# Bayesian Nash Equilibrium and Auctions

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# 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!

# Private-Value Auctions

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

Bidders simultaneously submit secret bids.

The highest bidder pays the amount of their own 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

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.

Private value auction: everyone has their own personal valuation of an object.

Common value: the object has an intrinsic value, but that value is unknown

## Common Value Auctions

Example: auctioning off land with an unknown amount of oil. Everyone can perform their own test (drill a hole somewhere on the land), and bids based on their private information from that test result.

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)

# Strategies and Strategy Spaces

A strategy is a  complete, contingent plan  of action for a player in a game.

This means that every player
must specify what action to take
at every decision node in the game tree
— including after moves of "nature"!

A strategy space is the set of all strategies available to a player.

## Strategy space

Dynamic game with complete and perfect information

Actions taken as a function of what has occurred previously in the game. Must specify what to do for every possible history!

Simultaneous game with
incomplete information

Actions taken as a function of private information. Must specify what to do for every possible value!

Simultaneous game with
complete information

Actions that can be taken

In every type of game, equilibrium is described as a strategy profile:
that is, the complete strategies chosen by each player, not just the actions taken!

## 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."

By Chris Makler

# Econ 51 | Fall 22 | 12 | Bayesian Nash Equilibrium and Auctions

Static Games of Incomplete Information

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