“Path Dependence,” S. Page (2006)

When we talk about strategic equilibrium, we can talk in a very formal sense, as many refinements with their well-known epistemic conditions have been proposed, the nature of uncertainty in such equilibria has been completely described, the problems of sequential decisionmaking are properly handled, etc. So when we do analyze history, we have a useful tool to describe how changes in parameters altered the equilibrium incentives of various agents. Path dependence, the idea that past realizations of history matter (perhaps through small events, as in Brian Arthur’s work) is widespread. A typical explanation given is increasing returns. If I buy a car in 1900, I make you more likely to buy a car in 1901 by, at the margin, lowering the production cost due to increasing returns to scale or lowering the operating cost by increasing incentives for gas station operators to operate.

This is quite informal, though; worse, the explanation of increasing returns is neither necessary nor sufficient for history-dependence. How can this be? First, consider that “history-dependence” may mean (at least) six different things. History can effect either the path of history, or its long-run outcome. For example, any historical process satisfying the assumptions of the ergodic theorem can be history-dependent along a path, yet still converge to the same state (in the network diffusion paper discussed here last week, a simple property of the network structure tells me whether an epidemic will diffuse entirely in the long-run, but the exact path of that eventual diffusion clearly depends on something much more complicated). We may believe, for instance, that the early pattern of railroads affected the path of settlement of the West without believing that this pattern had much consequence for the 2010 distribution of population in California. Next, history-dependence in the long-run or short-run can depend either on a state variable (from a pre-defined set of states), the ordered set of past realizations, or the unordered set of past realizations (the latter called path and phat dependence, respectively, since phat dependence does not depend on order). History matters in elections due to incumbent bias, but that history-dependence can basically be summed up by a single variable denoting who is the current incumbent, omitting the rest of history’s outcomes. Phat dependence is likely in simple technology diffusion: I adopt a technology as a function of which of my contacts has adopted it, regardless of the order in which they adopted. Path dependence comes up, for example, in models of learning following Aumann and Geanakoplos/Polemarchakis, consensus among a group can be broken if agents do not observe the time at which messages were sent between third parties.

Now consider increasing returns. For which types of increasing returns is this necessary or sufficient? It turns out the answer is, for none of them! Take again the car example, but assume there are three types of cars in 1900, steam, electric and gasoline. For the same reasons that gas-powered cars had increasing returns, steam and electric cars do as well. But the relative strength of the network effect for gas-powered cars is stronger. Page thinks of this as a biased Polya process. I begin with five balls, 3 G, 1 S and 1 E, in an urn. I draw one at random. If I get an S or an E, I return it to the urn with another ball of the same type (thus making future draws of that type more common, hence increasing returns). If I draw a G, I return it to the urn along with 2t more G balls, where t is the time which increments by 1 after each draw. This process converges to having arbitrarily close to all balls of type G, even though S and E balls also exhibit increasing returns.

Why about the necessary condition? Surely, increasing returns are necessary for any type of history-dependence? Well, not really. All I need is some reason for past events to increase the likelihood of future actions of some type, in any convoluted way I choose. One simple mechanism is complementarities. If A and B are complements (adopting A makes B more valuable, and vice versa), while C and D are also complements, then we can have the following situation. An early adoption of A makes B more valuable, increasing the probability of adopting B the next period which itself makes future A more valuable, increasing the probability of adopting A the following period, and so on. Such reasoning is often implicit in the rhetoric linking market-based middle class to a democratic political process: some event causes a private sector to emerge, which increases pressure for democratic politics, which increases protection of capitalist firms, and so on. As another example, consider the famous QWERTY keyboard, the best-known example of path dependence we have. Increasing returns – that is, the fact that owning a QWERTY keyboard makes this keyboard more valuable for both myself and others due to standardization – is not sufficient for killing the Dvorak or other keyboards. This is simple to see: the fact that QWERTY has increasing returns doesn’t mean that the diffusion of something like DVD players is history-dependent. Rather, it is the combination of increasing returns for QWERTY and a negative externality on Dvorak that leads to history-dependence for Dvorak. If preferences among QWERTY and Dvorak are Leontief, and valuations for both have increasing returns, then I merely buy the keyboard I value highest – this means that purchases of QWERTY by others lead to QWERTY lock-in by lowering the demand curve for Dvorak, not merely by raising the demand curve for QWERTY. (And yes, if you are like me and were once told to never refer to effects mediated by the market as “externalities”, you should quibble with the vocabulary here, but the point remains the same.)

All in all interesting, and sufficient evidence that we need a better formal theory and taxonomy of history dependence than we are using now.

Final version in the QJPS (No IDEAS version). The essay is written in a very qualitative/verbal manner, but more because of the audience than the author. Page graduated here at MEDS, initially teaching at Caltech, and his CV lists quite an all-star cast of theorist advisers: Myerson, Matt Jackson, Satterthwaite and Stanley Reiter!