Game Theory: Expectation, Variance, and Underdog Strategies

The offseason is upon us, and in the tradition of Heck's "Better Know A ..." series on Xs-and-Os strategy, I've decided to post some strategy articles of my own. But I'm much more of a math and statistics person than an Xs-and-Os guy, so these will be devoted to statistical analysis and game theory. (Yes, there will be math, though I'll try not to push into eyes-glaze-over territory. This one will probably be heavier on the math than the rest, if only because it's needed to set the stage.)

Let's start with the basics: expected value, variance, and standard deviation. Expected value (or expectation) is just a fancy way of saying "average" of a random variable. Variance is a measure of how wide the distribution of results is; specifically, if the expected value is M and x is the random variable, variance is the expected value of (M-x)^2. The standard deviation is the square root of the variance; standard deviation tends to be more directly useful (although both are used, depending on what you're doing with the numbers).

Variance and expectation aren't enough to tell the whole story about a distribution: a theoretical seven-sided die and a random process that gives 3 80% of the time and 8 20% of the time both have the same mean (4) and variance (also 4), but obviously the distributions as a whole are quite different. This is where I let you in on one of the convenient little secrets of statistics: add together enough independent random variables with the same distribution, and no matter what the initial distribution looks like, the sum of those variables begins to approximate a normal distribution. Some initial distributions approach this faster than others, but (with some rare exceptions for really bizarre distributions) all of them eventually do. It's called the Central Limit Theorem, and it means that for large numbers of trials we don't have to care what the real distribution looks like. Add together N independent variables, each with mean M and variance V, and (for sufficiently large N) the sum will be approximately normally distributed with mean M*N and variance V*N.

Why is being able to approximate everything as a normal distribution useful? Because a normal distribution is completely characterized by mean and variance; if you know those two parameters and you know the distribution is normal, you know everything about the distribution. The probability of an outcome greater or less than a particular value is a function of the ratio of the distance from the mean to the standard deviation of the distribution. This value (x-M)/S (where S = sqrt(V)) is often referred to as the "Z score" and maps a general normal distribution to the standard normal distribution (which has M = 0 and V = 1). Unfortunately, there is no simple calculation to determine the probability on either side of a given Z score (the integral involved has no closed-form solution and must be solved numerically), but tables of pre-computed values up to at least Z=3 are commonly available. (Going beyond Z=3 is rarely necessary; the probability of Z>3 is about 1 in 750, same for Z<-3.)

That about covers the math info dump for now. So what does any of this have to do with sports? Quite a bit, actually. Statistically, you can view a basketball game as the sum of about 65-70 random trials for each team which are approximately independent. (There are some dependencies - scoring in transition off a steal is usually significantly easier than scoring off an opponent's made basket - but the effects are relatively small and can be compensated for by combining one possession for each team and adding a little extra to the variance to account for the higher probability of scoring after an opponent fails to do so.) Strategic choices can shift the mean and standard deviation, but the distribution is still going to be approximately normal.

Football isn't modeled quite as easily this way. For one, there are fewer possessions (and thus the normal-distribution approximation isn't as accurate). But more importantly, possessions are decidedly not independent - field position makes a big difference. If you account for field position by looking at the average points for a drive starting there, however, you can get a reasonable estimate prior to endgame situations (though the accuracy of these estimates will probably be worse than for basketball). Hockey and soccer are probably better modeled with other distributions that handle the discrete (and binary, as opposed to football and basketball possessions which can end with varying numbers of points) nature of goals more accurately.

What does all of this suggest strategically?

• Underdogs may be better off taking "risky" strategies - giving up some points in expected value to increase the standard deviation. Take a situation where Team A is a 6-point favorite over Team B, with a standard deviation for the game equal to 12 points. Team B needs a situation where they come out ahead by more than half of a standard deviation in order to win - in other words, a Z score > 0.5. Outcomes of Z>0.5 happen about 31% of the time, so that's Team B's probability of winning. Now let's assume that Team B can choose between that strategy and a riskier one which costs them two points on average but raises the standard deviation to 20 points. Now they only need a Z score > 8/20 = 0.4, for a win probability of 34.5%. Doesn't sound like much of an improvement, but that same improvement in every game of a basketball season amounts to at least an extra win every year.
  The down side, of course, is that you're much more likely to get blown out by adopting such a strategy: a loss by 30 or more points becomes about six times as likely (from Z<-2, 2.3% to Z<-1.1, 13.6%).
  This is obvious in late-game situations: with a minute left, down six, even the most diehard puntosaur coaches won't kick a field goal on 4th and 15; it might be the right choice from a "maximize your score" perspective, but when it comes to winning games it's obviously wrong. But a serious underdog should be taking risks even early in the game; even when the scoreboard says 0-0, the underdog has to make up ground to balance the likely advantage the favorite will gain over the course of the game. Traditionally weak programs trying to improve should recognize this and accept an increased number of blowout losses in exchange for a better chance of pulling more upsets.
  
• Likewise, strong favorites can sometimes gain in terms of win probability by taking a more conservative strategy than the one that offers the highest average margin. Using the same initial strategy as in the previous scenario, if Team A can reduce the standard deviation from 12 to 6 at the cost of two points on average, they come out ahead (needing only Z>-0.67, 74.8%, instead of Z>-0.5, 69.1%). The favorites do give up a large part of their chances of achieving a blowout victory, however, going from Z>1.17 (12.2%) to win by 20 or more to Z>2.67 (0.4%). One need only look down the road to Lloyd Carr's tenure in Ann Arbor to see this theory in action.
• The above two points combined suggest one possible reason for coaches who are building up a weak program to plateau just above .500: the in-game strategy to take a team from bad to average, in terms of win/loss results, is the opposite of the one needed to take a team from good to great. A coach who's working with limited resources has to take risks to make up the gap with stronger programs, but once his program becomes strong taking those same risks makes them vulnerable to the same sort of upsets they pulled off when they were at the bottom of the pile. (This is by no means the only reason - no strategy is going to turn Toledo into Alabama overnight, or even within a decade - but it's a possible contributing factor.)
  
• A fast pace tends to favor the better team. If per-possession numbers are assumed to be independent of pace, the average margin scales linearly with pace while the standard deviation scales only as the square root of pace. The underdog has to sustain the same above-average level of performance per possession over a longer time, which is statistically less likely. This is especially true early in the game or when the game is close, as in these cases the expected margin to accumulate over the remainder of the game is larger than the current margin. Once a team has a large lead, it can pay off to slow down even if they were the favorite all along, since doing so offers fewer opportunities for the trailing team to close the deficit.
  An example: if a 64 possession game favors Team A by 6.4 points with a standard deviation of 8 (Z>-0.8, 78.8%), cranking up the tempo to 81 possessions makes them an 8.1 point favorite with a standard deviation of 9 (Z>-0.9, 81.6%).
  Another example, but later in the game: sometime in the second half of the game given above, Team A is up by five points. Ignoring the strategy changes that would occur in the endgame, if there are 16 possessions remaining, Team A is now a 6.6 point favorite with a standard deviation of 4 (Z>-1.65, 95.1%). Pushing the pace to 25 possessions remaining increases the projected margin to 7.5 points, but it also increases the standard deviation to 5, making an upset more likely (Team A wins for Z>-1.5, 93.9%). Up to a point (50 possessions, in this case), the extra time for Team B to make up the existing deficit outweighs the likelihood of that deficit growing during that time. Increasing the pace even further beyond that (again, assuming no strategy changes for the endgame and no impact on the per-possession stats for changes to the pace) would begin to increase Team A's win probability again.
• All else being equal, taking early risks tends to favor the team that can gain by adjusting their strategy based on the outcome. For instance, a football team down 15 late has to go for two on one of their two touchdowns in order to tie. If they go for it on the first one, they will know immediately whether they need one more score or two (situations which require significantly different strategy; if down two scores, it might be necessary to onside kick immediately while kicking away is safe if down only one score). Kick, and you do not know yet whether you need one more score or two; you might try to hurry on the assumption you might need a third score only to find that you've left your opponent too much time, or you might try to drain the clock and deny yourself even the unlikely hope of an onside kick and a third score when your conversion fails. Since your best strategy changes with the result of the two-point conversion (and the opponent's strategy doesn't change much - either way, they're trying to run out the clock and stop you from scoring), you should try to obtain that information as early as practical.

In sum, the key points for an underdog strategy are:

1. Take risks.
2. Take them early while you can still adjust your strategy to the results.
3. Slow the pace so your opponent's superior talent has less opportunity to overwhelm the risks you hope to gain by taking.

Doing all of the above may be difficult (VCU's famous Havoc defense, for instance, is high risk but tends to move the pace in the wrong direction for an underdog strategy), so the tradeoffs have to be examined in detail for each scenario, but any one of those elements is a bonus for an underdog if you can get them without sacrificing the others.

Over the course of the offseason, I'm planning to cover a few of the strategic questions that are relatively amenable to statistical analysis, such as:

• When to go for two or not
• How pace and score affect win probability, and how early to go into the clock-eating offense
• Fourth down decision-making
• Fouling up three in the waning seconds of a basketball game

If you have any other suggestions for topics to cover, let me know and I may be able to take a look at them.