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Game Theory Applications

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 OfferCorporate 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?

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# of shares sold to raider by others

200

180

210

249 499

Green is expected payoff if you sell

Red is expected payoff if you don’t sell

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Expected price is 210*250/(# that sell) +180*(# that sell-250)/(# that sell)

If you own a share of the stock, should you sell to the Corporate Raider?

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# of shares sold to raider by others

200

180

210

249 499

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Dominant Strategy is to Sell

Example 2: Ebay biddingOn-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 Auction7

Example 2: Ebay Game

Rd Valuation Bid Payoff

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2

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5

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AVE.

Rd Valuation Bid Payoff

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AVE

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TEAM 1 TEAM 2http://www.random.org/

Strategy in a Second Price Sealed Bid Auction when Bidders Valuation is V=100

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Highest Rival Bid

V=100

V=100B=90

What if Bid, B, which is less than V? For example, B=90.Payoff

Strategy in a Second Price Sealed Bid Auction when Bidders Valuation is V=100

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Highest Rival Bid

V=100

V=100 B=110

What if Bid, B, which is more than V? For Example, B=110Payoff

Strategy in a Second Price Sealed Bid Auction when Bidders Valuation is V=100

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Highest Rival Bid

V=100

V=B=100

What if Bid, B, which is equal to V? Say V=100 so B=100Payoff

Comparison Of Different StrategiesWhen 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.

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http://www.bing.com/videos/search?q=30+for+30+naismith+rules&FORM=VIRE1#view=detail&mid=94718A50722B

F95144FB94718A50722BF95144FB http://espn.go.com/video/clip?id=8395320

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.15

Example 3: 1st Price Sealed Bid Auction Game

Rd Valuation Bid Payoff

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2

3

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AVE.

Rd Valuation Bid Payoff

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AVE

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TEAM 1 TEAM 2

http://www.random.org/

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 RefusalMany 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 RefusalSuppose 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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I

R

P

PR

P

I

OI1

OR

OI2

Accept OR

Reject OI1Accept OI1

Accept OI1 Initially Reject OI1

Accept OI2

I, incumbent firmP, playerR, rival firm

Payoffs(incumbent, player, rival)

Reject

(10-OI1,OI1,0)

(0,OR,10-OR-.5)

(10-OI1,OI1,0)(0,0,0)

(0,0,-.5)(10-OI2,OI2,-.5)

No OfferMake Offer

RIGHT OF FIRST REFUSAL

Subgame Perfect – Backward InductionPlayer: 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: If make offer, offer OR<10

Rival Firm: No Offer

 

Player (if rival firm makes no offer): Accept Ol1 if Ol1>0

 

Player: Accept Ol1 if Ol1 >0

Reject if Ol1 <0

 

Incumbent: Offer Ol1 =0

Outcome – Incumbent offers 0 and Player Accepts23

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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I

R

P

PR

P

OI1

ORReject OI1

Accept OI1

Accept OI1

I, incumbent firmP, playerR, rival firm

Payoffs(incumbent, player, rival)

Reject

(10-OI1,OI1,0)

(0,OR,10-OR-.5)

(10-OI1,OI1,0)(0,0,0)

(0,0,-.5) (10-OI2,OI2,-.5)

No OfferMake Offer

NO RIGHT OF FIRST REFUSAL

Accept OR

Accept OI1Initially Reject OI1

Subgame Perfect – Backward InductionPlayer: 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 Accepts26

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?

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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.