Acting on
Private Information

Christopher Makler

Stanford University Department of Economics

 

Econ 51: Lecture 12

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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.

  • 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)
  • 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)

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

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?

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, 

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\(A^H\)

\(B^H\)

\(A^T\)

\(B^T\)

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)

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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?

What happens when both players have private information?

Bid or Fold

  • Players start with 6 chips. 
  • Both players are dealt a card. Ante is one chip per round.
  • Each simultaneously and independently chooses to bid one additional chip or fold
  • If both players fold, they lose the antes (payoff to each is \(-1\))
  • If one player folds and one bids, the bidder wins the pot
    (payoff is \(-1\) to folder, \(+1\) to bidder)
  • If both players bid, the higher card wins the pot
    (payoff is \(-2\) to loser, \(+2\) to winner); if tie, split the pot.
  • Chip total after 3 rounds added to HW7 score.

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.

v_i
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Your bid

Highest bid other than yours

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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
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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.

Common Value Auctions

Common Value Auctions

Suppose a plot of land is being auctioned off.

It sits on a certain amount of oil, \(Y \sim U [0,20]\).

There are two bidders. Each bidder is invited to drill test holes on half the land. This reveals how much oil there is on their half.

y_1
y_2
Y
y_1 \sim U[0,10]
y_2 \sim U[0,10]

\(Y\) = \(y_1\) + \(y_2\)

y_1
y_2
Y
y_1 \sim U[0,10]
y_2 \sim U[0,10]

\(Y\) = \(y_1\) + \(y_2\)

Given their signal \(y_1\), what is bidder 1's expected value for the entire parcel?

\mathbb{E}_1(Y\ |\ y_1) =
y_1 + 5

Suppose both bidders bid \(b_i = y_i + 5\).
When does bidder 1 win?

y_1 + 5 > y_2 + 5
b_1 > b_2
y_1 > y_2
\mathbb{E}[y_2] = 5
y_1 + \mathbb{E}[y_2]
=
y_1
y_2
Y
y_1 \sim U[0,10]
y_2 \sim U[0,10]

\(Y\) = \(y_1\) + \(y_2\)

Given their signal \(y_1\), what is bidder 1's expected value for the entire parcel?

\mathbb{E}_1(Y\ |\ y_1) =
y_1 + 5

Suppose both bidders bid \(b_i = y_i + 5\).
When does bidder 1 win?

y_1 > y_2

Conditional on winning, what is their expected value of the entire parcel...?

y_1 \sim U[0,10]
y_2 \sim U[0,10]

\(Y\) = \(y_1\) + \(y_2\)

Given their signal \(y_1\), what is bidder 1's expected value for the entire parcel?

\mathbb{E}_1(Y\ |\ y_1) =
y_1 + 5

Suppose both bidders bid \(b_i = y_i + 5\).
When does bidder 1 win?

y_1 > y_2

Conditional on winning, what is their expected value of the entire parcel...?

  • Suppose \(y_1 = 8\), so bidder 1 bids 13.
  • If bidder 1 wins, they know that \(y_2 < 8\).
  • But conditional on being less than 8, \(\mathbb{E}[y_2] = 4\)!
  • So, expected value of the total parcel is 8 + 4 = 12
  • Which means if they win with a bid of 13, expected payoff is \(12 - 13 = -1\)!
y_1 \sim U[0,10]
y_2 \sim U[0,10]

\(Y\) = \(y_1\) + \(y_2\)

It turns out that the BNE of this game is for each player to bid their signal: \(b_1 = y_1\), \(b_2 = y_2\)!

Summary and Next Steps

  • Today we looked at situations in which people had private information but acted simultaneously.
  • A strategy in this type of game maps each player's private information onto an action.
  • Next week, we'll introduce time back into the mix, and ask: can you deduce private information by how someone plays the game?