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经济代写|博弈论代考Game theory代写|ECN614

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经济代写|博弈论代考Game theory代写|Feasibility under Budget Balance

In many mechanism problems with several agents, the “principal” is not allowed to be a net source of funds to the agents. Moreover, the principal must raise enough revenue from the transfers to cover her cost. (In some applications, this cost is identically 0 .) This leads us to consider mechanisms that meet the additional constraint of budget balance:
(BB) $\sum_{i=1}^l t_i(\theta) \leq-C_0(x(\theta))$ for all $\theta .^{22}$
As in subsection 7.3.2, we say that an allocation $y=(x, t)$ is feasible if $x$ is implementable through $t$ and $y$ is individually rational; $y$ is feasible under budget balance if it satisfies BB as well.

One theme of this section will be that efficient allocations are typically not feasible under budget balance when there is incomplete information unless the individual-rationality constraints are very weak. (If budget balance is not required, individual-rationality constraints are irrelevant, as the principal can induce the agents to participate by giving them all very large positive transfers, and efficient allocations are usually feasible.) This kind of inefficiency is different from that in the monopoly-pricing example of section 7.1. There, the competitive outcome where price cquals the monopolist’s cost is both feasible and efficient; the monopolist’s optimal mechanism is inefficient because it is designed to maximize the monopolist’s profit and not social welfare. In contrast, the inefficiency results of this section pertain to all the allocations that are feasible under the budget-balance constraint.

经济代写|博弈论代考Game theory代写|Dominant Strategy vs. Bayesian Mechanisms

Mookherjee and Reichelstein (1989) identify a class of models in which dominant-strategy implementation involves no welfare loss relative to Bayesian implementation. ${ }^{23}$ Suppose that the agents have quasi-linear preferences, and that, for $i=1, \ldots, I$,
$$u_i(x, t, \theta)=V_i\left(x, \theta_i\right)+t_i,$$
where $t_i$ is the principal’s transfer to agent $i$. Mookherjec and Reichelstein allow $x$ to be multi-dimensional, but require that $V_i$ depend on $x$ only through a one-dimensional statistic $h_i(x)$ :
$$u_i(x, t, \theta)=V_i\left(h_i(x), \theta_i\right)+t_i .$$
They further assume that types are drawn independently, that the distribution $P_i(\cdot)$ of player $i$ ‘s type satisfies the monotone-hazard-rate condition $\left(p_i / 1-P_i\right)$ nondecrcasing) for cach $i$, and that preferences satisfy the sorting assumption $\lambda V_i / \partial \theta_i \partial h_i \geq 0$ and the condition that $\partial^2 V_i / \partial h_i \partial \theta_i$ is decreasing in $\theta_i$. Cnder these assumptions, they show that an allocation that maximizes the principal’s expected utility,
$$\mathrm{F}H\left(V_0(x, \theta)-\sum{i=1}^I t_i(\theta)\right),$$
subject to the constraints of Bayesian incentive compatibility (IC) and individual rationality,
(IR) $\mathrm{F}_\theta, u_i\left(y\left(\theta_i, \theta_i\right), \theta_i\right) \geq 0$ for all $\theta_i$.

博弈论代写

经济代写|博弈论代考Game theory代写|Feasibility under Budget Balance

(BB) $\sum_{i=1}^l t_i(\theta) \leq-C_0(x(\theta))$为所有$\theta .^{22}$

经济代写|博弈论代考Game theory代写|Dominant Strategy vs. Bayesian Mechanisms

Mookherjee和Reichelstein(1989)确定了一类模型，其中优势策略的实施相对于贝叶斯策略的实施没有福利损失。${ }^{23}$假设agent具有准线性偏好，对于$i=1, \ldots, I$，
$$u_i(x, t, \theta)=V_i\left(x, \theta_i\right)+t_i,$$

$$u_i(x, t, \theta)=V_i\left(h_i(x), \theta_i\right)+t_i .$$

$$\mathrm{F}H\left(V_0(x, \theta)-\sum{i=1}^I t_i(\theta)\right),$$

(IR) $\mathrm{F}_\theta, u_i\left(y\left(\theta_i, \theta_i\right), \theta_i\right) \geq 0$为所有$\theta_i$。

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