Game theory explains why AOC and other progressives should reject Nancy Pelosi and tank the Senate's bipartisan infrastructure bill

• The standoff between progressive and centrist Democrats resembles the classic prisoner's dilemma of game theory.
• Research has shown that the ideal strategy for this situation is to cooperate until you're betrayed.
• Centrists reneged on the plan to move both bills together, so game theory suggests progressives are right to threaten their own defection.

Democrats are at an impasse in negotiations over a pair of massive spending bills, and a classic problem in game theory can can clear up the nature of their squabbling.

The two bills at stake are a $1 trillion bipartisan, roads-and-bridges infrastructure deal favored by the centrist wing of the party and a $3.5 trillion social spending bill backed by the progressive faction.

Shortly after the bipartisan infrastructure bill passed the Senate in June, the Democratic leadership brokered a deal to couple it with the broader social spending reconciliation bill, which includes many of the progressives' biggest priorities. In other words, the two sides of the party promised their pet bills would move forward in tandem.

A vote on the bipartisan deal in the House of Representatives is scheduled for Thursday, but progressives like Rep. Alexandria Ocasio-Cortez and Sen. Bernie Sanders have made rumblings about opposing the bill in recent days after Speaker Nancy Pelosi appeared to uncouple the two spending packages under pressure from the centrist wing of the party.

The progressives' change of heart makes sense if you've ever played a board game - it's a classic strategy for a fundamental problem in game theory.

The prisoner's dilemma has 4 possible outcomes

There are basically four possible outcomes in a situation like this: you win and your opponent loses; you lose and your opponent wins; some compromise occurs where you both win; or, you both lose.

A possible ranking of those outcomes for both the centrists and the progressives might look like the following: Passing a complete version of one side's bill, without any compromise from the other side, would likely be the best outcome. Second-best would be a pair of compromises, in which your side gets most of what they want while the other side gets most of what they want. The third-best option for either side would be neither bill passing at the moment (which would theoretically leave open the option of ongoing negotiations). And the worst option would be the other side's bill passing while you get nothing and lose any further leverage.

That set of incentives leads to the classic game theory problem of the prisoner's dilemma. You have two parties who have to individually choose whether to cooperate with or betray the other. Both cooperating leads to a pretty good outcome for both, while both betraying leads to a pretty bad outcome for both. However, one side betraying the other gives them the best possible outcome while securing the worst outcome for their counterparty.

Given that basic structure, the logical strategy is to always betray your opponent

By betraying your opponent, you set yourself up to avoid the worst possible outcome. If your opponent sticks to the deal, you end up with the best outcome, but if they also betray you, at least you got some of what you wanted. That leads to the somewhat vexing conclusion that the most "stable" outcome of the game involves both parties betraying each other, even though they'd both end up with a better outcome by both cooperating.

A classic pop-culture example comes from the 2008 Batman movie "The Dark Knight." In the climax of the film, the Joker sets bombs on a boat of prisoners and a ferry of commuters, and gives the detonator for the opposite boat to each group. The nihilistic villain's hope is that both groups will blow the other up, but the people of Gotham City choose to do the right thing and cooperate, leaving both boats floating.

However, that conclusion that you're always better off betraying your opponent only works if you're in an isolated, one-time situation. Intuitively, if you're dealing with the same counterparty multiple times - as the two sides of the Democratic party are - there might be a stronger incentive towards cooperation.

This informs Progressives' 'tit-for-tat' approach

In 1980, a team of political and computer scientists tested out a variety of strategies for the iterated prisoner's dilemma, running a tournament simulating repeated encounters like the one above. They found that the most effective strategy when you keep running the prisoner's dilemma with the same people over and over again is what they called "tit-for-tat."

That strategy amounts to being friendly, but not a pushover: you begin by cooperating with a new opponent, and then on subsequent rounds, doing whatever they did on the previous round. Essentially, you keep cooperating with a cooperative partner, mutually gaining the modest reward for doing so, and then punish players who defect against you.

It looks an awful lot like the progressives have adopted a tit-for-tat strategy in negotiating over the spending bills.

Both sides initially agreed to the plan to vote for the bills in tandem, suggesting a happy mutual cooperation outcome in our prisoner's dilemma. But in late August, a number of centrists balked at the plan, instead pushing to pass the bipartisan infrastructure bill before considering the progressives' reconciliation bill.

That move by the centrists seems to represent a defection from the deal. Following the logic of tit-for-tat then, the progressives have no choice but to similarly defect and threaten to vote down the bipartisan bill. And so, the progressives balking at this week's vote are just doing what game theory tells them to.