# 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.

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].

## 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].

What is an optimal bidding strategy?

## 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.

# 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

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