How Game Theory Might Explain Joe Biden's Win- and His Tepid Support

After a long primary season, it now looks as if Vice President Joe Biden will be the Democratic nominee. From a diverse field, the presumptive nominee is a white man in his seventies—a surprise considering the options and the democratic electorate. Among the democrats once running there were both men and women of color as well as a glut of younger candidates—not least the rising star Pete Buttigieg, the first openly gay man to win a democratic primary.

And yet, Biden nearly swept the nomination after his South Carolina victory. He now stands at 1,435 delegates, just 556 from victory. Biden’s only real competitor, Bernie Sanders, has only 984 delegates and the remaining primaries will make it nearly impossible for him to catch up. More than 58% of the delegates have already been awarded, meaning Sanders would have to win 30% on average for the rest of the primary—a very unlikely feat. As it stands now, Biden has won the Democratic nomination

Pundits and the media have coalesced around a story which explains Biden’s surprising political comeback. After the South Carolina primary, moderate voters nervous about Bernie and unsure about the viability of Buttigieg or Klobuchar fell in line behind the former vice president. Further, Biden’s formidable lead among African American voters bolstered his polling numbers in future states and landed him hefty victories in subsequent primaries. These factors ensured that Biden’s Super Tuesday showing was indeed ‘super’ and that nearly all primaries after were to be easier wins.

Biden Courts Progressives

Now that Biden is the presumptive nominee, he is making moves to demonstrate his viability as a candidate. He is speaking to his executive experience, seemingly trying to remind Americans that he once sat in the White House and knows what it takes. He has been very outspoken about how he would respond to the novel coronavirus, how it would differ from Trump’s response, and why it matters. To solidify this distinction, he wrote an Op-Ed in the Washington Post, castigating Trump’s response and calling for faster, more effective action.

Biden is also courting the progressive wing of the party to broaden his coalition and avoid meeting the same fate as Clinton in 2016. He has amended his climate policy to resemble the Green New Deal and has become more vocal on issues of healthcare and workers’ rights. What is more, he has been vocal about picking a woman as his Vice President. Likely, she will be a woman of color.

Despite these overtures, progressives remain skeptical of Biden. To them, he represents a return to establishment politics- a call to doom the Overton window to its current neoliberal position. In fact, many in the party and in the electorate generally are unsure of Biden.

But why? If Biden is on track to win enough delegates to claim the nomination, why would he not also be the logical pick for democrats? In many ways, Biden’s victory does not make sense—many pundits expected a divided moderate wing and a Sanders victory.

Indeed, Sanders was ascendant in this race. After his 2016 run, the Democratic Socialist began immediately laying the groundwork for his return. To some extent, this strategy paid off with his victory in California, a vital state he lost in 2016. This time around, his support among young people and Latinos sealed the Sunshine State. In fact, both the Sanders campaign and some pundits believed that these groups would turn out in record numbers to deliver the nomination altogether. While the turnout did not match the Sanders campaign’s expectations, they still might have clinched the nomination with a divided opposition (similar to Trump in 2016). But the moderate candidates dropped out before Super Tuesday and threw their support behind the former vice president, blocking Sanders’ major pathway to victory.

The calculus behind this decision, for both the moderate candidates and the voters themselves, is quite peculiar, but not unreasonable. Indeed, it appears that both groups: the moderate candidates and the primary voters, were engaging in rational decision-making. These groups were using rational game theory to understand their options and to make choices about who to support.

The Electorate Game

There are decently strong polling trends to suggest that democrats are voting strategically. In both Maine and Minnesota voter preference shifted immensely depending on who was perceived by voters to be the favorite candidate. Biden and Sanders quickly became preferred in Maine over Bloomberg when voters realized that non-Bernie supporters would have to choose one candidate to prevent a Sanders victory in the state. A similar phenomenon occurred in Minnesota after Klobuchar—who was virtually tied with Sanders— dropped out, where Biden sprang from single-digit polling to front runner in only a week.

This seems to indicate that voters are playing a voting game, where non-Sanders voters want any candidate other than Sanders to win more than they prefer their own candidate to win. This game is a subset of a larger game where most of the electorate thinks any candidate besides Sanders has a better shot at winning the presidency, even if that means they are supporting a candidate who is not their first, or even second, choice. It is the pressure of the second game, imposed on the first game, which is inducing many voters to opt for Biden over a candidate they are enthusiastic about supporting.

Such a game might be represented in Game Theory by the following graph:

We see above that both candidates have an incentive to stay in to obtain a score of 2 if the other drops out. If neither drops out, Sanders wins and hence (according to moderate candidates) Trump wins the presidency (-100 points). However, if both candidates drop out then both get zero, which is not ideal, but is better than -2 or -100. You can do some relatively complicated math to understand how this strategy shakes-out in terms of statistics. But really what you need to know is the matrix above means each player will pick “Drop Out” 49 times out of 50 games— or 98% of the time.

The above game, as you may already know, is the classic Prisoner’s Dilemma. As you can see, both players—moderates #1 and #2 have an incentive to defect. In this case, defecting is voting for a candidate they would prefer over Biden. The matrix offers a higher payoff for defecting if your opponent votes for Biden, as they voted strategically, whereas you voted based on your preference and therefore derived higher utility. If both players vote for their preferred candidate, both get zero as the vote is split and neither of their candidates win, although it is not negative as they voted for their preferred candidate and hence derived a modicum of utility. If both players decide to vote for Biden, both obtain a higher utility than if they had both voted for their own candidate, but less than if only one had voted for their own candidate.

The unfortunate nature of this game is that players are incentivized to vote for their own candidate for the game to reach pareto-optimality (where no one benefits by moving to any other strategy). But in the real world, moderates largely chose to cooperate. This does not mean they were acting irrationally, but that there is something missing from the model. Likely, this variable is the external pressure of the general election. Since the moderates know that this game determines the outcome of a future game, they will alter their choices accordingly. In this case, moderates knew that cooperation would deliver a sub-pareto outcome now, for a more concrete benefit in the future.

Of course, the real situation is more complex than the simple game above. But it gives us an idea of how moderates may have thought about their options and why we saw the party coalesce around a candidate to block Sanders’ nomination—even when Biden was not their favorite choice.

The Candidate Game

For the candidates, a similar game obtains. On Super Tuesday two of Biden’s opponents for the moderates in the party dropped out, causing Biden to surge. These moderates dropped, ostensibly, to bolster Biden’s coalition and to secure the nomination for the moderate wing of the party in hopes of beating Trump.

The calculus was again similar in its goal: to beat Trump in November. As Amy Klobuchar said in the Feb. 7^th debate: “I think we are not going to be able to out-divide the divider-in-chief. And I think we need someone to head up this ticket that actually brings people with her instead of shutting them out.” This was about Sanders’ often virulent and caustic rhetoric toward the 1% and the Democratic establishment. At the time this was meant to bolster her own campaign, but Politico reported that her dropping out on Super Tuesday and swift endorsement of Joe Biden followed the same logic. Ultimately, Klobuchar, Buttigieg, Bloomberg, and other moderates dropped out and endorsed Biden to prevent a Sanders victory—to not lose both the subgame (primaries) and the larger game (presidential election).

While it is primarily in the self-interest of candidates to win the race, the secondary interest of not losing the presidential election is significant. Hence, the moderate candidate perspective is akin to the game of chicken. This game is commonly understood in popular culture, but it also has a Game Theory iteration which can be used to depict the choice facing moderate candidates:

This resembles what we saw in real life, most moderate candidates decided to drop out—especially once they saw the results of South Carolina, which told them Biden was a viable frontrunner. So, to avoid a Sanders victory and a (perceived) Trump victory, moderate candidates did what they thought was rational and cleared the way for Biden to become the nominee.

What Does this Mean for November?

As of March 30^th, there is a 12 point enthusiasm gap between Joe Biden and Donald Trump. Meaning that voters planning to vote for Trump reported higher levels of enthusiasm than did those for Biden. In fact, only 28% of Biden voters claimed to be very enthusiastic to cast a ballot for the former vice president in November.

From the perspective of game theory, this makes sense. Many voters chose Biden not because they wanted him as their candidate personally, but because they saw him as the best chance to beat Trump—and the candidate whom everyone else would vote for to beat Sanders.

Where does that leave us? We now have a candidate with many delegates, but lukewarm reception. By voting for the candidate we thought others would vote for, we have elected a candidate whom everyone thinks can beat Trump, but few are excited to vote for. We can only hope that such an enthusiasm gap does not spell disaster for the general election. The good news is that polls seem to show Biden well ahead of Trump. A crumbling economy and botched early coronavirus response can only improve Biden’s odds. Only time will tell who will win in November, one can only hope this strategic choice was prescient.