### Application 3: Auctions

Why study auctions? A lot of money at stake…

- Christie’s and Sotheby’s art auctions – $ billions annually
- Auctions for rights to natural resources (timber, oil, natural gas), government procurement, electricity and housing markets, etc
- eBay: $52 Billion worth of goods traded (in 2006)

Theory of auctions: the most *successful* application of Bayesian games

- “Rules of the game” and price formation are explicit, allowing for theoretical analysis
- Most relevant data can be mined, allowing for empirical work
- Auctions lend themselves to lab experiments
- Results on auctions may offer insight or intuition into behavior in less structured markets

### Auction as a Bayesian Game

Consider an auction where two bidders compete for one indivisible good

- Each risk-neutral bidder
*i*= 1, 2 has a valuation \(v_{i}\) for the good - If bidder
*i*wins and has to pay x for getting the good, then bidder*i*‘s payoff is

\(v_{i}\) – x

in X – U[0,900], Pr(X ≤ x) = x / 900

- \(v_{i}\) and \(v_{j}\) are independently and uniformly distributed on [0, 900]
- This is a special case of
*independent private*values:

Knowing the opponent’s valuation does *not* affect bidder *i*‘s own valuation

If the knowledge of \(v_{j}\) affects \(v_{i}\) (and correlated), then *interdependent* values

- We consider two different auction rules:

(1) **First-price** auction: bidder 1 wins if \(b_{i}\) > \(b_{2}\) and pays \(b_{1}\)

(2) **Second-price** auction: bidder 1 wins if \(b_{i}\) > \(b_{2}\) and pays \(b_{2}\)

### Second-price (Vickrey) Auctions

THEOREM 4.1. Bidding truthfully, \(b_{1} = v_{i}\), is the unique (weakly) dominant strategy in a second-price auction.

This implies that \(b_{i} = v_{i}\) satisfies

\(u_{i}(b_{i}, b_{j}) = u_{i}(v_{i}, b_{j}) ≥ u_{i}(b_{i}^{\prime}, b_{j})\) for all \(b_{j}\) and \(b_{i}^{\prime}\).

\(u_{i}(x, b_{i}) = \begin{cases} v_{i}-b_{j},\;if\;x>b_{j}\\ \frac{1}{2},\;if\;=b_{j}\\ 0,\;if\;x<b_{j} \end{cases}\)

William Vickrey, 1996 Nobel Laureate

Figure: Case: \(b_{j} < v_{i}\)

### First-price Auctions

To derive a BNE in first-price auctions, we take the second approach and assume

\(b_{1}(v) = b_{2}(v) = b(v)\) (**symmetry**)

Consider bidder 1’s expected payoff from making a bid of x:

\(u_{1}(x, b(v_{2})) = (v_{1} – x)Pr(b(v_{2}) < x) + 0 \cdot Pr(b(v_{2}) > x) = Pr(v_{2} < b^{-1}(x)) = (v_{1} – x) \cdot \frac{b^{-1}(x)}{900}\)

e.g., \((12 – q_{1} – q_{2})q_{1}, q_{1} = q_{2} = q^{*}, so (12 – 2q^{*})q^{*}\)

The environment is identical because bidders 1 and 2 are playing the same game at the same time, and their valuations both come from a uniform distribution between 0 and 900.

Therefore, in equilibrium, we can assume that bidders 1 and 2 use exactly the same strategy.

If we differentiate the above expression, 0 = \(-1 \cdot \frac{b^{-1}(x)}{900} + (v_{1} – x) \frac{1}{900b^{‘}(b^{-1}(x))}\)

\(b(b^{-1}(x)) = x\) ⇒ \(b^{‘}(b^{-1}(x)) \cdot \frac{\partial b^{-1}}{\partial x} = 1\)

in x = \(b(v_{1})\), 0 = \(-\frac{v_{1}}{900} + (v_{1} – b(v_{1})) \frac{1}{900b^{‘}(v_{1})}\)

\(v_{1}b^{‘}(v_{1}) = v_{1} – b(v_{1})\) ⇔ \(v_{1}b^{‘}(v_{1}) + b(v_{1}) = v_{1}\)

\(\int v_{1}b^{‘}(v_{1}) + b(v_{1}) = \int v_{1} dv_{1}\) ⇒ \(v_{1}b(v_{1}) = \frac{1}{2}v_{1}^{2} + C\)

\(v_{1}b(v_{1}) = \frac{1}{2}v_{1}^{2}\) so, \(b(v_{1}) = \frac{1}{2}v_{1}\)

Solving the last differential equation, we obtain

\(b(v_{1}) = \frac{v_{1}}{2}\)

eqbm bid

PROPOSITION 4.1. \(b_{i} = \frac{v_{i}}{2}\) constitutes a Bayesian Nash equilibrium in the first-price auction.

In contrast with SPA, the eqbm of FPA depends on the distribution of v

SPA: \(b^{”}(v)\) = v

FPA: \(b^{‘}(v) = \frac{v}{2}\)

Figure: Eqbm Bidding Strategies

1) Used car: What price to trade the car for, buyer has an incentive to lower the price, seller has an incentive to increase the price.

2) Public good: If I contribute, no one else does, if others contribute, I’m not left out.

3) Auction: **Efficiency** is guaranteed, It’s done aggressively because the person with the highest valuation gets the goods

### Revenue Equivalence

Observe that both bidding strategies are increasing in \(v_{i}\), implying that both auctions achieve allocative *efficiency* (the bidder who values the most would win in eqbm)

We compare performance of the two auctions in terms of the revenue

- In FPA, the expected profit accruing to the auctioneer amounts to

\(\frac{1}{2} \cdot \mathbb{E}[max(v_{1}, v_{2})]\) = 300

- In SPA,

\(\mathbb{E}[min(v_{1}, v_{2})]\) = 300

This revenue *equivalence* result is not a coincidence but holds in general

- Reference: Chang-Koo Chi, (33/50) Game Theory and Applications 9 – IPV Auctions, Jul 13, 2020, https://youtu.be/OyCeaDsbTMQ