From Continuous to Discrete: The Pains of Winning and Losing

Winning rocks. Losing sucks. But the difference between winning and losing is typically only a matter of inches. Often, losing sucks most when your team is within, I don’t know, one game of advancing to the conference finals.

Fundamentally, I think that the pain of losing comes down to the difference between continuous and discrete variables. The Bucks puts in an effort that falls somewhere on the continuum between garbage and divinity. No matter where they fall on that continuum, they are awarded a discrete outcome: a win or a loss. They could be other-worldly or merely world-beaters, but a win is a win. They could be off or offensively bad, but a loss is a loss. And yet, somewhere on that continuum, wins turn into losses and losses turn into wins.

Since it’s been a while, here’s a run-down on what comes next. I’ll introduce an idea from outside of the realm of basketball. In this case, it’s the Sorites Paradox, laboriously explained here and hopefully rendered more accessible below. I’ll then use that idea to provide perspective on the thoughts expressed above. When all is said and done, you can feel good about gaining a shred of insight into basketball, warm about learning something random from the real world, and / or frustrated that this was published at all.

The Sorites Paradox goes something like this. You have one grain of sand. You ask yourself: Is this a heap of sand? The answer is probably no; it’s a single grain of sand. You then add another grain of sand. Is this a heap of sand? Again, the answer is no; if so, it would be a pretty pathetic heap. And so on. After a while, it seems silly to even ask the question. At each step, only one grain of sand is added; how could one grain possibly tip the scales to heapiness? But if you repeat this step over and over, you end up with a million grains of sand while still denying its heapdom.

Conversely, let’s say that you start with those million grains of sand. You ask yourself: Is this a heap? The answer is probably yes. You then take away another grain of sand. Is this a heap? Again, the answer is yes. And so forth. The same trap emerges. It seems like one grain could not possibly deheapify your sand. And yet, repeating this step over and over leaves you with a single grain of sand, which you would adamantly defend as having heap status.

Taken together, it seems like somewhere between one and one million grains of sand there is a tipping point that shifts the sand from non-heap to heap. But on a continuum of sand grains, with only one added or subtracted each iteration, we seem cornered into accepting that a single grain of sand can make or break the heap.

Philosophers have taken a few cracks at this paradox that, in addition to the paradox itself, can also be circuitously applied to the Bucks.

The first and lamest solution is that terms like “heap” are vague. We are trying to apply formal logic (i.e., IF you have X grains of sand AND it is not a heap, THEN X+1 grains of sand is also not a heap) to a vague term. As anyone else who enjoys making playing card towers knows, the tower will collapse without a strong foundation. Still, vagueness is an inherent property of language. Words are our attempts to lasso physical reality with a verbal one; it’s no surprise that they come short.

The second, still pretty lame solution is that one of the premises is false, but we don't know which one. This is basically a bedtime story that parents tell their kids to calm them down. When the kids are freaking out that a single grain of sand could yield a heap from a non-heap, the parents calmly explain that since we don’t actually know which grain of sand tips the scales, there’s nothing to fuss over! The heapless sand train went into the tunnel and emerged 20 or so grains later as a heap: that’s it. Unfortunately, this still represents a fairly hand-wavy attempt to deal with the paradox.

The third, somewhat less lame solution is that the premises aren’t completely true or false. Is a single grain of sand a heap? No - mostly. Is one million grains of sand a heap? Yes - mostly. Essentially, heap is rendered continuous rather than discrete. At some point, a single grain of sand will turn a non-heap into a heap, but it’s more like turning a barely-not-a-heap into a barely-a-heap. That feels better, but it ducks the problem by inverting the premise of the paradox: that sand is continuous and heaps are discrete.

Now, instead of heaps, let’s talk about wins and losses, and instead of sand, let’s talk about vitamins (in the spirit of Coach Bud). If the Bucks play with the effort of a single vitamin, people will probably file out of Fiserv Forum early. If they play with the verve of another vitamin, it still won’t turn out pretty. Since adding additional vitamins doesn’t seem like it would flip the switch from a loss to a win, we would seem to be headed for the best odds in the draft lottery. We could do the same exercise starting from a million vitamins and working our way down - a far nicer endeavor, considering it would lead to the conclusion that the Bucks would instead be hurtling for the number one seed.

In both cases, the Bucks Paradox is admittedly more abstract than the Sorites Paradox, but the lesson is the same: quality of play is continuous and outcomes are discrete. It seems like a change in only one unit of quality - in this case, vitamins - is responsible for clinching a W rather than taking an L.

The solutions to the Sorites Paradox shed some interesting light on the Bucks Paradox. The lamest solution asks us to consider that wins are vague. This smacks a bit like a participation trophy - the wins are the fun you have along the way! - but actually rings more true in the basketball context. Wins are vague because the quality needed for a given win varies based on the quality of the opponent. In contrast, whether or not a certain number of sand grains is a heap is not affected by how poorly the Thunder play on a Tuesday night in OKC.

The pretty lame solution emphasizes that we don’t know at what point games are won or lost. A loss may have felt close, but it might actually have been a few vitamins away.

The somewhat less lame solution involves viewing different shades of wins and losses. Winning and losing may be discrete, but most of us (and our BrewHoop polls) distinguish between close losses and bad losses (and likewise good wins and close wins). It is healthy to acknowledge the variety of outcomes that are categorized under the category of “win;” however, it is this discrete categorization that ultimately dictates the standings.

There is really no way around this paradox. Wins and losses are tracked to order seeds, decide championships, and provide clarity to chaos. But that doesn’t mean that our evaluations should be similarly discrete. The Bucks can play well and lose and they can play poorly and win. It is healthier to cheer the good and jeer the bad, rather than rely on the distinction of whether their grains of sand became a heap.