City-CentreTraffic: N-Player PD?

I swear that there is an ongoing conspiracy to make me give up Krispy Cream for Crème Brule, Hamburgers for Pork Knuckles, Broadway for West End, Sex and the City for Coupling, etc. I swear, if they don’t stop soon, I may soon find myself at Heathrow instead of JFK Airport

But then again, why limit my options, I thought to myself, and thus, I decided to research Oxford’s Economics department. I didn’t find much useful material pertaining to the undergraduate application, but I did stumble across a collection of research papers written by the professors there. Talk about looking for a few grams of gold and finding the mother lode.

One paper which I found particularly interesting was a study which sought to disprove the representation of city-centre traffic as an n-player Prisoner Dilemma.

“Impressed by the elegance of the PD theory, social policy theorists have too casually assumed that the PD provides an adequate analogy for real-world social dilemmas, such as the traffic problem or, more recently, panic -buying of fuel (Hallsworth & Tolley, 2000).”

Let us first start with a two player game. Let us first assume that there are only two people in the city, Tom and I, who wish to enter the city centre, and that they can choose to drive or not to drive (take the public transport, walk, cycle, etc). We assume that the roads are so narrow that having two cars on the same road causes enough congestion to hamper both drivers.

Let X_Y be the term used to represent each person and his preferred mode of transport (X to represent the mode and Y to represent the person). For example, D_T would represent a case when Tom is driving and N_I would represent a case when I’m not driving.

In order for payoff,

For Tom: D_TN_I > N_TN_I > D_TD_I > N_TD_I
For Me: D_IN_T > N_IN_T > D_ID_T > N_ID_T

The best case for Tom (he gains the most utility) would occur when he drives and I don’t, since he would enjoy both the mobility of a car and not be boggled down with traffic congestion caused by two cars on the road.

The next best situation would be if both of us don’t drive since we would both reach the city centre without suffering the negative effects of private transport caused by others (congestion, pollution, etc).

The third best situation would be both of us driving. While relative to this situation, the two of us would prefer to cooperate and both not drive, if the other drives, our best option is to drive as well. This would be preferable, from Tom’s viewpoint, to a case when Tom takes public transport while I drive, the worst of the four possible outcomes.

The table below shows a more graphical representation of the case, where the first number refers to my payoff and the second number refers to Tom’s payoff. The higher the number, the greater the value of the payoff.

	Tom Doesn’t Drive	Tome drives
I Don’t Drive	(3,3)	(1,4)
I Drive	(4,1)	(2,2)

From the table, it can be inferred that if Tom doesn’t drive, then I can reap maximum benefits by driving. It can also be inferred that if Tom drives, I can reap maximum benefits by driving, although the benefits reaped in such a situation would be lesser than the previous one. Thus, mine dominant strategy is to drive since regardless of what Tom does, I reap maximum benefits.

Tom would think in a similar way, and thus choose to drive too. Thus, both of us would end up driving, reaping a (2,2) payoff even though both of us would prefer to cooperate and reap a (3,3) payoff. That is the irony of the PD, that when both of act rationally to maximise our payoff, the converse happens.

The same concept can be extended to a multiplayer game as long as driving remains a dominant strategy. Do note that that the payoff takes into account all factors including time taken, convenience, cost, cultural reasons, etc. Thus, while everyone has different preferences, we assume an averaged payoff for the general populace which encompasses all these factors.

Table 1.1

	Populace Doesn’t Drive	Populace Drives
I Don’t Drive	(3,3)	(1,4)
I Drive	(4,1)	(2,2)

This leads to traffic congestion as everyone chooses to drive. The same concept can be used to explain phenomena like why everybody rushes to withdraw money from the bank during a depression upon hearing rumours of its insolvency (which leads to its actual insolvency in the absence of vast amount of reserves or governmental intervention).

Or can it?

The authors of the paper go on to disprove the generalisation of n-player PD from 2-player PD in traffic conditions using vigorous statistical methods. I shan’t harp on that but shall instead write about my take on the issue.

In the two player game, we assumed that if both Tom and I drove, the congestion caused would lead to a diminished payoff, which would happen if the roads to the city centre were extremely narrow. Thus, we can infer that there must be a certain threshold congestion point, below which the populace can drive without causing significant diminished payoffs. What is the game like for such congestions?

Table 1.2

	Populace Doesn’t Drive	Populace Drives
I Don’t Drive	(2,2)	(1,4)
I Drive	(4,1)	(3, 3)

Driving remains a dominant strategy (Table 1.2) and this quickly leads to a build-up in traffic, bringing it to the equilibrium point (represented by the Table 1.1), and beyond. However, I assert that past the equilibrium point, the payoffs shown in Table 1.1 do not hold. Let us first consider another two player game between Tom and me, but this time when the roads to the city centre is already crowded.

Table 1.3

	Tom Doesn’t Drive	Tom Drives
I Don’t Drive	(4,4)	(3,2)
I Drive	(2,3)	(1,1)

In this case, I assume that if I’m driving, I’m made worse off if Tom drives as well since he adds to the congestion and pollution (the marginal effect is small but since I will be generalising to a n-player game, I think it’s best to include it). The same logic can be used to explain why if I don’t drive, I’m made worse off if Tom drives.

Unlike the game shown in Table 1.1, the dominant strategy when the roads are already congested is to not drive.

Let us now consider the n-player game. Assume that traffic conditions have been deteriorating since Monday, and yesterday (Thursday), word spread that it was much more convenient to take public transport. A rational person who has equal access to both his car and public transport would have a greater incentive to choose to take public transport, leading to a reduction of traffic on roads. In this way, a negative feedback loop is formed, ensuring that traffic congestion does not spiral out of control (in fact, it ought to hover near this congestion point) as predicted by an n-player PD.

But what good is this analysis, for if conditions hover near congestion point, it would already mean that cars are being delayed on roads. I’m not so sure myself (I only have a few hours of formal economics training) but, it seems systems are most efficient at equilibrium so by extrapolation, some kind of efficiency must be reached in this system as well. When I do find out, I’ll add it in.