Everyone using the service of the Tbilisi marshrutkas experiences one of two extreme cases: the marshrutka either moves tantalizingly slowly or excessively fast. How can this apparent paradox be explained? In search of an answer, let us turn to game theory, one of the appealings outgrows of mathematical economics.

GAME THEORY IN ACTION

A game is a situation in which different parties, usually called players, independently make decisions. Unlike in a unilateral decision problem, in a game, the payoff of each player depends on the actions taken by himself and by all other players. A central tool for analyzing a game is the so-called Nash equilibrium. John Nash, whose fascinating but tragic life was depicted in the Hollywood movie “A Beautiful Mind”, received the Nobel Prize in economics in 1994 for the rigorous formulation of this idea. But what is a Nash equilibrium?

Assume that each player has decided which action to take in the game. If none of them has an incentive to deviate unilaterally, then the combination of actions is a Nash equilibrium. Although it is so stunningly simple, Nash equilibrium has turned out to be one of the most powerful economic ideas of the 20^th century. Could it be the case that the puzzling behavior of Tbilisi marshrutka drivers is a Nash equilibrium?

Consider the players of our game to be the marshrutka drivers. In reality, the actions available to them are manifold – they can adjust the speed on their routes, they can wait longer or shorter at the bus stops, they can decide how many passengers to allow to enter their marshrutka. Yet as we want to shed light on the driving speed puzzle, let us assume that the only parameter they can decide on is the velocity. Moreover, let us assume that player wants to maximize the money earned through ticket sales per hour of work. In this setting, the hourly wage of a driver not only depends on his own choice of speed but also on the speed choices of the other drivers. So indeed, Tbilisi marshrutka drivers are playing a game!

The first thing to understand is that each marshrutka driver serving a route is only competing with the preceding marshrutka and the one that is coming next. If one drives too fast, being on the heels of the preceding marshrutka, one will reap only a few passengers – the preceding marshrutka collected everybody who was waiting for a bus stop and there is too little time for refilling the queue of customers in the meantime. Therefore every driver wants to maximize the distance to the preceding marshrutka.

If a driver goes too slow, however, the upcoming marshrutka may overtake – the worst possible case for each marshrutka driver, because now all the passengers that have accumulated down the road will be harvested by another driver. (In addition, as we heard, it is the worst humiliation of a marshrutka driver’s professional honor if he gets overtaken by another marshrutka!)

THE STRATEGIC PROBLEM OF A MARSHRUTKA DRIVER

A marshrutka driver can catch the more passengers the larger is the time interval between him and the preceding driver. What about the risk of being overtaken? Well, if we assume that the drivers are all equal in each and every relevant aspect, we can deduce a priori that in a Nash equilibrium every player will behave in exactly the same way. Differences in the chosen strategies would have to arise from asymmetries in the game, but there are no asymmetries – so we can conclude that in a Nash equilibrium, every driver will choose the same speed when driving in the outskirts. So nobody has to be afraid of being overtaken, because one could be overtaken only if the upcoming driver would go with a higher speed, but if everyone chooses the same speed, there will be no overtaking whatsoever.

But what is the speed chosen by all marshrutka drivers? Let us assume that there is a constant influx of people to the queue of each bus stop. Once a marshrutka passes, this queue is set back to 0 and then starts growing again. Under these assumptions, if it was just about collecting as many passengers as possible, the optimal speed of the marshrutkas would be close to 0. But if it is about maximizing the number of passengers per hour (because this number corresponds to the hourly wage that incurs for the driver), the optimal velocity does not go to 0. If the speed approaches 0, the number of passengers waiting down the road would go to infinity, but only a tiny part of them would be collected within an hour.

Taking both effects into account, the optimal speed will be such that the gain through a further reduction in speed would be exactly outweighed by the shorter distance that would be passed in the available time.

In reality, we observe marshrutkas in the outskirts moving at minimal speed, usually with less than 10 km/h, which is consistent with our observations. Is this a Nash equilibrium? Only if it would not be profitable for a driver to change the speed. This is the case: going slowly is definitely no improvement because the driver is going at that speed at which the additional passengers per hour generated through further slowing down would be outweighed through the reduction in kilometers passed. Going faster, on the other hand, would just reduce the distance to the preceding driver, implying fewer passengers waiting to be collected. Hence, we are indeed in a Nash equilibrium.

Once the marshrutka enters the inner city districts, many marshrutka routes are merging into one route. Now a reduction of speed would not lead to more passengers, because the distance to the preceding marshrutkas is only marginally determined by the speed. The probability that at a crossroad a marshrutka enters the gap to the preceding bus is increasing in the length of that gap. So the distance to the preceding marshrutka is largely independent of the speed chosen by the driver, and the only motivation that remains is to make as many kilometers as possible within each hour. As a result, the optimal choice is now to go as fast as possible. Again, this is the Nash equilibrium choice of each driver in the inner-city districts.

A BAD EQUILIBRIUM

The outcome of the strategic optimization of each driver is dissatisfied passengers, who have to wait long times at the bus stops in the outskirts, and car drivers who are hampered by slowly moving minibusses. Yet even the marshrutka drivers are dissatisfied with the equilibrium outcome. Assume that in the outskirts, every driver would speed up by one kilometer per hour. The number of passengers transported by the marshrutkas would not change through this, so the joint revenues of all marshrutka drivers would not be affected by such a change. Moreover, after the speed increase, every driver would go with the same speed, so the aggregate revenues would as before be shared in equal parts among all drivers. However, each driver would now need less time for transporting the same number of passengers, so the hourly wage would increase. For this reason, the best situation for every driver would be if drivers would go as fast as they can throughout their whole routes.

Unfortunately, this is no Nash equilibrium. If everybody would go as fast as possible, a driver would have an incentive to slow down and maximize the distance to the preceding driver (under the constraint that he does not let his follower overtake him). This unfortunate return to the bad equilibrium is illustrated graphically in the picture.

What we see here is one of the many examples in game theory, where uncoordinated strategic behavior of independent players leads to suboptimal outcomes. So do not blame the driver next time you are sitting in a marshrutka that moves at snail’s pace – he is just playing his Nash equilibrium strategy, and every rational person in his position would do that…