The author is an associate professor of economics and dean of the College of Business and Economics of De La Salle University.
Early one morning in late November, on my way to De La Salle University
for an important appointment, I hit a traffic jam that, hyperbole
aside, stretched for at least a mile on Zamora Bridge, that
flyover-bridge infrastructure complex that connects Valenzuela Street
in Sta. Mesa and Quirino Avenue in Pandacan. The cause of the
bumper-to-bumper snafu: gridlock at the Tejeron Street intersection,
where cars refused to give way to those going crosswise relative to
their direction. Ruing my choice of route for the day, as the minutes
ticked away I tried to make the best of a bad situation by devising an
economic model that offered an explanation for my predicament. What
follows is the game-theoretic parable that came out of my ruminations.
Consider a two-stage supergame (that is, a game that is iterated ad
infinitum) consisting of N players, where N is a fairly large number
(perhaps corresponding to the number of drivers in Metro Manila). At
the first stage of an iteration, the players each draw a card that
indicates whether or not they are in the game for that iteration, and
if so what their direction and rank in line are. For expositional
convenience, assume that there are only two directions, northbound and
eastbound, and that n, the number of players called on to play (which
varies randomly iteration by iteration) is much smaller than N. Thus,
players either draw a blank card, which means they are not playing, or
a card that says, say, 10 northbound, which means they are 10th in the
northbound line.
At the second stage of the iteration, the n players form two lines,
ordered by the direction and rank they have drawn. Players 1 northbound
and 1 eastbound then independently and secretly choose one of two
possible actions, give way or cross. Suppose that the two players
decide to give way. Then nothing happens and the players have to choose
their actions again. Suppose instead that both decide to cross. Then a
gridlock results (because only one player may cross at a time).
Finally, suppose that one of the players, say, 1 northbound, decides to
cross and the other player decides to give way. Then 1 northbound has
won the right-of-way (which is the objective of the players in each
iteration); 2 northbound moves up to take the space vacated, and play
is restarted between 1 eastbound and 2 northbound. This second stage
game is repeated over and over until one line vanishes and the players
on the other line can cross the intersection unimpeded.
To complete the description of the game, define the payoffs of the
outcomes as follows: If a player gives way, he receives 1 in terms of
some well-being index (however this may be measured). If he decides to
cross and wins the right-of-way, he receives 2. If he decides to cross
and gridlock ensues he receives -10. I assume that the payoffs are
additive over the games played so that if a player is able to cross
only after giving way once and “gridlocking” twice, he gets -19 , which
probably means that he is fuming mad as he moves away from the
intersection. Those who have studied game theory will recognize that
the structure of the game’s payoff matrix is that of prisoner’s
dilemma. The kink, however, is that a player cannot extricate himself
from an iteration until he has won the right of passage. In other
words, if the other line is long, it will not do to simply give way. At
some point, the player must risk crossing and causing a gridlock in
order to escape from the game. Indeed, the number of times that a
player may give way until he gets the gridlock-equivalent level of
well-being can be calculated from the payoff matrix. It is equal to 12,
On his 13th time “at bat,” so to speak, the player must win the
right-of-way if he is not to fall into a level of well-being that is
below even that at gridlock. And so when the players are left to their
own devices, i.e., when the convention is every man for himself, it is
almost inevitable that a gridlock will occur, since the cumulative cost
of repeatedly giving way will make a player increasingly insistent on
having his own turn at crossing the intersection. And unless he pairs
up with someone who is willing to give way at the very time that he
decides to cross, gridlock will occur.
‘disaster solution’
But “gridlocking” is a disaster solution, which charges a large penalty on the players. Can they do better?
In fact, institutions have been designed to improve on the gridlock
outcome. Two examples are the traffic light and the traffic cop
(whether duly authorized or not). The distinctive elements of both are
that (a) the right to assign who has the right-of-way is vested in a
device in the first case and in a person who is not himself one of the
drivers involved in the second case and (b) compliance from the drivers
is exacted by force of law or, in the case of civic-minded citizens who
direct the traffic flow without legal authority, by moral suasion.
What is unfortunate – and tragic – is that, in Metro Manila, no
institution has evolved that improves on every man for himself when
there is neither a traffic light nor a traffic cop. An example of such
an institution is the set of conventions that are in place in several
states in the US, which may be summarized as follows: (i) The first car
to arrive at an intersection has the right-of-way; (ii) In case two
cars arrive at the same time, the car on the right has the
right-of-way; and (iii) in case there is a line of cars at each of the
four points of an intersection, the right-of-way rotates among the
lines in a counterclockwise direction, with only one car allowed to
cross each time.
In the context of our game-theoretic model, the conventions have three
crucial features that allow them to solve the traffic coordination
problem. First, each player is made to bear a small cost – i.e., wait a
turn, or in our model, a well-being level of 1 , which is still way
above the payoff at gridlock – in exchange for earning the
right-of-way. Second, compliance is readily observed. (Think about how
much more difficult it would be to monitor compliance if the
right-of-way is given to, say, five cars at a time. Then it would be
more tempting for the sixth car to slip in with the other cars on the
pretext that its driver miscounted.) Third, continued compliance by
drivers is based on norms that are built on trust. This is why easy
verification of compliance is important. Because the more times drivers
observe that the conventions are complied with, the stronger the
conventions become as norms of behavior. Conversely, each time drivers
observe others crossing out of turn, trust in the conventions is eroded
and their status as norms of behavior becomes more tenuous. For these
conventions to have a chance to develop into norms in Metro Manila,
however, a crucial realization must dawn on all (N) drivers: This is
that, although the number of players, n, in a particular intersection
game is small relative to N, the intersection game is played infinitely
often. It is thus a virtual certainty that all drivers will be called
on to play the game at some point in their driving lives. Indeed, the
more often and the longer one drives (here think about drivers of
public utility vehicles), the higher is the probability of playing.
Moreover, the laws of probability ensure that a driver will win (i.e.,
cross) some and lose (i.e., gridlock) some. The problem is that the
payoff of winning is small (e.g., bragging rights), while the penalty
of losing is large (e.g., missed appointments and lost productivity).
Drivers therefore should not myopically look at the intersection game
as a one-shot deal. Instead, they should see it as it truly is: a
supergame whose penalties they can minimize with crossing norms based
on trust.
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