The Price of Anarchy

Interstate 5 stretches through California’s San Joaquin valley, straight and flat as far as the eye can see, giving drivers a unique opportunity to observe game theory in action. With only two lanes in each direction, most cars stay in the left lane, zipping past the slow moving trucks, which keep to the right. Every so often, an impatient truck driver lumbers into the left lane, blockading the road for several long minutes as he edges ahead of his even more sluggish comrades. Meanwhile, a long line of cars quickly builds up. Eventually, car drivers start to swerve into the right lane, zooming past the line of cars, only to duck back in at the front of the line. Each time a car cuts back in line, the cars behind brake, slowing traffic further. Soon, the right lane is clogged with impatient car drivers trying to advance, which, in turn, induces more truck drivers to try to pass on the left. Although some of the scofflaw traffic does get to Los Angeles in less than the average travel time, the average time increases. In other words, the selfish behavior of some individuals increases the congestion of the highway. This “highway game” is reminiscent of the well-known Prisoner’s Dilemma from economic game theory. There, each of two arrested suspects is given the choice of confessing or remaining silent. Each suspect knows the rules: If both confess, both get light (five-year) sentences for the crime. If neither confesses, both get six-month sentences, on a minor charge. If one confesses while the other remains silent, the one who confesses walks free while the silent one gets the full (ten-year) sentence. These rules yield a strange paradox: Each player improves his own lot by confessing, regardless of what the other player does, yet the optimal outcome for the pair occurs when both suspects remain silent. An instance of the game in which both suspects make the selfish choice is an example of a Nash equilibrium, a set of strategies such that even if each player knows what the other player’s strategy is, he has no incentive to change his own. A Nash equilibrium is a simple model of what happens in a competitive market. From a global perspective, as these examples show, competition often has a cost. Still, for most markets, achieving the global optimum is not a reasonable expectation, either because there is not enough information to compute it or because centralized control is undesirable for other reasons. In such situations, the Nash equilibrium may be a reasonable approximation to the optimum, and the best that can be hoped for. But how good an approximation is it? Computer scientists have found an answer.