Source
Arthur, W. B., “On Competing Technologies and Historical Small Events: The Dynamics of Choice under Increasing Returns”, IIASA WP-83-90 (1983), the precursor of the Economic Journal article (1989) in LIT-1314. Read in full in NOTE-1052. The derivation below is the record’s.
What was actually shown
Agents arrive one at a time, each an R or an S with probability one half, and adopt the technology with the higher return to them. With n_A and n_B previous adopters, an R-agent gets a_R + r n_A from A and b_R + r n_B from B, and an S-agent a_S + s n_A and b_S + s n_B, with a_R > b_R and a_S < b_S. Let d = n_A − n_B.
Switching gaps (increasing returns, r, s > 0). An R-agent chooses B iff b_R + r n_B > a_R + r n_A, that is d < −Δ_R with Δ_R = (a_R − b_R)/r. An S-agent chooses A iff d > Δ_S with Δ_S = (b_S − a_S)/s. Inside (−Δ_R, Δ_S) each type takes its own favourite, so d moves ±1 with probability one half: a fair walk. Once d passes Δ_S, both types choose A, d only rises, and it never returns. The same holds for B below −Δ_R. These are the paper’s absorbing barriers.
Lock-in and its odds. A fair walk between two absorbing barriers is absorbed with probability one, so the share of A goes to 0 or 1. By gambler’s ruin from d = 0, A wins with probability Δ_R/(Δ_R + Δ_S) = s(a_R − b_R)/(s(a_R − b_R) + r(b_S − a_S)). The observer knows this number and not the outcome.
Diminishing returns (r, s < 0). The inequalities reverse: an R-agent switches to B when A is far enough ahead, and an S-agent to A when B is. So both types choose the technology behind, the barriers reflect, d stays between constants, and d/n → 0. Shares go to one half from every sequence of arrivals: predictable and ergodic.
Inflexibility. After absorption, the gap in returns between the leader and the other grows linearly in n. A change of either returns function by a small ε shifts the barrier by ε/r and is passed, so it does not change any later choice.
Simulation, here. 2,000 runs of 4,000 arrivals each, with a_R = 1, b_R = 0, a_S = 0, b_S = 1. With r = s = 0.1, every run locked in, and A won 47% of them (predicted 50%). With b_S = 2, it won 30.5% (predicted 33.3%). With s = 0.3, it won 73% (predicted 75%). The shortfall is the discrete barriers and ties broken toward B, and it shrank with finer barriers (r = s = 0.04 and b_S = 2: 31.3% of 1,000 runs, with 3% not yet locked by 20,000 arrivals, predicted 33.3%; s = 0.12: 74.6%, predicted 75%). With r = s = −0.1 the share of A was 0.500 with a standard deviation of 0.0008. With r = s = 0 it was 0.500 with a standard deviation of 0.008.
What could have failed: the walk could have wandered for ever between the barriers, keeping both technologies in the market. It does not, because a fair walk on a finite interval is absorbed.
What this does not say
- Not that any real technology won this way. The paper’s QWERTY, rail-gauge and steam-car stories are illustrations it does not analyse.
- Not that the winner is worse. Only that it need not be better. With returns rising at different rates, “the adoption process may easily become inefficient”. The efficiency criterion is weak: no unadopted option better than an adopted one.
- Not beyond the model. Expectations, pricing, sponsors who can subsidize, and agents who can wait are excluded and named as open. The extensions to N technologies, unequal type shares and nonlinear returns are stated without proof. With returns that level off, the market can be shared.
- Not about the 1989 article’s model. That text was not read.
Connections
- CLAIM-408. Both are symmetric dynamics that select one side. They differ in what happens when the conditions are reversed. In the pitchfork, restoring κ > α restores the equal allocation. Here, lock-in is absorption in accumulated adoptions. Arthur: “Small subsidies to the excluded choice will not re-establish it; large subsidies, on the other hand, will swing the market and drive the dominant choice out.” The persistence is stored in a state variable that the reversal does not touch.
- CASE-112. Arthur’s advice to “‘tilt’ the process … toward the favored technology at the outset” is the tilt h applied before the symmetric state breaks. After lock-in a small tilt does nothing, as in CASE-112’s hysteresis in h.
- THEORY-429. North carries increasing returns to institutions.