First, I asked them to fully characterize the optimal strategy for player 1 in the following modified centipede game, assuming player 2 adopts their optimal strategy.
That second part is important, and I stressed it quite a bit. I particularly made sure it was clear that I wasn’t asking them to imagine that they were playing this game with a friend. I did this because I knew that some students would argue that their answers made more sense than the one I was looking for—which, in fairness, might well be true if we didn’t know how player 2 would behave. As others have noted, if player 1 believes player 2 will not behave optimally, it can in fact be optimal for player 1 to adopt a strategy that could not comprise a Nash equilibrium. That’s an interesting possibility, and in a different context, I’d consider it worth exploring. But my goal here was to figure out who’s thinking strategically, as they should be if they listened to the lecture on game theory, and who’s trying to get by on gut intuition. As I’ve said before, some of the activities I have planned are strictly meant to help students understand things better and my expectation is that everyone will receive all the points available for that day. But when I decided to flip the classroom, I knew it would be important to give the students a strong incentive not to wait until the night before the midterm to start listening to the lectures. The only way to do that is to make sure some of the activities are much more challenging to those who didn’t listen than those who did. This is one such activity.
It came as no surprise, then, that many students failed to identify the optimal strategy for player 1, which is take; take. (Actually, not a single student successfully characterized any full strategy. This is a bit disappointing, since I spent some time in the lecture pointing out that a strategy specifies what a player would do at every stage, regardless of whether they are called upon to do so in equilibrium, but I can’t say it’s especially surprising.) Some thought the game would end at the final node (which presumably means they think 1’s strategy is pass; pass) because that makes everyone better off. Others applied just enough strategic thinking to say that player 1 would take at the second opportunity, but evidently didn’t consider that 2 should anticipate this and so would not allow 1 the opportunity to do so, which in turn means that 1 has every incentive to take at the very beginning. But a decent number did correctly determine that player 1 would take right away, and I gave full credit to these students even though they didn’t specify what player 1 would do if called upon to make that second decision.
The second part, a version of the traveler’s dilemma, was harder.
Here, the correct answer is $2. And though some students figured that out (or guessed fortuitously), most did not. The most common answer was $99, though there were some more creative ones, like $51, and my absolute favorite, B. (Your guess is as good as mine.) I think the problem here was that students were treating player 2 as non-strategic. If player 2 wrote down $100, then player 1’s best response is indeed $99. But player 2 can anticipate this, and 2’s best response to 99 is 98. Of course, 1’s best response to 98 is 97. And so on. That logic drives both players down to $2.
Hopefully, this exercise convinced students that even narrowly self-interested actors are best served by trying to see things from the other person’s perspective. If nothing else, though, I think I might finally be convincing some of the slackers that this is not the sort of course one receives an A in without trying.
For those wondering, this completes the portion of the course that is intended to help students interpret the analysis I’ll be presenting in future lectures. The rest of the course is devoted to the study of cooperation and conflict, and the next activity will be designed to convince them that there are indeed lots of benefits to cooperation that are currently going unrealized.