Bernie Sanders suspended his campaign on April 9th of 2020. He will remain on the ballot.

What happened?

I think that there are some lessons from game theory that can apply here.

Disclaimer: this is not a political article, I have no opinion on any of the candidates below, and none of this is written in support of any candidate over another.

Also, please note two other things: 1) I am not a game theorist by trade and 2) I have simplified things as best as I can, and in that, I might have oversimplified.

If you have a question on politics or would like to run for office, please contact my friend Francis Wilson.

Game Theory of Politics

What is Game Theory?

Game theory is designed to address situations in which the outcome of a person’s decision depends not just on how they choose among several options, but also on the choices made by the people they are interacting with…. Game theory is concerned with situations in which decision-makers interact with one another, and in which the happiness of each participant with the outcome depends not just on his or her own decisions but on the decisions made by everyone.

Source: Cornell, David Easley and John Kleinberg

Game theory is dependent on the interaction effect that our decisions have. The ultimate outcome from this process does not occur in a vacuum. Decisions that are made by one group will impact another.

This is especially true in politics and voting.

So, what do we know about voting?

• A group of people are trying to reach a single decision that is representative of a large group, usually through a ranking process
• Voting is a subjective, as it is difficult to quantify people, policies, and situations

And what do we know about voting systems?

• It takes a complete and transitive individual preference and produces a group ranking
• Most often, it is majority rules, wherein the alternative that is preferred by the majority of voters is ranked first
• Voters can strategically choose candidates, misreporting their top ranked choice to favor a candidate with a better chance of winning
• Voting is subject to unavoidable trade-offs, and will exhibit certain forms of undesirable behaviors

The Bernie Situation

Suppose you’re a voter, and you have a choice of three candidates. There are two key assumptions we will make:

1. We can assume that you can only vote for one candidate
2. We can assume that you have you know how this candidate will perform under certain outcomes

There are three groups of voters. The first group is named Group M, and they have a 25% voter turnout. The second group, Group X, has a voter turnout of 35%. The third group, Group B, has a turnout of 40%. Thus, Group B has the highest voting power and Group M has the lowest voting power.

(Please note: you could have several voter groups and model this situation differently. I chose three groups to be representative of the three candidates)

There are three candidates in the race that the 3 groups are voting in (this is for illustrative purposes, I recognize that it would be either Bernie OR Biden in the race).

1. Bernie Sanders
2. Joe Biden
3. Donald Trump

Group M wants Bernie Sanders to win, with Biden as a second choice. Group X wants Joe Biden to win and is okay with Bernie as a second choice. Neither group wants Donald Trump. Group B wants Donald Trump, would vote for Biden if forced, and does not want Bernie.

Voting Profile for all three Groups:

• Percentage Voting for Group First Choice Candidate = 85%
• Percentage Voting for of Group Second Choice Candidate = 10%
• Percentage Voting for Group Third Choice Candidate = 5%

For example, the M Group is most likely to vote for Bernie, making up 85% of their voters. The M Group only has a 10% makeup for Biden, and a 5% group for Trump. 85% of Group X is going to vote for Biden, 10% for Bernie, and 5% for Trump.

Decision Tree for Voter Groups

If you multiply across the decision tree, you end up with the following results:

Bernie gets 26.8% of the votes. Biden gets 36.3% of the votes. And Trump gets 37% of the votes.

85% of the M Group voted for Bernie, but they only represented 25% of the total voter population. Same logic for the X Group and B Group. The larger turnout of B Group enabled them to win the election, despite the fact that their candidate was last choice for both the M Group and X Group.

When you change the voting profile of the groups (now 80% of Group M votes for Bernie, 15% vote for Biden, and 5% vote for Trump), this logically shifts the results. Biden is the winner because he is the least divise candidate.

Moderate collects votes from either extreme, and that shifts the final results.

Gamifying the Election

Here’s a gamified way to look at the above (skip this part if you aren’t too interested in game theory):

These three tables tell us what would happen if the B Group voted for Bernie, Biden, and Trump, respectively. The name that you see in the box is the name of the person that would win in each situation.

There are a few key things to note from this table:

• When all three groups align on a candidate, that a candidate wins. (Bernie, Bernie, Bernie, as shown in the very first cell in the first table: winner is Bernie)
• When the M group and the X Group both choose one candidate, they outweigh the choice of the B group as 25% + 35% = 60% which is > 40%
• However, when M and X are divided, they concede to the majority, which in this case is whatever the B group chose to vote for

But the B group does not want Bernie. He is their last choice. There is a very low probability of all three groups voting for Bernie.

Here are the payoff profiles for each of the groups, and how they feel when their chosen candidate is selected. It is shown as the payoff that that group receives for (Bernie, Biden, Trump)

• M Group: (2, 0, -2) = (payoff of 2 if Bernie is elected, payoff of 0 is Biden is elected, and payoff of -2 if Trump is elected)
• X Group: (0, 2, -2) = (payoff of 0 if Bernie is elected, payoff of 2 is Biden is elected, and payoff of -2 if Trump is elected)
• B Group: (-2, 0, 2) = (payoff of -2 if Bernie is elected, payoff of 0 is Biden is elected, and payoff of 2 if Trump is elected)

So if Bernie gets elected, the payoff profile for the three groups is (M, X, B) or (2, 0, -2) (Group M gets a payoff of 2, Group X gets a payoff of 0, and Group B gets a payoff of -2). If Biden is elected, the profile is (0, 2, 0). If Trump is elected, the profile is (-2, -2, 2).

These are for illustrative purposes, but please note that Biden has the most “positive” profile, as he nets zero between Group M and Group B. Trump has the most “negative” profile.

Here is how that looks like on the tables that I drew out previously. I have simply replaced the candidates name with their payoff profile. So where Bernie would have won, is now the profile of (2, 0, -2)

Now, we decipher the Nash Equilibrium for each scenario. So for the M Group, we compare across rows. For our X Group, we compare across columns. And for our B Group, we compare across tables.

Beginning with the M group in the B Group: Bernie table, I can see that this group receives an equal payoff of 2 whenever both the X Group and the B Group select Bernie (first column) regardless of what the M group themselves selects.

I have highlighted their payoff in yellow. It’s equal across the board, so they are indifferent among their own candidate selection, because they know both other groups are selecting Bernie.

For the X Group, we compare columns. They are in a similar situation, where they are indifferent among their own candidate selection as long as the M Group and the B Group are both selecting Bernie. This is highlighted in green below.

Now for the B Group, they are already choosing Bernie. They are unhappy. But we have to compare across tables to see where their payoffs are (the page player). So for the first column and first row, we can see that they have an equal payoff of -2 in any of the situations because the X Group and the M Group are already choosing Bernie. This is highlighted in blue below.

Best Responses and Nash Equilibriums

Now we will be looking for best responses and Nash Equilibriums:

Nash Equilibrium: This is not a concept that can be derived purely from rationality on the part of the players; instead, it is an equilibrium concept. The idea is that if the players choose strategies that are best responses to each other, then no player has an incentive to deviate to an alternative strategy — so the system is in a kind of equilibrium state, with no force pushing it toward a different outcome.

Source: Cornell, David Easley and John Kleinberg

The Nash Equilibrium will be the outcome where the groups exist at an equilibrium state – no better, or no worse off than anyone else. This also makes the assumption that each group already knows the voting habits of the others (which doesn’t quite apply to the real world).

Scenario 1) B Group Votes for Bernie: 5% Probability

What is the best response of the X Group to the decision of the M Group if they know that the B Group is voting for Bernie?

So in Table 1, shown above, the X Group has three choices: Bernie, Biden, and Trump. They want Biden to win, but they know that they need to be strategic. They know that the Trump group is planning on voting for Bernie, and if the M Group is also planning to vote for Bernie, they give in – they are indifferent. They’re fine with Bernie.

But if the M Group deviates, and votes for Biden, strategically, the X Group should vote for Biden too. They can overpower the B Group.

However, if the M Group deviates and votes for Trump, the X Group should vote for either Biden or Bernie to avoid Trump getting into office, avoiding a worst-case scenario.

Here is the X Group’s Best Response to the choice of the M Group:

Scenario 2) B Group Votes for Biden: 15% Probability

What is the best response of the X Group to the decision of the M Group if they know that the B Group is voting for Biden?

In Table 2, the X Group still wants Biden to win. Now, they have the majority group, the B Group, on their side. So if the M Group choses to vote for Bernie, the X Group should vote for Biden or Trump to get their desired outcome.

The reason that they would vote against Bernie in this scenario is because they know that the B Group is already voting for Biden – and if X Group deviates from the M Group (25% of votes go to Bernie, 35% votes go to Trump, and 40% go to Biden), Biden will still win. It was a strategic deviation, and that’s why the payoff is the same.

They are indifferent among choices if the M Group choses to vote for Biden. Biden will win regardless. They should vote for either Bernie or Biden if the M Group choses to vote for Trump (same logic as above – deviation from a candidate is essentially a vote for another in some cases)

Scenario 3) B Group Votes for Trump: 80% Probability

What is the best response of the X Group to the decision of the M Group if they know that the B Group is voting for Trump?

Here, the B Group is settled in. They are voting for Trump. How should X Group respond? Same logic as previously. If the M Group choses to vote for Bernie, the X Group should side with them, and also vote for Bernie to avoid Trump. Strategic deviation from a chosen candidate in order to avoid a potential “lesser of two evils” situation.

If M Group chooses to vote for Biden, X Group should vote for Biden. And if M Group votes for Trump, X Group gives a big sigh and is indifferent across all candidates.

Key Takeaways

• Divided voter groups concede to the majority
• Strategic deviation to a second choice candidate from a first choice candidate can avoid a worst case scenario
• A vote for one candidate does not always mean you are voting for that candidate (it might mean you are voting against another)

The Democratic National Convention

But as we know, in the real voting world, voters cannot choose from Bernie, Biden, and Trump on the national stage.

So here is a simulation of the decision making process for a nominee:

We can still make the same basic assumptions:

• The voter group will be divided 85 / 15 with 85% of their vote going to the group’s first candidate and 15% going to the second choice candidate
• M Group represents 45% of the Democratic population, X Group represents 55% of the population, and the B Group represents 5%
• Bernie will get 45% of the votes and Biden will get 55% of the votes

How does this play out? Bernie gets 44.8% of the votes, and Biden gets 55.3% of the votes. This makes sense.

The important part is the payoff that the candidate will receive from each of these groups. Here is a decision tree visualizing the payoff that each candidate will receive from targeting each respective group.

Multiplying across calculates to the following:

Bernie gets a 15.3% payoff from targeting Group M, knowing that he will get 45% of the votes, and knowing that they represent ~34% of the total voting base. Biden gets a 25.7% payoff from targeting Group X, knowing that they represent 46.8% of the total voting population. Both Bernie and Biden get relatively low payoffs from targeting the B Group, or the X Group and the M Group, respectively.

Key Takeaways

• Voter turnout really matters

The Nomination Process

So we know the respective payoffs that the candidates have for targeting each group. But how does that play out in calculating who is going to be nominated?

Scenario 1) Biden Wins Nomination

If Biden gets the nomination, he is theoretically less divisive in policy as compared to Bernie. So the M Group, who normally would have preferred Bernie, shifts into the X Group profile now in support of Biden.

In the least divisive scenario (most positive for Biden), Biden wins with a 55% share of votes against Trump. This assumes that Biden would receive 20% of the M Group vote. The rest of the M Group would be divided into the L Group, a nonvoting group.

In the “most divisive” scenario, Biden receives only 10% of the M Group, with the rest refusing to vote (L Group). That nonvoting group pushes the total win rate down, but Biden remains the winner of the election.

Scenario 2) Bernie Wins Nomination

What if Biden had dropped instead?

Bernie was theoretically more divisive than Biden. A lot of people disagreed with him, but a lot of people also agreed with him. The first simulation is adjusted so that Bernie captures 20% of Group X, with the rest moving to the nonvoting group.

This still provides him a win over Trump, primarily because he is able to capture 95% of this new group’s vote. Unity within his supporting group is something Bernie is good at.

However, in the most divisive situation, 30% of the group becomes nonvoting, and Bernie only captures 5% of Group X. That leads to a Trump win.

And that might have been why Bernie dropped out of the race.

Conclusion

This piece was probably one of the more subjective pieces about game theory out there. I don’t have exact numbers from the election, and I applied some liberty in the modeling process.

However, there are some key takeaways that apply here:

• Divided voter groups concede to the majority group
• Strategic deviation to a second-choice candidate from a first-choice candidate can avoid a worst-case scenario
• A vote for a candidate does not always mean you are truly voting for them
• When two opposing groups can come together, they can sway opinion
• Moderate collects votes from either extreme
• Voter turnout is important

Thanks for reading! This piece was fun to write, and I am open to feedback or questions about the process (I am open to any commentary about anything I write).

Stay safe.

Disclaimer: This was not political commentary.