P LAN DE N EGOCIOS Y C ALENDARIO DE I NVERSIONES Y D ESINVERSIONES Objetivo de Inversión

To better understand the scope of the results, it is worth briefly exploring the behavioral assumptions, and in particular SELF. Remember that this assumption states that, if all the prices realized at an institution z are weakly better for a trader k than the price p(k) at which that trader is currently trading, there is some positive probability that k leaves his current institu- tion to some active one, for instance to z. One natural possibility yielding alternative assumptions is to capture more optimistic or pessimistic behavior. Consider the following possibility.

Chapter 1 Trader Matching and the Selection of Market Institutions

OPT. Consider an arbitrary trader k and a state-matching pair (ω, γ). If k is matched and there is an institution z 6= ω(k) that features some price that is weakly better than p(k) (larger for k ∈ S, smaller for k ∈ B), then there is positive probability that k switches to some active institution other than ω(k). If k is unmatched, then k has positive probability of switching to every institution.

Under this (extreme) assumption, a trader can be interpreted as being optimistic because he focuses on the better prices at z, ignoring the ones worse than p(k). For instance, if he actually switches to z, one interpretation is that after switching the trader believes that he will be able to achieve the best observed outcome at the new institution even if, in the previous period, it was only obtained by some traders there. Obviously, OPT implies SELF and the results derived above hold. In particular, in the presence of D3, the characterization of stochastically stable institutions identified in Proposition 2 remains unchanged.

For general dynamics (in the absence of D3), Theorem 1 shows that, given ACT and SELF, a centralized institution is stochastically stable indepen- dently of the details of the dynamics, of which other institutions are avail- able and of what their specific characteristics are. If SELF is strengthened to OPT, it can be shown that all other matching-efficient but decentralized institutions are stochastically stable for all possible dynamics and alternative institutions. That is, a characterization as that in Theorem 2 holds.

Proposition 2. Let Z be an arbitrary set of institutions with z0 ∈ Z. Under

D1, D2, ACT, and OPT a non-trivial institution z ∈ Z is stochastically stable if and only if it is matching-efficient.

To gain some quick intuition, consider for example the double one-to- many institution zD (recall Example 3). This institution usually features two

prices which are highly asymmetric (a monopolistic price and a monopsonistic one). Hence, zD is rarely attractive for conservative traders as they require

both prices to be better than the one they trade at. However, for optimistic traders this asymmetry makes zD always appealing for at least one market

side. Thus OPT facilitates a transition towards the double one-to-many institution rendering it a long-run equilibrium for general learning dynamics. The result above is instructive for the general research agenda, since it does not rely on restrictions on the dynamics but rather concentrates on a subclass of behavioral rules. For instance, as long as behavioral rules fulfilling OPT are considered reasonable, Proposition 2 shows that there is no reasonable strengthening of the current assumptions which would render centralized institutions uniquely stochastically stable.

Results as Proposition 2 are of course less robust than Theorem 1, as they hinge on more restrictive assumptions. One can conceive other behavioral rules or dynamics for which the alternative institutions fail to be stable in the long run. Consider, for instance, a pessimistic behavioral rule as follows. PES. Consider an arbitrary trader k and a state-matching pair (ω, γ). If there exists an institution other than ω(k) that is attractive for k, then k switches to an institution z 6= ω(k) with positive probability if and only if z is attractive for k. If k is unmatched and all institutions other than ω(k) are inactive, then k has positive probability of switching to every institution.

Obviously, if a behavioral rule satisfies PES, it also fulfills SELF, but it must violate OPT (note that PES also implies ACT). A trader fulfilling this assumption can be interpreted as being overly cautious, since he will never switch to an institution where some realized price is worse than the one he is currently trading at. Since the double one-to-many institution is matching- efficient it follows from Theorem 2 that it is also stochastically stable under PES, provided the dynamics fulfills D3. However, under slower dynamics, this result is not true any more.

Proposition 3. Let Z = {z0, zD} and n, m > 3. Assume PES. Under asyn-

chronous learning, the double one-to-many institution zD is not stochastically

stable.

In general, the intuition is that for slow dynamics and pessimistic (or cautious) behavioral rules, the attractiveness of the double one-to-many in- stitution vanishes and this institution fails to be stochastically stable. Again,

Chapter 1 Trader Matching and the Selection of Market Institutions

this result is instructive. While for fast dynamics (fulfilling D3) a full, simple characterization of the class of stochastically stable institutions is feasible (as given by Theorem 2), the results above prove that for general dynamics such a simple characterization is impossible. Institutions as the double one-to- many example are stochastically stable for all dynamics and certain types of behavioral rules fulfilling SELF, but stop being stochastically stable for the same dynamics and other types of behavioral rules which, however, do fulfill SELF. Hence, there simply exists no characterization in the absence of as- sumptions on the speed of the dynamics beyond D1–D2 and in the absence of stronger assumptions on the behavioral rules.

1.5 Conclusion

This contribution is a parsimonious step in the study of the selection of market institutions by boundedly rational traders. Our results have been obtained in a setting which is as general as possible in some dimensions (dynamics, trader behavior) but remains necessarily stylized in others. Ac- cordingly, they pave the way for a number of possible extensions which are currently in our research agenda. First, the basic result can be used to study market design under asymmetric rationality as in Alós-Ferrer et al. (2010). Second, combining the results here with Alós-Ferrer and Kirchsteiger (2015) should allow to study more realistic institutions which combine restrictions on trader matching and price biases (rationing). Third, trader heterogeneity and multiple goods can be incorporated, either in buyer-seller models or in general equilibrium settings along the lines of Alós-Ferrer and Kirchsteiger (2010).