Now we have set the stage. We are considering social choice problems with transferrable utility. We want to achieve Pareto efficient outcomes which in this context is equivalent to utilitarianism.
Now we face the next problem. How do we know what the efficient policy is? It of course depends on the preferences of individuals and any institution must implicitly involve providing a medium through which preferences are communicated and mediated. In this lecture I introduce this idea in the context of a simple example.
Two roomates are condering purchasing an espresso machine. The machine costs $50. Each has a maximum willingness to pay, but each knows only his own willingness to pay and not the others. It is efficient to buy the machine if and only if the sum exceeds $50. They have to decide two things: whether or not to buy the machine and how to share the cost. I ask the class what they would do in this situation.
A natural proposal is to share the cost equally. I show that this is inefficient because it may be that one roomate has a high willingness to pay, say $40, and the other has a low willingness to pay, say $20. The sum exceeds $50 but one roomate will reject splitting the cost. This leads to discussion of how to improve the mechanism. Students propose clever mechanisms and we work out how each of them can be manipulated and we discover the conflict between efficiency and incentive-compatibility. There is scope for some very engaging class discussions here that create a good mindset for the coming more careful treatment.
At this stage I tell the students that these mechanisms create something like a game played by the roomates and if we are going to get a good handle on how institutions perform we need to start by developing a theory of how people play games like this. So we will take a quick detour into game theory.
For most of this class, very little game theory is necessary. So I begin by giving the basic notation and defining dominated and dominant strategies. I introduce all of these concepts through a hilarious video: The Golden Balls Split or Steal Game (which I have blogged here before.) I play the beginning video to setup the situation, then pause it and show how the game described in the video can be formally captured in our notation. Next I play the middle of the video where the two players engage in “pre-play communication.” I pause the video and have a discussion about what the players should do and whether they think that communication should matter. I poll the class on what they would do and what they predict the two players will do. Then I show them the dominant strategies.
Finally I play the conclusion of the video. Its a pretty fun moment. I conclude with a game to play in class. This year I had just started using Twitter and I came up with a fun game to play on Twitter. I blogged about this game previously.
(By the way this game is extremely interesting theoretically. I am pretty confident that this game would always succeed in implementing the desired outcome: getting the target number of players to sign up, but it is not easy to analyze because of the continuous time nature. The basic logic is this: if you think that the target will not be met, then you should sign up immediately. But then the target will be met.)
Here are the lecture slides.