A Game Theory Explanation of the Anti-Trump Movement

Recently, Mitt Romney encouraged Republican voters to not vote for their first choice candidate, but to vote for the candidate that has the best chance of beating Trump in their given state.  This suggestion prompted some outrage that the Republican establishment was telling voters how to vote.  A more precise point to make is that the Republican establishment was asking voters to change their voting strategy – while this may still provoke some outrage, it is always exciting to see political science theory in action, so indulge me in some hair splitting to show what Romney was saying.

In Game Theory, there are two ways to vote: naive voting and sophisticated voting, or what I will call preference voting and strategic voting, respectively.  Consider the figure below:

Slide1

This figure shows the (greatly simplified) choices for any given voter in any given time, and three hypothetical voters’ preferences.  Voter 1’s first choice is Cruz, second choice is Trump, and third choice is Rubio; Voter 2’s first choice is Rubio, second choice is Cruz, and third choice is Trump; Voter 3’s first choice is Trump, second choice is Rubio, and third choice is Cruz.  For the purposes of this exercise, they vote sequentially: first Voter 1, then Voter 2, then Voter 3.

In preference voting, each voter votes for their first-choice candidate.  So Voter 1 votes for Cruz, then Voter 2 votes for Rubio, and Voter 3 votes for Trump, and Trump wins the election.  This is the more intuitive way to think about elections, and how I think most voters approach the electoral process.

In strategic voting, Voter 1 first considers Voter 3’s choices, branch by branch.  Starting on the left hand side, Voter 3 is choosing between Trump and Rubio.  Since Trump is Voter 3’s first choice, given this decision Voter 3 votes Trump.  Knowing this, Voter 1 then considers Voter 2’s decisions.  Voter 2 knows that Voter 3 will choose Trump, so essentially a vote for Rubio is a vote for Trump.  In other words, Voter 2’s decision is really between Cruz and Trump.  Since Cruz is Voter 2’s second choice and Trump is the the last choice, Voter 2 votes for Cruz, i.e. not their first choice Rubio.  So, Voter 1 now knows that by voting for Cruz, the outcome will really be Cruz.  Now consider the right hand side, where Voter 3 has to choose between Cruz and Rubio.  Since Rubio is ranked higher in Voter 3’s preferences, Voter 3 votes for Rubio.  Knowing this, Voter 2 knows that the true choice is actually between Trump and Rubio.  Since Rubio is Voter 2’s first choice, they vote for Rubio.  Therefore, Voter 1 knows that a vote for Trump is really a vote for Rubio.  With this knowledge on hand, Voter 1 knows that the true choice is between Cruz and Rubio.  Since Cruz is Voter 1’s first choice, Voter 1 votes for Cruz, and the outcome of this mini-election is indeed Cruz.

This game is greatly simplified and makes several assumptions (such as complete knowledge about preferences, constant preferences, and complete rationality), but it does demonstrate Romney and the anti-Trump movement’s point: if you really want to beat Trump, you may have to strategically vote for someone other than your first preference.

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