We consider a robust version of the classic problem of optimal monopoly pricing with incomplete information. In the robust version, the seller faces model uncertainty and only knows that the true demand distribution is in the neighborhood of a given model distribution.
We characterize the optimal pricing policy under two distinct, but related, decision criteria with multiple priors: (i) maximin expected utility and (ii) minimax expected regret. The resulting optimal pricing policy under either criterion yields a robust policy to the model uncertainty.
While the classic monopoly policy and the maximin criterion yield a single deterministic price, minimax regret always prescribes a random pricing policy, or equivalently, a multi-item menu policy. Distinct implications of how a monopolist responds to an increase in uncertainty emerge under the two criteria.
We consider a robust version of the classic problem of optimal monopoly pricing with incomplete information. In the robust version of the problem the seller only knows that demand will be in a neighborhood of a given model distribution.
We characterize the optimal pricing policy under two distinct, but related, decision criteria with multiple priors: (i) maximin expected utility and (ii) minimax expected regret. While the classic monopoly policy and the maximin criterion yield a single deterministic price, minimax regret always prescribes a random pricing policy, or equivalently, a multi-item menu policy. The resulting optimal pricing policy under either criterion is robust to the model uncertainty. Finally we derive distinct implications of how a monopolist responds to an increase in ambiguity under each criterion.
We consider a robust version of the classic problem of optimal monopoly pricing with incomplete information. The robust version of the problem is distinct in two aspects: (i) the seller minimizes regret rather than maximizes revenue, and (ii) the seller only knows that the true distribution of the valuations is in a neighborhood of a given model distribution.
We characterize the robust pricing policy as the solution to a minimax problem for small and large neighborhoods. In contrast to the classic monopoly policy, which is a single deterministic price, the robust policy is always a random pricing policy, or equivalently, a multi-item menu policy. The responsiveness of the robust policy to an increase in risk is determined by the curvature of the static profit function.
We consider a robust version of the classic problem of optimal monopoly pricing with incomplete information. The robust version of the problem is distinct in two aspects: (i) the seller minimizes regret rather than maximizes revenue, and (ii) the seller only knows that the true distribution of the valuations is in a neighborhood of a given model distribution.
We characterize the robust pricing policy as the solution to a minimax problem for small and large neighborhoods. In contrast to the classic monopoly policy, which is a single deterministic price, the robust policy is always a random pricing policy, or equivalently, a multi-item menu policy. The responsiveness of the robust policy to an increase in risk is determined by the curvature of the static profit function.
We consider a robust version of the classic problem of optimal monopoly pricing with incomplete information. In the robust version of the problem the seller only knows that demand will be in a neighborhood of a given model distribution.
We characterize the optimal pricing policy under two distinct, but related, decision criteria with multiple priors: (i) maximin expected utility and (ii) minimax expected regret. While the classic monopoly policy and the maximin criterion yield a single deterministic price, minimax regret always prescribes a random pricing policy, or equivalently, a multi-item menu policy. The resulting optimal pricing policy under either criterion is robust to the model uncertainty. Finally we derive distinct implications of how a monopolist responds to an increase in ambiguity under each criterion.
A social choice function is robustly implemented if every equilibrium on every type space achieves outcomes consistent with a social choice function. We identify a robust monotonicity condition that is necessary and (with mild extra assumptions) sufficient for robust implementation. Robust monotonicity is strictly stronger than both Maskin monotonicity (necessary and almost sufficient for complete information implementation) and ex post monotonicity (necessary and almost sufficient for ex post implementation). It is equivalent to Bayesian monotonicity on all type spaces. It requires that there not be too much interdependence of types. We characterize robust monotonicity for some interesting economic environments. We identify conditions where, if robust implementation is possible, it is possible in a direct mechanism. We identify conditions where, if robust implementation is not possible, virtual robust implementation is not possible either.
A social choice function is robustly implemented if every equilibrium on every type space achieves outcomes consistent with a social choice function. We identify a robust monotonicity condition that is necessary and (with mild extra assumptions) sufficient for robust implementation. Robust monotonicity is strictly stronger than both Maskin monotonicity (necessary and almost sufficient for complete information implementation) and ex post monotonicity (necessary and almost sufficient for ex post implementation). It is equivalent to Bayesian monotonicity on all type spaces. It requires that there not be too much interdependence of types. We characterize robust monotonicity for some interesting economic environments. We identify conditions where, if robust implementation is possible, it is possible in a direct mechanism. We identify conditions where, if robust implementation is not possible, virtual robust implementation is not possible either.
We consider the optimal design of flexible use in a digital-rights-management policy. The basic model considers a single distributor of digital goods and a continuum of consumers. Each consumer can acquire the digital good either as a licensed product or an unlicensed copy. The availability of (or access to) unlicensed copies is increasing both in the number of licensed copies and in the flexibility accorded to licensed copies. We thus analyze the optimal design of flexibility in the presence of unlicensed distribution channels (the “greynet”).
We augment the basic model by introducing a “secure platform” that is required to use the digital good. We compare the optimal design of flexibility in the presence of a platform to the one without a platform. Finally, we analyze the equilibrium provision when platform and content are complimentary goods but are distributed and priced by different sellers.
We analyze the problem of fully implementing a social choice set in ex post equilibrium. We identify an ex post monotonicity condition that is necessary and — in economic environments — sufficient for full implementation in ex post equilibrium. We also identify an ex post monotonicity no veto condition that is sufficient.
Ex post monotonicity is satisfied in all single crossing environments with strict ex post incentive constraints. In many economically significant environments, ex post implementation can be achieved in the direct mechanism.
We show by means of two classic examples that ex post monotonicity does not imply nor is it implied by Maskin monotonicity (necessary and almost sufficient for complete information implementation). The single unit auction with interdependent valuations is shown to satisfy ex post monotonicity but not Maskin monotonicity. Ex post implementation in the direct mechanism is also possible in this case. We describe an example where the Pareto correspondence fails ex post monotonicity but Maskin monotonicity is satisfied.
We analyze the problem of fully implementing a social choice set in ex post equilibrium. We identify an ex post monotonicity condition that is necessary and — in economic environments — sufficient for full implementation in ex post equilibrium. We also identify an ex post monotonicity no veto condition that is sufficient.
Ex post monotonicity is satisfied in all single crossing environments with strict ex post incentive constraints. In many economically significant environments, ex post implementation can be achieved in the direct mechanism.
We show by means of two classic examples that ex post monotonicity does not imply nor is it implied by Maskin monotonicity (necessary and almost sufficient for complete information implementation). The single unit auction with interdependent valuations is shown to satisfy ex post monotonicity but not Maskin monotonicity. Ex post implementation in the direct mechanism is also possible in this case. We describe an example where the Pareto correspondence fails ex post monotonicity but Maskin monotonicity is satisfied.
Keywords: Ex post equilibrium, Implementation, Single crossing, Interdependent values
We consider the optimal design of flexible use in a digital-rights-management policy. The basic model considers a single distributor of digital goods and a continuum of consumers. Each consumer can acquire the digital good either as a licensed product or an unlicensed copy. The availability of (or access to) unlicensed copies is increasing both in the number of licensed copies and in the flexibility accorded to licensed copies. We thus analyze the optimal design of flexibility in the presence of unlicensed distribution channels (the “greynet”).
We augment the basic model by introducing a “secure platform” that is required to use the digital good. We compare the optimal design of flexibility in the presence of a platform to the one without a platform. Finally, we analyze the equilibrium provision when platform and content are complimentary goods but are distributed and priced by different sellers.
Keywords: Digital Rights Management, Platform, Flexibility, Piracy
We develop a dynamic model of experience goods pricing with independent private valuations. We show that the optimal paths of sales and prices can be described in terms of a simple dichotomy. In a mass market, prices are declining over time. In a niche market, the optimal prices are initially low followed by higher prices that extract surplus from the buyers with a high willingness to pay. We consider extensions of the model to integrate elements of social rather than private learning and turnover among buyers.
We develop a dynamic model of experience goods pricing with independent private valuations. We show that the optimal paths of sales and prices can be described in terms of a simple dichotomy. In a mass market, prices are declining over time. In a niche market, the optimal prices are initially low followed by higher prices that extract surplus from the buyers with a high willingness to pay. We consider extensions of the model to integrate elements of social rather than private learning and turnover among buyers.
We develop a dynamic model of experience goods pricing with independent private valuations. We show that the optimal paths of sales and prices can be described in terms of a simple dichotomy. In a mass market, prices are declining over time. In a niche market, the optimal prices are initially low followed by higher prices that extract surplus from the buyers with a high willingness to pay. We consider extensions of the model to integrate elements of social rather than private learning and turnover among buyers.
The mechanism design literature assumes too much common knowledge of the environment among the players and planner. We relax this assumption by studying implementation on richer type spaces.
We ask when ex post implementation is equivalent to interim (or Bayesian) implementation for all possible type spaces. The equivalence holds in the case of separable environments; examples of separable environments arise (1) when the planner is implementing a social choice function (not correspondence); and (2) in a quasilinear environment with no restrictions on transfers. The equivalence fails in general, including in some quasilinear environments with budget balance.
In private value environments, ex post implementation is equivalent to dominant strategies implementation. The private value versions of our results offer new insights into the relation between dominant strategy implementation and Bayesian implementation.