From 86aca79901db06988aa4d2d842270b5cd17094e8 Mon Sep 17 00:00:00 2001 From: John Stachurski Date: Wed, 16 Feb 2022 10:06:02 +1100 Subject: [PATCH] Fixed minor typos in Auction lecture (#204) * Fixed minor typos in Auction lecture * minor --- lectures/two_auctions.md | 27 ++++++++++++++------------- 1 file changed, 14 insertions(+), 13 deletions(-) diff --git a/lectures/two_auctions.md b/lectures/two_auctions.md index 20dcca8e1..3c9d5664c 100644 --- a/lectures/two_auctions.md +++ b/lectures/two_auctions.md @@ -42,7 +42,7 @@ and ```{youtube} eYTGQCGpmXI ``` -Anders Munk-Nielsen put his code on github here +Anders Munk-Nielsen put his code [on GitHub](https://github.com/GamEconCph/Lectures-2021/tree/main/Bayesian%20Games). Much of our Python code below is based on his. @@ -90,9 +90,9 @@ This means that bidders are in effect participating in a game in which players This is a **Bayesian game**, a Nash equilibrium of which is called a **Bayesian Nash equilibrium**. -To complete the specification of the situation, we'll assume that prospective buyers' valuations are independently and indentically distributed according to a probability distribution that is known by all bidders. +To complete the specification of the situation, we'll assume that prospective buyers' valuations are independently and identically distributed according to a probability distribution that is known by all bidders. -Bidder optimally chooses to bid less than $v_i$. +Bidder optimally chooses to bid less than $v_i$. ### Characterization of FPSB Auction @@ -112,7 +112,7 @@ $$ (eq:optbid2) -A proof for this assertion is available at this Wikepedia page about Vicker auctions (https://en.wikipedia.org/wiki/Vickrey_auction) +A proof for this assertion is available at the [Wikepedia page](https://en.wikipedia.org/wiki/Vickrey_auction) about Vickery auctions +++ @@ -122,13 +122,14 @@ A proof for this assertion is available at this Wikepedia page about Vicker auc **Protocols:** In a second-price sealed-bid (SPSB) auction, the winner pays the second-highest bid. -## Characterization of SPSB Auction. +## Characterization of SPSB Auction In a SPSB auction bidders optimally choose to bid their values. Formally, a dominant strategy profile in a SPSB auction with a single, indivisible item has each bidder bidding its value. -A proof is provided at this Wikepedia page about Vicker auctions (https://en.wikipedia.org/wiki/Vickrey_auction) +A proof is provided at [the Wikepedia + page](https://en.wikipedia.org/wiki/Vickrey_auction) about Vicker auctions +++ @@ -138,13 +139,13 @@ A proof is provided at this Wikepedia page about Vicker auctions (https://en.wik We assume valuation $v_{i}$ of bidder $i$ is distributed $v_{i} \stackrel{\text{i.i.d.}}{\sim} U(0,1)$. -Under this assumption, we can analytically compute probabilitiy distributions of prices bid in both FPSB and SPSB. +Under this assumption, we can analytically compute probability distributions of prices bid in both FPSB and SPSB. -We'll simulate outcomes and, by using a law of large numbers verify, that the simulated outcomes agree with analytical ones. +We'll simulate outcomes and, by using a law of large numbers, verify that the simulated outcomes agree with analytical ones. We can use our simulation to illustrate a **Revenue Equivalence Theorem** that asserts that on average first-price and second-price sealed bid auctions provide a seller the same revenue. -To read about the revenue equivalence theorem, see this Wikepdia page (https://en.wikipedia.org/wiki/Revenue_equivalence) +To read about the revenue equivalence theorem, see [this Wikepedia page](https://en.wikipedia.org/wiki/Revenue_equivalence) +++ @@ -170,7 +171,7 @@ $$ \end{aligned} $$ -and the PDF of $y$ is $\tilde{f}_{n-1}(y) = (n-1)y^{n-2}$. +and the PDF of $y_i$ is $\tilde{f}_{n-1}(y) = (n-1)y^{n-2}$. Then bidder $i$'s optimal bid in a **FPSB** auction is: @@ -343,7 +344,7 @@ The Revenue Equivalence Theorem lets us an optimal bidding strategy for a FPSB Let $b(v_{i})$ be the optimal bid in a FPSB auction. -The revenue equivlance theorem tells us that a bidder agent with value $v_{i}$ on average receives the same **payment** in the two types of auction. +The revenue equivalence theorem tells us that a bidder agent with value $v_{i}$ on average receives the same **payment** in the two types of auction. Consequently, @@ -359,8 +360,8 @@ It follows that an optimal bidding strategy in a FPSB auction is $b(v_{i}) = \ma +++ -In equations {eq}`eq:optbid1` and {eq}`eq:optbid1`, we displayed formulas for optimal bids in a symmetric Bayesian Nash Equilibrium of a -a FPSB auction" +In equations {eq}`eq:optbid1` and {eq}`eq:optbid1`, we displayed formulas for +optimal bids in a symmetric Bayesian Nash Equilibrium of a FPSB auction. $$ \mathbf{E}[y_{i} | y_{i} < v_{i}]