As the 12 graduate students in her Applied Game Theory course return from their five-minute break, finishing snacks and stowing cell phones, Monika Nalepa grabs a dry-erase marker to illustrate the concepts of strictly and weakly dominated actions.

“So imagine you’re driving a car or riding a bicycle in Hyde Park,” she begins. The associate professor of political science sketches the situation on one of the writable-surface walls in the first-floor Saieh Hall for Economics classroom. “You’re on a road with two lanes each way and you’re getting to an intersection, and there’s a car in the right lane and you want to go straight.” There’s no indication whether the car in the right lane is turning (he’s a Chicago driver so he doesn’t signal, says Nalepa).

The left lane is clearly the best choice for maximizing speed, Nalepa says, writing out formal notation to show that choosing the right lane would be a strictly dominated action—whether the other car will slow and turn or keep going straight, the left lane will always be fastest.

Nalepa then adds a car to the left lane of her drawing. Now, she explains, only if the right-lane car is turning will the left lane be the faster path through the intersection. She modifies the notation to show that here, choosing the right lane is a weakly dominated action—there are only some scenarios in which there’s a benefit to playing the other strategy—but a rational decision maker will still use the left lane.

As Nalepa continues to clarify strict and weak dominance, formal game theory notation proliferates across the whiteboard. These action profiles, utility functions, and payoff matrices are the foundation of the course; it’s an introduction to game theory, or the study of strategic decision making, for students in political and other social sciences.

Game theory has its roots in economics, devised as a way to logically predict people’s actions based on their individual costs and payoffs and the actual or presumed actions of others. Formal game theory models are increasingly used in the social sciences, says Nalepa, and she wants the students in her class to be able to understand, and question, proofs in academic journals.

Nalepa’s eyes light up when she talks about game theory—it’s a topic that’s fascinated her since she took decision theory as a philosophy undergraduate in Poland. “Basically, when I first started learning game theory, the whole world made sense,” she says. “It’s sort of like an aha moment—when you finally get a tool that allows you to understand a lot of social interactions that have been going on around you.” She proceeded to take every course that professor offered, add a major in sociology, and eventually earn a PhD in political science from Columbia University in 2005. She’s been teaching game theory ever since. She even knew her husband, Suyash Agrawal, JD’02, was special when his first words to her on their first date were about Arrow’s theorem, an important idea in social choice theory.

Nalepa wipes the board clean and starts to talk about voting. She introduces a situation, or game, where there’s an electorate with a large number of voting citizens, a majority of whom prefer Democrats over Republicans. “Now if I’m one of those citizens in the majority, if I prefer D over R,” says Nalepa, “how can I use this vocabulary that we just learned to describe the relationship between voting for D and voting for R? Is one of them weakly dominated by the other?”

A student puts down her pen to offer an answer: “Yeah, voting for R would be weakly dominated from the perspective of a person voting for D.”

“Exactly,” says Nalepa. In this situation, one Democrat pulling the lever for a Republican probably won’t matter—the Democratic candidate will likely win anyway. However, if other Democrats start doing the same, one voter could end up casting the pivotal vote and giving the election to the Republicans, making it an irrational choice. “Given this, don’t you think it would be reasonable to eliminate those weakly dominated actions from consideration?” asks Nalepa, erasing the strategy profiles that would have a voter casting a ballot for a nonpreferred candidate. After all, “who would ever vote for R if they preferred D, right?”

However, a game presented during last Thursday’s class showed there can be a strong incentive against voting at all. “The players are N citizens,” Nalepa said, and half support Party A while the other half support Party B. Players can either vote or abstain, and if a player’s preferred candidate wins or ties, there are payoffs for that player, but there’s also a (smaller) cost associated with voting, said Nalepa, like taking the time to go to the polls.

In the first scenario, there are just two citizens: one supporter of each party. Each player has an incentive to vote, no matter what the other player does—neither would want to potentially hand the election to the other party by staying at home. The class determined that both people voting is a Nash equilibrium, a stasis named after the Nobel Memorial Prize–winning mathematician John Nash in which no player has an incentive to change their course of action, regardless of what the other players do.

“What is annoying about this equilibrium?” asked Nalepa.

“They’re worse off than if they had just abstained,” observed one student.

“Right,” said Nalepa. “If they abstained, they could end up here,” pointing to the payoff matrix to show that if neither had voted, they still would have gotten the benefits of a tie without the cost of voting. “But it’s not an equilibrium, because here each of the voters has an incentive to deviate and vote. So what game does this remind you of?”

A chorus of students answered: the prisoner’s dilemma, a famous game in which two criminals are incentivized to serve as witnesses against each other, even though they both would be better off remaining silent.

It seems strange to compare a voting game to the prisoner’s dilemma “because it seems like they’re doing something good,” said Nalepa. “We’re so socialized to think of voting as our obligation,” not as an illogical use of time.

Nalepa then increased the population of the imagined electorate, keeping the number of supporters equal. When the election is tied or one candidate is winning by one vote, everyone has an incentive to go to the polls because a single ballot is still pivotal to the outcome, she explained. No one on the losing side would sit out the election if voting could improve his or her payoff by turning a loss to a tie or a tie to a win.

Things change when a candidate is winning by more than one vote. Nalepa drew out the scenario, increasing the number of voting Party A supporters and increasing the number of nonvoting Party B supporters. “Since Party A is winning,” said Nalepa, “none of the supporters has an incentive to change their strategy—”

Several students protested with variations of, “Yeah, they do!”

Nalepa quickly clarified. “This guy,” she said, indicating a nonvoting Party A supporter, “does not have an incentive to change,” as he’s getting the full payoff without the cost of voting. His fellow Party A supporters who are voting do have an incentive to not vote, she said, nodding to the students who had objected; they’re “overpaying” for their candidate’s win. But the number of voters for each party won’t start to even out, as one student then suggested—voting supporters of Party B are also being incentivized to stay home “because their party is losing and their vote is not pivotal,” said Nalepa. Voter participation drops in both parties.

These effects are even more pronounced when the electorate is not evenly divided between parties, or is as large as the United States electorate, said Nalepa. To get people to the polls, “the only thing we can count on is civic duty.”

Syllabus

The twice-per-week class uses Martin J. Osborne’s An Introduction to Game Theory (Oxford University Press, 2004), and grades are derived mostly from weekly problem sets—there are no exams. Students are also required to attend a session on formal modeling at the Midwest Political Science Association’s annual meeting in Chicago.

Recognizing that formal modeling can be daunting (students seemed relieved when teaching assistant Junyan Jiang, AM’10, offered to arrange a calculus review session), Nalepa has a strict “no student left behind” policy—she doesn’t move on until she’s sure everyone understands what was just covered. That often means the class doesn’t finish everything on the syllabus; right before Jiang offered the review session, Nalepa told her students to cross two topics off their syllabi. “I always make sure that there are some topics I can just omit” on the syllabus, she says. “It’s more important to me that, you know, someone who is lost at week four catches up by week five.”