-election-2016-president-214251
http://reason.com/blog/2016/06/03/gary-johnson-polls-debates-libertarian
The latest Quinnipiac poll:
https://www.qu.edu/news-and-events/quinnipiac-university-poll/national/release-detail?ReleaseID=2375