Bayes Nash Equilibrium and Auctions
Christopher Makler
Stanford University Department of Economics
Econ 51: Lecture 13
A
B
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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
A
B
X
Y
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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?
pollev.com/chrismakler
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,
2
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Y
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Y
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Y
X
Y
\(A^H\)
\(B^H\)
\(A^T\)
\(B^T\)
flips a coin, and picks which game is being played based on the coin flip.
Player 1 observes Nature's move, so they have to choose what to do if Nature flips Heads (\(A^H\) or \(B^H\)) and if Nature flips Tails (\(A^T\) or \(B^T\)).
Player 2 does not, so their information set spans the entire game: they are only choosing X or Y.
Nature
Heads
(1/2)
Tails
(1/2)
2
1
2
1
\(A^H\)
\(B^H\)
X
Y
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0
2
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0
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0
4
\(A^T\)
\(B^T\)
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Y
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0
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0
4
X
Y
X
Y
The Bayesian Normal Form representation of the game shows the expected payoffs for each of the strategies the players could play:
\(A^HA^T\)
\(A^HB^T\)
\(B^HA^T\)
\(B^HB^T\)
X
Y
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2
1
0
3
1
0
2
0
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2
4
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
-
who gets the object?
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
Dynamic Games of Incomplete Information
Christopher Makler
Stanford University Department of Economics
Econ 51: Lecture 14
- Complete information: everyone knows everything about the game
- Incomplete information: players have private information
(e.g., they know what their valuation of a good is)
- Incomplete information: players have private information
- Perfect information: everyone can perfectly observe everyone else's moves
- Imperfect information: players' moves are hidden from other players
(e.g. I cannot observe how much time you spend studying)
- Imperfect information: players' moves are hidden from other players
Information
Time
Information
Static
(Simultaneous)
Dynamic
(Sequential)
Complete
Incomplete
WEEK 5
WEEK 6
LAST TIME
TODAY
Prisoners' Dilemma
Cournot
Entry Deterrence
Stackelberg
Auctions
Job Market Signaling
Collusion
Cournot with Private Information
Poker
Time
Information
Static
(Simultaneous)
Dynamic
(Sequential)
Complete
Incomplete
Strategy: an action
Equilbirium: Nash Equilibrium
Strategy: a mapping from the history of the game onto an action.
Equilibrium: Subgame Perfect NE
Strategy: a plan of action that
specifies what to do after every possible history of the game, based on one's own private information and (updating) beliefs about other players' private information.
Equilibrium: Perfect Bayesian Equilibrium
Strategy: a mapping from one's private information onto an action.
Equilibrium: Bayesian NE
WEEK 5
WEEK 6
LAST TIME
TODAY
Prisoners' Dilemma
Cournot
Entry Deterrence
Stackelberg
Auctions
Job Market Signaling
Collusion
Cournot with Private Information
Poker
Example: Gift-Giving
She chooses to give one of three gifts:
X, Y, or Z.
1
X
Y
Z
Player 1 makes the first move.
Example: Gift-Giving
Twist: Gift X is unwrapped,
but Gifts Y and Z are wrapped.
(Player 1 knows what they are,
but player 2 does not.)
After each of player 1's moves,
player 2 has the move: she can either accept the gift or reject it.
2
Accept X
Reject X
2
1
X
Y
Z
We represent this by having an information set connecting
player 2's decision nodes
after player 1 chooses Y or Z.
2
2
Accept Y
Reject Y
Accept Z
Reject Z
Also: player 2 cannot make her action contingent on Y or Z; her actions must be "accept wrapped" or "reject wrapped"
Accept Wrapped
Reject Wrapped
Accept Wrapped
Reject Wrapped
Example: Gift-Giving
After player 2 accepts or rejects the gift, the game ends (terminal nodes) and payoffs are realized.
1
0
1
0
2
0
2
0
3
0
–1
0
2
2
1
X
Y
Z
,
,
,
,
,
,
Accept X
Reject X
Accept Wrapped
Reject Wrapped
Accept Wrapped
Reject Wrapped
In this game, both players get a payoff of
0 if any gift is rejected,
1 if gift X is accepted, and
2 if gift Y is accepted.
If gift Z is accepted, player 1 gets a payoff of 3, but player 2 gets a payoff of –1.
1
0
1
0
2
0
2
0
3
0
–1
0
2
2
1
X
Y
Z
,
,
,
,
,
,
Accept X
Reject X
Accept Wrapped
Reject Wrapped
Accept Wrapped
Reject Wrapped
Intuitively, what should the equilibrium of this game be? Why?
Player 2:
Player 1:
Accept X
Reject a Wrapped Gift
X
Player 2:
Player 1:
Accept everything
Z
This is a subgame. From this subgame,
we know player 2 will accept X if given X.
What about player 2's other strategy?
Why isn't this an equilibrium?
1
0
1
0
2
0
2
0
3
0
–1
0
2
2
1
X
Y
Z
,
,
,
,
,
,
Accept X
Reject X
Accept Wrapped
Reject Wrapped
Accept Wrapped
Reject Wrapped
We need some new concept to make player 2 reject a wrapped gift, if offered one.
Solution: we'll say that player 2 has to have beliefs about which gift was given, if it is wrapped.
\ p\
\ 1- p\
1
0
1
0
2
0
2
0
3
0
–1
0
2
2
1
X
Y
Z
,
,
,
,
,
,
Accept X
Reject X
Accept Wrapped
Reject Wrapped
Accept Wrapped
Reject Wrapped
Solution: we'll say that player 2 has to have beliefs about which gift was given, if it is wrapped.
\ p\
\ 1- p\
Player 2:
Player 1:
Accept X
Reject a Wrapped Gift
X
STRATEGIES
BELIEFS
Player 2:
\(p < {1 \over 3}\)
(Player 2 believes that if they're given a wrapped gift, it's probably Z.
Econ 51 | 13 | Static Games of Incomplete Information
By Chris Makler
Econ 51 | 13 | Static Games of Incomplete Information
Bayes Nash Equilibrium and Auctions
