Download Powerpoint: More Game Theory Applications

Game Theory Applications
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Example 1: Tender Offer
“Corporate Raiders” in the 1980s often used a
strategy of two-tiered tender offer such as the
following to take a public firm private.
Suppose the current stock price for the firm is
$200 and there are 500 shares. If the corporate
raider obtains more than 50% of the shares
(250 shares) and takes the firm private, the
remaining shareholders get $180 per share.
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Example 1: Tender Offer
Corporate Raider makes the following
tender offer to shareholders:
For the first 250 shares, the corporate raider
will pay a price of $210 and any share
over 250, corporate raider will pay $180.
For simplicity, assume all share holders make the
decision of whether or not to sell to the corporate raider
at the same time and if more than 250 shares are sold,
which ones obtain $210 for their share is random.
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If you own a share of the stock, should
you sell to the Corporate Raider?
Green is expected
payoff if you sell
210
Expected price is 210*250/(# that sell)
+180*(# that sell-250)/(# that sell)
200
195
Red is expected
payoff if you don’t sell
180
249
499
# of shares
sold to raider
by others
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If you own a share of the stock, should
you sell to the Corporate Raider?
210
Dominant Strategy is to Sell
200
195
180
249
499
# of shares
sold to raider
by others
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Example 2: Ebay bidding
On-line proxy bidding where bidders enter
an amount and the highest bidder obtains
the item and the price paid is based on the
second highest bid (second highest plus a
small increment).
Bidders often wait to the last possible
instant to bid (called sniping) so there is
little feedback about other bids at the time
an individual places a bid.
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Example 2: Model of Ebay
You know your valuation but know little about
other bidders.
 All bidders choose bid simultaneously.
 Highest bidder gets the item and pays the
second highest bid.

Referred to as a
Second-Price Sealed Bid Auction
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Example 2: Ebay Game
http://www.random.org/
TEAM 1
Rd
Valuation
Bid
TEAM 2
Payoff
Rd
1
1
2
2
3
3
4
4
5
5
6
6
AVE.
AVE
Valuation
Bid
Payoff
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Strategy in a Second Price Sealed Bid
Auction when Bidders Valuation is V=100
Payoff
What if Bid, B, which is less than V? For
example, B=90.
V=100
B=90
V=100
Highest Rival Bid
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Strategy in a Second Price Sealed Bid
Auction when Bidders Valuation is V=100
Payoff
What if Bid, B, which is more than V?
For Example, B=110
V=100
V=100
B=110
Highest Rival Bid
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Strategy in a Second Price Sealed Bid
Auction when Bidders Valuation is V=100
Payoff
What if Bid, B, which is equal to V? Say
V=100 so B=100
V=100
V=B=100
Highest Rival Bid
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Comparison Of Different Strategies
When B=V compared to B<V
 When
another bidder bids between B and V (90 and
100), Bidder wins if B=V and obtains a positive payoff
but losses when B<V.
 When another bidder does not bid between B and V,
then payoff is the same for both strategies.
When B=V compared to B>V
 When
another bidder bids between V and B (100 and
110), Bidder losses if B=V and wins when B>V. The
payoff of winning is negative.
 When another bidder does not bid between V and B,
then payoff is the same for both strategies.
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(Weakly) Dominant Strategy
In a Second Price Sealed Bid Auction,
the bidder does at least as well if she
bids her valuation (B = V) and strictly
better in some cases than if she bids
any other value. Therefore, bidding
her valuation is a weakly dominant
strategy.
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2nd Price Sealed Bid Auction and
English Auction
An English (or open outcry) auction is one where
bidders shout bids publicly. Auction ends when
there are no higher bids. Termed a “button
auction” in Japan. (like Sotheby does)
Strategies are the same in both auctions.
In English auction, bidder should drop out
when bid is greater than valuation. They
are strategically equivalent.
http://www.bing.com/videos/search?q=30+for+30+naismith+rules&FORM=VIR
E1#view=detail&mid=94718A50722BF95144FB94718A50722BF95144FB
http://espn.go.com/video/clip?id=8395320
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Example 3: 1st Price Sealed Bid Auction
An auction where bidders submit one bid in a
concealed fashion. The submitted bids are then
compared and the person with the highest bid
wins the award, and pays the amount of his bid.
(ex. refinancing credit, foreign exchange, gov’t
contracts, mining leases)
Strategically equivalent to a Dutch Auction
where the auctioneer begins with a high asking
price which is lowered until some participant is
willing to accept the auctioneer's price.
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Example 3: 1st Price Sealed Bid
Auction Game
TEAM 1
Rd
Valuation
Bid
TEAM 2
Payoff
Rd
1
1
2
2
3
3
4
4
5
5
6
6
AVE.
AVE
Valuation
http://www.random.org/
Bid
Payoff
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How Would Things Change if
there were Three Bidders?


In 2nd price sealed bid auction, weakly dominant
strategy is still for players to bid their valuations.
Expected revenue from auctioneer would increase
because second highest bid likely to be more.
In 1st price sealed bid auction, players would shade
their bids downward less when the number of bidders
increases. Expected revenue from auctioneer would
increase (the same amount as in the 2nd price sealed
bid auction).
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What Auction to Use?
The Second Price Sealed Bid Auction, the
First Price Sealed Bid Auction, the Dutch
Auction and the English Auction are
REVENUE EQUIVALENT. This means
that the expected revenue generated is
the same from these four auctions.
Punchline: As an auctioneer, rules of the
auction does not affect revenue much but
reservation price and “ease of entry” do.
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Example 4: Right of First Refusal
Many contracts contain a Right of First Refusal
clause – including the collective bargaining
agreement in the NBA pertaining to restricted
free agents.
http://www.nbpa.com/cba/2005
Article XI, Section 5 allows an incumbent firm
(current team) to retain the right of an
employee (player) by matching the best offer
made to that employee (player) by a rival firm
(another team).
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ESPN.com on July 17, 2014
According to ESPN's Chris Broussard, the Phoenix Suns and Eric
Bledsoe won't be agreeing to terms on a new contract any time soon:
The future of one of the most talented free agents left on the market
remains cloudy as Eric Bledsoe and the Phoenix Suns remain far apart
in contract talks, according to sources close to the situation.
Bledsoe's representatives have been engaged in discussions with
Phoenix, but the Suns' offer is far below what Bledsoe is looking for.
And while Bledsoe remains one of the premier names available from a
loaded class of free agents, Broussard notes a resolution may take
some time to manifest itself, for rival teams are wary of extending offer
sheets to the 24-year-old.
Per Broussard, "the belief that the Suns will match any offer Bledsoe
receives from a competing club has deterred teams from aggressively
pursuing the 24-year-old point guard."
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Example 4: Right of First Refusal
Suppose you are one of three players in the negotiations process:
current team, player and rival team.
The contract of the player is about to expire and the current team is
deciding what offer to make the player. The player is worth $10M
to the current team (based on their next best alterative). The
current team makes an initial offer to the player (OI1) which the
player can accept or initially reject. If the player rejects OI1, the
rival team decides whether to make an offer (OR) or not make an
offer. The rival team incurs a cost of $.5M when putting together
an offer. The player is worth $10M to the rival team. If the rival
team makes an offer, the current team can match this offer or not
(actually the current team can make any offer they want, OI2).
The player then decides whether to accept the current team’s
offer, the rival team’s offer or reject both. If the player does not
agree to a contract, the player must sit out the year and his payoff
would be zero.
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RIGHT OF FIRST REFUSAL
I
I, incumbent firm
P, player
R, rival firm
Payoffs
(incumbent, player, rival)
OI1
P
Accept OI1
Initially Reject OI1
R
Make Offer
No Offer
(10-OI1,OI1,0)
P
R
OR
I
Accept OI1
(10-OI1,OI1,0)
(0,0,0)
OI2
Accept
(0,OR,10-OR-.5)
OR
P
Reject
Reject OI1
Accept
OI2
(0,0,-.5)
(10-OI2,OI2,-.5)
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Subgame Perfect – Backward Induction
Player:
Accept OR if OR >O12 and OR >0
Accept Ol2 if O12 >ORand Ol2 >0
Reject if OR <0 and Ol2<0
Incumbent:
Offer Ol2 =OR if OR<10
Offer Ol2 <OR if OR>10
Rival Firm:
Rival Firm:
If make offer, offer OR<10
No Offer
Player (if rival firm makes no offer):
Player:
Accept Ol1 if Ol1 >0
Reject if Ol1 <0
Incumbent:
Offer Ol1 =0
Accept Ol1 if Ol1>0
Outcome – Incumbent offers 0 and Player Accepts
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What if there was not a
Right of First Refusal?
Suppose the game is the same as before but
now, if the player rejects OI1, and the rival firm
makes an offer OR, the incumbent firm cannot
make another offer.
What is the likely outcome to this game?
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NO RIGHT OF FIRST REFUSAL
I, incumbent firm
P, player
R, rival firm
I
Payoffs
(incumbent, player, rival)
OI1
P
Accept OI1
Initially Reject OI1
R
Make Offer
No Offer
(10-OI1,OI1,0)
P
R
OR
P
Accept
Accept OI1
(10-OI1,OI1,0)
(0,0,0)
OR
Reject
(0,OR,10-OR-.5)
Reject OI1
(0,0,-.5)
Accept
OI1
(10-OI2,OI2,-.5)
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Subgame Perfect – Backward Induction
Player:
Accept OR if OR >O11 and OR >0
Accept Ol1 if Ol1 >ORand Ol1 >0
Reject if OR <0 and Ol1<0
Rival Firm:
Offer OR =Ol1 if Ol1 <9.5
Offer OR <Ol1 if Ol1 >9.5
Rival Firm:
Make Offer if Ol1 <9.5
No Offer if Ol1 >9.5
Player (if rival firm makes no offer):
Accept Ol1 if Ol1>0
Player:
Accept Ol1 if Ol1 >9.5
Reject Ol1 if Ol1 <9.5
Incumbent:
Offer Ol1 =9.5
Outcome – Incumbent offers 9.5 and Player Accepts
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Strategy Depends on Sophistication
of Other Players
Rick Reilly article on Rock, Paper, Scissors
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Strategy Depends on Sophistication
of Other Players
Liars Poker:
http://wizardofodds.com/games/liars-poker/
http://quant.stackexchange.com/questions/4201/strategies-for-liars-poker
http://www.youtube.com/watch?v=AIxt2pJx31o
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Strategy Depends on Your Information and
what you know about Other Player’s Information
Alice and Bob are waiting for an interview with a potential
employer, each wearing a blue hat whose color they have
forgotten in the hustle of getting ready. The receptionist
announces that either or both of them may proceed to the
interview if they know the color of their hat (while
admonishing them not to look at their hats and not to speak).
Alice and Bob continue to sit, each looking at the other's blue
hat while paralyzed by uncertainty as to the color of their
own. After some time, the receptionist remarks that one of
their hats is blue. WHAT HAPPENS NEXT?
Lesson: In most games a player benefits from having more
information if other players in the game do not know he/she
has the information. Information can be detrimental if other
players know you have the information.
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