Dig Through the Trash. We'll Pay You.

My apartment building sent out a "community policies" document with the following edict:

TRASH CHUTES: ... You are not permitted to leave your trash in front of your apartment door or on the floor in the trash room. If this is found, the trash will be inspected and a $25.00 charge will be fined to the resident for improper trash disposal. Trash rooms open at 6:00 am and close at 10:00 pm.

I've encountered a locked trash room more than once. I stay up late, and I once had to throw the trash away early in the morning before a flight. Each time, I have left the trash bag outside the locked room and have yet to be fined.

This is a great example of the principal-agent problem, or a conflict of interest between owners and front-line employees. The owners may want to fine residents who don't observe the policy, but think of the situation facing the concierge (or resident) who finds the rogue trash bag.

Does he put on sanitary gloves, open and dig through the vile-smelling trash bag in search of a discarded envelope addressed to the offending resident, and file a report with the management against the resident, potentially drawing dirty looks from said resident whenever he passes by the concierge desk?

Or does he simply place the trash down the chute and avoid all the drama? It's clear that ignoring the letter of the law is the easier course of action for the agent.

Finally, I'm still mystified about why the trash rooms are locked in the first place. My only theory is that the management doesn't want homeless people to sleep in them overnight, but what are the chances that that would happen anyway?

Spongeworthiness: An Economic Model

Avinash Dixit, co-author of the outstanding "The Art of Strategy," recently published a paper exploring the Seinfeld episode in which Elaine must ration her remaining contraceptive sponges.

Dixit's model can give Elaine precise advice once she establishes how many sponges she has remaining and how she feels about the trade-offs between "pleasure" today and "pleasure" in the future (the discount rate).

If sponges are plentiful and Elaine's discount rate is fairly low, any man will do. But if her discount rate is high, she'll wait for a nearly perfect man no matter how many sponges she has.

I've finally had enough math to be able to follow the paper's model. I wondered if it made me a bad person that I labored through understanding this model when I usually just skim over models involving, say, optimal portfolio theory.

Hat tip to the Freakonomics blog.

How Has the Price of Madden Games Held Stable?

In the middle of last July, I preordered Madden NFL 10 for Wii, paying $46.99.

The Wii version was vastly inferior to the PS3 and X-Box versions, but that's another matter. As far as Wii football went, it was top of the line: new features and updated rosters. It was released right before the NFL season.

It is now the middle of June, 11 months later. The Super Bowl was played four months ago. A new version of Madden comes out in two months. There is probably less interest in football video gaming now, when football isn't in season.

Yet Madden NFL 10 still sells for ... $46.99.

Is anyone else surprised that the price hasn't dropped? Is Electronic Arts maximizing its profits with this price? People can evade the high price by buying used copies of the game or Madden games from prior years (Madden 09, for instance, is only $18.73).

Perhaps EA's stable-pricing policy encourages people to buy the game when it comes out, because they have seen year after year that the price isn't going to drop a few months later. These additional sales could more than offset the sales that EA is losing by not lowering its price in the interim months between seasons.

Can you think of any other goods that behave this way, with new versions coming out every year? What happens to the price of cars the month before the new models come out?

Game Theory and Natural Selection

Richard Dawkins shares a fascinating application of game theory in his 1976 book "The Selfish Gene," which he attributes to biologist J. Maynard Smith.

I won't get too far into the details, but a mixed-strategy Nash equilibrium in game theory is essentially the set of best strategies for each player in a game, given the other player's strategies. For instance, in soccer, the shooter shouldn't always kick to the right side of the goal, even if it's his strongest side, because the goalie would always dive to the same side and stop most of the shots. It may be the case, for example, that the shooter should kick right 5/7 of the time and left 2/7 of the time, while the goalie should dive right 3/4 of the time and left 1/4 of the time. If either player deviates from this strategy, he is worse off.

Dawkins applies this same idea to the evolution of genes for aggression:

An evolutionary stable strategy or ESS is defined as a strategy which, if most of the members of a population adopt it, cannot be bettered by an alternative strategy. It is a subtle and important idea. Another way of putting it is to say that the best strategy for an individual depends on what the majority of the population are doing. Since the rest of the population consists of individuals, each one trying to maximize his own success, the only strategy that persists will be one which, once employed, cannot be bettered by any deviant individual. Following a major environmental change there may be a brief period of evolutionary instability, perhaps even oscillation in the population. But once an ESS is achieved it will stay: selection will penalize deviation from it.

Dawkins imagines a species in which individuals could act as either "doves" or "hawks" (not the actual birds, but the attitudes toward aggression). Doves never pick fights and always run away when threatened. Hawks always fight as hard as they can, only surrendering when seriously injured. Dawkins assigns some arbitrary payouts to each strategy and concludes that neither can completely dominate: a hawk has a huge advantage in a population full of doves, and vice versa. However, a fixed proportion of the two strategies will eventually emerge and hold stable, when the payoffs to being a dove or being a hawk are equal. In Dawkins' numerical example, equilibrium is reached when the population consists of 5/12 doves and 7/12 hawks.

The Value of the Stolen Base: Two Economic Schools of Thought

Since the release of "Moneyball" in 2003, Sabermetrics has turned conventional baseball wisdom on its head.

Scouts' hunches, batting averages, and other long-cherished means of evaluating player talent have been replaced by complex statistical tools with names like VORP and OPS (and some that are much more obscure), many of which are founded on solid economic principles. The name of the game now is getting on base and hitting for power, while many traditional tactics and metrics have been criticized, including the stolen base.

The most illustrative critique of the stolen base I have seen comes from the Baseball Prospectus's "Baseball Between the Numbers: Why Everything You Know About the Game Is Wrong." A chapter compares the 1982 season of Rickey Henderson (in which he shattered the MLB record for stolen bases) and the 1986 season of Pete Incaviglia (a plodding outfielder). The book points out that while Rickey's steals helped his team, he got caught so often that he did a lot of damage, too:

The run-expectation tables from 1982 show that Henderson added an extra 22.2 runs to the A's offense with his 130 steals. But the 42 times he was caught cost the team 20.6 runs, meaning that for all that running, the A's gained a total of 1.6 runs for the season. In his first season, Incaviglia stole three bases and was caught twice. He cost his team about half a run. Because Henderson got caught so often, the difference between his base-stealing performance in 1982 and Incaviglia's in 1986 added up to about 2 runs.

Every base stolen slightly increases a team's chance of scoring in an inning, but every caught stealing drastically reduces it (for example, the team goes from a runner on first with one out to the bases empty with two outs). The raw number of bases that a player steals is irrelevant; if he steals successfully less than 75% of the time, he is costing his team runs, on average.

Professor Ben Polak, who taught a game theory course through Yale University's Open Yale program in 2007, has a decidedly different approach. The relevant discussion video is here (about 14:45 in) and the transcript is here (do a search for "baseball" to get to the right spot).

He analyzes base stealing in the context of mixed strategies. The theory behinds mixed strategies is rather involved, but it boils down to choosing one \action against an opponent some of the time and another action the rest of the time. For example, if a soccer player always kicked to the right on penalty shots, the goalie would always dive to the right and thus have a good chance of stopping many shots. So the player would score more points by mixing between kicking right sometimes and left others, even if he can kick right more skillfully.

In contrast, if kicking right always had a higher expected value, no matter what the goalie did, the player should always do it. This is known as a pure strategy.

So, back to base stealing, in which Polak talks to a member of the baseball team who is taking his class:

Professor Ben Polak: You're a pitcher, okay. So he's not going to get on base now, so he's not going to answer this. Suppose you did get on base, pitchers don't often get on base. Let's assume that happens, what might you randomize? There you are, you're standing on base, what might you randomize about?

Student: Whether to steal second or not.

Professor Ben Polak: Right, whether to steal or not, whether to try and steal or not. Stay up a second. So the decision whether to try and steal or not is likely to end up being random. If you're the pitcher, what can you do in response to that?

Student: You can either choose to try to pick them off or not.

Professor Ben Polak: What else? So one thing you can try and pick him off. What else?

Student: You can be quicker to the plate.

Professor Ben Polak: Quicker to the plate, what else?

Student: You can pitch out.

Professor Ben Polak: You can pitch out, what else? At least those three things, right?

Student: Yeah.

Professor Ben Polak: At least those three things okay, thank you. I have an expert here, I'm glad I had an expert. So in this case we can see there's randomization going on from the runner whether he attempts to steal the base or not, and by the pitcher on whether he throws the pitch out or whether he tries to throw, to get to the plate faster.

Because the other team is taking so many steps to prevent a player from stealing successfully, they are worse off in other dimensions. Polak provides the punchline (emphasis mine):

Student: The pitcher has to react differently in pitching when he knows that there's a fast guy on base.

Professor Ben Polak: Good, so our pitcher has to react differently. Let's talk to our pitcher again, so one thing our pitcher said was he wants to get to the plate faster. What does that mean getting to the plate faster? –Shout out so people can hear you.

Student: It means just getting the ball to the catcher as fast as possible so he has the best chance to throw out the runner.

Professor Ben Polak: Right, so you're going to pitch from, you're not going to do that funny windup thing, you're not, thank you, you're going to pitch from the stretch, I knew there was a term there somewhere. I'm learning American by being here. And you're more likely to throw a fast ball, there's some advantage in throwing a fast ball rather than a curve ball. Both actions of which, both having to move more towards fast balls and pitching to the stretch are actually costly for the pitcher.

From game theory's perspective, it's no surprise that base stealers don't add or subtract many runs from a team's total, but the costly actions it requires of the pitcher make a base-stealing threat very valuable. It helps players batting with the runner on base, who can expect more fastballs and pitchouts and thus have an advantage.