Virtually no one knows this world more deeply than Isaac Haxton, one of the world’s best cash-game poker players. In most sports it’s sufficient to be as good as possible, and the less self-conscious one is about one’s skills the better. But, Haxton explains, “In some ways the most important skill as a professional poker player is to be able to evaluate how good you are. If you’re anything short of the very best poker player in the world, you can be pretty much assured of going broke if you are endlessly willing to play people better than you.”
Haxton is a heads-up, no-limit specialist: “heads-up” meaning one-on-one poker, and “no-limit” meaning just that—the highest stakes, limited only by what they can bankroll and stomach. In multi-handed poker cash games, there will often be one weak player—a wealthy amateur, for instance—feeding a table full of professionals, who then don’t much care who among them is better than whom. In the world of heads-up, it’s different. “There has to be a disagreement between you and them about who’s better—or somebody has to be willingly losing.”
So what happens when there’s a fairly established consensus and no one’s willing to play anyone better than they are? You get something that looks a lot like players simply jockeying for seats. Most online poker sites have only a finite number of tables available. “So if you want to play heads-up no-limit, with blinds of fifty and one hundred dollars, there are only ten available tables for that,” says Haxton, “and so only the consensus ten best players who are out right now … sit and wait for someone to show up who wants to play.” And if a superior player arrives and sits down at one of these tables? If the person sitting isn’t willing to ante up, they scram.
“Imagine two monkeys,” says Christof Neumann. “One is sitting and feeding in its spot, very peacefully, and another one is coming up [to] where the other guy is sitting. And that guy would then stand up and leave.”
Neumann isn’t making a poker metaphor. He’s a behavioral biologist at the University of Neuchâtel who studies dominance in macaques. What he’s just described is known as displacement.
Displacement happens when an animal uses its knowledge of the hierarchy to determine that a particular confrontation simply isn’t worth it. In many animal societies, resources and opportunities—food, mates, preferred spaces, and so forth—are scarce, and somehow it must be decided who gets what. Establishing an order ahead of time is less violent than coming to blows every time a mating opportunity or a prime spot of grass becomes available. Though we may cringe when we see creatures turning their claws and beaks on each other, biologists tend to think of pecking orders as the violence that preempts violence.
Sound familiar? It’s the search-sort tradeoff.
The creation of a pecking order is a pugilistic solution to a fundamentally computational problem. For this reason, incidentally, debeaking chickens on farms may be a well-intentioned but counterproductive approach: it removes the authority of individual fights to resolve the order, and therefore makes it much harder for the flock to run any sorting procedure at all. So the amount of antagonism within the flock in many cases actually increases.
Looking at animal behavior from the perspective of computer science suggests several things. For one, it implies that the number of hostile confrontations encountered by each individual will grow substantially—at least logarithmically, and perhaps quadratically—as the group gets bigger. Indeed, studies of “agonistic behavior” in hens have found that “aggressive acts per hen increased as group size increased.” Sorting theory thus suggests that the ethical raising of livestock may include limiting the size of the flock or herd. (In the wild, feral chickens roam in groups of ten to twenty, far smaller than flock sizes on commercial farms.) The studies also show that aggression appears to go away after a period of some weeks, unless new members are added to the flock—corroborating the idea that the group is sorting itself.
The key to thinking about decentralized sorting in nature, argues Jessica Flack, codirector of the Center for Complexity and Collective Computation at UW–Madison, is that dominance hierarchies are ultimately information hierarchies. There’s a significant computational burden to these decentralized sorting systems, Flack points out. The number of fights in, say, a group of macaques is minimized only to the extent that every monkey has a detailed—and similar—understanding of the hierarchy. Otherwise violence will ensue.
If it comes down to how good the protagonists are at keeping track of the current order, we might expect to see fewer confrontations as animals become better able to reason and remember. And perhaps humans do come closest to optimally efficient sorting. As Haxton says of the poker world, “I’m one of the top heads-up, no-limit hold ’em players in the world, and in my head I have a fairly specific ranking of who I think the twenty or so best players are, and I think each of them has a similar ranking in their mind. I think there is a pretty high degree of consensus about what the list looks like.” Only when these rankings differ will cash games ensue.
A Race Instead of a Fight
We’ve now seen two separate downsides to the desire of any group to sort itself. You have, at minimum, a linearithmic number of confrontations, making everyone’s life more combative as the group grows—and you also oblige every competitor to keep track of the ever-shifting status of everyone else, otherwise they’ll find themselves fighting battles they didn’t need to. It taxes not only the body but the mind.
But it doesn’t have to be that way. There are ways of making order without the costs.
There’s one sporting contest, for instance, where tens of thousands of competitors are completely sorted within the time that it takes to hold just a single event. (A Round-Robin tournament with ten thousand players, on the other hand, would require a hundred million matchups.) The only caveat is that the time required for the event is determined by its slowest competitors. This sporting contest is the marathon, and it suggests something critical: a race is fundamentally different from a fight.
Consider the difference between boxers and skiers, between fencers and runners. An Olympic boxer must risk concussion O(log n) times, usually from 4 to 6, to make it to the podium; allowing a greater number of athletes into the games would imperil the health of all. But a skeleton racer or ski jumper or halfpipe specialist needs to make only a constant number of gambles with gravity, no matter the size of the field. A fencer puts herself at her opponent’s mercy O(log n) times, but a marathoner must endure only one race. Being able to assign a simple numerical measure of performance results in a constant-time algorithm for status.
This move from “ordinal” numbers (which only express rank) to “cardinal” ones (which directly assign a measure to something’s caliber) naturally orders a set without requiring pairwise comparisons. Accordingly, it makes possible dominance hierarchies that don’t require direct head-to-head matchups. The Fortune 500 list, to the extent that it creates a kind of corporate hierarchy, is one of these. To find the most valuable company in the United States, analysts don’t need to perform due diligence comparing Microsoft to General Motors, then General Motors to Chevron, Chevron to Walmart, and so forth. These seemingly apples-to-oranges contests (how many enterprise software installations equal how many oil futures?) become apples-to-apples in the medium of dollars. Having a benchmark—any benchmark—solves the computational problem of scaling up a sort.
In Silicon Valley, for instance, there’s an adage about meetings: “You go to the money, the money doesn’t come to you.” Vendors go to founders, founders go to venture capitalists, venture capitalists go to their limited partners. It’s possible for the individuals to resent the basis of this hierarchy, but not really to contest its verdict. As a result, individual pairwise interactions take place with a minimum of jockeying for status. By and large, any pair of people can tell, without needing to negotiate, who is supposed to show what level of respect to whom. Everyone