Friday, March 16, 2012

Obisidian's problem with reward cutoffs and continuity

There are tons of things in life which fall under the category of "If X is greater than Y, then you get this." For example, at my current job: I'm going to get laid off  eventually. I haven't been working here long, and by the time the layoff happens I will have been here less than a year. If I had been here longer such that when I get laid off I will have been employed for over a year, then my severance will be double  of what I will end up receiving. In simpler terms
  • Employed < 1 year: Severance=X
  • Employed ≥ 1 year: Severance=2X
Where X is a value that I'm not going to disclose. You've likely been in this sort of situation before if you've ever received a grade of 89, which for most school I've been to is the highest score you can get that isn't an A. You're so close, but not close enough, and it stings. I went to a Catholic high school where an A was 93 or higher. There were more situations that I'd like where my final grade was a 92. It was good, but not good enough, and at most other schools it would have been. I hated it.[1]

Obsidian, developers of Fallout: New Vegas, have found themselves in a similar situation recently. In their contract with Bethesda, Obsidian would only receive a bonus if the New Vegas received a Metacritic score of 85 or higher. The game has an 84. Therefore they receive no bonus, and this has likely led to their recent layoffs and the reported cancellation of an unnamed next-generation title. I'm guessing they needed that bonus to keep that team running. I feel bad for Obsidian, because as far as I'm aware they did pretty good work with Fallout: New Vegas. 

Now, I'm not here to discuss how using Metacritic as a measure of how good a game is quite flawed because Metacritic pulls from reliable, trustworthy sources and other, less reliable sources. I'm here to talk about the mathematics of the situation and how

Graphically, Obisidian's situations can be represented as such
We can all see this plainly, once they hit 85, they get the bonus. Otherwise, they don't. You see, the unfairness lies in the discontinuity in the graph. Does it make sense that they get nothing for an 84 but get it all for an 85? I argue that it's not. Instead of the function that is in their contract, I propose a different function. What if, starting at a score of 80 they receive a small bonus that increases until maxing out at 85.
This seems to make more sense to me, since it rewards different levels of accomplishment in varying amounts. Now, of course, this contract deal would obviously benefit Obsidian more than the previous contract, so if this were part of a discourse, a compromise would likely have to be struck. If might start at 82 instead of 80 and cap out at 87 instead of 85. I made a composite image of that with the payment model that was used.
As you can see, when comparing the two, my way benefits Obsidian if the score is from 82 to 85, It benefits Bethesda if the score is from 85 to 87, and it's the same everywhere else. Under this model, Bethesda still wouldn't have gotten the full bonus, but they would have gotten something. In fact, they would have received 40% of the bonus, which may not have been enough to save their game team, but it would have been something.

I think that people often underestimate the impact that a reward structure can have. Discontinuous ones such as the letter grade system, Obsidian's bonus, and my severance can have very harsh effects for falling just under the cutoff.

I don't think people understand just how important continuity is when we create functions (and these payoff systems are functions) for our life. Every real world phenomenon is continuous. Although things can seem sudden to the point of being an instantaneous difference, they never really are. Their change can be sharp, but it's always there. For example, the people who write the tax code understand this, although the marginal tax rate is a discontinuous function, it makes the actual amount of tax you pay a continuous (specifically, a continuous and increasing) function of your income. That is, until you start to talk about various tax exemptions and loopholes.

So what is it about continuous functions that makes them important? The important thing about a continuous function is that if you have a threshold on how much change you want in your output, there is some amount of change in your input that can be tolerated before breaching that threshold. To rephrase the problem with Obsidian's bonus payment function in these terms is that if you're sitting at a Metacritic score of 85, there is NO amount by which the score can go down and produce a small change in the bonus.[2]

[1] The grade scale at my high school was:
  • A=100-93,
  • B=92-85,
  • C=84-77,
  • D=76-70,
  • and F=69-0.
[2] The continuity of the function isn't the only important factor here. Frankly a continuous function, like the ones that I displayed, could have been made steep enough that it would emulate the bonus payment function that Obisidian had. It's not only the continuity of the function that's important, but also the continuity of all of it's derivatives. Continuity is just the first step at arriving to a function that would play more nicely. To note, a function for which all derivatives are continuous is called smooth.