CONGESTION TOLLS: EQUILIBRIUM AND OPTIMALITY

I

2 i f i f k , x k < 2 4 i f i # k , x k > 2

if i = k, x < 2 5 i f i = k , x k > 2

c,k(xk) = ' 3 k \

1

Robert W. Rosenthal

' for i = 1.2

Bell Telephone Laboratories, Inc. Murray Hill, New Jersey 07974

ABSTRACT

An example of a network with flow costs depending on congestion is presented for which no system of tolls and subsidies exists which can ensure that all equilibria in the game of route selection are Pareto optimal.

It has long been recognized that in transportation facilities subject to congestion delays, equilibrium flows may not be Pareto optimal. The imposition of tolls has been a much dis- cussed remedy (see, for example, the classical treatments by Pigou [41 and Knight [31 and the more recent discussions in Kahn [2] and Edelson [11). The purpose of this note is to show that an all-knowing central authority, using a system of tolls or a system of tolls and subsidies, can- not in general force an outcome which is Pareto optimal or even Pareto superior to a given nonoptimal equilibrium.

Thus, players 1 and 2 incur uniformly higher costs on arcs 1 and 2, respectively, while the costs for players 3 and 4 are 0, independent of route and traffic. It is easy to see that the Pareto optimal (costs being minimized) route selections result in the unique cost vector (2,2,0,0) and involve player 1 travelling on arc 2, player 2 travelling on arc 1, player 3 travelling on either arc, and player 4 travelling on the arc not traversed by 3. The set of Nash equilibria of the pure strategy game of route selection, however, consists of all the configurations of two players on each arc. The resulting cost vectors are (2,2,0,0), (2,3,0,0), (3,2,0,0) and (3,3,0,0).

23 1

232 R. W. ROSENTHAL

Suppose that a central authority, knowing the above data and fearing a nonoptimal equili- brium, wishes to force a Pareto optimal selection of routes for the players. Consider first a sys- tem of tolls on the arcs in the form of some otherwise worthless medium (say tokens) together with subsidies of tokens to the players. (In our usage a toll is a fixed charge which must be paid by every unit of flow using a particular arc. A subsidy is a gift to a player which is independent of route selection. Different players may receive different subsidies, and different arcs may be assigned different tolls; but different players pay the same toll if they use the same arc.) N o matter what the subsidies are (as long as all players can afford at least one of the arcs) there is no way to stop players 3 and 4 from travelling on the arc with the cheaper toll (or on the same arc, if both tolls are equal). This cannot result in a Pareto optimal selection.

Similarly, if the tolls and subsidies are in the form of money, 3 and 4 cannot be kept from travelling on the cheaper arc.

A trivial way out of the dilemma in general is to use a different kind of token on each arc in the network and to subsidize each player exactly the required number of units of each kind of token for the route desired. This method, of course, can be used to force every player to use the route deemed desirable for him. It does not appear to be practical in large networks, however.

ACKNOWLEDGMENT

I have benefitted from discussions with Carl Futia and William Taylor on this subject.

REFERENCES

ill Edelson, N.M., "Congestion Tolls Under Monopoly," American Economic Review 61, 873- 882 (1971).

[21 Kahn, A.E., The Economics of Regulation: Principles and Institutions, Vol. I , Chapter 4, Wiley, New York (1970).

[31 Knight, F.H. "Some Fallacies in the Interpretation of Social Cost," Quarterly Journal of Economics 38, 582-606 (1924). Reprinted in G.J. Stigler and K. Boulding (eds.) Readings in Price Theory, Irwin, Chicago, pp. 160-179 (1952).

[41 Pigou, A.C., The Economics of Welfare, Macmillan, London, 1920 edition only.