# 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:

Monopolies (wireless spectrum auctions);

Niche products (rare used items on eBay);

Product with very specific attributes (ad auctions).

Another situation where auctions are useful is when buyers have the computational power and patience to bid:

Expensive items (wireless spectrum);

Automated bidders (ad auctions).

There are many interactions between auctions and CS:

Auctions fund a lot of computer scientists (ad auctions);

Fast auctions require algorithmic bidders (ad auctions);

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**:

There is a set of bidders \(I\) with each bidder \(i\) having a value \(v_i\) for getting the item.

Each bidder knows their own value, the seller does not know bidders’ values but has some prior belief over values.

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:

(First-Price Auction)

First-Price Auction

Each bidder submits a bid \(b_i\), which is unobserved by other bidders;

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.

(All-Pay Auction)

All-Pay Auction

Each bidder submits a bid \(b_i\), which is unobserved by other bidders;

Highest bidder \(i^* = \arg\max_{i \in I}\{b_i\}\) gets the item;

*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.

(Second-Price Auction)

Second-Price Auction

Each bidder submits a bid \(b_i\), which is unobserved by other bidders;

Highest bidder \(i^* = \arg\max_{i \in I}\{b_i\}\) gets the item and pays the second highest price \(\max_{i \neq i^*} b_i\).

Second-price auctions are strategyproof ao truthful bidding is a dominant strategy and hence everyone bidding truthfully is an equilibrium.

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

There are two cases:

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.

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.

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 \(i\).

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 \(v_i - p_i \geq v_i - v_i = 0\).

In equilibrium, the second price auction allocates the good to the bidder with the highest value.

Proof. In equilibrium, \(v_i = b_i\) for all \(i\) so \(\arg\max_i b_i = \arg\max_i v_i\).

## Revenue Maximization#

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.

(Bayesian Nash Equilibrium)

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.

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

It turns out that these auctions generate the same revenue:

At equilibrium, expected payments are fully determined by the auction’s allocation rule.

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:

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 \(v_A < v_B\));

In an all-pay auction, there can be equilibria where the highest value bidder does not get the item;

In auctions with reserve prices, it could be the case that no bidder gets the item.