We develop a model of algorithmic pricing that shuts down every channel for explicit or implicit collusion while still generating collusive outcomes. We analyze the dynamics of a duopoly market where both firms use pricing algorithms consisting of a parameterized family of model specifications. The firms update both the parameters and the weights on models to adapt endogenously to market outcomes. We show that the market experiences recurrent episodes where both firms set prices at collusive levels. We analytically characterize the dynamics of the model, using large deviation theory to explain the recurrent episodes of collusive outcomes. Our results show that collusive outcomes may be a recurrent feature of algorithmic environments with complementarities and endogenous adaptation, providing a challenge for competition policy.
Lemma 5.5. 0 i,t is exponentially equivalent to N . By contrast, under M1, prices recurrently visit a neighborhood of the collusive price level pC. In Appendix B.3 we construct an unusual sequence of (1,t, 2,t) where the two shocks are almost perfectly positively correlated. While rare, such an event has a small but positive probability, and leads firms to increase their slope coefficient estimates 1 12,t simultaneously at the same rate. With a sufficient increase, the beliefs then enter a region where the expected direction of motion leads away from the stable point. As firms start to move prices together, they increase the estimated slope in the reaction function, which leads to a further increase in prices. What is crucial for such a result to persist is that the threshold to enter the self-reinforcing region is small enough. In Lemma B.2 in Appendix B.3 we show that the threshold gets smaller as 2, or the size of the experimentation gets smaller.
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11,t and 1 In general, as 2 0, the size of exploration decreases, and it becomes more difficult from the small neighborhood of the stable stationary point. In the case of specification M0, the lower bound of the rate function goes to infinity as 2 0, which is a crucial step in showing that 0 i,t is exponentially equivalent to N . In contrast, S(; 2), the large deviation rate function now emphasizing its dependence on the exploration variance, is uniformly bounded from above even if 2 0. Lemma 5.6. lim sup 20 S(; 2) < 5.3. Dynamics of i,t. Recall that i,t is the probability assessment of specification M1 at time t, which evolves according to i,t = i,t1 + t (cid:16) 1 I( i,t > 0 i,t) i,t1 (cid:17) . Since we know that 0 i,t is exponentially equivalent to N , we can write the recursive form as i,t = i,t1 + t (cid:16)
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I( 1 i,t > N ) i,t1 (cid:17) , whose associated ODE is i,t > N (cid:17) Our key result, proved in Appendix B.4, shows that in the limit each duopolist will put probability one on specification M1. This result is a consequence of our previous analysis, that profits are higher under the specification with feedback than the constant specification. (cid:16) 1 i = P i. COLLUSIVE OUTCOMES WITHOUT COLLUSION
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Proposition 5.7. For any 2 t 0, i,t > N (cid:17) Thus each duopolist, who chooses prices and experiments independently, and independently chooses a model specification, will in the long run choose the specification M1. Under this specification, the two firms will recurrently and in almost perfect unison, increase prices to near the cartelized, joint monopoly level, as first observed in Williams (2001). We have shut down all avenues for explicit or implicit collusion, and yet the pricing algorithms recurrently lead to prices near collusive levels. Thus the observation of parallel price increases to supra-competitive levels cannot on its own be used as evidence of collusion. In the remainder of the paper we further explore the key features of the model and its structure which underlie these results. (cid:16) 1 lim t P
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