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."