<p>Consider a set of players <span class="math-container">$i = 1,...,N$</span> who need to allocate some good. Each player <span class="math-container">$i$</span> valuates the good <span class="math-container">$v_i \in [0,1]$</span>, with <span class="math-container">$v_i \sim f_i$</span>. An individually rational direct mechanism <span class="math-container">$(p,c)$</span>, with <span class="math-container">$p : [0,1]^N \to \Delta(A)$</span>, <span class="math-container">$c : [0,1]^N \to \mathbb{R}_+^N$</span> is incentive compatible if and only if (i) the expected allocation probability function <span class="math-container">$\mathbb{E}_{t_{-i}}[p_i(x, t_{-i})]$</span> is weakly increasing and (ii) the expected transfer function
<span class="math-container">$$ \mathbb{E}_{t_{-i}}[c_i(x,t_{-i})] = \mathbb{E}_{t_{-i}}[p_i(x, t_{-i}) ] x - \int_0^x \mathbb{E}_{t_{-i}}[p_i(x, t_{-i}) dy $$</span></p>
<p>I understand the last expression algebraically as a sort of first-order condition where, by incentive compatibility,
<span class="math-container">$$ t_i = \arg\max_{x \in [0,1]} \mathbb{E}_{t_{-i}}[p_i(x, t_{-i})] x - \mathbb{E}_{t_{-i}}[c_i(x, t_{-i})] $$</span></p>
<p>But the expected transfer of individual with type <span class="math-container">$x$</span> also has a geometric interpretation: it is the difference between the rectangular region <span class="math-container">$\mathbb{E}_{t_{-i}}[p_i(x, t_{-i})] x$</span> (thanks to quasilinearity) and the integral of the expected probability function from 0 to <span class="math-container">$x$</span>.</p>
<p>Is there a clear way to appreciate the connection between these two interpretations?</p>

CC BY-SA 4.0
2023-02-01T21:22:45.203Z
could you provide some further detail, for example is this from a textbook or a paper?
Oliver Queen
CC BY-SA 4.0
2023-02-01T21:43:49.157Z
This is from some lecture notes, but I believe that the formula is, up to notation changes, an instance of the Myerson (1981) revenue equivalence principle, which says that the payment function (here transfer function) is uniquely determined by the payoff of the low types (here normalized to zero) and the assignment probability function.
raving-bandit
<game-theory><auctions><mechanism-design>
Geometric interpretation of transfer function in incentive-compatible direct mechanism
raving-bandit
# Geometric interpretation of transfer function in incentive-compatible direct mechanism

(Asked by: username_0 on 2023-02-01)

Consider a set of players $i = 1,...,N$ who need to allocate some good. Each player $i$ valuates the good $v\_i \in [0,1]$, with $v\_i \sim f\_i$. An individually rational direct mechanism $(p,c)$, with $p : [0,1]^N \to \Delta(A)$, $c : [0,1]^N \to \mathbb{R}\_+^N$ is incentive compatible if and only if (i) the expected allocation probability function $\mathbb{E}\_{t\_{-i}}[p\_i(x, t\_{-i})]$ is weakly increasing and (ii) the expected transfer function
$$ \mathbb{E}\_{t\_{-i}}[c\_i(x,t\_{-i})] = \mathbb{E}\_{t\_{-i}}[p\_i(x, t\_{-i}) ] x - \int\_0^x \mathbb{E}\_{t\_{-i}}[p\_i(x, t\_{-i}) dy $$

I understand the last expression algebraically as a sort of first-order condition where, by incentive compatibility,
$$ t\_i = \arg\max\_{x \in [0,1]} \mathbb{E}\_{t\_{-i}}[p\_i(x, t\_{-i})] x - \mathbb{E}\_{t\_{-i}}[c\_i(x, t\_{-i})] $$

But the expected transfer of individual with type $x$ also has a geometric interpretation: it is the difference between the rectangular region $\mathbb{E}\_{t\_{-i}}[p\_i(x, t\_{-i})] x$ (thanks to quasilinearity) and the integral of the expected probability function from 0 to $x$.

Is there a clear way to appreciate the connection between these two interpretations?

**Tags:** game-theory, auctions, mechanism-design

> **Comments:**
> **username_1**: could you provide some further detail, for example is this from a textbook or a paper?  
> **username_0**: This is from some lecture notes, but I believe that the formula is, up to notation changes, an instance of the Myerson (1981) revenue equivalence principle, which says that the payment function (here transfer function) is uniquely determined by the payoff of the low types (here normalized to zero) and the assignment probability function.  

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