This is just a reminder for those who see it that I will discuss Part III essays in my office D1.13 tomorrow, Tuesday, at 3.30pm.
Monday, 30 November 2015
Lecture 22
I appreciate with hindsight that there was a lot to take in this this lecture. The punchline was that for an optimal auction (maximizing the seller's expected revenue), the fact that we want a direct revelation mechanism forces:
\[ P_i(\theta_i) = \theta_iV_i(\theta_i)-\int_{\theta_i^*}^{\theta_i} V_i(w)dw\tag{1}
\]
where $P_i(\theta_i)$ is bidder $i$'s expected payment (and $V_i(\theta_i)$ is his probability of winning the item) given that his valuation for the item is $\theta_i$, and thus that
\[ \text{seller's expected revenue}=E\left[\sum_i\phi_i(\theta_i)g(\theta_i)v_i(\theta_1,\ldots,\theta_n)\right].
\] So the auctioneer should arrange that for every $\theta_1,\dotsc,\theta_n$, he maximizes $\sum_i\phi_i(\theta_i)g(\theta_i)v_i(\theta_1,\ldots,\theta_n)$. This simply means awarding the item to the bidder with the largest non-negative value of $g(\theta_i)$, and not awarding the item to anyone if no $g(\theta_i)$ is positive.
This becomes more interesting when agents are heterogeneous, so that $F_i$ differ. The anaysis alters only slightly, becoming
\[ \text{seller's expected revenue}=E\left[\sum_i\phi_i(\theta_i)g_i(\theta_i)v_i(\theta_1,\ldots,\theta_n)\right].
\] For example, if $n=2$ and $\theta_1,\theta_2$ are independent and distributed $U[0,1]$ and $U[0,2]$ then $g_1(\theta_1)=2\theta_1-1$ and $g_2(\theta_2)=2\theta_2-2$. So the optimal auction is one in which
(a) Bidder 1 wins the item if $2\theta_1-1\geq 2\theta_2-2$ and $\theta_1\geq 1/2$.
(b) Bidder 2 wins the item if $2\theta_2-2> 2\theta_1-1$ and $\theta_2\geq 1$.
(c) Otherwise the item is not won.
\[ P_i(\theta_i) = \theta_iV_i(\theta_i)-\int_{\theta_i^*}^{\theta_i} V_i(w)dw\tag{1}
\]
where $P_i(\theta_i)$ is bidder $i$'s expected payment (and $V_i(\theta_i)$ is his probability of winning the item) given that his valuation for the item is $\theta_i$, and thus that
\[ \text{seller's expected revenue}=E\left[\sum_i\phi_i(\theta_i)g(\theta_i)v_i(\theta_1,\ldots,\theta_n)\right].
\] So the auctioneer should arrange that for every $\theta_1,\dotsc,\theta_n$, he maximizes $\sum_i\phi_i(\theta_i)g(\theta_i)v_i(\theta_1,\ldots,\theta_n)$. This simply means awarding the item to the bidder with the largest non-negative value of $g(\theta_i)$, and not awarding the item to anyone if no $g(\theta_i)$ is positive.
This becomes more interesting when agents are heterogeneous, so that $F_i$ differ. The anaysis alters only slightly, becoming
\[ \text{seller's expected revenue}=E\left[\sum_i\phi_i(\theta_i)g_i(\theta_i)v_i(\theta_1,\ldots,\theta_n)\right].
\] For example, if $n=2$ and $\theta_1,\theta_2$ are independent and distributed $U[0,1]$ and $U[0,2]$ then $g_1(\theta_1)=2\theta_1-1$ and $g_2(\theta_2)=2\theta_2-2$. So the optimal auction is one in which
(a) Bidder 1 wins the item if $2\theta_1-1\geq 2\theta_2-2$ and $\theta_1\geq 1/2$.
(b) Bidder 2 wins the item if $2\theta_2-2> 2\theta_1-1$ and $\theta_2\geq 1$.
(c) Otherwise the item is not won.
Appropriate payments for the auction design have to be worked out using (1). However, another way to figure the payments is by the VCG mechanism. This boils down to agent $i$ paying, when he wins, the least $\theta_i$ for which he would still win. E.g. in case (a) bidder 1 should win, and pay $\max\{\theta_2-1/2,1/2\}$.
Friday, 27 November 2015
Lecture 21
I mentioned
The ultimate prize for a Game of Thrones fan: the chance to become one of the many, many characters killed off by George RR Martin in the books that are inspiring the TV series. And now two fans have paid $20,000 each in an online charity auction to receive the honour of a character being named after them in the next novel – and seeing them "certainly meet a grisly death". (from the Guardian 12/06/14)
and for the sale of the Roman empire by auction, see: An Early Example of the “Wmner's Curse” in an Auction.
The ultimate prize for a Game of Thrones fan: the chance to become one of the many, many characters killed off by George RR Martin in the books that are inspiring the TV series. And now two fans have paid $20,000 each in an online charity auction to receive the honour of a character being named after them in the next novel – and seeing them "certainly meet a grisly death". (from the Guardian 12/06/14)
and for the sale of the Roman empire by auction, see: An Early Example of the “Wmner's Curse” in an Auction.
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