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@@ -11,3 +11,6 @@ chapters:
 - file: notes/day5_file_sharing
 - file: notes/day6_market_eq
 - file: notes/day7_market_failures
+- file: notes/day8_single_auctions
+- file: notes/day9_sponsored_search
+

docs/notes/day8_single_auctions.md

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+# Single-Unit Auctions
+
+Auctions are useful in many situations. For instance, if the price of a
+good is unknown (a market for this product, for one reason or another,
+might not converge to some equilibrium price) auctions can be useful for
+price discovery. For instance:
+
+1.  Monopolies (wireless spectrum auctions);
+
+2.  Niche products (rare used items on eBay);
+
+3.  Product with very specific attributes (ad auctions).
+
+Another situation where auctions are useful is when buyers have the
+computational power and patience to bid:
+
+1.  Expensive items (wireless spectrum);
+
+2.  Automated bidders (ad auctions).
+
+There are many interactions between auctions and CS:
+
+1.  Auctions fund a lot of computer scientists (ad auctions);
+
+2.  Fast auctions require algorithmic bidders (ad auctions);
+
+3.  Complex auctions require algorithmic auctioneers.
+
+## Case of One Buyer
+
+Motivation: digital goods. Suppose there is a pool of users subscribed
+to a free product. What price should be charged to maximize revenue?
+
+Aggregating users' willingness to pay induces a demand curve for the
+good. Then, revenue is equal to the price multiplied by the number of
+people with willingness to pay more than that price.
+
+Roger Myerson (1981) showed that given some demand curve $D$, there is a
+formula $p(D)$ that maximizes revenue. This is the profit maximizing
+price.[^1]
+
+Another motivation: ad auctions with specialized advertisers so only one
+advertiser wants each ad spot. Sellers do not know exactly how much an
+advertiser is willing to pay, but have prior beliefs over the buyer's
+value. In this interpretation, the prior over values can be taken as a
+demand curve.
+
+## Case of Multiple Buyers
+
+**Model**:
+
+1.  There is a set of bidders $I$ with each bidder $i$ having a value
+    $v_i$ for getting the item.
+
+2.  Each bidder knows their own value, the seller does not know bidders'
+    values but has some prior belief over values.
+
+3.  Bidder $i$'s payoff while paying $p_i$ is $v_i - p_i$ if they get
+    the item and $-p_i$ otherwise.
+
+We will focus on sealed-bid auctions:
+
+```{prf:axiom}
+First-Price Auction
+
+1.  Each bidder submits a bid $b_i$, which is unobserved by other
+    bidders;
+
+2.  Highest bidder $i^* = \arg\max_{i \in I}\{b_i\}$ gets the item and
+    pays $b_{i^*}$.
+```
+
+Immediately, bidders will bid lower than their value $v_i$: for all
+$i \in I$ we have that $b_i < v_i$ in equilibrium. Intuitively, we
+expect bids to grow up as competition increases.
+
+```{prf:axiom}
+All-Pay Auction
+
+1.  Each bidder submits a bid $b_i$, which is unobserved by other
+    bidders;
+
+2.  Highest bidder $i^* = \arg\max_{i \in I}\{b_i\}$ gets the item;
+
+3.  *Every* bidder $i \in I$ pays $b_i$.
+```
+
+Not used to auctions things in practice, but used to model other
+situations (interest groups donating to a politician, animals hunting
+for food). In this model, bidders will still bid lower than their value
+and will generally bid lower than their first-price value.
+
+```{prf:axiom}
+Second-Price Auction
+
+1.  Each bidder submits a bid $b_i$, which is unobserved by other
+    bidders;
+
+2.  Highest bidder $i^* = \arg\max_{i \in I}\{b_i\}$ gets the item and
+    pays the second highest price $\max_{i \neq i^*} b_i$.
+```
+
+```{prf:theorem}
+Second-price auctions are strategyproof ao truthful bidding is a
+dominant strategy and hence everyone bidding truthfully is an
+equilibrium.
+```
+
+```{prf:proof}
+ Fix all bids of other people $b_j$ for $j \neq i$. We will show
+that bidding $b_i = v_i$ is optimal for bidder $i$. Let
+
+$$b^{(-i)} = \max_{j \neq i} b_j.$$ 
+
+There are two cases:
+
+1.  If $v_i < b^{(-i)}$ then $i$ prefers to not win than to pay
+    $b^{(-i)}$ so any $b_i < b^{(-i)}$ is optimal. In particular,
+    $b_i = v_i$ is an optimal bid.
+
+2.  If $v_i > b^{(-i)}$ then $i$ prefers to win and pay $b^{(-i)}$ than
+    not winning. Thus, any $v_i > b^{(-i)}$ is optimal and in
+    particular, $b_i = v_i$ is an optimal bid.
+
+ 
+```
+
+```{prf:theorem}
+The second price auction is individual rational in equilibrium: if
+$b_i = v_i$ for all $i$, then $v_i - p_i \geq 0$ for all $i$.
+```
+
+```{prf:proof}
+ If $i$ doesn't win, then their payoff is $0$. If they do win,
+then the price they pay is upper bounded by their valuation, so
+$v_i - p_i \geq v_i - v_i = 0$. 
+```
+
+```{prf:theorem}
+In equilibrium, the second price auction allocates the good to the
+bidder with the highest value.
+```
+
+```{prf:proof}
+ In equilibrium, $v_i = b_i$ for all $i$ so
+$\arg\max_i b_i = \arg\max_i v_i$. 
+```
+
+## Revenue Maximization
+
+```{prf:example}
+Suppose $A$ values the good at $1$ and $B$ values the good at $2$. In a
+second price auction, both bid truthfully and $B$ wins at a price of
+$1$. In a first price auction, the unique equilibrium is $A$ bids $1$
+and $B$ bids $1 + \epsilon$ and revenue is $1+\epsilon$ which is
+basically equivalent to $1$.
+```
+
+To analyze the case with uncertainty and Bayesian agents, we need to
+define a new notion of equilibrium.
+
+```{prf:definition}
+A Bayesian Nash Equilibrium is a strategy profile such that each player
+is maximizing their own payoff with respect to the posterior belief
+generated by others' strategies.
+```
+
+```{prf:example}
+Suppose $A$ and $B$ both have values drawn uniformly from $[0,1]$. In a
+second price auction, expected payoff is
+$\mathbb{E}\left[\min\{v_A,v_B\}|v_A,v_B \sim Uni[0,1]\right] = 1/3.$ In
+a first price auction, the equilibrium is $b_A = v_A/2, b_B = v_B/2$ and
+in expectation, revenue is
+
+$$\mathbb{E}[{\max\{b_A,b_B\}}] = \mathbb{E}[{\max\{v_A,v_B\}/2}] = 1/3.$$
+```
+
+It turns out that these auctions generate the same revenue:
+
+```{prf:theorem}
+At equilibrium, expected payments are fully determined by the auction's
+allocation rule.
+```
+
+```{prf:corollary}
+Taking the auction's allocation rule to be the one that gives the item
+to the bidder with the highest bid, first, second, and all-pay auctions
+generate the same revenue in Bayes-Nash Equilibria.
+```
+
+It is not always the case that auctions always give the good to the
+max-value bidder:
+
+1.  There can be overbidding in second price auctions ($A$ bids $v_B+1$,
+    $B$ bids $0$ is an equilibrium but doesn't give the good to the
+    max-value bidder if $v_A < v_B$);
+
+2.  In an all-pay auction, there can be equilibria where the highest
+    value bidder does not get the item;
+
+3.  In auctions with reserve prices, it could be the case that no bidder
+    gets the item.
+
+[^1]: In the case of $n$ buyers and each drawing a value from $D$
+    independently, the revenue-maximizing auction is a second price
+    auction with reserve price $p(D)$.