Braess' Paradox

Figure 1: An Example Of Braess' Paradox

Braess' paradox arises in transport economics, a field for applied research in economics. I was inspired by the example in Fujishige et al. (2017) for the example in this post. Under Braess' paradox, an improvement to a transport network, and thus an increase in the number of choices available to users of the network, results in decrease performance. In reliability engineering, one says such a transport network is not a coherent system.

A transport network, for a single mode (for example, air, rail, road, or water) can be specified by:

• A network, where a network consists of links between nodes. Links can be one-way or two way.
• A cost for traversing each link. The cost can be a function of the demand (that is, the amount of traffic traversing that link). Cost can have a stochastic component, such as a (perceived) standard deviation for the distribution of the time to traverse a link.
• Demands on the network, as specified by source nodes for users and the destination of each user.
• Objective functions for the users, such as the minimization of trip time or the maximization of the probability that total trip time will not exceed a given maximization. The probability for the latter objective function is known as trip reliability.

In my example (Figure 1), two road networks are specified. The network on the right differs from the one on the left in that an additional road, between nodes A and B has been added. All links are two ways. The cost for each link is specified as the number of minutes needed to travel across the link, where two links have a cost that depends on the traffic, thus modeling the effect of congestion. The parameters X_SA and X_SA denote the number of vehicles traversing the respective links. Thus, the number of minutes to travel across these links is proportional to the amount of traffic, with a proportionality constant of unity. The demand is assumed to be unchanged by the addition of the new link. One hundred users want to drive their vehicles from the source node S to the node destination node D. Each driver wants to minimize their total trip time.

Table 1: Costs for Each Link

Link	Cost
SA	XSA Minutes
SB	110 Minutes
AD	110 Minutes
BD	XBD Minutes
AB	Either infinity or 5 Minutes

Each user has a choice of two routes, ignoring purposeless cycles, in the network on the left. These routes pass through nodes S, A, and D, or through nodes S, B, and D. The addition of the "short-cut" provides two additional routes, through nodes S, A, B, and D, and through nodes S, B, A, and D.

My method of analysis is an equilibrium assignment of users to routes. John G. Waldrop created this notion of equilibrium, as I understand it. It is an application of Nash equilibrium to transport economics. Bell and Iida call this equilibrium a Deterministic User Equilibrium. The equilibrium assignments in the example are shown as green lines in the figure. On the left, 50 drivers choose each of the two routes, and each driver's trip requires 160 minutes. On the right, all 100 drivers choose the route S, A, B, and D. Each driver takes 205 minutes to complete their trip.

To see why these are equilibria, consider what happens if a single driver deviates from the equilibrium assignment. For example, suppose a driver of the left who has previously chosen the route S, A, and D selects the route S, B, D. The cost for the congested link BD will rise from 50 minutes to 51 minutes, and his total trip time will now be 161 minutes, an increase from the previous 160 minutes. In this model, a driven will not choose to be worse off in this way. Symmetrically, a driver assigned to the route S, B, and D will not decide to switch to the route S, A, and D.

Once the shortcut, AB, has been added, the analysis requires tabulating a few more trips. Suppose a driver swithes from the equilibrium route on the right to the route:

• S, A, and D or S, B, and D: In each case, the new route includes one congested link which all 100 drivers still traverse. The total trip time is 210 minutes, an undesirable increase over the equilibrium trip time of 205 minutes.
• S, B, A, and D: All links in this route have a fixed cost. Total trip time is 225 minutes, also an increase over the equilibrium trip time.

So here is a (long-established) case in which improvements to a transport network result in optimizing individuals becoming worse off.

References

• Satoru Fujishige, Michel X. Goemans, Tobias Harks, Britta Peis, and Rico Zenklusen (2017). Matroids are immune to Braess' Paradox. Mathematics of Operation Research. V. 42, Iss. 3: 745-761.
• M. G. H. Bell and Y. Iida (1997). Transportation Network Analysis. New York: John Wiley & Sons.

Structural Economic Dynamics with a Choice of Technique: A Numerical Example

A Bifurcation Diagram with Two Temporal Paths

I have a working paper with the post title. Here's the abstract:

This article illustrates the application of bifurcation analysis to structural economic dynamics with a choice of technique. A numerical example of the Samuelson-Garegnani model is presented in which technical change is introduced. Examples of temporal paths through the parameter space illustrate variations of the wage frontier. A single technique is initially uniquely cost-minimizing for all feasible rates of profits. Eventually, the technique for which coefficients of production decrease at the fastest rates is always cost-minimizing. During the transition between these positions, reswitching, the recurrence of techniques, and capital-reversing can arise. This example emphasizes the importance of fluke switch points and illustrates possible variations in the existence of Sraffa effects.

Elsewhere

• A July 24 Jonathan Schlefer article, "Market Parables and the Economics of Populism: When Experts are Wrong, People Revolt", in Foreign Affairs. Schlefer cites the Cambridge Capital Controversy as a demonstration that the neoliberal political project of remaking the world around unembedded markets is doomed to failure.
• A September 11 interview with Daniel Kahneman in which he basically credits Richard Thaler with inventing behavioral economics. (In his memoirs, Misbehaving, Thaler is also explicit about the disciplinary boundaries between economics and psychology.)
• Richard Thaler's anomaly columns in the Journal of Economic Perspectives
• I have not read Nancy Maclean's Democracy in Chains. Marshall Steinbaum reviews this book in Boston Review. Henry Farrell & Steven Teles respond.

Another ongoing brouhaha is about Alice and Wu's undergraduate paper documenting the sexism on Economic Job Market Rumors.

Others With Points Of View Like Sraffa's

In Production of Commodities by Means of Commodities, Sraffa writes:

"others have from time to time independently taken up points of view which are similar to one or other of those adopted in this paper and have developed them further or in different directions from those proposed here." -- P. Sraffa (1960): pp. vi - vii.

Who is Sraffa talking about? I suggest the following, and their works, at least:

• Tjalling C. Koopmans (1957). Three Essays on the State of Economic Science. New York: McGraw-Hill
• Wassily W. Leontief (1941). Structure of the American Economy, 1919-1929.
• Jacob T. Schwartz (1961). Lectures on the Mathematical Method in Economics. New York: Gordon & Breach.
• John Von Neumann (1945-1946). A Model of General Economic Equilibrium, A Model of General Economic Equilibrium. V. 13, No. 1: pp. 1-9.

I thought about listing David Hawkins and Herbert Simon, given how frequently the Hawkins-Simon condition is cited in expositions of Leontief input-output analysis. I might also mention Nicholas Georgescu-Roegen, the creator of the non-substitution theorem. The work of Ladislaus Bortkiewicz, Georg von Charasoff, Vladimir K. Dmitriev, and Robert Remak, as I understand it, mostly predates Sraffa's long period of preparation of his masterpiece.

Economic Impact Of Regional Disasters: A Job For Input-Output Analysis?

This post, unfortunately, is inspired by current events.

Economists can provide guidance on disaster recovery - for example, from earthquakes and hurricanes.

Economists, for a long time, have been developing input-output models of local economies and interactions between them. I think of Walter Isard as a pioneer here. Such models are of practical importance to my post topic.

Regional input-output models can describe disasters with either a supply-side or demand-side approach. In a supply-side approach, the output of an industry is reduced because the inputs into that industry are not available at the pre-disaster level. Some of the outputs of that industry are inputs into other industries. Other outputs satisfy final demands, for example, for household consumption. Input-output modeling can help trace these consequences.

In a demand-side approach, an industry's output is constrained because those who purchase its outputs cannot do so at the pre-disaster level. If those industries who purchase your products are suddenly reduced in size, you must need cut back your output, too.

Some issues arise here. Can supply-side and demand-side approaches be combined without double counting? How should one model the effects of external infusions of aid? Multiplier effects seem to sit comfortably with the demand side approach. The assumption of fixed coefficients in the Leontief input-output approach seems to be an important restriction here. When it comes to modeling resiliency, I think of the work of Adam Rose. Apparently, some use Computational General Equilibrium (CGE) models for this reason.

I do not know enough to have a firm opinion of CGE models. I have the impression that "Computational" is a misnomer; it does not relate to computational theory and Turing machines, as studied in computer science. I am also not sure that the GE in CGE is what I understand as GE. Anyways, practical considerations interact here with the ideological demands impacting economic theory. I like to think that economists are useful in the hard problem of what to do when disaster strikes.

Reference

• Walter Isard. 1951. Interregional and Regional Input-Output Analysis: A Model of a Space-Economy. Review of Economics and Statistics. Vol. 33, No. 4: pp. 318-328.