<p>Maybe using simple calculus makes things clearer. Assume that everything is sufficiently smooth. The payoff of <span class="math-container">$i$</span> stating their type to be <span class="math-container">$x$</span> when the real type is <span class="math-container">$t_i$</span> is
<span class="math-container">$$q_i(x,t_{-i})t_i-p_i(x,t_i).$$</span>
The first order condition for optimal behavior is
<span class="math-container">$$\frac{\partial q_i(x,t_{-i})}{\partial x}t_i-\frac{\partial p_i(x,t_i)}{\partial x}=0.$$</span> Under DSIC, truthfully stating one's type is optimal:
<span class="math-container">$$\frac{\partial q_i(t_i,t_{-i})}{\partial x}t_i-\frac{\partial p_i(t_i,t_i)}{\partial x}=0.$$</span>
This tells us that a higher probability of receiving the good must be compensated for with a higher payment. If <span class="math-container">$\partial q_i(t_i,t_{-i})/\partial x&lt;0$</span>, the agent with type <span class="math-container">$t_i$</span> could increase their chance of receiving the good while paying less by pretending to have a smaller type. This gives you the first condition.</p>
<p>To get the second condition, you use the logic of the envelope theorem: The marginal utility of each type has a direct effect since the probability of receiving the good is multiplied with the type. So the direct effect is simply the probability. There is also an indirect effect since one adjusts ones stated type optimally. But in an optimum, the marginal gain from adjustment is zero and can, therefore, be neglected. More formally,
<span class="math-container">$$u_i(t_i,t_{-i})=q_i(t_i,t_{-i})t_i-p_i(t_i,t_{-i})$$</span>
<span class="math-container">$$\frac{\partial u_i(t_i,t_{-i})}{\partial t_i}=\frac{\partial q_i(t_i,t_{-i})}{\partial x}t_i+q_i(t_i,t_{-i})-\frac{\partial p_i(x,t_i)}{\partial x}.$$</span>
But we already know that
<span class="math-container">$$\frac{\partial q_i(t_i,t_{-i})}{\partial x}t_i-\frac{\partial p_i(t_i,t_i)}{\partial x}=0,$$</span>
so <span class="math-container">$$\frac{\partial u_i(t_i,t_{-i})}{\partial t_i}=q_i(t_i,t_{-i}),$$</span>
which gives you the second condition. If you find the second condition intuitive, you can also use it to get the first one: Higher types must have higher utility. They can always have the probability and payment of a lower type, but get more out of winning. Since marginal utilities are the same as probabilities, higher types must get a higher probability.</p>

CC BY-SA 4.0
2022-01-13T15:55:51.360Z
This is so much clear. Thanks!!
virtue_signal
CC BY-SA 4.0
2022-01-12T22:59:52.077Z
2022-01-12T22:59:52.077Z
Michael Greinecker
<p>Myerson (1981) states that in a single object allocation problem with quasilinear preferences, where <span class="math-container">$q_i(.)$</span> is an allocation function, <span class="math-container">$p_i(.)$</span> a payment function and <span class="math-container">$u_i(.)$</span> a utility function for individual <span class="math-container">$i$</span>, then a mechanism <span class="math-container">$(q,p)$</span> is DSIC if and only if:</p>
<ol>
<li><span class="math-container">$q_i(t_i,t_{-i})$</span> is increasing in <span class="math-container">$t_i$</span>, for all <span class="math-container">$t_{-i} \in T_{-i}$</span></li>
<li><span class="math-container">$u(t_i, t_{-i}) = u(0, t_{-i}) + \int_{0}^{t_i}q_i(x,t_{-i})dx$</span></li>
</ol>
<p>Is there any intuition behind why the above holds? I understand the proof, but unable to figure out how misreporting one's type becomes beneficial if the allocation rule becomes decreasing in some interval.</p>
<p>Property 1 is often called Myerson's monotonicity. Is it possible to have interpretation of Myerson's monotonocity through the lens of Maskin's monotonicity, given that Maskin's monotonicity is a little bit intuitive?</p>

CC BY-SA 4.0
2022-01-11T10:36:31.947Z
What is the proof you are familiar with?
Michael Greinecker
CC BY-SA 4.0
2022-01-12T16:55:25.463Z
The one where we show that utility function of agents is convex under quasilinear setup, and then from DSIC constraints, identifying that function $q_i(t_i,t_{i})$ is a subgradient of utility function. From the property of subgradient of convex functions, points 1 and 2 follows.
virtue_signal
<mechanism-design>
Intuition behind Myerson's characterization of DSIC Single Object Allocation Mechanism
virtue_signal
# Intuition behind Myerson's characterization of DSIC Single Object Allocation Mechanism

(Asked by: username_0 on 2022-01-11)

Myerson (1981) states that in a single object allocation problem with quasilinear preferences, where $q\_i(.)$ is an allocation function, $p\_i(.)$ a payment function and $u\_i(.)$ a utility function for individual $i$, then a mechanism $(q,p)$ is DSIC if and only if:

1. $q\_i(t\_i,t\_{-i})$ is increasing in $t\_i$, for all $t\_{-i} \in T\_{-i}$
2. $u(t\_i, t\_{-i}) = u(0, t\_{-i}) + \int\_{0}^{t\_i}q\_i(x,t\_{-i})dx$

Is there any intuition behind why the above holds? I understand the proof, but unable to figure out how misreporting one's type becomes beneficial if the allocation rule becomes decreasing in some interval.

Property 1 is often called Myerson's monotonicity. Is it possible to have interpretation of Myerson's monotonocity through the lens of Maskin's monotonicity, given that Maskin's monotonicity is a little bit intuitive?

**Tags:** mechanism-design

> **Comments:**
> **username_1**: What is the proof you are familiar with?  
> **username_0**: The one where we show that utility function of agents is convex under quasilinear setup, and then from DSIC constraints, identifying that function $q_i(t_i,t_{i})$ is a subgradient of utility function. From the property of subgradient of convex functions, points 1 and 2 follows.  

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## 1 answers

### Answer by username_1 on 2022-01-12. Score: 5 | Accepted answer

Maybe using simple calculus makes things clearer. Assume that everything is sufficiently smooth. The payoff of $i$ stating their type to be $x$ when the real type is $t\_i$ is
$$q\_i(x,t\_{-i})t\_i-p\_i(x,t\_i).$$
The first order condition for optimal behavior is
$$\frac{\partial q\_i(x,t\_{-i})}{\partial x}t\_i-\frac{\partial p\_i(x,t\_i)}{\partial x}=0.$$ Under DSIC, truthfully stating one's type is optimal:
$$\frac{\partial q\_i(t\_i,t\_{-i})}{\partial x}t\_i-\frac{\partial p\_i(t\_i,t\_i)}{\partial x}=0.$$
This tells us that a higher probability of receiving the good must be compensated for with a higher payment. If $\partial q\_i(t\_i,t\_{-i})/\partial x<0$, the agent with type $t\_i$ could increase their chance of receiving the good while paying less by pretending to have a smaller type. This gives you the first condition.

To get the second condition, you use the logic of the envelope theorem: The marginal utility of each type has a direct effect since the probability of receiving the good is multiplied with the type. So the direct effect is simply the probability. There is also an indirect effect since one adjusts ones stated type optimally. But in an optimum, the marginal gain from adjustment is zero and can, therefore, be neglected. More formally,
$$u\_i(t\_i,t\_{-i})=q\_i(t\_i,t\_{-i})t\_i-p\_i(t\_i,t\_{-i})$$
$$\frac{\partial u\_i(t\_i,t\_{-i})}{\partial t\_i}=\frac{\partial q\_i(t\_i,t\_{-i})}{\partial x}t\_i+q\_i(t\_i,t\_{-i})-\frac{\partial p\_i(x,t\_i)}{\partial x}.$$
But we already know that
$$\frac{\partial q\_i(t\_i,t\_{-i})}{\partial x}t\_i-\frac{\partial p\_i(t\_i,t\_i)}{\partial x}=0,$$
so $$\frac{\partial u\_i(t\_i,t\_{-i})}{\partial t\_i}=q\_i(t\_i,t\_{-i}),$$
which gives you the second condition. If you find the second condition intuitive, you can also use it to get the first one: Higher types must have higher utility. They can always have the probability and payment of a lower type, but get more out of winning. Since marginal utilities are the same as probabilities, higher types must get a higher probability.

> **Comments:**
> **username_0**: This is so much clear. Thanks!!  

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