Problem:

Nigel and Sophie both like to play a single-player casual computer game. (They don't play together. They just happen to both like the same game.) It's the kind of game in which there are multiple rounds, or boards, or levels, like Minsweeper, Angry Birds, or Blendoku. The essential characteristic of this kind of game, for our purposes, is that once you decide to start playing, the decision of when to stop (i.e. how long to spend playing in total) is determined by a series of stop/don't-stop decisions made at the end of each round. The motivation for this problemand the research agenda it is based onis the so-called "one more game" phenomenon that so many gamers experience, in which players find themselves repeatedly saying "I'll just play one more game." This problem is intended to explore how we might model that phenomenon, and how naivete vs. sophistication are implicated.

Let's start by constructing a (very) simplified model of the decisions involved in playing a casual game. It's a four-period model, t = 1,2,3,4. In each of the first three periods, the player decides whether or not to play the game. Thus, in period one they are deciding whether to start playing, and in periods two and three they decide whether to play another round. Not playing is an absorbing state, which means that once you decide to not play, you can't start again. In other words, deciding to not play in period two means that you played for one round and then stopped. Period 4 comes after they stop playing, and is intended to capture any long-term or delayed costs or benefits of playing in periods 1 through 3. One way to think of this is that you play the game in the evening, and the time you spend playing is time you could have been sleeping, so period 4 is the next day, when you are too tired to perform well on an exam. (Note that in this simplified model we are going to set = 1 so it doesn't matter that periods 1 through 3 are much shorter than period 4, because from the perspective of any prior period, all period-4 costs and benefits are just going to be discounted by .)

For each period that a player plays the game they receive an enjoyment benefit of btwhich they experience immediately, and they pay a delayed cost of c, which they experience in period four. (As above, c represents something like decreased performance on an exam the next day. The total delayed cost depends on how long they played the game. Nigel and Sophie have identical preferences over the gamemeaning identical costs, benefits, and discountingas follows: c = 4,b1 = 5,b2 = 3,b3 = 1, = 1/2, = 1

Thus, if a player plays in period one they receive a benefit of b1 = 5 in period one, and a cost of c = 4 in period four. If a player plays in both period one and period two, they receive a benefit of b1 = 5 in period one, a benefit of b2 = 3 in period two, and a total cost of 2 c = 8 in period four. Etc.

However, Nigel is a naif, while Sophie is a sophisticate.

(a) How many rounds will Nigel play? [Hint: this is possible without a game tree, by simply computing Nigel's utility maximization problem for playing versus not playing, given what he believes his future selves will do. But you should be able to draw the appropriate game tree, and you will need to for part b anyway.]

(b) How many rounds will Sophie play? [Hint: This time it will really help to start by drawing a game tree. It should have three players, each of which represents Sophie in a different time period. Think carefully about how to represent the utility outcomes of each choice Sophie makes for each of her three "selves."]

Now assume that there is a period before the beginning of the game, t = 0. In period zero players cannot play the game, but they can pay a flat-rate price, p, for a binding commitment device that allows them to decide in advance how many rounds they want to limit themselves to, and prevents them from playing more than that.

(c) In period zero, how many rounds of play will Sophie choose to limit herself to?

(d) [Harder] Assuming that all costs and benefits are measured in dollars, what is the maximum p that Sophie will pay to use the commitment device?