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