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

Heads

(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

Heads

(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?

Auctions

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].

0
100
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
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.

Your bid

Highest bid other than yours

80

b_i

90

b_{-i}

65

90

65

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.

Your bid

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.

Your bid

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.

Your bid

Highest bid other than yours

80

b_i

90

b_{-i}

85

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.

Your bid

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.

Your bid

Highest bid other than yours

80

b_i

70

b_{-i}

95

90

65

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.

Your bid

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.

Your bid

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.

Your bid

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.

Your bid

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.

Your bid

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.

Your bid

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.

Your bid

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

Your bid

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)

Common Value Auctions

Next Time

What happens when we have dynamic games with incomplete information?