Bayesian Nash Equilibrium and Auctions
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Today's Agenda
Part 1: Bayesian Nash Equilibrium
Part 2: Auctions
Which game are we playing?
Cournot with unknown costs
Private-Value Auctions
Common-Value Auctions
Which Game Are We Playing?
One-Shot Bayesian Game
Nature reveals private information to one or more of the players:
e.g., a firm's cost, the state of demand, a person's valuation of a good
The players take simultaneous actions (e.g., submit bids, produce a good)
Payoffs are revealed
Critical feature: there is no opportunity for information to be revealed through play;
we get to that next time with Perfect Bayesian Equilibria!
What is Luke's strategy space? What is Darth's?
Draw the Bayesian normal form of this game,
and solve for the Bayesian Nash Equilibrium.
Cournot with Unknown Costs
Market demand: \(p = 10 - Q\)
Firm 1's costs: \(c_1(q_1) = 0\)
Firm 2's costs: \(c_2(q_2) = \begin{cases}0 & \text{ w/prob }\frac{1}{2}\\4q_2 & \text{ w/prob }\frac{1}{2}\end{cases}\)
Firm 1 is best responding to two potential quantities (\(q_2^L\) or \(q_2^H\)) Firm 2 would choose in different states of the world.
\pi_1(q_1|q_2^L, q_2^H) =
\pi_2^H(q_2^H|q_1) =
\pi_2^L(q_2^L|q_1) =
Firm 2 is best responding to the single quantity \(q_1\) it anticipates Firm 1 will choose, but will choose differently based on its costs.
Private-Value Auctions
Private-Value Auction
A single item is being auctioned off.
Each bidder knows their valuation of the good and the probability distribution of other bidders' valuations.
Optimal bidding strategy depends on the structure of the auction:
Sealed-bid vs. open bid
First-price vs. second-price
Because we're studying simultaneous-move games today, we'll concentrate on sealed-bid auctions and compare strategies in first-price and second-price auctions.
Second-Price, Sealed-Bid Auction
Bidders simultaneously submit secret bids.
The highest bidder pays the amount of the second-highest bid.
What is an optimal bidding strategy?
Nature reveals private valuations \(v_i\), uniformly distributed along [0, 100].
v
0
100
First--Price, Sealed-Bid Auction
Two bidders simultaneously submit secret bids.
The highest bidder pays the amount of their own bid.
Nature reveals private valuations \(v_i\), uniformly distributed along [0, 100].
0
100
What is an optimal bidding strategy?
v
First--Price, Sealed-Bid Auction
Nature reveals private valuations \(v_i\), uniformly distributed along [0, 100].
Suppose you believe player 2 is bidding some fraction \(a\) of their valuation.
What is the distribution of their bid? What is your probability of winning if you bid \(b_1\)?
Suppose you believe player 2 is bidding some fraction \(a\) of their valuation.
What is the distribution of their bid? What is your probability of winning if you bid \(b_1\)?
u(b_i) = (v_i - b_i) \times \frac{b_i}{100a}
If the other bidder is bidding fraction \(a\) of their valuation, and their valuation is
uniformly distributed over [0, 100], what's your optimal bid if your valuation is \(v_i\)?
u'(b_i) = \frac{v_i - 2b_i}{100a} = 0 \Rightarrow b_i^* = \frac{1}{2}v_i
PAYOFF IF WIN
PROBABILITY OF WINNING
OPTIMAL TO BID HALF YOUR VALUE
Aside: Order Statistics
Two bidders: expected value of higher value is \(\frac{2}{3}\overline v\), lower value is \(\frac{1}{3}\overline v\)
Nature reveals private valuations \(v_i\), uniformly distributed along \([0, \overline v]\).
0
\overline v
\frac{2}{3}\overline v
What is the expected revenue from a second-price, sealed-bid auction? From a first-price auction?
\frac{1}{3}\overline v
Revenue equivalence theorem: for certain economic environments, the expected revenue and bidder profits for a broad class of auctions will be the same provided that bidders use equilibrium strategies.
Common-Value Auctions
Suppose I were to auction off this jar of coins.
Who would win the auction?
Suppose everyone gets a signal about the value of the coins in the jar, and that the signal is unbiased: its mean is the true value.
The winner's curse says that
in a common value auction,
then if you win the auction,
you've almost certainly overpaid.
(we won't do the math on this, it's just cool so we mention it)
Conclusions and Next Steps
They then played a simultaneous game -- so there was no opportunity for learning.
Next week: we'll look at situations in which players play games over time,
so they can update their beliefs as the game is played.
Today we looked at situations in which different agents had different information,
revealed by "nature."
Copy of Econ 51 | 15 | Bayesian Nash Equilibrium and Auctions
By Chris Makler
Copy of Econ 51 | 15 | Bayesian Nash Equilibrium and Auctions
Static Games of Incomplete Information
