Introduction
Centralized College Admissions under Application Constraints
Introduction
Among the students considered by the school, the school accepts the most desirable candidates up to its capacity. If she's interested, she'll apply to schools with a specific major and get a benefit equal to the quality of the school she's finally accepted to.
Model
In the first round, students' application files go to the schools at the top of their preference lists. In the second round, students' files go to the second school on their preference lists if they are rejected by their reported preferred schools.
Equilibrium strategies
Under BA, the student can be accepted by Sj under Pˆ, but is rejected by both Si and Sj under Pˆ0. If she is not accepted by S2, her file goes to S1, and she will compete with the student provisionally accepted by S1 in the first round.
Efficiency
Otherwise, under BA, the matching outcome is more complicated and not necessarily assortative, but the bottom school and bottom student will not be left unmatched. Under BA, efficiency loss is largely attributed to the possibility that the middle student is unmatched.
Conclusion and Discussion
Appendix
In addition, receiving the grant itself is an honor for the scientist (personal gain). Each proponent's utility comes from the private value of his project and the grant payment if it is awarded. This mechanism generates the same welfare for the grant issuer if she is faced with a flexible budget of the same size.
If all proposers truthfully report their private values, the grantor's utility function is This paper discusses optimal mechanism design problems related to the grant issuing procedure under different environments.
Optimal mechanisms with Type-Dependent Outside Options
Introduction
It is natural to ask: what is the optimal mechanism for a seller when potential buyers have valued outside option heterogeneously. This form of the optimal mechanism is surprisingly simple considering the complexity of the set of mechanisms that the seller can consider. Nevertheless, I show that the optimal mechanism's allocation and payment rules are independent of the set of visitors.
The optimal mechanism is generally intractable due to the complicated interactions between sellers and buyers. In the present paper, I explore the optimal mechanism given a specific setting of external options without imposing any.
Model
The entry fee is paid from the bidders to the auctioneer, and aims to screen out high cost bidders. According to the revelation principle, it is without loss of generality to focus on incentive-compatible direct mechanisms. With any given mechanism, each buyer must make two decisions: (1) to participate in the seller's sale or to exercise the outside option (participation strategy), and (2) if participating, which type to sell to the seller report ( reporting strategy).
In an equilibrium, no buyer can obtain a higher expected profit by deviating in either the participation strategy or the reporting strategy. An optimal mechanism conducts a second-price auction among the seller's visitors with either a reserve price or a display fee.
Solve for optimal mechanisms
Compared to the previous buyer, this buyer's surplus due to visiting the seller increases by approximately q(u0)k∆, where q(·) ≤ 1 is the probability that the buyer will get the item from the seller. Now with endogenous participation, the seller may have an incentive to pay a strictly positive appearance fee to induce higher value buyers to participate in the sale. In order to keep them in the sale, the seller has to make additional sacrifices, so the loss of these buyers due to the abandonment of the external option is well compensated.
Note that it can be interpreted as the reserve price, which is set by the seller to screen out low-value buyers. From (3.4), it is optimal for the seller to assign the item to a buyer with the highest adjusted virtual value.
Conclusion and discussion
The resolution of the tension between "price" and "quantity" depends on the specific shape of the distribution of private values, as well as on the value of the parameters and p. A general feature of the reserve price is that it never exceeds Myerson's optimal reserve price. With the type-dependent outside option, a low reserve price has the merit of increasing sales participation.
Since the selling participants all have a zero-value outside option, the optimal reserve price coincides with Myerson's optimal reserve price, given that the buyers' private values are distributed according to the original truncated atp distribution. More future work is required to obtain a detailed characterization of the optimal mechanism in this configuration.
Appendix
To capture the essence of the grant issuance process, this paper considers the following model. Specifically, if the grant maker faces a flexible budget, the optimal mechanism does not depend on the specification of individual rationality. Furthermore, the higher the public value a project generates, the larger the amount offered by the grant issuer.
If the grant issuer is faced with a fixed budget, the shape of the optimal mechanisms varies with different specifications of individual rationality. This value is an adjustment of the private value of each proposal due to the budget constraint and the sacrifice made by the grant issuer to induce a true ratio.
Optimal Grant-issuing Mechanisms
Introduction
To apply for a grant, applicants must simultaneously submit their project proposals to the grant issuer. Such flexibility would allow the grant issuer to adjust overall costs according to the quality of the proposals submitted. In the first scenario, there is no contract or legal obligation for the grant applicant to carry out the project after receiving the grant.
In this case, the grant provider does not have to worry about the grant being rejected. This article studies the subsidy granting process in a more general setting and focuses on optimal mechanisms.
Model
The proponent will decide whether or not to complete the project without grant funding. To encourage as many applications as possible, the grant mechanism must satisfy the individual rationality constraint, i.e. Since the calculation of a proposer's utility from applying for the grant varies in different scenarios, different forms of constraints on individual rationality must be taken into account.
Under this constraint, the utility that each proposer can obtain outside the mechanism does not exceed his expected utility from applying for the grant. For simplicity, assumesi >−νi for alli, so that all proposals for the grant issuer are desirable.
Grants with ex ante budget constraint
In contrast, the posterior budget constraint requires that the total payment does not exceed the budget for every possible payment realization. When the budget increases, so that B > −¯νi, it is possible for the grantor to support this highly beneficial project. For example, if all proposals have private values of exactlyπi, the total payments exceed the budget sincePN.
In this problem, the budget constraint is replaced with an additional term in the objective function −λPN. The objective function in Problem 3 and the Lagrangian of Problem 2 that incorporates the budget constraint are very similar.
Grants with ex post budget constraint
Note that if the selection rule is such that a grant is either awarded with probability 1 or not awarded at all, the ex post weak budget constraint coincides with the ex post budget constraint. To this end, the constructed ex post payoff rule preserves the expected payoff, but the ex post payoff is adjusted by adding terms with zero expectation in order to satisfy the ex post budget constraint. Then we apply the selection rule in Proposition 4 and adjust the payment rule to satisfy the ex post weak budget constraint.
It is easy to see that this mechanism satisfies the ex post budget constraint and the ex post individual rationality. Moreover, this mechanism generates higher welfare than any marginal choice rules that satisfy the ex post individual rationality and the ex post budget constraint.
Conclusion and discussion
As a result, even if proponents do not have information about public values, they can draw conclusions from the form of the mechanism announced by the grant issuer. In reality, selected proponents may put much less effort into the project after receiving the grant and thus provide lower merit than the original estimate. This case can be excluded when both of the following conditions are met: first, there is no noise in the project quality measurement; and two, the cost of breaking their promises is high enough that no proposer is willing to take the risk.
For grants provided by agencies such as the National Institutes of Health (NIH), the benefit at the end of the grant period is usually very specific. For other grants, especially basic science grants, it is difficult to say whether project failure stems from a lack of effort or from the risk of the project itself.
Appendix
Note that proposers with vi > 0 will realize their private value regardless of the selection outcome, so they have an incentive to underreport (overreport) if the utility from participating in the mechanism grows at a rate lower (higher) than 1 To solve for the optimal thresholds by means of Proposition 6, replace the cutoff selection rule and the fixed payment rule with the grantor's utility function (4.5) and prior budget constraint (4.17), and the problem becomes. As λ increases while keeping πi the same, the left side also increases, since by previous discussion (1−Fi(πi)−πifi(πi)) ≥ 0.
When λ increases, keeping πi the same, the left-hand side also increases and the above condition may fail. Payoff equivalence between rational Bayesian and ex post dominant strategic mechanisms? In: Economic Theory 13.1, pg.
