Arthur's (1989) model of competing technologies is a simple model which is often cited in Economics to illustrate several interesting concepts such as increasing returns to adoption, lock-in phenomena and path dependence. This model has been widely cited in Evolutionary Economics and Computational Economics to emphasise the complex evolving dimension of various economic phenomena –in sharp contrast to the equilibrium approach of neoclassical economics. Leydesdorff (2001) provides an interesting extension: a mixture model of cultural diffusion of technological innovations that combines Arthur's model with Axelrod's (1997) model of dissemination of culture (which is also analysed in this appendix).
In the model, two competing technologies -labelled A and B- compete in a market of indecisive adopters. The process of technology adoption is modelled as a random sequential arrival of agents at the market. On arriving, each agent must decide which technology to adopt. To make this decision, the agent considers her natural (a priori) preference and also the number of adopters of each technology at that moment. In the presence of increasing (decreasing) returns to adoption, the more adopters a technology has, the more (less) attractive it becomes. Thus, assuming increasing returns to scale, an agent may -in principle- have a preference for technology A, but decide to adopt technology B because the number of adopters of B is much greater than the number of adopters of A.
To be precise, there are two types of newcomers, R and S, who show different natural preferences towards each technology: in the absence of returns to adoption, R agents prefer technology A, whereas S agents prefer technology B. Table 1 shows the payoff matrix that governs the agents' decision once returns to adoption are taken into account: each agent chooses the technology that provides her with the greater payoff. (We assume that ties are resolved randomly.)
Technology A | Technology B | |
R agent | a_{R} + r·n_{A}(t) | b_{R} + r·n_{B}(t) |
S agent | a_{S} + s·n_{A}(t) | b_{S} + s·n_{B}(t) |
Table 1. Payoff matrix of technology adoption for each type of agent (R or S), given the numbers n_{A}(t) and n_{B}(t) of current adopters of each technology (A and B) in time-step t. R agents show a natural preference for technology A (a_{R} > b_{R}), whereas S agents' natural preference is for technology B (a_{S} < b_{S}). Parameters r and s are used to set the type of returns to adoption: {r > 0 & s > 0} induce increasing returns, {r < 0 & s < 0} denote diminishing returns, and null values {r = 0 & s = 0} imply constant returns. |
Arthur (1989) analyses a model where the arriving agent is equally likely to be of either type (p_{R} = p_{S} = 0.5). In particular, in the presence of increasing returns, Arthur (1989) proves that sooner or later the system locks in one of the two competing technologies, i.e. there is a point after which every newcomer arriving at the market chooses the same technology regardless of her natural preference. Moreover, Arthur points out that the order in which a certain collection of historical events take place (e.g. the order in which a certain set of agents arrive at the market) influences the final outcome. Arthur's model is useful to understand the mechanisms that have led to some lock-ins in the real world, e.g. the QWERTY typewriter and the VHS video recording system. In these two cases the market locked in an inferior technology despite the subsequent arrival of superior technologies.
Arthur uses the time series of the market share of technology A ( n_{A}(t)/(n_{A}(t)+n_{B}(t)) ) to study the system. For constant and diminishing returns to adoption, the model does not yield very interesting results: for constant returns the competing process can be formalised as an unbounded random walk whilst assuming diminishing returns introduces two reflecting barriers in the random walk. In either case the market share tends to 0.5 in the long term (the market is split 50-50).
Here we focus on the model with increasing returns to adoption and analyse it within the Markov chain framework. To do that, we define d(t) as the difference in the number of adopters.
d(t) = n_{A}(t) - n_{B}(t)
Note that the system reaches an A-lock-in state when an S agent arriving at the market adopts technology A instead of following her natural preference. It is clear that from that point onwards every agent will choose technology A regardless of her natural preference. Taking into account the payoff matrix of technology adoption, this A-lock-in occurs when:
a_{S} + s·n_{A}(t) > b_{S} + s·n_{B}(t)
d(t) = n_{A}(t) - n_{B}(t) > (b_{S} - a_{S})/s = S_{0}
This equation represents an absorbing barrier, a point of no return where technology A becomes dominant. Similarly, we can derive the other absorbing barrier in the system, i.e. the B-lock-in state:
d(t) = n_{A}(t) - n_{B}(t) < (b_{R} - a_{R})/r = R_{0}
For convenience -given that d(t) takes integer values only- we define the absorbing values n_{S} and n_{R} as:
n_{S} = IntegerPart(S_{0}) + 1
n_{R} = IntegerPart(R_{0}) - 1
Note that n_{R} is negative because a_{R} > b_{R}. We can now define a new variable d^{*}(t) which fully characterises the state of the system in time-step t:
d^{*}(t) = n_{R} | if d(t) ≤ n_{R} | |
d^{*}(t) = d(t) | if n_{R} < d(t) < n_{S} | |
d^{*}(t) = n_{S} | if d(t) ≥ n_{S} |
Thus, the space state of this Markov chain representation of Arthur's model is the set of possible values that d^{*}(t) may take: {n_{R}, n_{R}+1, ..., -1, 0, 1, ..., n_{S}-1, n_{S}}. In Arthur's model, the transition probabilities between non-absorbing consecutive states are 0.5, because the arrivals of either type of agent are equiprobable (p_{R} = p_{S} = 0.5). Thus, the competing process can be described as a 1-dimensional random walk with absorbing barriers. In the more general case where p_{R} ≠ 0.5 ≠ p_{S}, the transition probabilities from non-absorbing states (i.e. n_{R} < d(t) < n_{S}) are:
Pr{ d^{*}(t+1) = d^{*}(t) + 1 } = p_{R} |
Pr{ d^{*}(t+1) = d^{*}(t) - 1 } = p_{S} = 1 - p_{R} |
This THMC has two absorbing states ( d^{*}(t)=n_{S} and d^{*}(t)=n_{R} ) and n_{S}+n_{R}-1 states belonging to the same non-closed communicating class. (Note that it is possible to reach either absorbing state from any non-absorbing state.) We can easily evaluate the transient distribution in any time-step t using Proposition 1 in our paper:
a^{(t)} = a^{(0)}·P^{t}
As for the asymptotic dynamics of the model, using Proposition 3 we can state that sooner or later the system will end up in one of the two absorbing states (i.e. all the non-absorbing states are transient). The probability of ending up in each particular absorbing state depends on the initial conditions (and -naturally- on the value of p_{R}, which determines the transition probabilities). In fact, these probabilities are not difficult to derive analytically. In the unbiased case where p_{R} = p_{S} = 0.5:
Pr{ A-lock-in | d^{*}(0)=x } = (n_{S}-x) / (n_{S}-n_{R})
= 1 - Pr{ B-lock-in | d^{*}(0)=x }
In the general case where p_{R} > 0.5 > p_{S}:
Pr{ A-lock-in | d^{*}(0)=x } = (1-(p_{S}/p_{R})^{x-nR}) / (1-(p_{S}/p_{R})^{nS-nR})
= 1 - Pr{ B-lock-in | d^{*}(0)=x }
The proof of this formula can be found in the literature looking for "gambler's ruin".
The analysis of the model with diminishing returns is analogous to the one explained here. It is not difficult to see that the system under this alternative regime does not have any absorbing states (the barriers of the random walk are now reflecting rather than absorbing). The new THMC is irreducible and periodic (with period 2) and, as such, has a unique occupancy distribution independent of the initial conditions. This periodic THMC does not have a unique limiting distribution; instead, it cycles through 2 probability functions depending on the initial conditions.
ARTHUR W (1989) Competing Technologies, Increasing Returns, and Lock-In By Historical Events. Economic Journal 99(394), pp. 116-131.
AXELROD R (1997) The dissemination of culture - A model with local convergence and global polarization. Journal of Conflict Resolution 41(2), pp.203-226.
LEYDESDORFF L (2001) Technology and culture: The dissemination and the potential 'lock-in' of new technologies. Journal of Artificial Societies and Social Simulation 4(3)5. http://jasss.soc.surrey.ac.uk/4/3/5.html