# Bayes Nash Equilibrium and Auctions

Christopher Makler

Stanford University Department of Economics

Econ 51: Lecture 15

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Most of the really big mistakes you'll make in your life
aren't because you play the game wrong,
but because you don't know the game you're playing.

# Games of Incomplete Information

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Suppose one of these
two games is being played.

Both players know there is an equal probability of each game.

Only player 1 knows which game is being played right now.

What is player 1's strategy space?
Player 2's?

Nature

(1/2)

Tails

(1/2)

Both players know there is an equal probability of each game.

Only player 1 knows which game is being played right now.

We can model this "as if" there is a nonstrategic player called Nature who moves first, flipping a coin, and picks which game is being played based on the coin flip.

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$$A^H$$

$$B^H$$

$$A^T$$

$$B^T$$

Nature

(1/2)

Tails

(1/2)

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$$A^H$$

$$B^H$$

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$$A^T$$

$$B^T$$

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The Bayesian Normal Form representation of the game shows the expected payoffs for each of the strategies the players could play:

$$A^HA^T$$

$$A^HB^T$$

$$B^HA^T$$

$$B^HB^T$$

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Bayes Nash Equilibrium is the NE of this game. It maps private information onto (simultaneously taken) actions.

## 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!

# 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 2 knows its own costs; Firm 1 knows that firm 2's costs are 0 and 4q with equal probability.

pollev.com/chrismakler

What is a strategy for firm 1?

What is a strategy for firm 2?

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's strategy is a single quantity ($$q_1$$), since it doesn't know firm 2's costs.
Its expected profit is based on its profit if firm 2 has low cost and produces $$q_2^L$$, or high cost and produces $$q_2^H$$.

\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's strategy must be to choose an amount to produce if it has low costs ($$q^L$$),
and an amount to produce if it is has high costs ($$q^H$$). It will choose the amount, knowing its own cost.

(10 - q_1 - q_2^H)q_2^H - 4q_2^H
(10 - q_1 - q_2^L)q_2^L
={1 \over 2}\left[(10 - q_1 - q_2^L)q_1\right] + {1 \over 2}\left[(10 - q_1 - q_2^H)q_1\right]

(if $$MC_2 = 0$$)

(if $$MC_2 = 4$$)

Bayes Nash Equilibrium will specify:
$$q_1$$ which is a best response to firm 2 playing $$q_2^L$$ and $$q_2^H$$ with equal probability;
and $$q_2^L$$ and $$q_2^H$$ which are each best responses to $$q_1$$ in their respective states of the world.

\Pr\{q_2 = q_2^L\} \times \pi_1(q_1,q_2^L)
\Pr\{q_2 = q_2^H\} \times \pi_1(q_1,q_2^H)
+
\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) =
(10 - q_1 - q_2^H)q_2^H - 4q_2^H
(10 - q_1 - q_2^L)q_2^L
={1 \over 2}\left[(10 - q_1 - q_2^L)q_1\right] + {1 \over 2}\left[(10 - q_1 - q_2^H)q_1\right]

Calculate Best Responses

{\partial \pi_2^L(q_2^L|q_1) \over \partial q_2^L} =
10 - q_1 - 2q_2^L
=0
q_2^L(q_1) = 5 - {1 \over 2}q_1

Firm 2's best response to $$q_1$$ if MC = 0

{\partial \pi_2^H(q_2^H|q_1) \over \partial q_2^H} =
10 - q_1 - 2q_2^H - 4
=0
q_2^H(q_1) = 3 - {1 \over 2}q_1

Firm 2's best response to $$q_1$$ if MC = 4

=(10 - q_1 - ({1 \over 2}q_2^L + {1 \over 2}q_2^H))q_1
{\partial \pi_1(q_1|q_2^L, q_2^H) \over \partial q_1} =
10 - 2q_1 - ({1 \over 2}q_2^L + {1 \over 2}q_2^H)
=0
q_1(q_2^L,q_2^H)=5 - {1 \over 2}({1 \over 2}q_2^L + {1 \over 2}q_2^H)

Firm 1's best response to firm 2 playing $$q_2^L$$ and $$q_2^H$$ with equal probability

Solve for Nash Equilibrium

q_2^L = 5 - {1 \over 2}q_1
q_2^H = 3 - {1 \over 2}q_1
q_1=5 - {1 \over 2}({1 \over 2}q_2^L + {1 \over 2}q_2^H)
q_1=5 - {1 \over 2}({1 \over 2}(5 - {1 \over 2}q_1) + {1 \over 2}(3 - {1 \over 2}q_1))
q_1=5 - {1 \over 2}(4 - {1 \over 2}q_1)
q_1=5- 2 + {1 \over 4}q_1
{3 \over 4}q_1=3
q_1=4
q_2^L = 5 - {1 \over 2} \times 4
q_2^L = 3
q_2^H = 3 - {1 \over 2} \times 4
q_2^H = 1

Interestingly, the firm with unknown costs produces less (and therefore makes less profits) than the firm with known costs, even when they both have no costs.

Can you figure out why?

# Private-Value Auctions

A single object is being auctioned off. Rules of the auction:

• each player $$i$$ submits bid $$b_i$$
• a rule determines who gets the object, and how much they pay,
as a function of the vector of bids $$b = (b_1, b_2, \cdots, b_n)$$
• examples of rules:
• who gets the object?
• today: always the highest bidder
• sealed-bid vs. open bid: are bids public information?
• today: sealed bid, so the game is a simultaneous game
• first-price vs. second-price: what does the winner pay, based on the bids?
• first-price: the winner pays their own bid
• second-price: the winner pays the second-highest bid

# Auction Payoffs

u_i(v_i,b) = v_i - b

Each player $$i$$ knows their own valuation of the object, $$v_i$$.

(We can think of this as a move by nature that occurs before the game begins.)

If you win the auction and pay some amount $$b$$, your payoff is

If you lose, your payoff is zero.

We assume there is no additional emotional payoff from the fact that you won or lost.

## 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].

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v_i

## Second-Price, Sealed-Bid Auction

Bidders simultaneously submit secret bids.

The highest bidder pays the amount of the second highest bid.

v_i
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100

Suppose your valuation is $$v_i = 80$$. Let's make a payoff matrix based on your bid and the highest bid other than yours.

Highest bid other than yours

80

b_i

90

b_{-i}

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90

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15

If you bid 90 and the highest other bid is 65, you win the object and pay 65; payoff is 80 - 65 = 15

## Second-Price, Sealed-Bid Auction

Bidders simultaneously submit secret bids.

The highest bidder pays the amount of the second highest bid.

v_i
0
100

Suppose your valuation is $$v_i = 80$$. Let's make a payoff matrix based on your bid and the highest bid other than yours.

Highest bid other than yours

80

b_i

90

b_{-i}

65

90

65

15

## Second-Price, Sealed-Bid Auction

Bidders simultaneously submit secret bids.

The highest bidder pays the amount of the second highest bid.

v_i
0
100

Suppose your valuation is $$v_i = 80$$. Let's make a payoff matrix based on your bid and the highest bid other than yours.

Highest bid other than yours

80

b_i

90

b_{-i}

75

90

65

15

75

5

## Second-Price, Sealed-Bid Auction

Bidders simultaneously submit secret bids.

The highest bidder pays the amount of the second highest bid.

v_i
0
100

Suppose your valuation is $$v_i = 80$$. Let's make a payoff matrix based on your bid and the highest bid other than yours.

Highest bid other than yours

80

b_i

90

b_{-i}

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90

65

15

75

5

85

-5

If you bid 90 and the highest other bid is 85, you win the object and pay 85; payoff is 80 - 85 = -5

## Second-Price, Sealed-Bid Auction

Bidders simultaneously submit secret bids.

The highest bidder pays the amount of the second highest bid.

v_i
0
100

Suppose your valuation is $$v_i = 80$$. Let's make a payoff matrix based on your bid and the highest bid other than yours.

Highest bid other than yours

80

b_i

90

b_{-i}

95

90

65

15

75

5

85

-5

95

0

If you bid 90 and the highest other bid is 95, you don't win the object and your payoff is 0.

## Second-Price, Sealed-Bid Auction

Bidders simultaneously submit secret bids.

The highest bidder pays the amount of the second highest bid.

v_i
0
100

Suppose your valuation is $$v_i = 80$$. Let's make a payoff matrix based on your bid and the highest bid other than yours.

Highest bid other than yours

80

b_i

70

b_{-i}

95

90

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15

75

5

85

-5

95

0

70

0

## Second-Price, Sealed-Bid Auction

Bidders simultaneously submit secret bids.

The highest bidder pays the amount of the second highest bid.

v_i
0
100

Suppose your valuation is $$v_i = 80$$. Let's make a payoff matrix based on your bid and the highest bid other than yours.

Highest bid other than yours

80

b_i

70

b_{-i}

85

90

65

15

75

5

85

-5

95

0

70

0

0

## Second-Price, Sealed-Bid Auction

Bidders simultaneously submit secret bids.

The highest bidder pays the amount of the second highest bid.

v_i
0
100

Suppose your valuation is $$v_i = 80$$. Let's make a payoff matrix based on your bid and the highest bid other than yours.

Highest bid other than yours

80

b_i

70

b_{-i}

75

90

65

15

75

5

85

-5

95

0

70

0

0

0

## Second-Price, Sealed-Bid Auction

Bidders simultaneously submit secret bids.

The highest bidder pays the amount of the second highest bid.

v_i
0
100

Suppose your valuation is $$v_i = 80$$. Let's make a payoff matrix based on your bid and the highest bid other than yours.

Highest bid other than yours

80

b_i

70

b_{-i}

65

90

65

15

75

5

85

-5

95

0

70

0

0

0

15

## Second-Price, Sealed-Bid Auction

Bidders simultaneously submit secret bids.

The highest bidder pays the amount of the second highest bid.

b_i=v_i
0
100

Suppose your valuation is $$v_i = 80$$. Let's make a payoff matrix based on your bid and the highest bid other than yours.

Highest bid other than yours

80

b_{-i}

65

90

65

15

75

5

85

-5

95

0

70

0

0

0

15

80

15

## Second-Price, Sealed-Bid Auction

Bidders simultaneously submit secret bids.

The highest bidder pays the amount of the second highest bid.

b_i=v_i
0
100

Suppose your valuation is $$v_i = 80$$. Let's make a payoff matrix based on your bid and the highest bid other than yours.

Highest bid other than yours

80

b_{-i}

75

90

65

15

75

5

85

-5

95

0

70

0

0

0

15

80

15

5

## Second-Price, Sealed-Bid Auction

Bidders simultaneously submit secret bids.

The highest bidder pays the amount of the second highest bid.

b_i=v_i
0
100

Suppose your valuation is $$v_i = 80$$. Let's make a payoff matrix based on your bid and the highest bid other than yours.

Highest bid other than yours

80

b_{-i}

85

90

65

15

75

5

85

-5

95

0

70

0

0

0

15

80

15

5

0

## Second-Price, Sealed-Bid Auction

Bidders simultaneously submit secret bids.

The highest bidder pays the amount of the second highest bid.

b_i=v_i
0
100

Suppose your valuation is $$v_i = 80$$. Let's make a payoff matrix based on your bid and the highest bid other than yours.

Highest bid other than yours

80

b_{-i}

95

90

65

15

75

5

85

-5

95

0

70

0

0

0

15

80

15

5

0

0

## Second-Price, Sealed-Bid Auction

Bidders simultaneously submit secret bids.

The highest bidder pays the amount of the second highest bid.

v_i
0
100

Highest bid other than yours

80

90

65

15

75

5

85

-5

95

0

70

0

0

0

15

80

15

5

0

0

Bidding your true valuation is sometimes
better than underbidding, and never worse

Bidding your true valuation is sometimes better than overbidding, and never worse.

Bidding your true valuation is a
weakly dominant strategy!

## 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)

## Next Time

What happens when we have dynamic games with incomplete information?

By Chris Makler

# Econ 51 | 14 | Static Games of Incomplete Information

Uncertainty and Risk Aversion - Presentation

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