# 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].

## 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].

## 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 \(b_1\)?

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 \(v_i\)?

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 \([0, \overline v]\).

What is the expected revenue from a second-price, sealed-bid auction? From a first-price auction?

**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)

## Common Value Auctions

# 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.

## Type of game

## 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."

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

By Chris Makler

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

Static Games of Incomplete Information

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